Physical Science

Lori Mayberry, Walid Shihabi, and Jennifer Campbell

Physical Science by Tulsa Community College is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

Contents

- Introduction

- Part I. Chapter 1

- The Real Process of Science

- A Blueprint for Scientific Investigation

- Units of Measurement

- Significant Figures and Order of Magnitude CC BY-SA (Attribution ShareAlike)

- Dimensional Analysis CC BY (Attribution)

- Part II. Chapter 2

- Basics of Kinematics CC BY (Attribution)

- Speed and Velocity CC BY (Attribution)

- Acceleration CC BY (Attribution)

- Free-Falling Objects CC BY (Attribution)

- Projectile Motion CC BY (Attribution)

- Part III. Chapter 3

- Introduction CC BY (Attribution)

- Force and Mass CC BY (Attribution)

- Newton’s Laws CC BY (Attribution)

- Other Examples of Forces CC BY (Attribution)

- Introduction CC BY (Attribution)

- Conservation of Momentum CC BY (Attribution)

- Part IV. Introduction to Chemistry

- 1. Discoveries Leading to Nuclear Atom Model CC BY (Attribution)

- 2. History of Atomic Structure CC BY (Attribution)

- 3. The Structure of the Atom CC BY (Attribution)

- 4. The Periodic Table CC BY (Attribution)

- 5. The Ionic Bond CC BY (Attribution)

- 6. The Covalent Bond CC BY (Attribution)

- 7. Overview of Chemistry CC BY (Attribution)

- 8. Classification of Matter CC BY (Attribution)

- 9. Physical and Chemical Properties of Matter CC BY (Attribution)

- 10. Units of Measurement CC BY (Attribution)

- Part V. Acids and Bases

- 11. Acids and Bases CC BY (Attribution)

- Part VI. Physics

- 12. Energy Conservation CC BY (Attribution)

- 13. Introduction CC BY (Attribution)

- 14. Work Done by a Constant Force CC BY (Attribution)

- 15. Power CC BY (Attribution)

- 16. Temperature and Temperature Scales CC BY (Attribution)

- 17. Phase Changes CC BY (Attribution)

- 18. Waves CC BY (Attribution)

- 19. Sound Intensity and Level CC BY (Attribution)

- 20. Overview CC BY (Attribution)

- 21. Coulomb’s Law CC BY (Attribution)

- 22. Electric Current CC BY (Attribution)

- 23. Resistance and Resistors CC BY (Attribution)

- 24. Electric Power and Energy CC BY (Attribution)

- 25. Resistors in Series and Parallel CC BY (Attribution)

- 26. Magnetism and Magnetic Fields CC BY (Attribution)

- 27. Magnetic Flux, Induction, and Faraday’s Law CC BY (Attribution)

- 28. The Electromagnetic Spectrum CC BY (Attribution)

- 29. Reflection, Refraction, and Dispersion CC BY (Attribution)

- 30. Lenses CC BY (Attribution)

- 31. Mirrors CC BY (Attribution)

- Part VII. Heat and Heat Transfer

- 32. Introduction CC BY (Attribution)

- 33. Specific Heat CC BY (Attribution)

- Appendix

1

This book explores the concepts of Physical Science.

I

Chapter 1

- The Real Process of Science (Web link)

- A Blueprint for Scientific Investigation (Web link)

- Units of Measurement

- Significant Figures and Magnitude

- Dimensional Analysis

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The Real Process of Science

This portion of the textbook will navigate to an external link. Please use the information in the linked portion to learn about the real process of science. (Opens in a new tab)

The Real Process of Science

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A Blueprint for Scientific Investigation

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A Blueprint for Scientific Investigation

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Units of Measurement

Units of Measurement

Standard Units (SI Units)

The International System of Units (abbreviated SI) is the metric system used in science, industry, and medicine.

LEARNING OBJECTIVES

Recognize SI units and their importance for measurement

KEY TAKEAWAYS

Key Points

- Every field of science involves taking measurements, understanding them, and communicating them to others. In other words, we all have to speak the same basic language.

- The SI system, also called the metric system, is used around the world.

- There are seven basic units in the SI system: the meter (m), the kilogram (kg), the second (s), the kelvin (K), the ampere (A), the mole (mol), and the candela (cd).

Key Terms

- SI system: A series of units that is accepted and used throughout the scientific world.

The Need for a Common Language

Every field of science involves taking measurements, understanding them, and communicating them to others. In other words, we all have to speak the same basic language. Whether you are a chemist, a physicist, a biologist, an engineer, or even a medical doctor, you need a consistent way of communicating size, mass, shape, temperature, time, amount, energy, power, and speed.

Consider the screen on which you’re reading this text right now. It might be an LCD screen, which is made up of liquid crystals. The chemist developing a specific formulation for a liquid crystal has to meaningfully communicate information to an engineer so that the engineer knows how to manufacture it. The engineer, in turn has to be able to communicate with other engineers, physicists, and chemists to design the circuit boards, display screens, and electronic interfaces of the rest of the computer. If these people don’t all speak the same language, the enterprise will never get off the ground.

The International System of Units (abbreviated SI, from the French Système international d’unités) is the metric system used in science, industry, and medicine. Depending on your age and geographic location, you might be very familiar with the “imperial” system, which includes units such as gallons, feet, miles, and pounds. The imperial system is used for “everyday” measurements in a few places, such as the United States. But in most of the world (including Europe) and in all scientific circles, the SI system is in common use.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=36

Scientific Units the SI and Metric Units: Mr. Causey teaches scientific units of the SI system, the metric system, and the CGS system. Mr. Causey also shares the major prefixes and their meanings. Science measurements are based on the metric system, so it is important that you know the metric base units and prefixes.

Units of the SI System

There are seven base units in the SI system:

- the kilogram (kg), for mass

- the second (s), for time

- the kelvin (K), for temperature

- the ampere (A), for electric current

- the mole (mol), for the amount of a substance

- the candela (cd), for luminous intensity

- the meter (m), for distance

The Seven SI Units: This figure displays the fundamental SI units and the combinations that lead to more complex units of measurement.

It should be apparent that the move into modern times has greatly refined the conditions of measurement for each basic unit in the SI system, making the measurement of, for example, the luminous intensity of a light source a standard measurement in every laboratory in the world. A light source made to produce 20 cd will be the same regardless of whether it is made in the United States, in the UK, or anywhere else. The use of the SI system provides all scientists and engineers with a common language of measurement.

History of the SI System

The SI units of measurement have an interesting history. Over time they have been refined for clarity and simplicity.

- The meter (m), or metre, was originally defined as 1/10,000,000 of the distance from the Earth’s equator to the North Pole measured on the circumference through Paris. In modern terms, it is defined as the distance traveled by light in a vacuum over a time interval of 1/299,792,458 of a second.

- The kilogram (kg) was originally defined as the mass of a liter (i.e., of one thousandth of a cubic meter). It is currently defined as the mass of a platinum-iridium kilogram sample maintained by the Bureau International des Poids et Mesures in Sevres, France.

- The second (s) was originally based on a “standard day” of 24 hours, with each hour divided in 60 minutes and each minute divided in 60 seconds. However, we now know that a complete rotation of the Earth actually takes 23 hours, 56 minutes, and 4.1 seconds. Therefore, a second is now defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

- The ampere (A) is a measure of the amount of electric charge passing a point in an electric circuit per unit time. 6.241×10^18 electrons, or one coulomb, per second constitutes one ampere.

- The kelvin (K) is the unit of the thermodynamic temperature scale. This scale starts at 0 K. The incremental size of the kelvin is the same as that of the degree on the Celsius (also called centigrade) scale. The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (exactly 0.01 °C, or 32.018 °F).

- The mole (mol) is a number that relates molecular or atomic mass to a constant number of particles. It is defined as the amount of a substance that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12.

- The candela (cd) was so named to refer to “candlepower” back in the days when candles were the most common source of illumination (because so many people used candles, their properties were standardized). Now, with the prevalence of incandescent and fluorescent light sources, the candela is defined as the luminous intensity in a given direction of a source that emits monochromatic radiation of frequency 540X10^12 . Hertz and that has a radiant intensity in that direction of 1/683 watts per steradian.

SI Unit Prefixes

The basic SI units can be expressed as fractions and multiples of basic units by using a set of simple prefixes.

LEARNING OBJECTIVES

Convert between SI units

KEY TAKEAWAYS

Key Points

- The set of prefixes is simple and easy to use.

- Prefixes cannot be combined.

- The set of prefixes is universal.

Key Terms

- prefix: One or more letters or syllables added to the beginning of a word to modify its meaning; for example, kilo can be added to gram to create kilogram

- fraction: a part of a whole, especially a comparatively small part

SI Unit Prefixes

Now that we know about the SI system and what it provides to the scientist and engineer, we can examine some aspects of actual measurement. The SI system utilizes a standard system of prefixes to the basic units that allow them to be more relevant to and descriptive of relative magnitude.

For example, when reading about chemical kinetics, you may encounter the terms “ms” or “ns,” meaning “millisecond” and “nanosecond” respectively. Once you’ve become accustomed to the practice of using the prefixes, you will immediately know that the millisecond is 1/1000 of one second and is 1 million times larger than a nanosecond, which is 1/1000000000 of one second, or 10-9 seconds.

Briefly review the basic SI units before you study the prefixes.

Unit name

Unit symbol

Quantity name

Quantity symbol

Dimension symbol

meter

m

length

l, x, r

L

kilogram[1]

kg

mass

m

M

second

s

time

t

T

ampere

A

electric current

I

I

kelvin

K

thermodynamic temperature

T

Θ

candela

cd

luminous intensity

Iv

J

mole

mol

amount of substance

n

N

There are 20 accepted prefixes. A prefix may be used to identify multiples of the original unit or fractions of the original unit. For example, kilo- denotes a multiple of a thousand, so there are one thousand meters in a kilometer. Milli- denotes a thousandth; therefore, there are one thousand millimeters in a meter.

Prefixes for SI units: The prefixes redefine the measurement as either a multiple or a fraction of the basic unit.

Keep in mind that prefixes should never be combined. Thus a millionth of a meter is a micrometer, not a millimillimeter, and a millionth of a kilogram is a milligram, not a microkilogram.

In older usage, a micron (a measurement often encountered in physics and engineering) is the same as a micrometer, 10-6 meters. Another older form of usage, the millimicron, is one thousandth of a micrometer, or 1 thousandth of 10-6 meters, or 10-9 meter, now called a nanometer. While these older terms are not in common usage, they are often encountered in older publications, and knowing their modern equivalents is an advantage.

Volume and Density

Density and volume are two common measurements in chemistry.

LEARNING OBJECTIVES

Describe the relationship between density and volume

KEY TAKEAWAYS

Key Points

- The volume of a substance is related to the quantity of the substance present at a defined temperature and pressure.

- The volume of a substance can be measured in volumetric glassware, such as the volumetric flask and the graduated cylinder.

- Density indicates how much of a substance occupies a specific volume at a defined temperature and pressure. The density of a substance can be used to define the substance.

- Water is unusual because when water freezes, its solid form (ice) is less dense than liquid water, and thus floats on top of liquid water.

Key Terms

- volume: A unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric.

- density: A measure of the amount of matter contained in a given volume.

Volume and Density

The properties of a material may be described in many ways. Any amount of any substance will have a volume. If you have two containers of water that are different sizes, they each hold a different amount, or volume, of water. The unit for volume is a unit derived from the SI unit of length and is not a fundamental SI measurement.

If two water samples have different volumes, they still share a common measurement: the density. Density is another measurement derived from SI basic units. The density of a material is defined as its mass per unit volume. In this example, each volume of water is different and therefore has a specific and unique mass. The mass of water is expressed in grams (g) or kilograms (kg), and the volume is measured in liters (L), cubic centimeters (cm^3), or milliliters (mL). Density is calculated by the dividing the mass by the volume, so that density is measured as units of mass/volume, often g/mL. If both water samples are at the same temperature, their densities should be identical, regardless of the samples’ volume.

Measurement Tools

The measuring cup: The measuring cup is a common household utensil used for measuring the volumes of liquids.

If you have ever cooked in a kitchen, you have probably seen some sort of measuring cup, which allows the user to measure liquid volumes with reasonable accuracy. The measuring cup expresses liquid volume in the standard SI units of liters and milliliters. Most American measuring cups also measure liquid in the older system of cups and ounces.

Volumetric Glassware

Scientists who work in a laboratory must be familiar with typical laboratory glassware, often called volumetric glassware. These may include beakers, a volumetric flask, an Erlenmeyer flask, and a graduated cylinder. Each of these containers is used in a laboratory setting to measure liquid volumes for different purposes.

Laboratory volumetric glassware: Glassware, such as these beakers, is commonly used in a laboratory setting to conveniently measure and separate different volumes of liquids.

Density of Water

Different substances have different densities, so density is often used as a method to identify a material. Comparing the densities of two materials can also predict how substances will interact. Water is used as the common standard to substances, and it has a density of 1000 kg/m^3 at Standard Temperature and Pressure (called STP).

Using Water as a Density Comparison

When an object is placed in water, the object’s relative density determines whether it floats or sinks. If the object has a lower density than water, it will float to the top of the water. An object with a higher density will sink. For example, cork has a density of 240 kg/m^3, so it will float. Air has a density of approximately 1.2 kg/m^3, so it rises immediately to the top of a water column. The metals sodium (970 kg/m^3) and potassium (860 kg/m^3) will both float on water, while lead (11,340 kg/m^3) will sink.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=36

Density: A Story of Archimedes and the Gold Crown: Is the crown made of pure gold? An ancient Greek king needs to know if he’s been cheated by a goldsmith. He summons Archimedes, who decides to use density to determine the metal. But how can he determine the volume of the crown?

Liquids tend to form layers when added to water. The sugar alcohol glycerol (1,261 kg/m^3) will sink into the water and form a separate layer until it is thoroughly mixed (glycerol is soluble in water). Vegetable oil (approx. 900 kg/m^3) will float on water, and no matter how vigorously mixed, will always return as a layer on the water surface (oil is not soluble in water).

The Variable Density of Water

Water itself is a complicated and unique molecule. Even if the pressure is consistent, water’s density will change based on the temperature. Recall that the three basic forms of matter are solid, liquid and gas (ignore plasma for the time being). As a rule of thumb, almost all materials are more dense in their solid or crystalline form than in their liquid form; place the solid form of almost any material on the surface of its liquid form, and it will sink. Water, on the other hand, does something very special: ice (the solid form of water) floats on liquid water.

Look carefully at the relationship between water’s temperature and its density. Beginning at 100 °C, the density of water steadily increases, as far as 4 °C. At that point, the density trend reverses. At 0 °C, water freezes to ice and floats.

This table lists the densities of water at different temperatures and constant pressure.

The density of water at constant pressure

Temp (ºC)

Density (kg/m^3)

100

958.4

80

971.8

60

983.2

40

992.2

30

995.6502

25

997.0479

22

997.7735

20

998.2071

15

999.1026

10

999.7026

4

999.9720

0

999.8395

−10

998.117

−20

993.547

−30

983.854

The values below 0ºC refer to super cooled water

The implications of this simple fact are enormous: when a lake freezes, ice crusts at the surface and insulates the liquid below from freezing, while at the same time allowing the colder water (with a temp of approx. 4 °C and a high density) to sink to the bottom. If ice did not float, it would sink to the bottom, allowing more ice to form and sink, until the lake froze solid! Scuba divers and swimmers often encounter these water temperature gradients, and they might even encounter a water layer at the very bottom of a lake with a temperature of approximately 4 °C. That’s just about as cold as the lake will get at the bottom; as soon as the water gets colder, the liquid water becomes less dense and rises.

Layers of water in a winter lake: During the winter months of seasonal climates, the warmest water in most lakes and rivers is only 4°C. This 4°C water has the highest density and sinks to the bottom of the lake. As the water becomes colder (<4°C), it becomes less dense and rises to form ice on the surface of the lake. As a result, liquid water always exists in lakes and rivers during the winter months. This unique property of water enables animals and plants to survive under the frozen lake or winter, ensuring that all freshwater life does not go extinct each winter.

Temperature

The ability to measure temperature accurately was a major scientific advancement, putting absolute numbers on an observable phenomenon.

LEARNING OBJECTIVES

State the key developments in the history of temperature measurement

KEY TAKEAWAYS

Key Points

- The measurement of temperature is accurate and reproducible.

- The measurement of temperature should be against accepted standards.

- Temperature can be calibrated to several scales including Celsius, Fahrenheit, and kelvin.

- Converting between different temperature scales is easy using the conversion equations.

- Kinetic energy arises from the motion of atoms and molecules. It is postulated that at absolute 0, there is no motion and therefore, no kinetic energy.

Key Terms

- temperature: A measure of cold or heat, often measurable with a thermometer.

- kelvin: A unit of measurement for temperature. It is one of the seven base units in the International System of Units (SI).

- Fahrenheit: An unit of measurement for temperature used most commonly in the United States.

- Celsius: A scale and unit of measurement for temperature where 0 °C is the freezing point of water. Also known as centigrade.

- Celsius: A scale and unit of measurement for temperature where 0 °C is the freezing point of water. Also known as centigrade.

How hot was it last summer? Will it be cold enough for skiing next week? Each of these questions forces quantification to a routine experience. Whether we’re talking about the weather, cooking food, or conducting a scientific experiment, we need to know how hot or how cold something is. To know this, one must be able to put some sort of accurate number on the concept. Although there is an entire field of study devoted to measuring temperature (thermometry), the focus of this section is on the fundamental measurements of temperature.

Monthly mean temperature: Temperature enables us to accurately measure and compare climates in different parts of the world.

The History of Measuring Temperature

To people in the 21st century, measuring temperature is a quick and easy thing to do. Thousands of years ago, however, things were different. Temperature-related phenomena were always being observed. Snow fell and collected in cold weather, and melted into liquid water when the air warmed in spring. Liquid water fell as rain when the air was warm. Ice melted when placed near a source of heat, and water completely boiled out of a pot on a hot stove. However, these are all qualitative observations. They do not generate a number: they do not tell us that water freezes at 0 °C, or that it boils at 100 °C. All we learn from observation is that heat and cold do something to water, or that water behaves differently when it is heated or cooled.

In the 16th and 17th centuries, scientists refined the observations and experiments of the Byzantines and Greeks to produce rudimentary devices relating the amount of “hotness” or “coldness” in the air. The devices they built were called thermoscopes. These basic measuring tools utilized the expansion and contraction of air and water when heated and cooled.

The concept was remarkable, but thermoscopes did not have a numeric scale. The thermoscope could not answer the question, “How hot is it today?” with a number, but it could give a relative measurement. The thermoscope was often a simple tube of gas over liquid. Thermoscopes also served as barometers (which measure pressure ). That made it difficult to use them as thermometers, but they reacted to both pressure and temperature. Even when early thermometers did have a numeric scale, the scales were not standardized.

The dawn of the 18th century saw great change in thermometers, thanks to the work of Isaac Newton, Anders Celsius, and Daniel Fahrenheit.

- Isaac Newton proposed a thermometer with a scale of 12 degrees between the freezing and boiling points of water.

- Fahrenheit was working with tubes filled with mercury, which has a very high coefficient of thermal expansion. This, combined with the quality and accuracy of Fahrenheit’s work, led to much greater sensitivity, and his thermometer was standardized against a brine solution and universally adopted, with the Fahrenheit scale being named in his honor.

- Anders Celsius proposed a 100 degree scale for the difference between freezing and boiling of water, and after a few minor adjustments, the Celsius, or centigrade, system was also widely adopted.

Thermometer calibrated with the Celsius scal: Celsius is a scale and unit of measurement for temperature where 0 °C is the freezing point of water. Our ability to accurately measure temperature enables us to measure the weather, cook food accurately, or conduct a scientific experiment.

Further advances led to faster-acting thermometers, which were useful in medicine and chemistry. Early thermometers did not record or hold the temperature they were measuring: if you removed the thermometer from the substance being measured, its reading would change. Scientists invented new thermometers that would maintain their reading, at least for a limited period of time, to reduce measurement errors and make it easier to record the temperature. Dial thermometers using bimetallic strips were also developed. The bimetallic strips are made from two dissimilar metals bonded together, with each metal having a different coefficient of thermal expansion. Upon heating or cooling, the two metals expand or contract at different rates, causing a bending or curvature to appear in the strip. This bending is useful as a transducer for the temperature reading; it can control a thermostatted circuit or drive a simple dial thermometer.

Absolute Zero

Through the development of temperature measurement, however, one question remained unanswered: “How cold can it really get? How cold is absolute 0?”

The trivial answer is “0 degrees,” but what exactly does that mean? Temperature itself is the measurement of the average kinetic energy of a substance. The kinetic energy arises from the motion of atoms and molecules, and it is postulated that at absolute 0, there is no motion and therefore, no kinetic energy. Therefore, the temperature must be “absolute 0.”

The question remains: how much colder is absolute 0 than 0 °C?

In 1848, Lord Kelvin (William Thomson) wrote a paper entitled “On An Absolute Thermometric Scale” about the need to seek out a thermodynamic zero temperature. Using the Celsius system for its measurement of degrees, Lord Kelvin calculated the ultimate cold temperature to be -273 °C. Today that is referred to as 0 K on the Kelvin thermodynamic temperature scale. Modern methods have refined the measurement to -273.16 °C.

Types of Temperature Scales

Temperature can be measured and represented in many different ways. The fundamental requirements of the practice involve accuracy, a standard, linearity, and reproducibility. The SI unit, chosen for its simplicity and relationship to thermodynamics, is the kelvin, named in honor of Lord Kelvin. While incrementally equal to the Celsius scale, the temperature in kelvins is a true representation of the kinetic energy in a thermodynamic sense. Chemistry and physics require many calculations involving temperature. Those calculations are always made in kelvins.

Comparison of temperature scales: Temperatures of some common events and substances in different units.

A comparison of temperature scales table illustrates a variety of temperature scales, some of which are no longer used. It is interesting to see the temperatures of commonly occurring events over these scales, and to imagine the great hurdles that were overcome in developing modern thermometry.

Conversion to and from kelvin: Use the equations in this table to calculate temperatures using the kelvin measurement system.

Although in most cases scientists are equipped with some sort of electronic calculator, there might be times when a conversion from one scale to another is required. Conversion tables can be used to convert a measurement to any scale from any other temperature scale, such as kelvin or Celsius.

Conversion to and from degrees Celsius: Use the equations in this table to convert temperatures to the Celsius measurement system.

1. Despite the prefix “kilo-,” the kilogram is the base unit of mass. The kilogram, not the gram, is used in definitions of derived units. Nonetheless, units of mass are named as if the gram were the base unit. ↵

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- Chemistry – Scientific Units the SI and Metric Units – YouTube. Located at: http://www.youtube.com/watch?v=WuaxXcgX6Rc. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license

- SI Units. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/SI_Units. License: Public Domain: No Known Copyright

- SI Units. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/SI_Units. License: Public Domain: No Known Copyright

- Lab glassware. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/File:Lab_glassware.jpg. License: CC BY-SA: Attribution-ShareAlike

- Simple Measuring Cup. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/File:Simple_Measuring_Cup.jpg. License: CC BY-SA: Attribution-ShareAlike

- Density: A Story of Archimedes and the Gold Crown. Located at: http://www.youtube.com/watch?v=KMNwXUCXLdk. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license

- Anomalous expansion of water Summer Winter. Provided by: Wikimedia. Located at: http://commons.wikimedia.org/wiki/File:Anomalous_expansion_of_water_Summer_Winter.svg. License: CC BY-SA: Attribution-ShareAlike

- Density. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Density. License: Public Domain: No Known Copyright

- Thermodynamic temperature. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Thermodynamic_temperature. License: CC BY-SA: Attribution-ShareAlike

- Thermometer. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Thermometer. License: CC BY-SA: Attribution-ShareAlike

- General Chemistry/Thermodynamics/Introduction. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/General_Chemistry/Thermodynamics/Introduction. License: CC BY-SA: Attribution-ShareAlike

- Kelvin. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Kelvin. License: CC BY-SA: Attribution-ShareAlike

- Temperature measurement. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Temperature_measurement. License: CC BY-SA: Attribution-ShareAlike

- Fahrenheit. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/Fahrenheit. License: CC BY-SA: Attribution-ShareAlike

- kelvin. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/kelvin. License: CC BY-SA: Attribution-ShareAlike

- temperature. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/temperature. License: CC BY-SA: Attribution-ShareAlike

- SI base unit. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/File:SI_base_unit.svg. License: CC BY-SA: Attribution-ShareAlike

- Chemistry – Scientific Units the SI and Metric Units – YouTube. Located at: http://www.youtube.com/watch?v=WuaxXcgX6Rc. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license

- SI Units. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/SI_Units. License: Public Domain: No Known Copyright

- SI Units. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/SI_Units. License: Public Domain: No Known Copyright

- Lab glassware. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/File:Lab_glassware.jpg. License: CC BY-SA: Attribution-ShareAlike

- Simple Measuring Cup. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/File:Simple_Measuring_Cup.jpg. License: CC BY-SA: Attribution-ShareAlike

- Density: A Story of Archimedes and the Gold Crown. Located at: http://www.youtube.com/watch?v=KMNwXUCXLdk. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license

- Anomalous expansion of water Summer Winter. Provided by: Wikimedia. Located at: http://commons.wikimedia.org/wiki/File:Anomalous_expansion_of_water_Summer_Winter.svg. License: CC BY-SA: Attribution-ShareAlike

- Density. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Density. License: Public Domain: No Known Copyright

- Temperature conversion. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Temperature_conversion. License: Public Domain: No Known Copyright

- Pakkanen. Provided by: Wikimedia. Located at: http://en.wikipedia.org/wiki/Celsius%23mediaviewer/File:Pakkanen.jpg. License: Public Domain: No Known Copyright

- MonthlyMeanT. Provided by: Wikimedia. Located at: http://en.wikipedia.org/wiki/Temperature%23mediaviewer/File:MonthlyMeanT.gif. License: CC BY-SA: Attribution-ShareAlike

- Temperature measurement. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Temperature_measurement. License: Public Domain: No Known Copyright

Significant Figures and Order of Magnitude

Scientific Notation

Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form.

Learning Objectives

Convert properly between standard and scientific notation and identify appropriate situations to use it

Key Takeaways

Key Points

- Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of 10.

- In scientific notation all numbers are written in the form of a • 10b (a times ten raised to the power of b).

- Each consecutive exponent number is ten times bigger than the previous one; negative exponents are used for small numbers.

Key Terms

- exponent: The power to which a number, symbol or expression is to be raised. For example, the 3 in $x^3$.

- Scientific notation: A method of writing, or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10

Scientific Notation: A Matter of Convenience

Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers.

In scientific notation, all numbers are written in the form of a x 10b(a multiplied by the power of ten raised to the power of b), where the exponent b is an integer and the coefficient a is any real number.

Scientific Notation: There are three parts to writing a number in scientific notation: the coefficient, the base, and the exponent.

Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. Thomas Young’s discovery that light was a wave preceded the use of scientific notation, and he was obliged to write that the time required for one vibration of the wave was “ of a millionth of a millionth of a second”; an inconvenient way of expressing the point. Scientific notation is a less awkward and wordy way to write very large and very small numbers such as these.

A Simple System

Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten.

For instance,

, and so forth…

Each number is ten times bigger than the previous one. Since is ten times smaller than , it makes sense to use the notation to stand for one, the number that is in turn ten times smaller than . Continuing on, we can write to stand for 0.1, the number ten times smaller than . Negative exponents are used for small numbers:

Scientific notation displayed calculators can take other shortened forms that mean the same thing. For example, (written notation) is the same as (notation on some calculators) and (notation on some other calculators).

Round-off Error

A round-off error is the difference between the calculated approximation of a number and its exact mathematical value.

Learning Objectives

Explain the impact round-off errors may have on calculations, and how to reduce this impact

Key Takeaways

Key Points

- When a sequence of calculations subject to rounding error is made, these errors can accumulate and lead to the misrepresentation of calculated values.

- Increasing the number of digits allowed in a representation reduces the magnitude of possible round-off errors, but may not always be feasible, especially when doing manual calculations.

- The degree to which numbers are rounded off is relative to the purpose of calculations and the actual value.

Key Terms

- approximation: An imprecise solution or result that is adequate for a defined purpose.

Round-off Error

A round-off error, also called a rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations, algorithms, or both, especially when using finitely many digits to represent real numbers. When a sequence of calculations subject to rounding errors is made, errors may accumulate, sometimes dominating the calculation.

Calculations rarely lead to whole numbers. As such, values are expressed in the form of a decimal with infinite digits. The more digits that are used, the more accurate the calculations will be upon completion. Using a slew of digits in multiple calculations, however, is often unfeasible if calculating by hand and can lead to much more human error when keeping track of so many digits. To make calculations much easier, the results are often ’rounded off’ to the nearest few decimal places.

For example, the equation for finding the area of a circle is . The number (pi) has infinitely many digits, but can be truncated to a rounded representation of as 3.14159265359. However, for the convenience of performing calculations by hand, this number is typically rounded even further, to the nearest two decimal places, giving just 3.14. Though this technically decreases the accuracy of the calculations, the value derived is typically ‘close enough’ for most estimation purposes.

However, when doing a series of calculations, numbers are rounded off at each subsequent step. This leads to an accumulation of errors, and if profound enough, can misrepresent calculated values and lead to miscalculations and mistakes.

The following is an example of round-off error:

Rounding these numbers off to one decimal place or to the nearest whole number would change the answer to 5.7 and 6, respectively. The more rounding off that is done, the more errors are introduced.

Order of Magnitude Calculations

An order of magnitude is the class of scale of any amount in which each class contains values of a fixed ratio to the class preceding it.

Learning Objectives

Choose when it is appropriate to perform an order-of-magnitude calculation

Key Takeaways

Key Points

- Orders of magnitude are generally used to make very approximate comparisons and reflect very large differences.

- In the field of science, it is often sufficient for an estimate to be within an order of magnitude of the value in question.

- When estimating area or volume, you are much better off estimating linear dimensions and computing volume from those linear dimensions.

Key Terms

- Order of Magnitude: The class of scale or magnitude of any amount, where each class contains values of a fixed ratio (most often 10) to the class preceding it. For example, something that is 2 orders of magnitude larger is 100 times larger; something that is 3 orders of magnitude larger is 1000 times larger; and something that is 6 orders of magnitude larger is one million times larger, because 102 = 100, 103 = 1000, and 106 = one million

Orders of Magnitude

An order of magnitude is the class of scale of any amount in which each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount scaled is 10, and the scale is the exponent applied to this amount (therefore, to be an order of magnitude greater is to be 10 times, or 10 to the power of 1, greater). Such differences in order of magnitude can be measured on the logarithmic scale in “decades,” or factors of ten. It is common among scientists and technologists to say that a parameter whose value is not accurately known or is known only within a range is “on the order of” some value. The order of magnitude of a physical quantity is its magnitude in powers of ten when the physical quantity is expressed in powers of ten with one digit to the left of the decimal.

Orders of magnitude are generally used to make very approximate comparisons and reflect very large differences. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale — the larger value is less than ten times the smaller value.

It is important in the field of science that estimates be at least in the right ballpark. In many situations, it is often sufficient for an estimate to be within an order of magnitude of the value in question. Although making order-of-magnitude estimates seems simple and natural to experienced scientists, it may be completely unfamiliar to the less experienced.

An Example Estimation

Some of the mental steps of estimating in orders of magnitude are illustrated in answering the following example question: Roughly what percentage of the price of a tomato comes from the cost of transporting it in a truck?

Guessing the Number of Jelly Beans: Can you guess how many jelly beans are in the jar? If you try to guess directly, you will almost certainly underestimate. The right way to do it is to estimate the linear dimensions and then estimate the volume indirectly.

Incorrect solution: Let’s say the trucker needs to make a profit on the trip. Taking into account her benifits, the cost of gas, and maintenance and payments on the truck, let’s say the total cost is more like 2000. You might guess about 5000 tomatoes would fit in the back of the truck, so the extra cost per tomato is 40 cents. That means the cost of transporting one tomato is comparable to the cost of the tomato itself.

The problem here is that the human brain is not very good at estimating area or volume — it turns out the estimate of 5000 tomatoes fitting in the truck is way off. (This is why people have a hard time in volume-estimation contests, such as the one shown below.) When estimating area or volume, you are much better off estimating linear dimensions and computing the volume from there.

So, here’s a better solution: As before, let’s say the cost of the trip is $2000. The dimensions of the bin are probably 4m by 2m by 1m, for a volume of . Since our goal is just an order-of-magnitude estimate, let’s round that volume off to the nearest power of ten: . The shape of a tomato doesn’t follow linear dimensions, but since this is just an estimate, let’s pretend that a tomato is an 0.1m by 0.1m by 0.1m cube, with a volume of . We can find the total number of tomatoes by dividing the volume of the bin by the volume of one tomato: tomatoes. The transportation cost per tomato is  per tomato. That means that transportation really doesn’t contribute very much to the cost of a tomato. Approximating the shape of a tomato as a cube is an example of another general strategy for making order-of-magnitude estimates.

Dimensional Analysis

Converting from One Unit to Another

Converting units using dimensional analysis makes working with large and small measurements more convenient.

Learning Objectives

Describe the purpose of unit analysis

Key Takeaways

Key Points

- Dimensional analysis is the process of converting between units.

- The International System of Units (SI) specifies a set of seven base units from which all other units of measurement are formed. Derived units are based on those seven base units.

- Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension.

- Dimensional analysis involves using conversion factors, which are ratios of related physical quantities expressed in the desired units.

Key Terms

- dimensional analysis: A method of converting from one unit to another. It is also sometimes called unit conversion.

Base and Derived Units

For most quantities, a unit is absolutely necessary to communicate values of that physical quantity. Imagine you need to buy some rope to tie something onto the roof of a car. How would you tell the salesperson how much rope you need without using some unit of measurement?

However, not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Therefore, only a small set of units is required. These units are called base units, and other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units.

Different systems of units are based on different choices of base units. The most widely used system of units is the International System of Units, or SI. There are seven SI base units, and all other SI units can be derived from these base units.

The seven base SI units are: [Physical Quantity: unit symbol (unit name)]

- Length: m (meter)

- Mass: kg (kilogram)

- Time: s (second)

- Electric Current: A (Ampere)

- Thermodynamic Temperature: K (degrees Kelvin)

- Amount of Substance: mol ( mole )

- Luminous Intensity: cd (candela)

The base units of SI are actually not the smallest set possible; smaller sets have been defined. For example, there are unit sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic fields are actually different manifestations of the same phenomenon.

Derived units are based on units from the SI system of units. For example, volume is a derived unit because volume is based on length. To calculate the volume of something, you multiply the width x length x height, all in meters. Therefore, the derived unit for volume is m^3. Here is a list of some commonly derived units:

- Area: m^2

- Volume: m^3

- Velocity: m/s

- Acceleration: m/s^2

- Density: g/mL or g/cm^3

- Force: , or the Newton (N)

- Energy: , or the Joule (J)

Dimensional Analysis

Sometimes, it is necessary to deal with measurements that are very small (as in the size of an atom ) or very large (as in numbers of atoms). In these cases, it is often necessary to convert between units of metric measurement. For example, a mass measured in grams may be more convenient to work with if it was expressed in mg (10–3 g). Converting between metric units is called unit analysis or dimensional analysis.

Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension. Algebraically, we know that any number multiplied by one will be unchanged. If, however, the number has units, and we multiply it by a ratio containing units, the units in the number will multiply and divide by the units of the ratio, giving the original number (remember you are multiplying by one) but with different units.

This method can be generalized as: multiply or divide a given number by a known ratio to find your answer. The given number is a numerical quantity (with its units). The ratios used are based upon the units and are set up so that the units in the denominator of the ratio match the numerator units of the given and the units in the numerator of the ratio match those in either the next ratio or the final answer. When these are multiplied, the given number will now have the correct units for your answer.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=142

Converting Units with Conversion Factors – YouTube: How to convert units using conversion factors and canceling units.

Example 1

For example, say you were trying to convert 3.41 grams of He to a number of atoms of He. You would identify 3.41 grams as the given. The first step is always to place the given out front of your equation. Then find a ratio that will help you convert the units of grams to atoms. As you probably have already guessed, you need to use a couple of ratios to help you in this problem. The ratio that 4.002 g of He = 1 mole (molar mass) will help you in this problem. Avogadro’s number, 6.022 x 10^23 atoms = 1 mole, will also help you in this problem. Then you set up your ratios so that your units will cancel successfully (the same unit must be in the numerator of the equation and also in the denominator of the equation). Lastly, multiply through to get your final answer. As always, your final answer should contain the correct number of sig figs and the correct units.

Example 2

If you had a sample of a substance with a mass of 0.0034 grams, and you wanted to express that mass in mg, you could use the following dimensional analysis. The given quantity is the mass of 0.0034 grams. The quantity that you want to find is the mass in mg, and we know that 1 mg = 10-3 g. Expressing this as a proportion or ratio, there is one mg per 10-3 grams, or 1000 mg/1 g.

Therefore, 0.0034g x (1000 mg/1 g) = 3.4 mg

Strategy for General Problem Solving

To convert a measured quantity to a different unit of measure without changing the relative amount, use a conversion factor.

Learning Objectives

Apply knowledge of dimensional analysis to convert between units in chemistry problems

Key Takeaways

Key Points

- Chemistry, along with other sciences and engineering, makes use of many different units.

- In mathematics and chemistry, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount.

- Units behave just like numbers in products and quotients—they can be multiplied and divided.

Key Terms

- conversion factor: A conversion factor changes one unit to a new unit.

Dimensional Analysis

Chemistry, along with other sciences and engineering, makes use of many different units. Some of the common ones include mass (ton, pounds, ounces, grains, grams); length (yard, feet, inches, meters); and energy (Joule, erg, kcal, eV). Since there are so many different units that can be used, it is necessary to be able to convert between the various units. To do this, one uses a conversion factor.

In mathematics, specifically algebra, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount. To accomplish this, a ratio ( fraction ) is established that equals one (1). In the ratio, the conversion factor is a multiplier that, when applied to the original unit, converts the original unit into a new unit, by multiplication with the ratio.

When doing dimensional analysis problems, follow this list of steps:

1. Identify the given (see previous concept for additional information).

2. Identify conversion factors that will help you get from your original units to your desired unit.

3. Set up your equation so that your undesired units cancel out to give you your desired units. A unit will cancel out if it appears in both the numerator and the denominator during the equation.

4. Multiply through to get your final answer. Don’t forget the units and sig figs!

Example Problem 1

Here is an example problem: How many hours are in 3 days?

1. Identify the given: 3 days

2. Identify conversion factors that will help you get from your original units to your desired unit:

3. Set up your equation so that your undesired units cancel out to give you your desired units:

4. Multiply through to get your final answer: 72 hours

Flipping the Conversion Factor

Don’t forget that if need be, you can flip a conversion factor. After all, if a = b, then a/b = 1 and b/a = 1. For example, days are converted to hours by multiplying the days by the conversion factor of 24. The conversion can be reversed by dividing the hours by 24 to get days. The reciprocal 1/24 could be considered the reverse conversion factor for an hours-to-days conversion. The term “conversion factor” is the multiplier, not divisor, which yields the result.

Consider the relationship between feet and inches.

1 foot = 12 inches

1 foot/12 inches = 1 = 12 inches/1 foot.

Both fractions are equal to 1. If the units are ignored, the quotients do not numerically equal 1, but 1/12 or 12. However, with the inclusions of the units, both the numerators and denominators describe the exact same length, so the quotients are equal to 1. Since the two quotients are equal to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1. It does not change the equation, only the relative numerical values within the various units.

Example Problem 2

You can also use these quotients to convert from inches to feet or from feet to inches. For example, how many inches are in 5 feet?

1. The given is 5 feet.

2. The conversion factor is

3. Set up the equation:

4. Multiply through: 60 inches

Another example is: how many feet are in 30 inches?

If there is confusion regarding which quotient to use in the conversion, just make sure the units cancel out correctly. In the first equation, the unit (feet) is in both the numerator and denominator of the expression, so they cancel. The units behave just like numbers in products and quotients—they can be multiplied and divided.

Converting Between Moles and Grams

You can also use dimensional analysis to convert between moles and grams. For example:

Knowing that one molecule of H_2O contains two hydrogens (2 g/mol) and one oxygen (16g/mol), you can calculate the molecular weight of water equal to 18 g/mol. Then use this number in the equation to get 1.24 mol H_2O in 22.34g. The equation is set up correctly because all the units cancel out to give moles.

II

Chapter 2

- Basics of Kinematics

- Speed and Velocity

- Acceleration

- Free-Falling Objects

- Projectile Motion

Basics of Kinematics

Defining Kinematics

Kinematics is the study of the motion of points, objects, and groups of objects without considering the causes of its motion.

Learning Objectives

Define kinematics

Key Takeaways

Key Points

- To describe motion, kinematics studies the trajectories of points, lines and other geometric objects.

- The study of kinematics can be abstracted into purely mathematical expressions.

- Kinematic equations can be used to calculate various aspects of motion such as velocity, acceleration, displacement, and time.

Key Terms

- kinematics: The branch of mechanics concerned with objects in motion, but not with the forces involved.

Kinematics is the branch of classical mechanics that describes the motion of points, objects and systems of groups of objects, without reference to the causes of motion (i.e., forces ). The study of kinematics is often referred to as the “geometry of motion.”

Objects are in motion all around us. Everything from a tennis match to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects there is continuous motion in the vibrations of atoms and molecules. Interesting questions about motion can arise: how long will it take for a space probe to travel to Mars? Where will a football land if thrown at a certain angle? An understanding of motion, however, is also key to understanding other concepts in physics. An understanding of acceleration, for example, is crucial to the study of force.

To describe motion, kinematics studies the trajectories of points, lines and other geometric objects, as well as their differential properties (such as velocity and acceleration). Kinematics is used in astrophysics to describe the motion of celestial bodies and systems; and in mechanical engineering, robotics and biomechanics to describe the motion of systems composed of joined parts (such as an engine, a robotic arm, or the skeleton of the human body).

A formal study of physics begins with kinematics. The word “kinematics” comes from a Greek word “kinesis” meaning motion, and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. The study of kinematics can be abstracted into purely mathematical expressions, which can be used to calculate various aspects of motion such as velocity, acceleration, displacement, time, and trajectory.

Kinematics of a particle trajectory: Kinematic equations can be used to calculate the trajectory of particles or objects. The physical quantities relevant to the motion of a particle include: mass m, position r, velocity v, acceleration a.

Reference Frames and Displacement

In order to describe an object’s motion, you need to specify its position relative to a convenient reference frame.

Learning Objectives

Evaluate displacement within a frame of reference.

Key Takeaways

Key Points

- Choosing a frame of reference requires deciding where the object’s initial position is and which direction will be considered positive.

- Valid frames of reference can differ from each other by moving relative to one another.

- Frames of reference are particularly important when describing an object’s displacement.

- Displacement is the change in position of an object relative to its reference frame.

Key Terms

- displacement: A vector quantity that denotes distance with a directional component.

- frame of reference: A coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it.

In order to describe the motion of an object, you must first describe its position — where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of objects related to its position to or from Earth. Mathematically, the position of an object is generally represented by the variable x.

Frames of Reference

There are two choices you have to make in order to define a position variable x. You have to decide where to put x = 0 and which direction will be positive. This is referred to as choosing a coordinate system, or choosing a frame of reference. As long as you are consistent, any frame is equally valid. But you don’t want to change coordinate systems in the middle of a calculation. Imagine sitting in a train in a station when suddenly you notice that the station is moving backward. Most people would say that they just failed to notice that the train was moving — it only seemed like the station was moving. But this shows that there is a third arbitrary choice that goes into choosing a coordinate system: valid frames of reference can differ from each other by moving relative to one another. It might seem strange to use a coordinate system moving relative to the earth — but, for instance, the frame of reference moving along with a train might be far more convenient for describing things happening inside the train. Frames of reference are particularly important when describing an object’s displacement.

FRAMES OF REFERENCE by Professor Hume and Professor Donald Ivey of the University of Toronto

In this classic film, Professors Hume and Ivey cleverly illustrate reference frames and distinguish between fixed and moving frames of reference.

https://youtu.be/b4xmHNgiYtA

Frames of Reference (1960) Educational Film: Frames of Reference is a 1960 educational film by Physical Sciences Study Committee. The film was made to be shown in high school physics courses. In the film University of Toronto physics professors Patterson Hume and Donald Ivey explain the distinction between inertial and nonintertial frames of reference, while demonstrating these concepts through humorous camera tricks. For example, the film opens with Dr. Hume, who appears to be upside down, accusing Dr. Ivey of being upside down. Only when the pair flip a coin does it become obvious that Dr. Ivey — and the camera — are indeed inverted. The film’s humor serves both to hold students’ interest and to demonstrate the concepts being discussed. This PSSC film utilizes a fascinating set consisting of a rotating table and furniture occupying surprisingly unpredictable spots within the viewing area. The fine cinematography by Abraham Morochnik, and funny narration by University of Toronto professors Donald Ivey and Patterson Hume is a wonderful example of the fun a creative team of filmmakers can have with a subject that other, less imaginative types might find pedestrian. Producer: Richard Leacock Production Company: Educational Development Corp. Sponsor: Eric Prestamon

Displacement

Displacement is the change in position of an object relative to its reference frame. For example, if a car moves from a house to a grocery store, its displacement is the relative distance of the grocery store to the reference frame, or the house. The word “displacement” implies that an object has moved or has been displaced. Displacement is the change in position of an object and can be represented mathematically as follows:

where Δx is displacement, xf is the final position, and x_0 is the initial position.

shows the importance of using a frame of reference when describing the displacement of a passenger on an airplane.

Displacement in Terms of Frame of Reference: A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by x. The -4.0m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far).

Introduction to Scalars and Vectors

A vector is any quantity that has both magnitude and direction, whereas a scalar has only magnitude.

Learning Objectives

Distinguish the difference between scalars and vectors

Key Takeaways

Key Points

- A vector is any quantity that has magnitude and direction.

- A scalar is any quantity that has magnitude but no direction.

- Displacement and velocity are vectors, whereas distance and speed are scalars.

Key Terms

- scalar: A quantity that has magnitude but not direction; compare vector.

- vector: A directed quantity, one with both magnitude and direction; the between two points.

What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined by magnitude alone. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=68

Scalars and Vectors: Mr. Andersen explains the differences between scalar and vectors quantities. He also uses a demonstration to show the importance of vectors and vector addition.

In mathematics, physics, and engineering, a vector is a geometric object that has a magnitude (or length) and direction and can be added to other vectors according to vector algebra. The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (−) sign. A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, as shown in.

Vector representation: A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B.

Some physical quantities, like distance, either have no direction or no specified direction. In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations. It is any quantity that can be expressed by a single number and has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person’s 1.8 m height, and a distance of 2.0 m are all scalars, or quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows. (A comparison of scalars vs. vectors is shown in. )

Scalars vs. Vectors: A brief list of quantities that are either scalars or vectors.

Speed and Velocity

Average Velocity: A Graphical Interpretation

Average velocity is defined as the change in position (or displacement) over the time of travel.

Learning Objectives

Contrast speed and velocity in physics

Key Takeaways

Key Points

- Average velocity can be calculated by determining the total displacement divided by the total time of travel.

- The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point.

- Average velocity is different from average speed in that it considers the direction of travel and the overall change in position.

Key Terms

- velocity: A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.

In everyday usage, the terms “speed” and “velocity” are used interchangeably. In physics, however, they are distinct quantities. Speed is a scalar quantity and has only magnitude. Velocity, on the other hand, is a vector quantity and so has both magnitude and direction. This distinction becomes more apparent when we calculate average speed and velocity.

Average speed is calculated as the distance traveled over the total time of travel. In contrast, average velocity is defined as the change in position (or displacement) over the total time of travel.

Average Velocity: The kinematic formula for calculating average velocity is the change in position over the time of travel.

The SI unit for velocity is meters per second, or m/s, but many other units (such as km/h, mph, and cm/s) are commonly used. Suppose, for example, an airplane passenger took five seconds to move -4 m (the negative sign indicates that displacement is toward the back of the plane ). His average velocity would be:

v = Δx/t = -4m/5s = -0.8 m/s

The minus sign indicates that the average velocity is also toward the rear of the plane.

The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he gets to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.

To illustrate the difference between average speed and average velocity, consider the following additional example. Imagine you are walking in a small rectangle. You walk three meters north, four meters east, three meters south, and another four meters west. The entire walk takes you 30 seconds. If you are calculating average speed, you would calculate the entire distance (3 + 4 + 3 + 4 = 14 meters) over the total time, 30 seconds. From this, you would get an average speed of 14/30 = 0.47 m/s. When calculating average velocity, however, you are looking at the displacement over time. Because you walked in a full rectangle and ended up exactly where you started, your displacement is 0 meters. Therefore, your average velocity, or displacement over time, would be 0 m/s.

Average Speed vs. Average Velocity: If you started walking from one corner and went all the way around the rectangle in 30 seconds, your average speed would be 0.47 m/s, but your average velocity would be 0 m/s.

Instananeous Velocity: A Graphical Interpretation

Instantaneous velocity is the velocity of an object at a single point in time and space as calculated by the slope of the tangent line.

Learning Objectives

Differentiate instantaneous velocity from other ways of determining velocity

Key Takeaways

Key Points

- When velocity is constantly changing, we can estimate our velocity by looking at instantaneous velocity.

- Instantaneous velocity is calculated by determining the slope of the line tangent to the curve at the point of interest.

- Instantaneous velocity is similar to determining how many meters the object would travel in one second at a specific moment.

Key Terms

- instantaneous: (As in velocity)—occurring, arising, or functioning without any delay; happening within an imperceptibly brief period of time.

Typically, motion is not with constant velocity nor speed. While driving in a car, for example, we continuously speed up and slow down. A graphical representation of our motion in terms of distance vs. time, therefore, would be more variable or “curvy” rather than a straight line, indicating motion with a constant velocity as shown below. (We limit our discussion to one dimensional motion. It should be straightforward to generalize to three dimensional cases.)

Motion with Changing Velocity: Motion is often observed with changing velocity. This would result in a curvy line when graphed with distance over time.

To calculate the speed of an object from a graph representing constant velocity, all that is needed is to find the slope of the line; this would indicate the change in distance over the change in time. However, changing velocity it is not as straightforward.

Since our velocity is constantly changing, we can estimate velocity in different ways. One way is to look at our instantaneous velocity, represented by one point on our curvy line of motion graphed with distance vs. time. In order to determine our velocity at any given moment, we must determine the slope at that point. To do this, we find a line that represents our velocity in that moment, shown graphically in. That line would be the line tangent to the curve at that point. If we extend this line, we can easily calculate the displacement of distance over time and determine our velocity at that given point. The velocity of an object at any given moment is the slope of the tangent line through the relevant point on its x vs. t graph.

Determining instantaneous velocity: The velocity at any given moment is defined as the slope of the tangent line through the relevant point on the graph

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Instantaneous Velocity, Acceleration, Jerk, Slopes, Graphs vs. Time: This is how kinematics begins.

In calculus, finding the slope of curve f(x) at x=x_0 is equivalent to finding the first derivative:

.

One interpretation of this definition is that the velocity shows how many meters the object would travel in one second if it continues moving at the same speed for at least one second.

Acceleration

Graphical Interpretation

The graphical representation of acceleration over time can be derived through the graph of an object’s position over time.

Learning Objectives

Distinguish the difference between how to plot a velocity graph and how to plot an acceleration graph

Key Takeaways

Key Points

- Acceleration is the rate at which the velocity of a body changes with time.

- Acceleration is a vector that points in the same direction as the change in velocity, though it may not always be in the direction of motion.

- Because acceleration is velocity in m/s divided by time in s, we can derive a graph of acceleration from a graph of an object’s speed or position.

Key Terms

- acceleration: The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).

- velocity: A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.

- position: A place or location.

In physics, acceleration is the rate at which the velocity of a body changes with time. It is a vector quantity with both magnitude and direction. Acceleration is accompanied by a force, as described by Newton’s Second Law; the force, as a vector, is the product of the mass of the object being accelerated and the acceleration (vector), or . The SI unit of acceleration is the meter per second squared:

Acceleration is a vector that points in the same direction as the change in velocity, though it may not always be in the direction of motion. For example, when an object slows down, or decelerating, its acceleration is in the opposite direction of its motion.

The motion of an object can be depicted graphically by plotting the position of an object over time. This distance-time graph can be used to create another graph that shows changes in velocity over time. Because acceleration is velocity in divided by time in s, we can further derive a graph of acceleration from a graph of an object’s speed or position.

is a graph of an object’s position over time. This graph is similar to the motion of a car. In the beginning, the object’s position changes slowly as it gains speed. In the middle, the speed is constant and the position changes at a constant rate. As it slows down toward the end, the position changes more slowly. From this graph, we can derive a velocity vs time graph.

Position vs Time Graph: Notice that the object’s position changes slowly at the beginning of the journey, then more and more quickly as it picks up speed. Its position then changes more slowly as it slows down at the end of the journey. In the middle of the journey, while the velocity remains constant, the position changes at a constant rate.

This shows the velocity of the object over time. The object’s velocity increases in the beginning as it accelerates at the beginning, then remains constant in the middle before it slows down toward the end. Notice that this graph is a representation of the slope of the previous position vs time graph. From this graph, we can further derive an acceleration vs time graph.

Velocity vs Time: The object’s velocity increases as it accelerates at the beginning of the journey. It remains the same in the middle of the journey (where there is no acceleration). It decreases as the object decelerates at the end of the journey.

To do this, we would also plot the slope of the velocity vs time graph. In this graph, the acceleration is constant in the three different stages of motion. As we noted earlier, the object is increasing speed and changing positions slowly in the beginning. The acceleration graph shows that the object was increasing at a positive constant acceleration during this time. In the middle, when the object was changing position at a constant velocity, the acceleration was 0. This is because the object is no longer changing its velocity and is moving at a constant rate. Towards the end of the motion, the object slows down. This is depicted as a negative value on the acceleration graph. Note that in this example, the motion of the object is still forward (positive), but since it is decelerating, the acceleration is negative.

Acceleration vs Time Graph: The object has positive acceleration as it speeds up at the beginning of the journey. It has no acceleration as it travels at constant velocity in the middle of the journey. Its acceleration is negative as it slows down at the end of the journey.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=72

Graphing Motion: A brief introduction to particle diagrams and motion graphs.

Motion with Constant Acceleration

Constant acceleration occurs when an object’s velocity changes by an equal amount in every equal time period.

Learning Objectives

Describe how constant acceleration affects the motion of an object

Key Takeaways

Key Points

- Assuming acceleration to be constant does not seriously limit the situations we can study and does not degrade the accuracy of our treatment.

- Due to the algebraic properties of constant acceleration, there are kinematic equations that can be used to calculate displacement, velocity, acceleration, and time.

- Calculations with constant acceleration can be done in relation to one-dimensional motion as well as two-dimensional motion.

Key Terms

- acceleration: The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).

- kinematic: of or relating to motion or kinematics

One-Dimensional Motion: When you drop an object, it falls vertically toward the center of the earth due to the constant acceleration of gravity.

An object experiencing constant acceleration has a velocity that increases or decreases by an equal amount for any constant period of time. Acceleration can be derived easily from basic kinematic principles. It is defined as the first time derivative of velocity (so the second derivative of position with respect to time):

Assuming acceleration to be constant does not seriously limit the situations we can study and does not degrade the accuracy of our treatment, because in a great number of situations, acceleration is constant. When it is not, we can either consider it in separate parts of constant acceleration or use an average acceleration over a period of time.

The motion of falling objects is a simple, one-dimensional type of projectile motion in which there is no horizontal movement. For example, if you held a rock out and dropped it, the rock would fall only vertically downward toward the earth. If you were to throw the rock instead of just dropping it, it would follow a more projectile-like pattern, similar to the one a kicked ball follows.

Projectile motion is the motion of an object thrown or projected into the air and is subject only to the acceleration of gravity. The object thrown is called a projectile, and the object’s path is called its trajectory. In two-dimensional projectile motion, there is both a vertical and a horizontal component.

Due to the algebraic properties of constant acceleration, there are kinematic equations that relate displacement, initial velocity, final velocity, acceleration, and time. A summary of these equations is given below.

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Constant Acceleration Explained with Vectors and Algebra: This video answers the question “what is acceleration? “.

Free-Falling Objects

Free-Falling Objects

Free fall is the motion of a body where its weight is the only force acting on an object.

Learning Objectives

Solve basic problems concerning free fall and distinguish it from other kinds of motion

Key Takeaways

Key Points

- The acceleration of free-falling objects is called the acceleration due to gravity, since objects are pulled towards the center of the earth.

- The acceleration due to gravity is constant on the surface of the Earth and has the value of 9.80 .

Key Terms

- acceleration: The amount by which a speed or velocity changes within a certain period of time (and so a scalar quantity or a vector quantity).

Free Fall

The motion of falling objects is the simplest and most common example of motion with changing velocity. If a coin and a piece of paper are simultaneously dropped side by side, the paper takes much longer to hit the ground. However, if you crumple the paper into a compact ball and drop the items again, it will look like both the coin and the paper hit the floor simultaneously. This is because the amount of force acting on an object is a function of not only its mass, but also area. Free fall is the motion of a body where its weight is the only force acting on an object.

Free Fall: This clip shows an object in free fall.

Galileo also observed this phenomena and realized that it disagreed with the Aristotle principle that heavier items fall more quickly. Galileo then hypothesized that there is an upward force exerted by air in addition to the downward force of gravity. If air resistance and friction are negligible, then in a given location (because gravity changes with location), all objects fall toward the center of Earth with the same constant acceleration, independent of their mass, that constant acceleration is gravity. Air resistance opposes the motion of an object through the air, while friction opposes motion between objects and the medium through which they are traveling. The acceleration of free-falling objects is referred to as the acceleration due to gravity . As we said earlier, gravity varies depending on location and altitude on Earth (or any other planet), but the average acceleration due to gravity on Earth is 9.8 . This value is also often expressed as a negative acceleration in mathematical calculations due to the downward direction of gravity.

Equations

The best way to see the basic features of motion involving gravity is to start by considering straight up and down motion with no air resistance or friction. This means that if the object is dropped, we know the initial velocity is zero. Once the object is in motion, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration, . The kinematic equations for objects experiencing free fall are:

where , , , and .

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=74

Free Fall Motion – YouTube: Describes how to calculate the time for an object to fall if given the height and the height that an object fell if given the time to fall.

Examples

Some examples of objects that are in free fall include:

- A spacecraft in continuous orbit. The free fall would end once the propulsion devices turned on.

- An stone dropped down an empty well.

- An object, in projectile motion, on its descent.

Projectile Motion

Basic Equations and Parabolic Path

Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory.

Learning Objectives

Assess the effect of angle and velocity on the trajectory of the projectile; derive maximum height using displacement

Key Takeaways

Key Points

- Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path.

- The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection.

- When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward.

- The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object.

Key Terms

- trajectory: The path of a body as it travels through space.

- symmetrical: Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.

Projectile Motion

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e. and ).

Initial Velocity

The initial velocity can be expressed as x components and y components:

In this equation, stands for initial velocity magnitude and refers to projectile angle.

Time of Flight

The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, depends on the initial velocity magnitude and the angle of the projectile:

Acceleration

In projectile motion, there is no acceleration in the horizontal direction. The acceleration, , in the vertical direction is just due to gravity, also known as free fall:

Velocity

The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time, , the velocity is:

You can also use the Pythagorean Theorem to find velocity:

Displacement

At time, t, the displacement components are:

The equation for the magnitude of the displacement is .

Parabolic Trajectory

We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:

Maximum Height

The maximum height is reached when . Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height

where stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height

Range

The range of the motion is fixed by the condition . Using this we can rearrange the parabolic motion equation to find the range of the motion:

.

Range of Trajectory: The range of a trajectory is shown in this figure.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=76

Projectiles at an Angle: This video gives a clear and simple explanation of how to solve a problem on Projectiles Launched at an Angle. I try to go step by step through this difficult problem to layout how to solve it in a super clear way. 2D kinematic problems take time to solve, take notes on the order of how I solved it. Best wishes. Tune into my other videos for more help. Peace.

Solving Problems

In projectile motion, an object moves in parabolic path; the path the object follows is called its trajectory.

Learning Objectives

Identify which components are essential in determining projectile motion of an object

Key Takeaways

Key Points

- When solving problems involving projectile motion, we must remember all the key components of the motion and the basic equations that go along with them.

- Using that information, we can solve many different types of problems as long as we can analyze the information we are given and use the basic equations to figure it out.

- To clear two posts of equal height, and to figure out what the distance between these posts is, we need to remember that the trajectory is a parabolic shape and that there are two different times at which the object will reach the height of the posts.

- When dealing with an object in projectile motion on an incline, we first need to use the given information to reorientate the coordinate system in order to have the object launch and fall on the same surface.

Key Terms

- reorientate: to orientate anew; to cause to face a different direction

We have previously discussed projectile motion and its key components and basic equations. Using that information, we can solve many problems involving projectile motion. Before we do this, let’s review some of the key factors that will go into this problem-solving.

What is Projectile Motion?

Projectile motion is when an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning, after which the only influence on the trajectory is that of gravity.

What are the Key Components of Projectile Motion?

The key components that we need to remember in order to solve projectile motion problems are:

- Initial launch angle,

- Initial velocity,

- Time of flight,

- Acceleration,

- Horizontal velocity,

- Vertical velocity,

- Displacement,

- Maximum height,

- Range,

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=76

How To Solve Any Projectile Motion Problem (The Toolbox Method): Introducing the “Toolbox” method of solving projectile motion problems! Here we use kinematic equations and modify with initial conditions to generate a “toolbox” of equations with which to solve a classic three-part projectile motion problem.

Now, let’s look at two examples of problems involving projectile motion.

Examples

Example 1

Let’s say you are given an object that needs to clear two posts of equal height separated by a specific distance. Refer to for this example. The projectile is thrown at m/s at an angle of 45°. If the object is to clear both posts, each with a height of 30m, find the minimum: (a) position of the launch on the ground in relation to the posts and (b) the separation between the posts. For simplicity’s sake, use a gravity constant of 10. Problems of any type in physics are much easier to solve if you list the things that you know (the “givens”).

Diagram for Example 1: Use this figure as a reference to solve example 1. The problem is to make sure the object is able to clear both posts.

Solution: The first thing we need to do is figure out at what time  the object reaches the specified height. Since the motion is in a parabolic shape, this will occur twice: once when traveling upward, and again when the object is traveling downward. For this we can use the equation of displacement in the vertical direction, :

We substitute in the appropriate variables:

Therefore:

We can use the quadratic equation to find that the roots of this equation are 2s and 3s. This means that the projectile will reach 30m after 2s, on its way up, and after 3s, on its way down.

Example 2

An object is launched from the base of an incline, which is at an angle of 30°. If the launch angle is 60° from the horizontal and the launch speed is 10 m/s, what is the total flight time? The following information is given: ; °; .

Diagram for Example 2: When dealing with an object in projectile motion on an incline, we first need to use the given information to reorient the coordinate system in order to have the object launch and fall on the same surface.

Solution: In order to account for the incline angle, we have to reorient the coordinate system so that the points of projection and return are on the same level. The angle of projection with respect to the direction is , and the acceleration in the direction is . We replace with and with :

Zero Launch Angle

An object launched horizontally at a height travels a range during a time of flight .

Learning Objectives

Explain the relationship between the range and the time of flight

Key Takeaways

Key Points

- For the zero launch angle, there is no vertical component in the initial velocity.

- The duration of the flight before the object hits the ground is given as T = \sqrt{\frac{2H}{g}}.

- In the horizontal direction, the object travels at a constant speed v_0 during the flight. The range R (in the horizontal direction) is given as: .

Key Terms

- trajectory: The path of a body as it travels through space.

Projectile motion is a form of motion where an object moves in a parabolic path. The path followed by the object is called its trajectory. Projectile motion occurs when a force is applied at the beginning of the trajectory for the launch (after this the projectile is subject only to the gravity).

One of the key components of the projectile motion, and the trajectory it follows, is the initial launch angle. The angle at which the object is launched dictates the range, height, and time of flight the object will experience while in projectile motion. shows different paths for the same object being launched at the same initial velocity and different launch angles. As illustrated by the figure, the larger the initial launch angle and maximum height, the longer the flight time of the object.

Projectile Trajectories: The launch angle determines the range and maximum height that an object will experience after being launched.This image shows that path of the same object being launched at the same speed but different angles.

We have previously discussed the effects of different launch angles on range, height, and time of flight. However, what happens if there is no angle, and the object is just launched horizontally? It makes sense that the object should be launched at a certain height (), otherwise it wouldn’t travel very far before hitting the ground. Let’s examine how an object launched horizontally at a height travels. In our case is when is 0.

Projectile motion: Projectile moving following a parabola.Initial launch angle is , and the velocity is .

Duration of Flight

There is no vertical component in the initial velocity () because the object is launched horizontally. Since the object travels distance in the vertical direction before it hits the ground, we can use the kinematic equation for the vertical motion:

Here, is the duration of the flight before the object its the ground. Therefore:

Range

In the horizontal direction, the object travels at a constant speed during the flight. Therefore, the range (in the horizontal direction) is given as:

General Launch Angle

The initial launch angle (0-90 degrees) of an object in projectile motion dictates the range, height, and time of flight of that object.

Learning Objectives

Choose the appropriate equation to find range, maximum height, and time of flight

Key Takeaways

Key Points

- If the same object is launched at the same initial velocity, the height and time of flight will increase proportionally to the initial launch angle.

- An object launched into projectile motion will have an initial launch angle anywhere from 0 to 90 degrees.

- The range of an object, given the initial launch angle and initial velocity is found with: .

- The maximum height of an object, given the initial launch angle and initial velocity is found with:.

- The time of flight of an object, given the initial launch angle and initial velocity is found with: .

- The angle of reach is the angle the object must be launched at in order to achieve a specific distance: .

Key Terms

- trajectory: The path of a body as it travels through space.

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning of the trajectory, after which the only interference is from gravity.

One of the key components of projectile motion and the trajectory that it follows is the initial launch angle. This angle can be anywhere from 0 to 90 degrees. The angle at which the object is launched dictates the range, height, and time of flight it will experience while in projectile motion. shows different paths for the same object launched at the same initial velocity at different launch angles. As you can see from the figure, the larger the initial launch angle, the closer the object comes to maximum height and the longer the flight time. The largest range will be experienced at a launch angle up to 45 degrees.

Launch Angle: The launch angle determines the range and maximum height that an object will experience after being launched. This image shows that path of the same object being launched at the same velocity but different angles.

The range, maximum height, and time of flight can be found if you know the initial launch angle and velocity, using the following equations:

Where R – Range, h – maximum height, T – time of flight, vi – initial velocity, θi – initial launch angle, g – gravity.

Now that we understand how the launch angle plays a major role in many other components of the trajectory of an object in projectile motion, we can apply that knowledge to making an object land where we want it. If there is a certain distance, d, that you want your object to go and you know the initial velocity at which it will be launched, the initial launch angle required to get it that distance is called the angle of reach. It can be found using the following equation:

Key Points: Range, Symmetry, Maximum Height

Projectile motion is a form of motion where an object moves in parabolic path. The path that the object follows is called its trajectory.

Learning Objectives

Construct a model of projectile motion by including time of flight, maximum height, and range

Key Takeaways

Key Points

- Objects that are projected from and land on the same horizontal surface will have a path symmetric about a vertical line through a point at the maximum height of the projectile.

- The time it takes from an object to be projected and land is called the time of flight. It depends on the initial velocity of the projectile and the angle of projection.

- The maximum height of the projectile is when the projectile reaches zero vertical velocity. From this point the vertical component of the velocity vector will point downwards.

- The horizontal displacement of the projectile is called the range of the projectile and depends on the initial velocity of the object.

- If an object is projected at the same initial speed, but two complementary angles of projection, the range of the projectile will be the same.

Key Terms

- gravity: Resultant force on Earth’s surface, of the attraction by the Earth’s masses, and the centrifugal pseudo-force caused by the Earth’s rotation.

- trajectory: The path of a body as it travels through space.

- bilateral symmetry: the property of being symmetrical about a vertical plane

What is Projectile Motion ?

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In this atom we are going to discuss what the various components of an object in projectile motion are, we will discuss the basic equations that go along with them in another atom, “Basic Equations and Parabolic Path”

Key Components of Projectile Motion:

Time of Flight, T:

The time of flight of a projectile motion is exactly what it sounds like. It is the time from when the object is projected to the time it reaches the surface. The time of flight depends on the initial velocity of the object and the angle of the projection, . When the point of projection and point of return are on the same horizontal plane, the net vertical displacement of the object is zero.

Symmetry:

All projectile motion happens in a bilaterally symmetrical path, as long as the point of projection and return occur along the same horizontal surface. Bilateral symmetry means that the motion is symmetrical in the vertical plane. If you were to draw a straight vertical line from the maximum height of the trajectory, it would mirror itself along this line.

Maximum Height, H:

The maximum height of a object in a projectile trajectory occurs when the vertical component of velocity, , equals zero. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decelerate. Eventually the vertical velocity will reach zero, and the projectile is accelerated downward under gravity immediately. Once the projectile reaches its maximum height, it begins to accelerate downward. This is also the point where you would draw a vertical line of symmetry.

Range of the Projectile, R:

The range of the projectile is the displacement in the horizontal direction. There is no acceleration in this direction since gravity only acts vertically. shows the line of range. Like time of flight and maximum height, the range of the projectile is a function of initial speed.

Range: The range of a projectile motion, as seen in this image, is independent of the forces of gravity.

III

Chapter 3

- Introduction

- Force and Mass

- Newton’s Laws

- Other Examples of Forces

- Introduction of Linear Momentum and Collision

- Conservation of Momentum

Media Attributions

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Media Attributions

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Introduction

Newton and His Laws

There are three laws of motion that describe the relationship between forces, mass, and acceleration.

Learning Objectives

Apply three Newton’s laws of motion to relate forces, mass, and acceleration

Key Takeaways

Key Points

- Acceleration of an object is proportional to the force on it.

- Force causes an object to move.

- Objects with more mass require more force to move.

Key Terms

- force: Any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.

Newton’s laws of motion describe the relationship between the forces acting on a body and its motion due to those forces. For example, if your car breaks down and you need to push it, you must exert a force with your hands on the car in order for it to move. The laws of motion will tell you how quickly the car will move from your pushing. There are three laws of motion:

First law: If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero). For example, if you don’t push the car (no force), then it doesn’t move.

Second law: The acceleration  of a body is parallel and directly proportional to the net force acting on the body, is in the direction of the net force, and is inversely proportional to the mass of the body:

or

For example, if you push the car with a greater force it will accelerate more. But, if the car is more massive ( is larger) then it won’t accelerate as much from the same size force as a lighter car.

Third law: When a first body exerts a force on a second body, the second body simultaneously exerts a force on the first body. This means that and are equal in magnitude and opposite in direction. For example, when you push a car, if it is exerting the same force on you that you are exerting on it, you might wonder why you don’t move backwards? The answer is there are also forces from the ground on your feet pushing you forward. So, in fact, the car is pushing a force back on you that is of the same magnitude that you are using to push it forward.

In the figure below there are some practical examples illustrating the concept of force:

- Strain: by using a machine known as pulley you can easily raise or lower a massive body

- Gravitational Force: a massive body is attracted downward by the gravitational force practiced by the Earth

- Magnetic Force: two magnets repel each other when the same poles get closer

Examples of Force: Some situations in which forces are at play.

Force and Mass

Force

Force is any influence that causes an object to change, either concerning its movement, direction, or geometrical construction.

Learning Objectives

Develop the relationship between mass and acceleration in determining force

Key Takeaways

Key Points

- Force is stated as a vector quantity, meaning it has elements of both magnitude and direction. Mass and acceleration respectively.

- In layman’s terms, force is a push or pull that can be defined in terms of various standards.

- Dynamics is the study of the force that causes objects and systems to move or deform.

- External forces are any outside forces that act on a body, and internal forces are any force acting within a body.

Key Terms

- force: A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.

- velocity: A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.

- vector: A directed quantity, one with both magnitude and direction; the between two points.

Overview of Forces

In physics, a force is any influence that causes an object to undergo a certain change, either concerning its movement, direction, or geometrical construction. It is measured with the SI unit of Newtons. A force is that which can cause an object with mass to change its velocity, i.e., to accelerate, or which can cause a flexible object to deform. Force can also be described by intuitive concepts such as a push or pull. A force has both magnitude and direction, making it a vector quantity.

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What is a force?: Describes what forces are and what they do.

Qualities of Force

The original form of Newton’s second law states that the net force acting upon an object is equal to the rate at which its momentum changes. This law is further given to mean that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.

As we mentioned, force is a vector quantity. A vector is a one dimensional array with elements of both magnitude and direction. In a force vector, the mass, , is the magnitude component and the acceleration, , is the directional component. The equation for force is written:

Related concepts to force include thrust, which increases the velocity of an object; drag, which decreases the velocity of an object; and torque which produces changes in rotational speed of an object. Forces which do not act uniformly on all parts of a body will also cause mechanical stresses, a technical term for influences which cause deformation of matter. While mechanical stress can remain embedded in a solid object, gradually deforming it, mechanical stress in a fluid determines changes in its pressure and volume.

Dynamics

Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. Our intuitive definition of force — that is, a push or a pull —is a good place to start. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in each regard.

shows a few examples of the “push-pull” nature of force. The top left example is that of a pulley system. The force that someone would have to pull down on the cable would have to equal and exceed the force made by the mass the object and the effects of gravity on those object in order for the system to move up. The top right example shows that any object resting on a surface will still exert force on that surface. The bottom example is that of two magnets being attracted to each other due to magnetic force.

Examples of Force: Some situations in which forces are at play.

Mass

Mass is a physical property of matter that depends on size and shape of matter, and is expressed as kilograms by the SI system.

Learning Objectives

Justify the significance of understanding mass in physics

Key Takeaways

Key Points

- Mass is defined as a quantitative measure of an object’s resistance to acceleration.

- According to Newton’s second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m.

- Mass is central in many concepts of physics, including:weight, momentum, acceleration, and kinetic energy.

- According to Newton’s second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m.

Key Terms

- mass: The quantity of matter which a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.

What is Mass?

All elements have physical properties whose values can help describe an elements physical state. Changes to these properties can describe elemental transformations. Physical properties do not change the chemical nature of matter. The physical property we are covering in this atom is called mass.

Mass is defined as a quantitative measure of an object’s resistance to acceleration. The terms mass and weight are often interchanged, however it is incorrect to do so. Weight is a different property of matter that, while related to mass, is not mass, but rather the amount of gravitational force acting on a given body of matter. Mass is an intrinsic property that never changes.

Units of Mass

In order to measure something, a standard value must be established to use in relation to the object of measurement. This relation is called a unit. The International System of Units (SI) measures mass in kilograms, or kg. There are other units of mass, including the following (only the first two are accepted by the SI system):

- t – Tonne; 1t = 1000kg

- u – atomic mass unit; 1u ~= 1.66×10-27kg

- sl – slug

- lb – pound

Concepts Using Mass

- Weight – see

- Newtons Second Law – mass has a central role in determining the behavior of bodies. Newtons Second Law relates force f, exerted in a body of mass m, to the body’s acceleration a: F=ma

- Momentum – mass relates a body’s momentum, p, to its linear velocity, v: p = mv

- Kinetic Energy – mass relates kinetic energy, K to velocity, v:

Newton’s Laws

The First Law: Inertia

Newton’s first law of motion describes inertia. According to this law, a body at rest tends to stay at rest, and a body in motion tends to stay in motion, unless acted on by a net external force.

Learning Objectives

Define the First Law of Motion

Key Takeaways

Key Points

- Newton’s three laws of physics are the basis for mechanics.

- The first law states that a body at rest will stay at rest until a net external force acts upon it and that a body in motion will remain in motion at a constant velocity until acted on by a net external force.

- Net external force is the sum of all of the forcing acting on an object.

- Just because there are forces acting on an object doesn’t necessarily mean that there is a net external force; forces that are equal in magnitude but acting in opposite directions can cancel one another out.

- Friction is the force between an object in motion and the surface on which it moves. Friction is the external force that acts on objects and causes them to slow down when no other external force acts upon them.

- Inertia is the tendency of a body in motion to remain in motion. Inertia is dependent on mass, which is why it is harder to change the direction of a heavy body in motion than it is to change the direction of a lighter object in motion.

Key Terms

- inertia: The property of a body that resists any change to its uniform motion; equivalent to its mass.

- friction: A force that resists the relative motion or tendency to such motion of two bodies in contact.

- uniform motion: Motion at a constant velocity (with zero acceleration). Note that an object in motion will not change its velocity unless an unbalanced force acts upon it.

History

Sir Isaac Newton was an English scientist who was interested in the motion of objects under various conditions. In 1687, he published a work called Philosophiae Naturalis Principla Mathematica, which described his three laws of motion. Newton used these laws to explain and explore the motion of physical objects and systems. These laws form the basis for mechanics. The laws describe the relationship between forces acting on a body and the motions experienced due to these forces. The three laws are as follows:

1. If an object experiences no net force, its velocity will remain constant. The object is either at rest and the velocity is zero or it moves in a straight line with a constant speed.

2. The acceleration of an object is parallel and directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.

3. When a first object exerts a force on a second object, the second object simultaneously exerts a force on the first object, meaning that the force of the first object and the force of the second object are equal in magnitude and opposite in direction.

The First Law of Motion

You have most likely heard Newton’s first law of motion before. If you haven’t heard it in the form written above, you have probably heard that “a body in motion stays in motion, and a body at rest stays at rest.” This means that an object that is in motion will not change its velocity unless an unbalanced force acts upon it. This is called uniform motion. It is easier to explain this concept through examples.

Examples

If you are ice skating, and you push yourself away from the side of the rink, according to Newton’s first law you will continue all the way to the other side of the rink. But, this won’t actually happen. Newton says that a body in motion will stay in motion until an outside force acts upon it. In this and most other real world cases, this outside force is friction. The friction between your ice skates and the ice is what causes you to slow down and eventually stop.

Let’s look at another situation. Refer to for this example. Why do we wear seat belts? Obviously, they’re there to protect us from injury in case of a car accident. If a car is traveling at 60 mph, the driver is also traveling at 60 mph. When the car suddenly stops, an external force is applied to the car that causes it to slow down. But there is no force acting on the driver, so the driver continues to travel at 60 mph. The seat belt is there to counteract this and act as that external force to slow the driver down along with the car, preventing them from being harmed.

Newton’s First Law: Newton’s first law in effect on the driver of a car

Inertia

Sometimes this first law of motion is referred to as the law of inertia. Inertia is the property of a body to remain at rest or to remain in motion with constant velocity. Some objects have more inertia than others because the inertia of an object is equivalent to its mass. This is why it is more difficult to change the direction of a boulder than a baseball.

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Doc Physics – Newton: Newton’s first law is hugely counterintuitive. You may have learned it in gradeschool, though. Let’s see it for the mind-blowing conclusion it really is.

The Second Law: Force and Acceleration

The second law states that the net force on an object is equal to the rate of change, or derivative, of its linear momentum.

Learning Objectives

Define the Second Law of Motion

Key Takeaways

Key Points

- Newton’s three laws of motion explain the relationship between forces acting on an object and the motion they experience due to these forces. These laws act as the basis for mechanics.

- The second law explains the relationship between force and motion, as opposed to velocity and motion. It uses the concept of linear momentum to do this.

- Linear momentum , is the product of mass , and velocity : .

- The second law states that the net force is equal to the derivative, or rate of change of its linear momentum.

- By simplifying this relationship and remembering that acceleration is the rate of change of velocity, we can see that the second law of motion is where the relationship between force and acceleration comes from.

Key Terms

- net force: The combination of all the forces that act on an object.

- momentum: (of a body in motion) the product of its mass and velocity.

- acceleration: The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).

English scientist Sir Isaac Newton examined the motion of physical objects and systems under various conditions. In 1687, he published his three laws of motion in Philosophiae Naturalis Principla Mathematica. The laws form the basis for mechanics—they describe the relationship between forces acting on a body, and the motion experienced due to these forces. These three laws state:

1. If an object experiences no net force, its velocity will remain constant. The object is either at rest and the velocity is zero, or it moves in a straight line with a constant speed.

2. The acceleration of an object is parallel and directly proportional to the net force acting on the object, is in the direction of the net force and is inversely proportional to the mass of the object.

3. When a first object exerts a force on a second object, the second object simultaneously exerts a force on the first object, meaning that the force of the first object and the force of the second object are equal in magnitude and opposite in direction.

The first law of motion defines only the natural state of the motion of the body (i.e., when the net force is zero). It does not allow us to quantify the force and acceleration of a body. The acceleration is the rate of change in velocity; it is caused only by an external force acting on it. The second law of motion states that the net force on an object is equal to the rate of change of its linear momentum.

Linear Momentum

Linear momentum of an object is a vector quantity that has both magnitude and direction. It is the product of mass and velocity of a particle at a given time:

where, , , and . From this equation, we see that objects with more mass will have more momentum.

The Second Law of Motion

Picture two balls of different mass, traveling in the same direction at the same velocity. If they both collide with a wall at the same time, the heavier ball will exert a larger force on the wall. This concept, illustrated below, explains Newton’s second law, which emphasizes the importance of force and motion, over velocity alone. It states: the net force on an object is equal to the rate of change of its linear momentum. From calculus we know that the rate of change is the same as a derivative. When we the linear momentum of an object we get:

Force and Mass: This animation demonstrates the connection between force and mass.

where, F = Force and t = time. From this we can further simplify the equation:

where, . As we stated earlier, acceleration is the rate of change of velocity, or velocity divided by time.

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Newton’s Three Laws of Mechanics – Second Law – Part 1: Here we’ll see how many people can confuse your understanding of Newton’s 2nd law of motion through oversight, sloppy language, or cruel intentions.

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Newton’s Three Laws of Mechanics – Second Law – Part Two: Equilibrium is investigated and Newton’s 1st law is seen as a special case of Newton’s 2nd law!

The Third Law: Symmetry in Forces

The third law of motion states that for every action, there is an equal and opposite reaction.

Learning Objectives

Define the Third Law of Motion

Key Takeaways

Key Points

- If an object A exerts a force on object B, object B exerts an equal and opposite force on object A.

- Newton’s third law can be seen in many everyday circumstances. When you walk, the force you use to push off the ground backwards makes you move forward.

- Thrust is an application of the third law of motion. A helicopter uses thrust to push the air under the propeller down, and therefore lift off the ground.

Key Terms

- symmetry: Exact correspondence on either side of a dividing line, plane, center or axis.

- thrust: The force generated by propulsion, as in a jet engine.

Sir Isaac Newton was a scientist from England who was interested in the motion of objects under various conditions. In 1687, he published a work called Philosophiae Naturalis Principla Mathematica, which contained his three laws of motion. Newton used these laws to explain and explore the motion of physical objects and systems. These laws form the bases for mechanics. The laws describe the relationship between forces acting on a body, and the motion is an experience due to these forces. Newton’s three laws are:

1. If an object experiences no net force, its velocity will remain constant. The object is either at rest and the velocity is zero or it moves in a straight line with a constant speed.

2. The acceleration of an object is parallel and directly proportional to the net force acting on the object, is in the direction of the net force and is inversely proportional to the mass of the object.

3. When a first object exerts a force on a second object, the second object simultaneously exerts a force on the first object, meaning that the force of the first object and the force of the second object are equal in magnitude and opposite in direction.

Newton’s Third Law of Motion

Newton’s third law basically states that for every action, there is an equal and opposite reaction. If object A exerts a force on object B, because of the law of symmetry, object B will exert a force on object A that is equal to the force acted on it:

In this example, FA is the action and FB is the reaction. You have undoubtedly witnessed this law of motion. For example, take a swimmer who uses her feet to push off the wall in order to gain speed. The more force she exerts on the wall, the harder she pushes off. This is because the wall exerts the same force on her that she forces on it. She pushes the wall in the direction behind her, therefore the wall will exert a force on her that is in the direction in front of her and propel her forward.

Newton’s Third Law of Motion: When a swimmer pushes off the wall, the swimmer is using the third law of motion.

Take as another example, the concept of thrust. When a rocket launches into outer space, it expels gas backward at a high velocity. The rocket exerts a large backward force on the gas, and the gas exerts and equal and opposite reaction force forward on the rocket, causing it to launch. This force is called thrust. Thrust is used in cars and planes as well.

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Newton’s Third Law: The most fundamental statement of basic physical reality is also the most often misunderstood. As your mom if she’s clear on Newton’s Third. Then ask her why things can move if every force has a paired opposite force all the time, forever.

Other Examples of Forces

Weight

Weight is taken as the force on an object due to gravity, and is different than the mass of an object.

Learning Objectives

Infer what factors other than gravity will contribute to the apparent weight of an object

Key Takeaways

Key Points

- Weight is taken to be the force on an object due to gravity.

- Weight and mass are not the same thing!

- The weight of a given mass will be different when the acceleration due to gravity is different.

- /Apparent weight can change because of the effect of buoyancy.

- The strength of gravity is almost the same everywhere on the surface of the Earth.

Key Terms

- Gravitational acceleration: Gravitational acceleration is the acceleration that an object undergoes due solely to gravity

In physics, it is important to differentiate the weight of an object from its mass. The mass of an object is an intrinsic quantity, independent of the location of the object. On the other hand, the weight of an object is an extrinsic quantity. It is considered as the force on an object due to gravity. Since gravitational acceleration changes depending on the location in the universe, weight does as well.

Mathematically, the weight of an object (W) can be found by multiplying its mass (m) by the acceleration due to gravity (g): . The strength of gravity varies very little over the surface of the Earth. In fact, the greatest percent difference in the value of the acceleration due to gravity on Earth is 0.5%.

For most calculations involving the weight of an object on Earth, it is sufficient to assume that .

The weight of an object has the same SI unit as force—the Newton ().

In US customary units, the weight of an object can be expressed in pounds. Keep in mind that in US units the pound is either a unit of force or of mass. If one must find the weight (as opposed to the mass) of an object in US units, it can be calculated in terms of pounds of force.

It is important to note that the apparent weight of an object (i.e., the weight of an object determined by a scale) will vary if forces other than gravity are acting upon the object. For example, if you weigh a given mass underwater you will find a different result than if you weigh that mass in air. In this case, the weight of the object varies due to the force of buoyancy. While the mass is in the water it displaces fluid, resulting in an upward force upon it. This upward force affects the net force that the mass exerts on the scale, and thus alters its “apparent” weight.

Spring Scale: A spring scale measures weight by finding the extent to which a spring is compressed.This is proportional to the force that a mass exerts on the scale due to its weight.

Normal Forces

The normal force comes about when an object contacts a surface; the resulting force is always perpendicular to the surface of contact.

Learning Objectives

Evaluate Newton’s Second and Third Laws in determining the normal force on an object

Key Takeaways

Key Points

- The normal force, , comes about when an object contacts a surface.

- The normal force exists because for every force, there is always an equal and opposite force.

- The normal force is always perpendicular to the plane that the object contacts or rests on.

Key Terms

- perpendicular: at or forming a right angle (to).

- normal: A line or vector that is perpendicular to another line, surface, or plane.

Overview

The normal force, , comes about when an object contacts a surface. According to Newton’s third law, when one object exerts a force on a second object, the second object always exerts a force that is equal in magnitude and opposite in direction on the first object. This is the reason that the normal force exists.

A common situation in which a normal force exists is when a person stands on the ground. Because of Newton’s third law, the ground exerts a force on the person that is equal in magnitude to the person’s weight. In this simple case, the weight of the person and the opposing normal force are the only two forces considered on the person. The person remains still because the forces due to weight and the normal force create a net force of zero on the person.

Forces on Inclined Planes

A more complex example of a situation in which a normal force exists is when a mass rests on an inclined plane. In this case, the normal force is not in the exact opposite direction as the force due to the weight of the mass. This is because the mass contacts the surface at an angle. By taking this angle into account, the magnitude of the normal force () can be found from:

,

where:

- is the mass under consideration,

- is the acceleration due to gravity,

- and is the angle between the inclined surface and the horizontal.

Inclined Plane: A mass rests on an inclined plane that is at an angle to the horizontal. The following forces act on the mass: the weight of the mass (),the force due to friction (),and the normal force ().

Another interesting example involving normal forces is when a person stands in an elevator. When the elevator goes up, the normal force is actually greater than the force due to gravity. In this situation there are only two forces acting on the person. The first is the force of gravity on the person, which does not change. The second is the normal force. By summing the forces and setting them equal to (utilizing Newton’s second law), we find:

where:

- is the normal force,

- is the force due to gravity,

- is the mass of the person,

- and is the acceleration.

Since acceleration is positive, the normal force must actually be greater than the force due to gravity (the weight of the person).

Introduction

Linear Momentum

Linear momentum is the product of the mass and velocity of an object, it is conserved in elastic and inelastic collisions.

Learning Objectives

Calculate the momentum of two colliding objects

Key Takeaways

Key Points

- Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude.

- Momentum, like energy, is important because it is a conserved quantity.

- The momentum of a system of particles is the sum of their momenta. If two particles have masses m_1 and m_2, and velocities v_1 and v_2, the total momentum is .

Key Terms

- inelastic: (As referring to an inelastic collision, in contrast to an elastic collision. ) A collision in which kinetic energy is not conserved.

- elastic collision: An encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.

- conservation: A particular measurable property of an isolated physical system does not change as the system evolves.

In classical mechanics, linear momentum, or simply momentum (SI unit kg m/s, or equivalently N s), is the product of the mass and velocity of an object. Mathematically it is stated as:

.

(Note here that p and v are vectors. ) Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude. Linear momentum is particularly important because it is a conserved quantity, meaning that in a closed system (without any external forces ) its total linear momentum cannot change.

Because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Momentum is conserved in both inelastic and elastic collisions. (Kinetic energy is not conserved in inelastic collisions but is conserved in elastic collisions. ) It important to note that if the collision takes place on a surface with friction, or if there is air resistance, we would need to account for the momentum of the bodies that would be transferred to the surface and/or air.

Let’s take a look at a simple, one-dimensional example: The momentum of a system of two particles is the sum of their momenta. If two particles have masses m_1 and m_2, and velocities v_1 and v_2, the total momentum is:

.

Keep in mind that momentum and velocity are vectors. Therefore, in 1D, if two particles are moving in the same direction, v1 and v2 have the same sign. If the particles are moving in opposite directions they will have opposite signs.

If two particles were moving on a plane we would choose our xy-plane to be on the plane of motion. We can then write the x and y component of the total momentum as:

If the 2D momentum vector is decomposed into two components, the equations for each component are reduced to its 1D equivalents.

Momentum, like energy, is important because it is conserved. “Newton’s cradle” shown in is an example of conservation of momentum. As we will discuss in the next concept (on Momentum, Force, and Newton’s Second Law ), in classical mechanics, conservation of linear momentum is implied by Newton’s laws. Only a few physical quantities are conserved in nature. Studying these quantities yields fundamental insight into how nature works.

Newton’s Cradle: Total momentum of the system (or Cradle) is conserved. (neglecting frictional loss in the system. )

Momentum, Force, and Newton’s Second Law

In the most general form, Newton’s 2nd law can be written as .

Learning Objectives

Relate Newton’s Second Law to momentum and force

Key Takeaways

Key Points

- In a closed system, without any external forces, the total momentum is constant.

- The familiar equation is a special case of the more general form of the 2nd law when the mass of the system is constant.

- Momentum conservation holds (in the absence of external force) regardless of the nature of the interparticle (or internal) force, no matter how complicated the force is between particles.

Key Terms

- closed system: A physical system that doesn’t exchange any matter with its surroundings and isn’t subject to any force whose source is external to the system.

In a closed system (one that does not exchange any matter with the outside and is not acted on by outside forces), the total momentum is constant. This fact, known as the law of conservation of momentum, is implied by Newton’s laws of motion. Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that

or

Therefore, total momentum (p_1+p_2) is constant. If the velocities of the particles are u_1 and u_2 before the interaction, and afterwards they are v_1 and v_2, then

This law holds regardless of the nature of the interparticle (or internal) force, no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero.

Newton’s Second Law

Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is

,

where is the net external force, Δp is the change in momentum, and Δt is the change in time.

This statement of Newton’s second law of motion includes the more familiar as a special case. We can derive this form as follows. First, note that the change in momentum Δp is given by Δp=Δ(mv). If the mass of the system is constant, then Δ(mv)=mΔv. So for constant mass, Newton’s second law of motion becomes

.

Because Δv/Δt=a, we get the familiar equation when the mass of the system is constant. Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass.

Momentum in a Closed System: In a game of pool, the system of entire balls can be considered a closed system. Therefore, the total momentum of the balls is conserved.

Impulse

Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts.

Learning Objectives

Explain the relationship between change in momentum and the amount of time a force acts

Key Takeaways

Key Points

- A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important.

- A force produces an acceleration, and the greater the force acting on an object, the greater its change in velocity and, hence, the greater its change in momentum. However, changing momentum is also related to how long a time the force acts.

- In case of a time-varying force, impulse can be calculated by integrating the force over the time duration. .

Key Terms

- momentum: (of a body in motion) the product of its mass and velocity.

- impulse: The integral of force over time.

Impulse

Forces produce either acceleration or deceleration on moving bodies, and the greater the force acting on an object, the greater its change in velocity and, hence, the greater its change in momentum. However, changing momentum is also related to how long a time the force acts. If a brief force is applied to a stalled automobile, a change in its momentum is produced. The same force applied over an extended period of time produces a greater change in the automobile’s momentum. The quantity of impulse is force × time interval, or in shorthand notation:

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Momentum & Impulse: A brief overview of momentum and impulse for high school physics students.

,

where F is the net force on the system, and Δt is the duration of the force.

From Newton’s 2nd law:

(Δp: change in momentum),

change in momentum equals the average net external force multiplied by the time this force acts.

.

Therefore, impulse as defined in the previous paragraph is simply equivalent to p.

A force sustained over a long time produces more change in momentum than does the same force applied briefly. A small force applied for a long time can produce the same momentum change as a large force applied briefly because it is the product of the force and the time for which it is applied that is important. Impulse is always equal to change in momentum and is measured in Ns (Newton seconds), as both force and the time interval are important in changing momentum.

Our definition of impulse includes an assumption that the force is constant over the time interval Δt. Forces are usually not constant. Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force Feff that produces the same result as the corresponding time-varying force. shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times t1 and t2. That area is equal to the area inside the rectangle bounded by Feff, t1, and t2. Thus, the impulses and their effects are the same for both the actual and effective forces. Equivalently, we can simple find the area under the curve F(t) between t1 and t2 to compute the impulse in mathematical form:

Force vs. Time: A graph of force versus time with time along the x-axis and force along the y-axis for an actual force and an equivalent effective force. The areas under the two curves are equal.

.

Conservation of Momentum

Internal vs. External Forces

Net external forces (that are nonzero) change the total momentum of the system, while internal forces do not.

Learning Objectives

Contrast the effects of external and internal forces on linear momentum and collisions

Key Takeaways

Key Points

- External forces are forces caused by external agent outside of the system.

- Internal forces are forces exchanged by the objects in the system.

- To determine what part should be considered external and internal, mechanical system should be clearly defined.

Key Terms

- inelastic: (As referring to an inelastic collision, in contrast to an elastic collision. ) A collision in which kinetic energy is not conserved.

- elastic: referring to elastic collision, in contrast to inelastic collision. A collision in which kinetic energy is conserved

Linear Momentum and Collisions

Newton’s 2nd law, applied to an isolated system composed of particles, indicates that the total momentum of the entire system should be constant in the absence of net external forces. Forces external to the system may change the total momentum when their sum is not 0, but internal forces, regardless of the nature of the forces, will not contribute to the change in the total momentum. To analyze a mechanical system, it is important to recognize which forces are internal and which are external. Once a mechanical system is clearly defined, it’s not hard to understand what part should be considered external.

Newton’s Cradle: Total momentum of the system (or Cradle) is conserved. (neglecting frictional loss in the system. )

- External forces: forces caused by external agent outside of the system.

- Internal forces: forces exchanged by the particles in the system.

To give you a better idea, let’s consider a simple example. We have two hockey pucks sliding across a frictionless surface, and we neglect air resistance for simplicity. They collide with each other at t=0.

Let’s first list all the forces present in the system. There are mainly three kinds of forces: Gravity, normal force (between ice & pucks), and frictional forces during the collision between the pucks

How should we define our system? In most cases, we would be interested in the motion of the pucks (and nothing else). Therefore, our system consists of two pucks (and nothing else). All the rest of the universe becomes external. With this in mind, we can see that gravity and normal forces are external, while the frictional forces between pucks are internal. Since all the external forces cancel out with each other, there are no net external forces. (Gravity and normal force on each puck have the same magnitude, but are in the opposite directions) Therefore, we conclude that the total momentum of the two pucks should be a conserved quantity.

- In the previous example, it is worthwhile to note that we didn’t assume anything about the nature of the collision between the two pucks. Without knowing anything about the internal forces (frictional forces during contact), we learned that the total momentum of the system is a conserved quantity (p1 and p2 are momentum vectors of the pucks. ) In fact, this relation holds true both in elastic or inelastic collisions. Whether the total kinetic energy of the pucks is conserved or not, total momentum is conserved.

- Also note that, in the previous example, if we include the rest of the Earth in our system, the gravity and normal forces themselves become internal.

IV

Introduction to Chemistry

1

Discoveries Leading to Nuclear Atom Model

Cathode Rays

Cathode rays are electron beams or streams of electrons that were observed for the first time in Crookes tubes (vacuum tubes).

Learning Objectives

Define a cathode ray

Key Takeaways

Key Points

- Electrons accelerated to high velocities travel in straight lines through an empty cathode ray tube and strike the glass wall of the tube, causing excited atoms to fluoresce or glow.

- Researchers realized that something was traveling from the anode when objects placed in the tube in front of it could cast a shadow on the glowing wall. Cathode rays carry electronic currents through the tube. Electrons were first discovered as the constituents of cathode rays.

- J.J. Thomson used the cathode ray tube to determine that atoms had small negatively charged particles inside of them, which he called “electrons.”

Key Terms

- cathode rays: Streams of electrons observed in vacuum tubes

- crookes tube: An early experimental electrical discharge tube, invented by English physicist William Crookes and others around 1869-1875, in which cathode rays, streams of electrons, were discovered

Cathode Rays

Cathode rays (also called an electron beam or an e-beam) are streams of electrons observed in vacuum tubes. If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass opposite the negative electrode is observed to glow from electrons emitted from the cathode. Electrons were first discovered as the constituents of cathode rays. The image in a classic television set is created by focused beam of electrons deflected by electric or magnetic fields in cathode ray tubes (CRTs).

Cathode rays are so named because they are emitted by the negative electrode, or cathode, in a vacuum tube. To release electrons into the tube, they must first be detached from the atoms of the cathode. The early cold cathode vacuum tubes, called Crookes tubes, used a high electrical potential between the anode and the cathode to ionize the residual gas in the tube. The electric field accelerated the ions and the ions released electrons when they collided with the cathode.

Modern vacuum tubes use thermionic emission, in which the cathode is made of a thin wire filament that is heated by a separate electric current passing through it. The increased random heat motion of the filament atoms knocks electrons out of the atoms at the surface of the filament and into the evacuated space of the tube. Since the electrons have a negative charge, they are repelled by the cathode and attracted to the anode. They travel in straight lines through the empty tube. The voltage applied between the electrodes accelerates these low mass particles to high velocities.

Cathode rays are invisible, but their presence was first detected in early vacuum tubes when they struck the glass wall of the tube, exciting the atoms of the glass and causing them to emit light—a glow called fluorescence. Researchers noticed that objects placed in the tube in front of the cathode could cast a shadow on the glowing wall, and realized that something must be traveling in straight lines from the cathode. After the electrons reach the anode, they travel through the anode wire to the power supply and back to the cathode, so cathode rays carry electric current through the tube.

History of Cathode Rays

In 1838, Michael Faraday passed a current through a rarefied air-filled glass tube and noticed a strange light arc with its beginning at the cathode (negative electrode) and its end almost at the anode (positive electrode).

Crookes Tubes

In the 1870s, British physicist William Crookes and others were able to evacuate rarefied tubes to a pressure below 10−6 atm. These were called Crookes tubes. Faraday had been the first to notice a dark space just in front of the cathode, where there was no luminescence. This came to be called the cathode dark space, Faraday dark space, or Crookes dark space.

Crookes found that as he pumped more air out of the tubes, the Faraday dark space spread down the tube from the cathode toward the anode, until the tube was totally dark. But at the anode (positive) end of the tube, the glass of the tube itself began to glow. What was happening was that as more air was pumped from the tubes, the electrons could travel farther, on average, before they struck a gas atom. By the time the tube was dark, most of the electrons could travel in straight lines from the cathode to the anode end of the tube without a collision. With no obstructions, these low mass particles were accelerated to high velocities by the voltage between the electrodes. These were the cathode rays. When they reached the anode end of the tube, they were traveling so fast that, although they were attracted to it, they often flew past the anode and struck the back wall of the tube. When they struck atoms in the glass wall, they excited their orbital electrons to higher energy levels, causing them to fluoresce.

Crookes tube: A Crookes tube is a rarefied tube evacuated to a pressure below 10−6 atm. It was used in discovery of cathode rays.

Later researchers painted the inside back wall with fluorescent chemicals such as zinc sulfide, to make the glow more visible. Cathode rays themselves are invisible, but this accidental fluorescence allowed researchers to notice that objects in the tube in front of the cathode, such as the anode, cast sharp-edged shadows on the glowing back wall. In 1869, German physicist Johann Hittorf was first to realize that something must be traveling in straight lines from the cathode to cast the shadows. Eugene Goldstein named them cathode rays.

J.J. Thomson’s Experiment

J.J. Thomson studied cathode ray tubes and came up with the idea that the particles in the cathode beams must be negative because they were repelled by negatively charged items (either the cathode or a negatively charged plate in the cathode ray tube) and attracted by positively charged items (either the anode or the positively charged plate in the cathode ray tube). He called these super tiny pieces of the atom, “electrons.” Through his experiments, Thomson disproved Dalton’s atomic theory, because Dalton’s atomic theory stated that atoms were the smallest piece of the matter in the universe and they were indivisible. Clearly, the presence of electrons negated these portions of Dalton’s atomic theory.

Interactive: Crookes Tube: Connect the two electrodes to a high voltage source and see them produce cathode rays. J.J. Thomson used a similar experimental setup to discover the first sub-atomic particle. See what you can determine about the stream of particles that were called cathode rays.

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Discovery of the Electron: Cathode Ray Tube Experiment – YouTube: J.J. Thompson discovered the electron, the first of the subatomic particles, using the cathode ray tube experiment. Thompson’s work disproved John Dalton’s theory of the atom.

Millikan’s Oil Drop Experiment

The oil drop experiment calculated the charge of an electron using charged oil droplets suspended in an electric field.

Learning Objectives

Describe the major findings of Millikan’s oil drop experiment

Key Takeaways

Key Points

- Millikan’s oil drop experiment measured the charge of an electron. Before this experiment, existence of subatomic particles was not universally accepted.

- Millikan’s apparatus contained an electric field created between a parallel pair of metal plates, which were held apart by insulating material. Electrically charged oil droplets entered the electric field and were balanced between two plates by altering the field.

- When the charged drops fell at a constant rate, the gravitational and electric forces on it were equal. Therefore, the charge on the oil drop was calculated using formula Q = Millikan found that the charge of a single electron was 1.6 x 10-19 C.

Key Terms

- electron: The subatomic particle having a negative charge and orbiting the nucleus; the flow of electrons in a conductor constitutes electricity.

- oil drop experiment: Experiment performed by Robert Millikan and Harvey Fletcher in 1909 to measure the charge of the electron.

The Oil Drop Experiment

In 1909, Robert Millikan and Harvey Fletcher conducted the oil drop experiment to determine the charge of an electron. They suspended tiny charged droplets of oil between two metal electrodes by balancing downward gravitational force with upward drag and electric forces. The density of the oil was known, so Millikan and Fletcher could determine the droplets’ masses from their observed radii (since from the radii they could calculate the volume and thus, the mass). Using the known electric field and the values of gravity and mass, Millikan and Fletcher determined the charge on oil droplets in mechanical equilibrium. By repeating the experiment, they confirmed that the charges were all multiples of some fundamental value. They calculated this value to be 1.5924 × 10−19 Coulombs (C), which is within 1% of the currently accepted value of 1.602176487 × 10−19 C. They proposed that this was the charge of a single electron.

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Charge of an Electron: Millikan: How did scientists discover how much negative charge an electron had? Robert Millikan and Harvey Fletcher used the oil drop experiment.

How did the process work?

The figure below shows a simplified scheme of Millikan’s oil drop experiment. The apparatus incorporated a pair of metal plates and a specific type of oil. Millikan and Fletcher discovered it was best to use an oil with an extremely low vapor pressure, such as one designed for use in a vacuum apparatus. Ordinary oil would evaporate under the heat of the light source, causing the mass of the oil drop to change over the course of the experiment.

By applying a potential difference across a parallel pair of horizontal metal plates, a uniform electric field was created in the space between them. A ring of insulating material was used to hold the plates apart. Four holes were cut into the ring—three for illumination by a bright light and another to allow viewing through a microscope. A fine mist of oil droplets was sprayed into a chamber above the plates. The oil drops became electrically charged through friction with the nozzle as they were sprayed. Alternatively, charge could be induced by including an ionizing radiation source (such as an X-ray tube).

Simplified scheme of Millikan’s oil-drop experiment: This apparatus has a parallel pair of horizontal metal plates. A uniform electric field is created between them. The ring has three holes for illumination and one for viewing through a microscope. A specific type of oil is sprayed into the chamber, where drops become electrically charged. The droplets enter the space between the plates and can be controlled by changing the voltage across the plates.

The droplets entered the space between the plates and, because they were charged, they could be controlled by changing the voltage across the plates. Initially, the oil drops were allowed to fall between the plates with the electric field turned off. They quickly reached terminal velocity due to friction with the air in the chamber. The field was turned on and, if it was large enough, some of the drops (the charged ones) would start to rise. This is because the upwards electric force, FE, is greater for them than the downwards gravitational force, g. (A charged rubber rod can pick up bits of paper in the same way.) A likely looking drop was selected and kept in the middle of the field of view by alternately switching off the voltage until all the other drops fell. The experiment was continued with this single drop.

Millikan’s experiment was meant to have the drops fall at a constant rate. At this constant rate, the force of gravity on the drop and the force of the electric field on the drop are equal:

Fup = Fdown

Fup = QE Fdown = m

Q is the charge of an electron, E is the electric field, m is mass of the droplet, and g is gravity.

QE = mg

Q =

One can see how Millikan calculated the charge of an electron. Millikan found that all drops had charges which were multiples of 1.6 x 10-19 C.

At the time of Millikan and Fletcher’s oil drop experiments, the existence of subatomic particles was not universally accepted. Experimenting with cathode rays in 1897, J. J. Thomson discovered negatively charged “corpuscles” with a mass about 1,840 times smaller than that of a hydrogen atom. George FitzGerald and Walter Kaufmann found similar results. In 1923, Millikan won the Nobel Prize in physics in part because of this experiment.

Aside from discerning an electron’s charge, the beauty of the oil drop experiment lies in its simple and elegant demonstration that charge is actually quantized. The experiment has since been repeated by generations of physics students—although it is rather expensive and difficult to do properly.

Oil drop apparatus: A very similar oil drop apparatus was used by Millikan.

2

History of Atomic Structure

Early Ideas about Atoms

The concept of the atom as an indivisible building block of matter was recorded as early as the 5th century BCE.

Learning Objectives

Describe the early developments leading to the modern concept of the atom

Key Takeaways

Key Points

- The ancient Greek philosophers Democritus and Leucippus recorded the concept of the atomos, an indivisible building block of matter, as early as the 5th century BCE.

- The idea of an indivisible particle was further elaborated upon and explored by a number of scientists and philosophers, including Galileo, Newton, Boyle, Lavoisier, and Dalton.

- John Dalton, an English chemist and meteorologist, is credited with the first modern atomic theory based on his experiments with atmospheric gases.

Key Terms

- atom: The smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.

- law of multiple proportions: The law stating that reactants will always combine in set whole number ratios.

Early History of the Atom

Matter is composed of indivisible building blocks. This idea was recorded as early as the fifth century BCE by Leucippus and Democritus. The Greeks called these particles atomos, meaning indivisible, and the modern word “atom” is derived from this term. Democritus proposed that different types and combinations of these particles were responsible for the various forms of matter. However, these ideas were largely ignored at the time, as most philosophers favored the Aristotelian perspective.

The concept of the atom was revisited and elaborated upon by many scientists and philosophers, including Galileo, Newton, Boyle, and Lavoisier. In 1661, Boyle presented a discussion of atoms in his The Sceptical Chymist. However, the English chemist and meteorologist John Dalton is credited with the first modern atomic theory, as explained in his ANew Systemof Chemical Philosophy.

John Dalton’s A New System of Chemical Philosophy: Chemical structures from Dalton’s A New System of Chemical Philosophy.

Dalton’s experiments with gases led to some of the earliest measurements of atomic masses and a concept of atomic structure and reactivity. Dalton’s atomic theory contained the following ideas:

- All atoms of a given element are identical.

- The atoms of different elements vary in mass and size.

- Atoms are indestructible. Chemical reactions may result in their rearrangement, but not their creation or destruction.

Dalton also outlined a law of multiple proportions, which described how reactants will combine in set ratios. Like the early philosophers, Dalton’s theories were not popularly accepted for much of the 19th century, but his ideas have since been accepted, with amendments addressing subatomic particles and the interconversion of energy and mass.

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Models of the Atom Timeline – YouTube: This video is about the different ways that scientists have pictured the atoms over the years. It starts with Democritus and Leucippus, the first philosophers to discuss atoms. The video also covers the work of Dalton, Thompson, Rutherford, Niels Bohr, and Schrodinger.

The Law of Conservation of Mass

The law of conservation of mass states that mass in an isolated system is neither created nor destroyed.

Learning Objectives

Define the law of conservation of mass

Key Takeaways

Key Points

- The law of conservation of mass states that mass in an isolated system is neither created nor destroyed by chemical reactions or physical transformations.

- According to the law of conservation of mass, the mass of the products in a chemical reaction must equal the mass of the reactants.

- The law of conservation of mass is useful for a number of calculations and can be used to solve for unknown masses, such the amount of gas consumed or produced during a reaction.

Key Terms

- law of conservation of mass: A law that states that mass cannot be created or destroyed; it is merely rearranged.

- product: A chemical substance formed as a result of a chemical reaction.

- reactant: Any of the participants present at the start of a chemical reaction. Also, a molecule before it undergoes a chemical change.

History of the Law of the Conservation of Mass

Antoine Lavoisier: A portrait of Antoine Lavoisier, the scientist credited with the discovery of the law of conservation of mass.

The ancient Greeks first proposed the idea that the total amount of matter in the universe is constant. However, Antoine Lavoisier described the law of conservation of mass (or the principle of mass/matter conservation) as a fundamental principle of physics in 1789.

This law states that, despite chemical reactions or physical transformations, mass is conserved—that is, it cannot be created or destroyed—within an isolated system. In other words, in a chemical reaction, the mass of the products will always be equal to the mass of the reactants.

The Law of Conservation of Mass- Energy

This law was later amended by Einstein in the law of conservation of mass-energy, which describes the fact that the total mass and energy in a system remain constant. This amendment incorporates the fact that mass and energy can be converted from one to another. However, the law of conservation of mass remains a useful concept in chemistry, since the energy produced or consumed in a typical chemical reaction accounts for a minute amount of mass.

We can therefore visualize chemical reactions as the rearrangement of atoms and bonds, while the number of atoms involved in a reaction remains unchanged. This assumption allows us to represent a chemical reaction as a balanced equation, in which the number of moles of any element involved is the same on both sides of the equation. An additional useful application of this law is the determination of the masses of gaseous reactants and products. If the sums of the solid or liquid reactants and products are known, any remaining mass can be assigned to gas.

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Conservation of Atoms – YouTube: This video explains how atoms are conserved in a chemical reaction.

The Law of Definite Composition

The law of definite composition states that chemical compounds are composed of a fixed ratio of elements as determined by mass.

Learning Objectives

Define the law of definite composition

Key Takeaways

Key Points

- The law of definite composition was proposed by Joseph Proust based on his observations on the composition of chemical compounds.

- Proust proposed that a compound is always composed of the same proportions of elements by mass.

- Though initially controversial, the law of definite composition was supported by Dalton’s atomic theory.

Key Terms

- element: Any one of the simplest chemical substances that cannot be decomposed in a chemical reaction or by any chemical means, and are made up of atoms all having the same number of protons.

- law of definite composition: A law that states that chemical compounds are formed of constant and defined ratios of elements as determined by mass.

History of the Law of Definite Composition or Proportions

Joseph Proust: Portrait of Joseph Proust

French chemist Joseph Proust proposed the law of definite composition or proportions based on his experiments conducted between 1798 and 1804 on the elemental composition of water and copper carbonate.

In 1806, Proust summarized his observations in what is now called Proust’s Law. It stated that chemical compounds are formed of constant and defined ratios of elements, as determined by mass. For example, carbon dioxide is composed of one carbon atom and two oxygen atoms. Therefore, by mass, carbon dioxide can be described by the fixed ratio of 12 (mass of carbon):32 (mass of oxygen), or simplified as 3:8.

At the time, Proust’s theory was a controversial one and disputed by a number of chemists, most notably another French chemist, Claude Louis Berthollet. Berthollet supported the concept that elements could mix in any ratio. However, the English chemist John Dalton’s formulation of atomic theory supported Proust’s idea at an atomic level, as Dalton proposed that chemical compounds were composed of set formulations of atoms from different elements. Dalton’s law of multiple proportions expanded on the law of definite composition to postulate that, in situations in which elements can combine to form multiple combinations, the ratio of the elements in those compounds can be expressed as small whole numbers.

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John Dalton and the Law of Definite Proportions – YouTube: This video examines the law of definite proportions and the law of multiple proportions.

Applications of the Law of Definite Composition or Proportions

The law of definite composition has applications to both molecular compounds with a fixed composition and ionic compounds as they require certain ratios to achieve electrical neutrality. There are some exceptions to the law of definite composition. These compounds are known as non-stoichometric compounds, and examples include ferrous oxide. In addition, the law of definite composition does not account for isotopic mixtures.

The Law of Multiple Proportions

The law of multiple proportions states that elements combine in small whole number ratios to form compounds.

Learning Objectives

Define the law of multiple proportions.

Key Takeaways

Key Points

- The law of multiple proportions is a rule of stoichiometry.

- John Dalton formulated the law of multiple proportions as part of his theory that atoms formed the basic indivisible building block of matter.

- The law of multiple proportions says that when elements form compounds, the proportions of the elements in those chemical compounds can be expressed in small whole number ratios.

- The law of multiple proportions is an extension of the law of definite composition, which states that compounds will consist of defined ratios of elements.

Key Terms

- law of multiple proportions: A law stating that if two elements form a compound, then the ratio of the mass of the second element and the mass of the first element will be small whole number ratios.

- atom: The smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.

Dalton’s Law

The law of multiple proportions, also known as Dalton’s law, was proposed by the English chemist and meteorologist John Dalton in his 1804 work, A New Systemof Chemical Philosophy. It is a rule of stoichiometry. The law, which was based on Dalton’s observations of the reactions of atmospheric gases, states that when elements form compounds, the proportions of the elements in those chemical compounds can be expressed in small whole number ratios.

For example, the reaction of the elements carbon and oxygen can yield both carbon monoxide (CO) and carbon dioxide (CO_2). In CO_2, the ratio of the amount of oxygen compared to the amount of carbon is a fixed ratio of 1:2, a ratio of simple whole numbers. In CO, the ratio is 1:1.

In his theory of atomic structure and composition, Dalton presented the concept that all matter was composed of different combinations of atoms, which are the indivisible building blocks of matter. Dalton’s law of multiple proportions is part of the basis for modern atomic theory, along with Joseph Proust’s law of definite composition (which states that compounds are formed by defined mass ratios of reacting elements) and the law of conservation of mass that was proposed by Antoine Lavoisier. These laws paved the way for our current understanding of atomic structure and composition, including concepts like molecular or chemical formulas.

John Dalton and Atomic Theory

Dalton introduced a theory that proposed that elements differed due to the mass of their atoms.

Learning Objectives

Identify the main points of Dalton’s atomic theory

Key Takeaways

Key Points

- Dalton’s atomic theory proposed that all matter was composed of atoms, indivisible and indestructible building blocks. While all atoms of an element were identical, different elements had atoms of differing size and mass.

- Dalton’s atomic theory also stated that all compounds were composed of combinations of these atoms in defined ratios.

- Dalton also postulated that chemical reactions resulted in the rearrangement of the reacting atoms.

Key Terms

- atom: The smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.

- atomic mass unit: The standard unit that is used for indicating mass of an atom.

History of Dalton’s Atomic Theory

Although the concept of the atom dates back to the ideas of Democritus, the English meteorologist and chemist John Dalton formulated the first modern description of it as the fundamental building block of chemical structures. Dalton developed the law of multiple proportions (first presented in 1803) by studying and expanding upon the works of Antoine Lavoisier and Joseph Proust.

Proust had studied tin oxides and found that their masses were either 88.1% tin and 11.9% oxygen or 78.7% tin and 21.3% oxygen (these were tin(II) oxide and tin dioxide respectively). Dalton noted from these percentages that 100g of tin will combine either with 13.5g or 27g of oxygen; 13.5 and 27 form a ratio of 1:2. Dalton found an atomic theory of matter could elegantly explain this common pattern in chemistry – in the case of Proust’s tin oxides, one tin atom will combine with either one or two oxygen atoms.

Dalton also believed atomic theory could explain why water absorbed different gases in different proportions: for example, he found that water absorbed carbon dioxide far better than it absorbed nitrogen. Dalton hypothesized this was due to the differences in the mass and complexity of the gases’ respective particles. Indeed, carbon dioxide molecules (CO_2) are heavier and larger than nitrogen molecules (N_2).

Dalton proposed that each chemical element is composed of atoms of a single, unique type, and though they cannot be altered or destroyed by chemical means, they can combine to form more complex structures (chemical compounds). Since Dalton reached his conclusions by experimentation and examination of the results in an empirical fashion, this marked the first truly scientific theory of the atom.

John Dalton’s A New System of Chemical Philosophy: This image from Dalton’s A New System of Chemical Philosophy, published in 1808, depicts various atoms and molecules.

Dalton’s Atomic Theory

The main points of Dalton’s atomic theory are:

1. Everything is composed of atoms, which are the indivisible building blocks of matter and cannot be destroyed.

2. All atoms of an element are identical.

3. The atoms of different elements vary in size and mass.

4. Compounds are produced through different whole-number combinations of atoms.

5. A chemical reaction results in the rearrangement of atoms in the reactant and product compounds.

Atomic theory has been revised over the years to incorporate the existence of atomic isotopes and the interconversion of mass and energy. In addition, the discovery of subatomic particles has shown that atoms can be divided into smaller parts. However, Dalton’s importance in the development of modern atomic theory has been recognized by the designation of the atomic mass unit as a Dalton.

3

The Structure of the Atom

Overview of Atomic Structure

Atoms are made up of particles called protons, neutrons, and electrons, which are responsible for the mass and charge of atoms.

Learning Objectives

Discuss the electronic and structural properties of an atom

Key Takeaways

Key Points

- An atom is composed of two regions: the nucleus, which is in the center of the atom and contains protons and neutrons, and the outer region of the atom, which holds its electrons in orbit around the nucleus.

- Protons and neutrons have approximately the same mass, about 1.67 × 10-24 grams, which scientists define as one atomic mass unit (amu) or one Dalton.

- Each electron has a negative charge (-1) equal to the positive charge of a proton (+1).

- Neutrons are uncharged particles found within the nucleus.

Key Terms

- atom: The smallest possible amount of matter which still retains its identity as a chemical element, consisting of a nucleus surrounded by electrons.

- proton: Positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element. It weighs 1 amu.

- neutron: A subatomic particle forming part of the nucleus of an atom. It has no charge. It is equal in mass to a proton or it weighs 1 amu.

An atom is the smallest unit of matter that retains all of the chemical properties of an element. Atoms combine to form molecules, which then interact to form solids, gases, or liquids. For example, water is composed of hydrogen and oxygen atoms that have combined to form water molecules. Many biological processes are devoted to breaking down molecules into their component atoms so they can be reassembled into a more useful molecule.

Atomic Particles

Atoms consist of three basic particles: protons, electrons, and neutrons. The nucleus (center) of the atom contains the protons (positively charged) and the neutrons (no charge). The outermost regions of the atom are called electron shells and contain the electrons (negatively charged). Atoms have different properties based on the arrangement and number of their basic particles.

The hydrogen atom (H) contains only one proton, one electron, and no neutrons. This can be determined using the atomic number and the mass number of the element (see the concept on atomic numbers and mass numbers).

Structure of an atom: Elements, such as helium, depicted here, are made up of atoms. Atoms are made up of protons and neutrons located within the nucleus, with electrons in orbitals surrounding the nucleus.

Atomic Mass

Protons and neutrons have approximately the same mass, about 1.67 × 10-24 grams. Scientists define this amount of mass as one atomic mass unit (amu) or one Dalton. Although similar in mass, protons are positively charged, while neutrons have no charge. Therefore, the number of neutrons in an atom contributes significantly to its mass, but not to its charge.

Electrons are much smaller in mass than protons, weighing only 9.11 × 10-28 grams, or about 1/1800 of an atomic mass unit. Therefore, they do not contribute much to an element’s overall atomic mass. When considering atomic mass, it is customary to ignore the mass of any electrons and calculate the atom’s mass based on the number of protons and neutrons alone.

Electrons contribute greatly to the atom’s charge, as each electron has a negative charge equal to the positive charge of a proton. Scientists define these charges as “+1” and “-1. ” In an uncharged, neutral atom, the number of electrons orbiting the nucleus is equal to the number of protons inside the nucleus. In these atoms, the positive and negative charges cancel each other out, leading to an atom with no net charge.

Protons, neutrons, and electrons: Both protons and neutrons have a mass of 1 amu and are found in the nucleus. However, protons have a charge of +1, and neutrons are uncharged. Electrons have a mass of approximately 0 amu, orbit the nucleus, and have a charge of -1.

Exploring Electron Properties: Compare the behavior of electrons to that of other charged particles to discover properties of electrons such as charge and mass.

Volume of Atoms

Accounting for the sizes of protons, neutrons, and electrons, most of the volume of an atom—greater than 99 percent—is, in fact, empty space. Despite all this empty space, solid objects do not just pass through one another. The electrons that surround all atoms are negatively charged and cause atoms to repel one another, preventing atoms from occupying the same space. These intermolecular forces prevent you from falling through an object like your chair.

Interactive: Build an Atom: Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!

Atomic Number and Mass Number

The atomic number is the number of protons in an element, while the mass number is the number of protons plus the number of neutrons.

Learning Objectives

Determine the relationship between the mass number of an atom, its atomic number, its atomic mass, and its number of subatomic particles

Key Takeaways

Key Points

- Neutral atoms of each element contain an equal number of protons and electrons.

- The number of protons determines an element’s atomic number and is used to distinguish one element from another.

- The number of neutrons is variable, resulting in isotopes, which are different forms of the same atom that vary only in the number of neutrons they possess.

- Together, the number of protons and the number of neutrons determine an element’s mass number.

- Since an element’s isotopes have slightly different mass numbers, the atomic mass is calculated by obtaining the mean of the mass numbers for its isotopes.

Key Terms

- mass number: The sum of the number of protons and the number of neutrons in an atom.

- atomic number: The number of protons in an atom.

- atomic mass: The average mass of an atom, taking into account all its naturally occurring isotopes.

Atomic Number

Neutral atoms of an element contain an equal number of protons and electrons. The number of protons determines an element’s atomic number (Z) and distinguishes one element from another. For example, carbon’s atomic number (Z) is 6 because it has 6 protons. The number of neutrons can vary to produce isotopes, which are atoms of the same element that have different numbers of neutrons. The number of electrons can also be different in atoms of the same element, thus producing ions (charged atoms). For instance, iron, Fe, can exist in its neutral state, or in the +2 and +3  ionic states.

Mass Number

An element’s mass number (A) is the sum of the number of protons and the number of neutrons. The small contribution of mass from electrons is disregarded in calculating the mass number. This approximation of mass can be used to easily calculate how many neutrons an element has by simply subtracting the number of protons from the mass number. Protons and neutrons both weigh about one atomic mass unit or amu. Isotopes of the same element will have the same atomic number but different mass numbers.

Atomic number, chemical symbol, and mass number: Carbon has an atomic number of six, and two stable isotopes with mass numbers of twelve and thirteen, respectively. Its average atomic mass is 12.11.

Scientists determine the atomic mass by calculating the mean of the mass numbers for its naturally-occurring isotopes. Often, the resulting number contains a decimal. For example, the atomic mass of chlorine (Cl) is 35.45 amu because chlorine is composed of several isotopes, some (the majority) with an atomic mass of 35 amu (17 protons and 18 neutrons) and some with an atomic mass of 37 amu (17 protons and 20 neutrons).

Given an atomic number (Z) and mass number (A), you can find the number of protons, neutrons, and electrons in a neutral atom. For example, a lithium atom (Z=3, A=7 amu) contains three protons (found from Z), three electrons (as the number of protons is equal to the number of electrons in an atom), and four neutrons (7 – 3 = 4).

Isotopes

Isotopes are various forms of an element that have the same number of protons, but a different number of neutrons.

Learning Objectives

Discuss the properties of isotopes and their use in radiometric dating

Key Takeaways

Key Points

- Isotopes are atoms of the same element that contain an identical number of protons, but a different number of neutrons.

- Despite having different numbers of neutrons, isotopes of the same element have very similar physical properties.

- Some isotopes are unstable and will undergo radioactive decay to become other elements.

- The predictable half-life of different decaying isotopes allows scientists to date material based on its isotopic composition, such as with Carbon-14 dating.

Key Terms

- isotope: Any of two or more forms of an element where the atoms have the same number of protons, but a different number of neutrons within their nuclei.

- half-life: The time it takes for half of the original concentration of an isotope to decay back to its more stable form.

- radioactive isotopes: an atom with an unstable nucleus, characterized by excess energy available that undergoes radioactive decay and creates most commonly gamma rays, alpha or beta particles.

- radiocarbon dating: Determining the age of an object by comparing the ratio of the 14C concentration found in it to the amount of 14C in the atmosphere.

What is an Isotope?

Isotopes are various forms of an element that have the same number of protons but a different number of neutrons. Some elements, such as carbon, potassium, and uranium, have multiple naturally-occurring isotopes. Isotopes are defined first by their element and then by the sum of the protons and neutrons present.

- Carbon-12 (or 12C) contains six protons, six neutrons, and six electrons; therefore, it has a mass number of 12 amu (six protons and six neutrons).

- Carbon-14 (or 14C) contains six protons, eight neutrons, and six electrons; its atomic mass is 14 amu (six protons and eight neutrons).

While the mass of individual isotopes is different, their physical and chemical properties remain mostly unchanged.

Isotopes do differ in their stability. Carbon-12 (12C) is the most abundant of the carbon isotopes, accounting for 98.89% of carbon on Earth. Carbon-14 (14C) is unstable and only occurs in trace amounts. Unstable isotopes most commonly emit alpha particles (He2+) and electrons. Neutrons, protons, and positrons can also be emitted and electrons can be captured to attain a more stable atomic configuration (lower level of potential energy ) through a process called radioactive decay. The new atoms created may be in a high energy state and emit gamma rays which lowers the energy but alone does not change the atom into another isotope. These atoms are called radioactive isotopes or radioisotopes.

Radiocarbon Dating

Carbon is normally present in the atmosphere in the form of gaseous compounds like carbon dioxide and methane. Carbon-14 (14C) is a naturally-occurring radioisotope that is created from atmospheric 14N (nitrogen) by the addition of a neutron and the loss of a proton, which is caused by cosmic rays. This is a continuous process so more 14C is always being created in the atmosphere. Once produced, the 14C often combines with the oxygen in the atmosphere to form carbon dioxide. Carbon dioxide produced in this way diffuses in the atmosphere, is dissolved in the ocean, and is incorporated by plants via photosynthesis. Animals eat the plants and, ultimately, the radiocarbon is distributed throughout the biosphere.

In living organisms, the relative amount of 14C in their body is approximately equal to the concentration of 14C in the atmosphere. When an organism dies, it is no longer ingesting 14C, so the ratio between 14C and 12C will decline as 14C gradually decays back to 14N. This slow process, which is called beta decay, releases energy through the emission of electrons from the nucleus or positrons.

After approximately 5,730 years, half of the starting concentration of 14C will have been converted back to 14N. This is referred to as its half-life, or the time it takes for half of the original concentration of an isotope to decay back to its more stable form. Because the half-life of 14C is long, it is used to date formerly-living objects such as old bones or wood. Comparing the ratio of the 14C concentration found in an object to the amount of 14C in the atmosphere, the amount of the isotope that has not yet decayed can be determined. On the basis of this amount, the age of the material can be accurately calculated, as long as the material is believed to be less than 50,000 years old. This technique is called radiocarbon dating, or carbon dating for short.

Application of carbon dating: The age of carbon-containing remains less than 50,000 years old, such as this pygmy mammoth, can be determined using carbon dating.

Other elements have isotopes with different half lives. For example, 40K (potassium-40) has a half-life of 1.25 billion years, and 235U (uranium-235) has a half-life of about 700 million years. Scientists often use these other radioactive elements to date objects that are older than 50,000 years (the limit of carbon dating). Through the use of radiometric dating, scientists can study the age of fossils or other remains of extinct organisms.

4

The Periodic Table

The Periodic Table

The periodic table shows all the elements and their physical properties; it is arranged based on atomic numbers and electron configurations.

Learning Objectives

Identify the common periodic trends that can be deduced from the periodic table of elements

Key Takeaways

Key Points

- The periodic table lists all the elements, with information about their atomic weights, chemical symbols, and atomic numbers.

- The arrangement of the periodic table leads us to visualize certain trends among the atoms.

- The vertical columns (groups) of the periodic table are arranged such that all its elements have the same number of valence electrons. All elements within a certain group thus share similar properties.

Key Terms

- atomic number: The number, equal to the number of protons in an atom, that determines its chemical properties. Symbol: Z

- group: A vertical column in the periodic table, which signifies the number of valence shell electrons in an element’s atom.

- period: A horizontal row in the periodic table, which signifies the total number of electron shells in an element’s atom.

Element Symbols

The periodic table is structured as an 18 X 7 grid, positioned above a smaller double row of elements. The periodic table only lists chemical elements, and includes each isotope of each element within one cell. In the typical periodic table, each element is listed by its element symbol and atomic number. For example, “H” denotes hydrogen, “Li” denotes lithium, and so on. Most elements are represented by the first letter or first two letters of their English name, but there are some exceptions. Two notable exceptions include silver and mercury. The symbol for silver is “Ag” from Latin argentum, which means “gray” or “shining.” The symbol for mercury is “Hg” from the Latinized Greek hydrargyrum, which means “water-silver.” Many periodic tables include the full name of element as well and color-code the elements based on their phase at room temperature (solid, liquid, or gas).

The periodic table: The periodic table is a tabular display of all the chemical elements. The atoms are grouped in order of increasing atomic number.

Rows and Periods

The element symbol is always almost accompanied by other information such as atomic number and atomic weight. Atomic number describes the number of protons in one atom of that element. For example, an atom of oxygen contains 8 protons. Elements are listed in order of increasing atomic number from left to right. Each row of the periodic table is called a period and each column of the periodic table is called a group (or family). Some groups have specific names like the halogens or noble gases. Elements within the same period or group have similar properties.

Determining Chemical Properties using the Periodic Table

Chemical properties of each element are determined by the element’s electronic configuration, and particularly by its outermost valence electrons. An element’s location in the periodic table is largely dependent on its electrons; the number of valence shell electrons determines its group, and the type of orbital in which the valence electrons lie in determines the element’s block. In addition, the total number of electron shells an atom determines which period it belongs to. Because of its structure, the periodic table has become an extremely useful tool for assessing and predicting elemental and chemical trends.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=54

Chemistry 3.1 Introduction to the Periodic Table – YouTube: An introduction to the periodic table which identifies metals,,nonmetals, and metalloids by location and compares and contrasts the physical properties of metals and nonmetals.

Molecules

Molecules are electrically neutral compounds made of multiple atoms bound together by chemical bonds.

Learning Objectives

Recognize the general properties of molecules

Key Takeaways

Key Points

- Molecules are neutral and do not carry any charge.

- A molecule may consist of nonmetallic atoms of a single chemical element, as with oxygen (O_2), or of different elements, as with water (H_2O).

- The geometry and composition of a molecule will determine its chemical and physical properties.

- Isomers are molecules with the same atoms in different geometric arrangements.

Key Terms

- covalent: when 2 or more nonmetallic atoms are bound together by sharing electrons.

- isomer: Molecules with the same number of atoms in different geometric arrangements.

- compound: when two or more different atoms are held together by a covalent bond. All compounds are molecules but not all molecules are compounds.

- molecule: Two or more atoms that are held together by a chemical covalent bond.

Atoms and Molecules

An atom is defined as a basic unit of matter that contains a centralized dense nucleus surrounded by an electron cloud. When two or more atoms are held together by a chemical covalent bond, this new entity is known as a molecule. The word “molecule” is a loose term, and it colloquially carries different meanings across different fields of study. For example, the term “molecules” is used in the kinetic theory of gases, referring to any gaseous particle regardless of its composition.

Most often, the term “molecules” refers to multiple atoms; a molecule may be composed of a single chemical element, as with oxygen (O_2), or of multiple elements, such as water (H_2O). Molecules are neutral and carry no charge; this property distinguishes them from polyatomic ions, such as nitrate (NO_3–).

caffeine molecule: Caffeine is a complicated molecule, composed of many atoms bonded to each other in a specific arrangement.

Molecular size varies depending on the number of atoms that make up the molecule. Most molecules are too small to be seen with the naked eye. The smallest molecule is diatomic hydrogen (H_2), with a bond length of 0.74 angstroms. Macromolecules are large molecules composed of smaller subunits; this term from biochemistry refers to nucleic acids, proteins, carbohydrates, and lipids. Some macromolecules may be observed by specialized microscopes.

Often, a compound ‘s composition can also be denoted by an empirical formula, which is the simplest integer ratio of its constituent chemical elements. However, this empirical formula does not always describe the specific molecule in question, since it provides only the ratio of its elements. The full elemental composition of a molecule can be precisely represented by its molecular formula, which indicates the exact number of atoms that are in the molecule.

Example

- C_6H_12O_6 = molecular formula for glucose

- CH_2O = empirical (simplified ratio) formula for glucose

Isomers

Isomers are molecules with the same atoms in different geometric arrangements. Because of these different arrangements, isomers often have very different chemical and physical properties. In the picture below, 1-propanol is mostly used in the synthesis of other compounds and has a less offensive odor, whereas 2-propanol is the common household alcohol.

propanol structural isomers: The chemical formula for propanol (C_3H_7OH) describes several different molecules, which vary by the position of the alcohol (OH). Each molecule is a structural isomer of the other.

Ions

An ion is an atom or molecule that has a net electrical charge because its total number of electrons is not equal to its number of protons.

Learning Objectives

Compare the different classes of ions

Key Takeaways

Key Points

- Ions are formed when the number of protons in an atom does not equal the number of electrons. If more protons are present, the ion is positive and is known as a cation; if more electrons are present, the ion is negative and referred to as an anion.

- Ions are highly reactive species. They are generally found in a gaseous state and do not occur in abundance on Earth. They are repelled by like electric charges and are attracted to opposite charges.

- The electron cloud of an atom determines the size of the atom; added electrons (anions) increase the electron repulsion, increasing the ion’s size, while cations (with less electrons) are smaller than the atom because there are fewer electrons in the cloud to repel each other.

Key Terms

- ion: An atom or group of atoms bearing an electrical charge, such as the sodium and chlorine atoms in a salt solution.

- anion: Ions that are negatively charged because they have more electrons than protons.

- cation: Ions that are positively charged because they have more protons than electrons.

An atom is a basic unit of matter that consists of a dense nucleus composed of positively charged protons and neutral neutrons, which is surrounded by a cloud of negatively charged electrons. If an atom has the same number of protons and electrons, it is electronically neutral. However, if the total number of electrons does not equal the number of protons, the atom has a net electrical charge.

Any atom or molecule with a net charge, either positive or negative, is known as an ion. An ion consisting of a single atom is a monoatomic ion; an ion consisting of two or more atoms is referred to as a polyatomic ion. The positive electric charge of a proton is equal in magnitude to the negative charge of an electron; therefore, the net electric charge of an ion is equal to its number of protons minus its number of electrons.

Ions are highly reactive species. They are generally found in a gaseous state and do not occur in abundance on Earth. Ions in the liquid or solid state are produced when salts interact with their solvents. They are repelled by like electric charges and are attracted to opposite charges.

Types of Ions

There are specialized types of ions. Anions have more electrons than protons and so have a net negative charge. Cations have more protons than electrons and so have a net positive charge. Zwitterions are neutral and have both positive and negative charges at different locations throughout the molecule. Anions are generally larger than the parent molecule or atom, because the excess electrons repel each other and add to the physical size of the electron cloud. Cations are generally smaller than their parent atom or molecule due to the smaller size of their electron clouds.

Hydrogen ions: The relationship between a molecule, its cation, and its anion is shown.

An ion is denoted by writing its net negative charge in superscript immediately after the chemical structure for the atom/molecule. Conventionally the net charge is written with the magnitude before the sign; the magnitude of singly charged molecules/atoms is generally omitted. Monoatomic ions are sometimes also represented by Roman numerals, which designate the formal oxidation state of the element, whereas the superscripted numerals denote the net charge. For example, Fe2+ can be referred to as Fe(II). These representations can be thought of as equivalent for monoatomic ions, but the Roman numerals cannot be applied to polyatomic ions.

Forming Ions

Ions can be formed by ionization, which is the process of a neutral atom losing or gaining electrons. Generally, the electrons are either added to or lost from the valence shell of an atom; the inner-shell electrons are more tightly bound to the positively charged nucleus and so do not participate in this type of chemical interaction.

Ionization generally involves a transfer of electrons between atoms or molecules. The process is motivated by the achievement of more stable electronic configurations, such as the octet rule, which states that most stable atoms and ions have eight electrons in their outermost (valence) shell. Polyatomic and molecular ions can also be formed, generally by gaining or losing elemental ions, such as H+, in neutral molecules. Polyatomic ions are generally very unstable and reactive.

An common example of an ion is Na+. Sodium has a +1 charge because sodium has eleven electrons. However, according to the octet rule, sodium would be more stable with 10 electrons (2 in its inner most shell, 8 in its outermost shell). Therefore, sodium tends to lose an electron to become more stable. On the other hand, chlorine tends to gain an electron to become Cl–. Chlorine naturally has 17 electrons but it would be more stable with 18 electrons (2 in its inner most shell, 8 in its second shell, and 8 in its valence shell). Therefore, chlorine will take an electron from another atom to become negatively charged.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=54

Periodic Properties: Part 3, Ionic Radius, Predicting Ionic Charges – YouTube: A continuation of the discussion of periodic properties, including ionic radius and how to predict ionic charges.

5

The Ionic Bond

Ionic Bonding and Electron Transfer

An ionic bond results from the transfer of an electron from a metal atom to a non-metal atom.

Learning Objectives

Identify the key features of ionic bonds

Key Takeaways

Key Points

- Ionic bonds are formed between cations and anions.

- A cation is formed when a metal ion loses a valence electron while an anion is formed when a non-metal gains a valence electron. They both achieve a more stable electronic configuration through this exchange.

- Ionic solids form crystalline lattices, or repeating patterns of atoms, with high melting points, and are typically soluble in water.

Key Terms

- electrolyte: An ionic compound which dissolves in H2O, making the resulting solution capable of conducting electricity.

- electronegativity: The tendency of an atom to attract electrons to itself.

- cation: A positively charged ion.

- anion: A negatively charged ion.

Ionic Bonds

Ionic bonding is a type of chemical bond in which valence electrons are lost from one atom and gained by another. This exchange results in a more stable, noble gas electronic configuration for both atoms involved. An ionic bond is based on attractive electrostatic forces between two ions of opposite charge.

Cations and Anions

Ionic bonds involve a cation and an anion. The bond is formed when an atom, typically a metal, loses an electron or electrons, and becomes a positive ion, or cation. Another atom, typically a non-metal, is able to acquire the electron(s) to become a negative ion, or anion.

One example of an ionic bond is the formation of sodium fluoride, NaF, from a sodium atom and a fluorine atom. In this reaction, the sodium atom loses its single valence electron to the fluorine atom, which has just enough space to accept it. The ions produced are oppositely charged and are attracted to one another due to electrostatic forces.

Formation of NaF: An electron is transferred from Na to F. The resulting Na+ and F– ions are electrically attracted to each other.

At the macroscopic scale, ionic compounds form lattices, are crystalline solids under normal conditions, and have high melting points. Most of these solids are soluble in H_2O and conduct electricity when dissolved. The ability to conduct electricity in solution is why these substances are called electrolytes. Table salt, NaCl, is a good example of this type of compound.

Ionic bonds differ from covalent bonds. Both types result in the stable electronic states associated with the noble gases. However, in covalent bonds, the electrons are shared between the two atoms. All ionic bonds have some covalent character, but the larger the difference in electronegativity between the two atoms, the greater the ionic character of the interaction.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=56

Ionic Bonding – YouTube: In this video, Paul Andersen explains how ionic solids form when cations and anions are attracted.

Lattice Energy

Lattice energy is a measure of the bond strength in an ionic compound.

Learning Objectives

Describe lattice energy and the factors that affect it

Key Takeaways

Key Points

- Lattice energy is defined as the energy required to separate a mole of an ionic solid into gaseous ions.

- Lattice energy cannot be measured empirically, but it can be calculated using electrostatics or estimated using the Born-Haber cycle.

- Two main factors that contribute to the magnitude of the lattice energy are the charge and radius of the bonded ions.

Key Terms

- exothermic reaction: A process which releases heat into its surroundings.

- lattice energy: The amount of energy released upon formation of a crystalline ionic solid from gaseous ions.

Definition of Lattice Energy

Lattice energy is an estimate of the bond strength in ionic compounds. It is defined as the heat of formation for ions of opposite charge in the gas phase to combine into an ionic solid. As an example, the lattice energy of sodium chloride, NaCl, is the energy released when gaseous Na+ and Cl– ions come together to form a lattice of alternating ions in the NaCl crystal.

The negative sign of the energy is indicative of an exothermic reaction.

Alternatively, lattice energy can be thought of as the energy required to separate a mole of an ionic solid into the gaseous form of its ions (that is, the reverse of the reaction shown above).

NaCl Crystalline Lattice: Sodium ions (Na+) and chloride(Cl–) ions, depicted in purple and green respectively, alternate in the crystal lattice of solid NaCl.

Alternatively, lattice energy can be thought of as the energy required to separate a mole of an ionic solid into the gaseous form of its ions (that is, the reverse of the reaction shown above).

Lattice energy cannot be determined experimentally due to the difficulty in isolating gaseous ions. The energy value can be estimated using the Born-Haber cycle, or it can be calculated theoretically with an electrostatic examination of the crystal structure.

Factors Affecting Lattice Energy

In 1918, Born and Lande presented the following model for lattice energy:

In this equation, NA is Avogadro’s constant; M is the Madelung constant, which depends on the crystal geometry; z+ is the charge number of the cation; z– is the charge number of the anion; e is the elementary charge of the electron; n is the Born exponent, a characteristic of the compressibility of the solid; is the permittivity of free space; and r_0 is the distance to the closest ion.

This model emphasizes two main factors that contribute to the lattice energy of an ionic solid: the charge on the ions, and the radius, or size, of the ions. The effect of those factors is:

- as the charge of the ions increases, the lattice energy increases

- as the size of the ions increases, the lattice energy decreases

Lattice energies are also important in predicting the solubility of ionic solids in H_2O. Ionic compounds with smaller lattice energies tend to be more soluble in H_2O.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=56

Lattice Energies – Chemistry Tutorial: This tutorial covers lattice energy and how to compare the relative lattice energies of different ionic compounds.

Formulas of Ionic Compounds

Ionic formulas must satisfy the noble gas configurations for the constituent ions and the product compound must be electrically neutral.

Learning Objectives

Apply knowledge of ionic bonding to predict the formula of ionic compounds

Key Takeaways

Key Points

- The charge on the cations and anions in an ionic compound can be determined by the loss or gain of valence electrons necessary in order to achieve stable, noble gas electronic configurations.

- The number of cations and anions that are combined in an ionic compound is the simplest ratio of whole integers that can be combined to reach electrical neutrality.

- The cation precedes the anion in both the written form and the formula.

Key Terms

- noble gas: Any of the elements of group 18 of the periodic table, which are monatomic and, with very limited exceptions, inert, or non-reactive.

- electrically neutral: A net charge of zero, which occurs when an atom or molecule/compound has no surplus or deficit of electrons.

- empirical formula: The simplest whole-number ratio between elements in a formula of a compound.

- polyatomic ion: An ion composed of several atoms.

Ionic Compounds

An ionic bond is formed through the transfer of one or more valence electrons, typically from a metal to a non-metal, which produces a cation and an anion that are bound together by an attractive electrostatic force. On a macroscopic scale, ionic compounds, such as sodium chloride (NaCl), form a crystalline lattice and are solids at normal temperatures and pressures.

Crystalline Lattice: Sodium chloride crystal lattice

The charge on the cations and anions is determined by the number of electrons required to achieve stable noble gas electronic configurations. The ionic composition is then defined by the requirement that the resulting compound be electrically neutral overall.

For example, to combine magnesium (Mg) and bromine (Br) to get an ionic compound, we first note the electronic configurations of these atoms (valence level in indicated in italics):

Mg: 1s^22s^22p^63s2

Br: 1s^22s^22p^63s^23p^63d^104s24p5

In order to achieve noble gas configurations, the magnesium atom needs to lose its two valence electrons, while the bromine atom, which has 7 valence electrons, requires one additional electron to fill its outer shell. Therefore, for the resulting compound to be neutral, two bromine anions must combine with one magnesium cation to form magnesium bromide (MgBr_2). In addition, though any ratio of 2 bromine atoms to 1 magnesium atom will satisfy the two requirements above, the formula for ionic compounds is typically presented as the empirical formula, or the simplest whole-number ratio of atoms with positive integers.

Note that the cation always precedes the anion both in written form and in formulas. In the written form, while the cation name is generally the same as the element, the suffix of single-atom anions is changed to –ide, as in the case of sodium chloride. If the anion is a polyatomic ion, its suffix can vary, but is typically either –ate or –ite,as in the cases of sodium phosphate and calcium nitrite, depending on the identity of the ion.

More examples:

- lithium fluoride: Li+ and F– combine to form LiF

- calcium chloride: Ca2+ and Cl– combine to form CaCl_2

- iron (II) oxide: Fe2+ and O2- combine to form FeO

- aluminum sulfide: Al3+ and S2- combine to form Al_2S_3

- sodium sulfate: Na+ and SO_42- combine to form Na_2SO_4

- ammonium phosphate: NH4+ and PO_43- combine to form (NH_4)3PO_4

- potassium chlorite: K+ and ClO_2– combine to form KClO_2

Video Summary

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=56

Chemistry 5.3 Formula Writing: Ionic Compounds – YouTube: A lesson on writing formulas for binary ionic compounds as well as ionic compounds containing polyatomic ions. The cross-over method is demonstrated.

Ionic vs Covalent Bond Character

Ionic bonds can have some covalent character.

Learning Objectives

Discuss the idea that, in nature, bonds exhibit characteristics of both ionic and covalent bonds

Key Takeaways

Key Points

- Ionic bonding is presented as the complete transfer of valence electrons, typically from a metal to a non-metal.

- In reality, electron density remains shared between the constituent atoms, meaning all bonds have some covalent character.

- The ionic or covalent nature of a bond is determined by the relative electronegativities of the atoms involved.

Key Terms

- polar covalent bond: A covalent bond that has a partial ionic character to it, as a result of the difference in electronegativity between the two bonding atoms.

- electronegativity: A measure of the tendency of an atom to attract electrons to itself.

- covalent character: The partial sharing of electrons between atoms that have an ionic bond.

Ionic vs Covalent Bonding

Chemical compounds are frequently classified by the bonds between constituent atoms. There are multiple kinds of attractive forces, including covalent, ionic, and metallic bonds. Ionic bonding models are generally presented as the complete loss or gain of one or more valence electrons from a metal to a nonmetal, resulting in cations and anions that are held together by attractive electrostatic forces.

Ionic bonding: The formation of an ionic bond between lithium and fluorine to form LiF.

Example of a polar covalent bond: When a carbon atom forms a bond with fluorine, they share a pair of electrons. However, because fluorine is more highly electronegative than carbon, it attracts that shared electron pair closer to itself and thus creates an electric dipole. The lowercase greek delta written above the atoms is used to indicate the presence of partial charges. This bond is considered to have characteristics of both covalent and ionic bonds.

In reality, the bond between these atoms is more complex than this model illustrates. The bond formed between any two atoms is not a purely ionic bond. All bonding interactions have some covalent character because the electron density remains shared between the atoms. The degree of ionic versus covalent character of a bond is determined by the difference in electronegativity between the constituent atoms. The larger the difference, the more ionic the nature of the bond. In the conventional presentation, bonds are designated as ionic when the ionic aspect is greater than the covalent aspect of the bond. Bonds that fall in between the two extremes, having both ionic and covalent character, are classified as polar covalent bonds. Such bonds are thought of as consisting of partially charged positive and negative poles.

Though ionic and covalent character represent points along a continuum, these designations are frequently useful in understanding and comparing the macroscopic properties of ionic and covalent compounds. For example, ionic compounds typically have higher boiling and melting points, and they are also usually more soluble in water than covalent compounds.

6

The Covalent Bond

Comparison between Covalent and Ionic Compounds

Covalent and ionic compounds have distinct physical properties.

Learning Objectives

Identify element pairs which are likely to form ionic or covalent bonds

Key Takeaways

Key Points

- Ionic compounds are formed from strong electrostatic interactions between ions, which result in higher melting points and electrical conductivity compared to covalent compounds.

- Covalent compounds have bonds where electrons are shared between atoms. Due to the sharing of electrons, they exhibit characteristic physical properties that include lower melting points and electrical conductivity compared to ionic compounds.

Key Terms

- valence electrons: Electrons in the outermost principal energy (valence) level of an atom that can participate in the formation of chemical bonds with other atoms.

- octet rule: Atoms lose, gain, or share electrons in order to have a full valence level of eight electrons. Hydrogen and helium are exceptions because they can hold a maximum of two valence electrons.

- electronegativity: The tendency of an atom or molecule to attract electrons and form bonds.

Two Classes of Compounds

Compounds are defined as substances containing two or more different chemical elements. They have distinct chemical structures characterized by a fixed ratio of atoms held together by chemical bonds. Here, we discuss two classes of compounds based on the bond type that holds the atoms together: ionic and covalent.

Covalent Compounds

Covalent bonds are characterized by the sharing of electrons between two or more atoms. These bonds mostly occur between nonmetals or between two of the same (or similar) elements.Two atoms with similar electronegativity will not exchange an electron from their outermost shell; the atoms instead share electrons so that their valence electron shell is filled.

Examples of compounds that contain only covalent bonds are methane (CH_4), carbon monoxide (CO), and iodine monobromide (IBr).

Covalent bonding between hydrogen atoms: Since each hydrogen atom has one electron, they are able to fill their outermost shells by sharing a pair of electrons through a covalent bond.

Ionic Compounds

Ionic bonding occurs when there is a large difference in electronegativity between two atoms. This large difference leads to the loss of an electron from the less electronegative atom and the gain of that electron by the more electronegative atom, resulting in two ions. These oppositely charged ions feel an attraction to each other, and this electrostatic attraction constitutes an ionic bond.

Ionic bonding occurs between a nonmetal, which acts as an electron acceptor, and a metal, which acts as an electron donor. Metals have few valence electrons, whereas nonmetals have closer to eight valence electrons; to easily satisfy the octet rule, the nonmetal will accept an electron donated by the metal. More than one electron can be donated and received in an ionic bond.

Some examples of compounds with ionic bonding include NaCl, KI, MgCl_2.

Formation of sodium fluoride (NaF): The transfer of an electron from a neutral sodium atom to a neutral fluorine atom creates two oppositely charge ions: Na+ and F–. Attraction of the oppositely charged ions is the ionic bond between Na and F.

Effect on Physical Properties

Covalent and ionic compounds can be differentiated easily because of their different physical properties based on the nature of their bonding. Here are some differences:

1. At room temperature and normal atmospheric pressure, covalent compounds may exist as a solid, a liquid, or a gas, whereas ionic compounds exist only as solids.

2. Although solid ionic compounds do not conduct electricity because there are no free mobile ions or electrons, ionic compounds dissolved in water make an electrically conductive solution. In contrast, covalent compounds do not exhibit any electrical conductivity, either in pure form or when dissolved in water.

3. Ionic compounds exist in stable crystalline structures. Therefore, they have higher melting and boiling points compared to covalent compounds.

Single Covalent Bonds

Single covalent bonds are sigma bonds, which occur when one pair of electrons is shared between atoms.

Learning Objectives

Identify the four orbital types used in covalent bond formation

Key Takeaways

Key Points

- Covalent bonds occur when electrons are shared between two atoms. A single covalent bond is when only one pair of electrons is shared between atoms.

- A sigma bond is the strongest type of covalent bond, in which the atomic orbitals directly overlap between the nuclei of two atoms.

- Sigma bonds can occur between any kind of atomic orbitals; the only requirement is that the atomic orbital overlap happens directly between the nuclei of atoms.

Key Terms

- sigma bond: A covalent bond whose electron density is concentrated in the region directly between the nuclei.

- covalent bond: A type of chemical bond where two atoms are connected to each other by the sharing of two or more electrons.

- atomic orbital: A region in space around the atom’s nucleus where there is a probability of finding an electron.

Hierarchical Structure of the Atom

There are four hierarchical levels that describe the position and energy of the electrons an atom has. Here they are listed along with some of the possible values (or letters) they can have:

1. Principal energy levels (1, 2, 3, etc.)

2. Sublevels (s, p, d, f)

3. Orbitals

4. Electrons

Principal energy levels are made out of sublevels, which are in turn made out of orbitals, in which electrons are found.

Atomic Orbitals

An atomic orbital is defined as the probability of finding an electron in an area around an atom’s nucleus. Generally, orbital shapes are drawn to describe the region in space in which electrons are likely to be found. This is referred to as “electron density.”

Atomic orbitals: The shapes of the first five atomic orbitals are shown in order: 1s, 2s, and the three 2p orbitals. Both blue and orange-shaded regions represent regions in space where electrons can be found ‘belonging’ to these orbitals.

Sigma Bonds

Covalent bonding occurs when two atomic orbitals come together in close proximity and their electron densities overlap. The strongest type of covalent bonds are sigma bonds, which are formed by the direct overlap of orbitals from each of the two bonded atoms. Regardless of the atomic orbital type, sigma bonds can occur as long as the orbitals directly overlap between the nuclei of the atoms.

Orbital overlaps and sigma bonds: These are all possible overlaps between different types of atomic orbitals that result in the formation of a sigma bond between two atoms. Notice that the area of overlap always occurs between the nuclei of the two bonded atoms.

Single covalent bonds occur when one pair of electrons is shared between atoms as part of a molecule or compound. A single covalent bond can be represented by a single line between the two atoms. For instance, the diatomic hydrogen molecule, H_2, can be written as H—H to indicate the single covalent bond between the two hydrogen atoms.

Sigma bond in the hydrogen molecule: Higher intensity of the red color indicates a greater probability of the bonding electrons being localized between the nuclei.

Double and Triple Covalent Bonds

Double and triple bonds, comprised of sigma and pi bonds, increase the stability and restrict the geometry of a compound.

Learning Objectives

Describe the types of orbital overlap that occur in single, double, and triple bonds

Key Takeaways

Key Points

- Double and triple covalent bonds are stronger than single covalent bonds and they are characterized by the sharing of four or six electrons between atoms, respectively.

- Double and triple bonds are comprised of sigma bonds between hybridized orbitals, and pi bonds between unhybridized p orbitals. Double and triple bonds offer added stability to compounds, and restrict any rotation around the bond axis.

- Bond lengths between atoms with multiple bonds are shorter than in those with single bonds.

Key Terms

- bond strength: Directly related to the amount of energy required to break the bond between two atoms. The more energy required, the stronger the bond is said to be.

- bond length: The distance between the nuclei of two bonded atoms. It can be experimentally determined.

- orbital hybridization: The concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties and geometries.

- atomic orbitals: The physical region in space around the nucleus where an electron has a probability of being.

Double and Triple Covalent Bonds

Covalent bonding occurs when electrons are shared between atoms. Double and triple covalent bonds occur when four or six electrons are shared between two atoms, and they are indicated in Lewis structures by drawing two or three lines connecting one atom to another. It is important to note that only atoms with the need to gain or lose at least two valence electrons through sharing can participate in multiple bonds.

Bonding Concepts

Hybridization

Double and triple bonds can be explained by orbital hybridization, or the ‘mixing’ of atomic orbitals to form new hybrid orbitals. Hybridization describes the bonding situation from a specific atom’s point of view. A combination of s and p orbitals results in the formation of hybrid orbitals. The newly formed hybrid orbitals all have the same energy and have a specific geometrical arrangement in space that agrees with the observed bonding geometry in molecules. Hybrid orbitals are denoted as spx, where s and p denote the orbitals used for the mixing process, and the value of the superscript x ranges from 1-3, depending on how many p orbitals are required to explain the observed bonding.

Hybridized orbitals: A schematic of the resulting orientation in space of sp^3 hybrid orbitals. Notice that the sum of the superscripts (1 for s, and 3 for p) gives the total number of formed hybrid orbitals. In this case, four orbitals are produced which point along the direction of the vertices of a tetrahedron.

Pi Bonds

Pi, or , bonds occur when there is overlap between unhybridized p orbitals of two adjacent atoms. The overlap does not occur between the nuclei of the atoms, and this is the key difference between sigma and pi bonds. For the bond to form efficiently, there has to be a proper geometrical relationship between the unhybridized p orbitals: they must be on the same plane.

Pi bond formation: Overlap between adjacent unhybridized p orbitals produces a pi bond. The electron density corresponding to the shared electrons is not concentrated along the internuclear axis (i.e., between the two atoms), unlike in sigma bonds.

Multiple bonds between atoms always consist of a sigma bond, with any additional bonds being of the π type.

Examples of Pi Bonds

The simplest example of an organic compound with a double bond is ethylene, or ethene, C_2H_4. The double bond between the two carbon atoms consists of a sigma bond and a π bond.

Ethylene bonding: An example of a simple molecule with a double bond between carbon atoms. The bond lengths and angles (indicative of the molecular geometry) are indicated.

From the perspective of the carbon atoms, each has three sp^2 hybrid orbitals and one unhybridized p orbital. The three sp^2 orbitals lie in a single plane at 120-degree angles. As the carbon atoms approach each other, their orbitals overlap and form a bond. Simultaneously, the p orbitals approach each other and form a bond. To maintain this bond, the p orbitals must stay parallel to each other; therefore, rotation is not possible.

A triple bond involves the sharing of six electrons, with a sigma bond and two bonds. The simplest triple-bonded organic compound is acetylene, C_2H_2. Triple bonds are stronger than double bonds due to the the presence of two bonds rather than one. Each carbon has two sp hybrid orbitals, and one of them overlaps with its corresponding one from the other carbon atom to form an sp-sp sigma bond. The remaining four unhybridized p orbitals overlap with each other and form two bonds. Similar to double bonds, no rotation around the triple bond axis is possible.

Observable Consequences of Multiple Bonds

Bond Strength

Covalent bonds can be classified in terms of the amount of energy that is required to break them. Based on the experimental observation that more energy is needed to break a bond between two oxygen atoms in O_2 than two hydrogen atoms in H_2, we infer that the oxygen atoms are more tightly bound together. We say that the bond between the two oxygen atoms is stronger than the bond between two hydrogen atoms.

Experiments have shown that double bonds are stronger than single bonds, and triple bonds are stronger than double bonds. Therefore, it would take more energy to break the triple bond in N_2 compared to the double bond in O_2. Indeed, it takes 497 kcal/mol to break the O_2 molecule, while it takes 945 kJ/mol to do the same to the N_2 molecule.

Bond Length

Another consequence of the presence of multiple bonds between atoms is the difference in the distance between the nuclei of the bonded atoms. Double bonds have shorter distances than single bonds, and triple bonds are shorter than double bonds.

Physical Properties of Covalent Molecules

The covalent bonding model helps predict many of the physical properties of compounds.

Learning Objectives

Discuss the qualitative predictions of covalent bond theory on the boiling and melting points, bond length and strength, and conductivity of molecules

Key Takeaways

Key Points

- The Lewis theory of covalent bonding says that the bond strength of double bonds is twice that of single bonds, which is not true.

- General physical properties that can be explained by the covalent bonding model include boiling and melting points, electrical conductivity, bond strength, and bond length.

Key Terms

- bond length: The distance between the nuclei of two bonded atoms. It can be experimentally determined.

- intermolecular forces: Attractive forces or interactions between different molecules in a sample of a substance. The strength of these interactions is an important factor that determines the substance’s physical properties.

- bond strength: Directly related to the amount of energy required to break the bond between two atoms. The more energy required, the stronger the bond is said to be.

- octet rule: Atoms lose, gain, or share electrons in order to have a full valence shell of eight electrons. Hydrogen is an exception because it can hold a maximum of two electrons in its valence level.

First described by Gilbert Lewis, a covalent bond occurs when electrons of different atoms are shared between the two atoms. These cases of electron sharing can be predicted by the octet rule. The octet rule is a chemical rule that generalizes that atoms of low atomic number (< 20) will combine in a way that results in their having 8 electrons in their valence shells. Having 8 valence electrons is favorable for stability and is similar to the electron configuration of the inert noble gases. In a covalent bond, the shared electrons contribute to each atom’s octet and thus enhance the stability of the compound.

The Lewis bonding theory can explain many properties of compounds. For example, the theory predicts the existence of diatomic molecules such as hydrogen, H_2, and the halogens (F_2, Cl_2, Br_2, I_2). A H atom needs one additional electron to fill its valence level, and the halogens need one more electron to fill the octet in their valence levels. Lewis bonding theory states that these atoms will share their valence electrons, effectively allowing each atom to create its own octet.

Several physical properties of molecules/compounds are related to the presence of covalent bonds:

- Covalent bonds between atoms are quite strong, but attractions between molecules/compounds, or intermolecular forces, can be relatively weak. Covalent compounds generally have low boiling and melting points, and are found in all three physical states at room temperature.

- Covalent compounds do not conduct electricity; this is because covalent compounds do not have charged particles capable of transporting electrons.

- Lewis theory also accounts for bond length; the stronger the bond and the more electrons shared, the shorter the bond length is.

However, the Lewis theory of covalent bonding does not account for some observations of compounds in nature. The theory predicts that with more shared electrons, the bond between the two atoms should be stronger. According to the theory, triple bonds are stronger than double bonds, and double bonds are stronger than single bonds. This is true. However, the theory implies that the bond strength of double bonds is twice that of single bonds, which is not true. Therefore, while the covalent bonding model accounts for many physical observations, it does have its limitations.

7

Overview of Chemistry

The Study of Chemistry

Chemistry is the study of the composition, structure, properties, and transformation of matter.

Learning Objectives

State the general aims and subject matter of chemistry

Key Takeaways

Key Points

- Chemistry is sometimes called “the central science,” because it bridges physics with other natural sciences, such as geology and biology.

- Chemistry is the study of matter and its properties.

- Sub-domains of chemistry include: analytical chemistry, biochemistry, inorganic chemistry, organic chemistry, physical chemistry, and biophysical chemistry.

Key Terms

- matter: Something that has mass and takes up space (has volume) and makes up almost everything in the world.

- chemistry: The branch of science that deals with the composition and constitution of substances and the changes that they undergo as a consequence of alterations in the constitution of their molecules.

Antoine-Laurent de Lavoisier: Antoine-Laurent de Lavoisier is considered the “father of modern chemistry” for his work on the principle of conservation of mass and for developing a new system of chemical nomenclature.

Chemistry is the study of matter and the chemical reactions between substances. Chemistry is also the study of matter’s composition, structure, and properties. Matter is essentially anything in the world that takes up space and has mass. Chemistry is sometimes called “the central science,” because it bridges physics with other natural sciences, such as geology and biology.

History of Chemistry

A basic chemical hypothesis first emerged in Classical Greece when Aristotle defined the four elements of fire, air, earth, and water. It was not until the 17th and 18th centuries when scientists such as Robert Boyle (1627-1691) and Antoine Lavoisier (1743-1794) began to reshape the old alchemical traditions into a rigorous scientific discipline.

As one of the natural sciences, chemistry provides scientists with insight into other physical sciences and powerful analytical tools for engineering applications. The biological sciences and their offshoots, such as psychology, are rooted in biochemistry, and scientists are only now beginning to understand how the different levels of organization influence each other. For example, the basis of modern medicine is the biochemical processes of the human body.

Chemistry and the Natural World

Chemistry has the power to explain innumerable phenomena in the world, from the ordinary to the bizarre. Why does iron rust? What makes propane such an efficient, clean-burning fuel? How can soot and diamond be so different in appearance yet so chemically similar? Chemistry has the answer to these questions and many more. Understanding chemistry is the key to understanding the world as we know it.

Chemistry: Chemistry is the study of the properties, composition, and transformation of matter.

Different Branches of Chemistry

The study of chemistry can be organized into distinct branches that emphasize subsets of chemical concepts. Analytical chemistry seeks to determine the exact chemical compositions of substances. Biochemistry is the study of chemicals found in living things (such as DNA and proteins). Inorganic chemistry studies substances that do not contain carbon. Organic chemistry studies carbon-based substances. Physical chemistry is the study of the physical properties of chemicals. Biophysical chemistry is the application of physical chemistry in a biological context.

The Scientific Method

The scientific method is a process by which observations are questioned; hypotheses are created and tested; and the results are analyzed.

Learning Objectives

Discuss hypotheses and the components of a scientific experiment as part of the scientific method

Key Takeaways

Key Points

- In the scientific method, observations lead to questions that require answers.

- In the scientific method, the hypothesis is a testable statement proposed to answer a question.

- In the scientific method, experiments (often with controls and variables) are devised to test hypotheses.

- In the scientific method, analysis of the results of an experiment will lead to the hypothesis being accepted or rejected.

Key Terms

- scientific method: a way of discovering knowledge based on making falsifiable predictions (hypotheses), testing them, and developing theories based on collected data

- hypothesis: an educated guess that usually is found in an “if…then…” format

- control group: a group that contains every feature of the experimental group except it is not given the manipulation that is hypothesized

The Scientific Method

Biologists study the living world by posing questions about it and seeking science -based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626) who set up inductive methods for scientific inquiry. The scientific method can be applied to almost all fields of study as a logical, rational, problem-solving method.

Sir Francis Bacon: Sir Francis Bacon (1561–1626) is credited with being the first to define the scientific method.

The scientific process typically starts with an observation (often a problem to be solved) that leads to a question. Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. A teenager notices that his friend is really tall and wonders why. So his question might be, “Why is my friend so tall? ”

The Scientific Method: The scientific method consists of a series of well-defined steps. If a hypothesis is not supported by experimental data, a new hypothesis can be proposed.

Proposing a Hypothesis

Recall that a hypothesis is an educated guess that can be tested. Hypotheses often also include an explanation for the educated guess. To solve one problem, several hypotheses may be proposed. For example, the student might believe that his friend is tall because he drinks a lot of milk. So his hypothesis might be “If a person drinks a lot of milk, then they will grow to be very tall because milk is good for your bones.” Generally, hypotheses have the format “If…then…” Keep in mind that there could be other responses to the question; therefore, other hypotheses may be proposed. A second hypothesis might be, “If a person has tall parents, then they will also be tall, because they have the genes to be tall. ”

Once a hypothesis has been selected, the student can make a prediction. A prediction is similar to a hypothesis but it is truly a guess. For instance, they might predict that their friend is tall because he drinks a lot of milk.

Testing a Hypothesis

A valid hypothesis must be testable. It should also be falsifiable, meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. This step—openness to disproving ideas—is what distinguishes sciences from non-sciences. The presence of the supernatural, for instance, is neither testable nor falsifiable. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains every feature of the experimental group except it is not given the manipulation that is hypothesized. For example, a control group could be a group of varied teenagers that did not drink milk and they could be compared to the experimental group, a group of varied teenagers that did drink milk. Thus, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation rather than some outside factor. To test the first hypothesis, the student would find out if drinking milk affects height. If drinking milk has no affect on height, then there must be another reason for the height of the friend. To test the second hypothesis, the student could check whether or not his friend has tall parents. Each hypothesis should be tested by carrying out appropriate experiments. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted. It simply eliminates one hypothesis that is not valid. Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.

While this “tallness” example is based on observational results, other hypotheses and experiments might have clearer controls. For instance, a student might attend class on Monday and realize she had difficulty concentrating on the lecture. One hypothesis to explain this occurrence might be, “If I eat breakfast before class, then I am better able to pay attention.” The student could then design an experiment with a control to test this hypothesis.

The scientific method may seem too rigid and structured. It is important to keep in mind that although scientists often follow this sequence, there is flexibility. Many times, science does not operate in a linear fashion. Instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests.

8

Classification of Matter

Three States of Matter

The three states of matter are the distinct physical forms that matter can take: solid, liquid, and gas.

Learning Objectives

Describe the three states of matter

Key Takeaways

Key Points

- Matter can exist in one of three main states: solid, liquid, or gas.

- Solid matter is composed of tightly packed particles. A solid will retain its shape; the particles are not free to move around.

- Liquid matter is made of more loosely packed particles. It will take the shape of its container. Particles can move about within a liquid, but they are packed densely enough that volume is maintained.

- Gaseous matter is composed of particles packed so loosely that it has neither a defined shape nor a defined volume. A gas can be compressed.

Key Terms

- liquid: A substance that flows and keeps no definite shape because its molecules are loosely packed and constantly moving. It takes the shape of its container but maintains constant volume.

- gas: A substance that can only be contained if it is fully surrounded by a container (or held together by gravitational pull); a substance whose molecules have negligible intermolecular interactions and can move freely.

- solid: A substance that retains its size and shape without a container; a substance whose molecules cannot move freely except to vibrate.

The three states of matter are the three distinct physical forms that matter can take in most environments: solid, liquid, and gas. In extreme environments, other states may be present, such as plasma, Bose-Einstein condensates, and neutron stars. Further states, such as quark-gluon plasmas, are also believed to be possible. Much of the atomic matter of the universe is hot plasma in the form of rarefied interstellar medium and dense stars.

Historically, the states of matter were distinguished based on qualitative differences in their bulk properties. Solid is the state in which matter maintains a fixed volume and shape; liquid is the state in which matter adapts to the shape of its container but varies only slightly in volume; and gas is the state in which matter expands to occupy the volume and shape of its container. Each of these three classical states of matter can transition directly into either of the other two classical states.

The states of matter: This diagram shows the nomenclature for the different phase transitions.

Solids

A solid’s particles are packed closely together. The forces between the particles are strong enough that the particles cannot move freely; they can only vibrate. As a result, a solid has a stable, definite shape and a definite volume. Solids can only change shape under force, as when broken or cut.

In crystalline solids, particles are packed in a regularly ordered, repeating pattern. There are many different crystal structures, and the same substance can have more than one structure. For example, iron has a body-centered cubic structure at temperatures below 912 °C and a face-centered cubic structure between 912 and 1394 °C. Ice has fifteen known crystal structures, each of which exists at a different temperature and pressure.

A solid can transform into a liquid through melting, and a liquid can transform into a solid through freezing. A solid can also change directly into a gas through a process called sublimation.

Liquids

A liquid is a fluid that conforms to the shape of its container but that retains a nearly constant volume independent of pressure. The volume is definite (does not change) if the temperature and pressure are constant. When a solid is heated above its melting point, it becomes liquid because the pressure is higher than the triple point of the substance. Intermolecular (or interatomic or interionic) forces are still important, but the molecules have enough energy to move around, which makes the structure mobile. This means that a liquid is not definite in shape but rather conforms to the shape of its container. Its volume is usually greater than that of its corresponding solid (water is a well-known exception to this rule). The highest temperature at which a particular liquid can exist is called its critical temperature.

A liquid can be converted to a gas through heating at constant pressure to the substance’s boiling point or through reduction of pressure at constant temperature. This process of a liquid changing to a gas is called evaporation.

Gases

Gas molecules have either very weak bonds or no bonds at all, so they can move freely and quickly. Because of this, not only will a gas conform to the shape of its container, it will also expand to completely fill the container. Gas molecules have enough kinetic energy that the effect of intermolecular forces is small (or zero, for an ideal gas), and they are spaced very far apart from each other; the typical distance between neighboring molecules is much greater than the size of the molecules themselves.

A gas at a temperature below its critical temperature can also be called a vapor. A vapor can be liquefied through compression without cooling. It can also exist in equilibrium with a liquid (or solid), in which case the gas pressure equals the vapor pressure of the liquid (or solid).

A supercritical fluid (SCF) is a gas whose temperature and pressure are greater than the critical temperature and critical pressure. In this state, the distinction between liquid and gas disappears. A supercritical fluid has the physical properties of a gas, but its high density lends it the properties of a solvent in some cases. This can be useful in several applications. For example, supercritical carbon dioxide is used to extract caffeine in the manufacturing of decaffeinated coffee.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=42

Phase Changes: What does a phase change look like at the molecular level? This video takes a look at the molecular structure of solids, liquids, and gases and examines how the kinetic energy of the particles changes. The video also discusses melting, vaporization, condensation, and freezing.

Substances and Mixtures

Substances are composed of pure elements or chemically bonded elements, whereas mixtures are composed of non-bonded substances.

Learning Objectives

Distinguish chemical substances from mixtures

Key Takeaways

Key Points

- Matter can be broken down into two categories: pure substances and mixtures. Pure substances are further broken down into elements and compounds. Mixtures are physically combined structures that can be separated into their original components.

- A chemical substance is composed of one type of atom or molecule.

- A mixture is composed of different types of atoms or molecules that are not chemically bonded.

- A heterogeneous mixture is a mixture of two or more chemical substances where the various components can be visually distinguished.

- A homogeneous mixture is a type of mixture in which the composition is uniform and every part of the solution has the same properties.

- Various separation techniques exist in order to separate matter, including include distillation, filtration, evaporation and chromatography. Matter can be in the same phase or in two different phases for this separation to take place.

Key Terms

- mixture: Something that consists of diverse, non-bonded elements or molecules.

- element: A chemical substance that is made up of a particular kind of atom and cannot be broken down or transformed by a chemical reaction.

- substance: A form of matter that has constant chemical composition and characteristic properties. It is composed of one type of atom or molecule.

Chemical Substances

In chemistry, a chemical substance is a form of matter that has constant chemical composition and characteristic properties. It cannot be separated into components without breaking chemical bonds. Chemical substances can be solids, liquids, gases, or plasma. Changes in temperature or pressure can cause substances to shift between the different phases of matter.

An element is a chemical substance that is made up of a particular kind of atom and hence cannot be broken down or transformed by a chemical reaction into a different element. All atoms of an element have the same number of protons, though they may have different numbers of neutrons and electrons.

A pure chemical compound is a chemical substance that is composed of a particular set of molecules or ions that are chemically bonded. Two or more elements combined into one substance through a chemical reaction, such as water, form a chemical compound. All compounds are substances, but not all substances are compounds. A chemical compound can be either atoms bonded together in molecules or crystals in which atoms, molecules or ions form a crystalline lattice. Compounds made primarily of carbon and hydrogen atoms are called organic compounds, and all others are called inorganic compounds. Compounds containing bonds between carbon and a metal are called organometallic compounds.

Chemical substances are often called ‘pure’ to set them apart from mixtures. A common example of a chemical substance is pure water; it always has the same properties and the same ratio of hydrogen to oxygen whether it is isolated from a river or made in a laboratory. Other chemical substances commonly encountered in pure form are diamond (carbon), gold, table salt (sodium chloride), and refined sugar (sucrose). Simple or seemingly pure substances found in nature can in fact be mixtures of chemical substances. For example, tap water may contain small amounts of dissolved sodium chloride and compounds containing iron, calcium, and many other chemical substances. Pure distilled water is a substance, but seawater, since it contains ions and complex molecules, is a mixture.

Chemical Mixtures

A mixture is a material system made up of two or more different substances, which are mixed but not combined chemically. A mixture refers to the physical combination of two or more substances in which the identities of the individual substances are retained. Mixtures take the form of alloys, solutions, suspensions, and colloids.

Naturally occurring sulfur crystals: Sulfur occurs naturally as elemental sulfur, sulfide, and sulfate minerals and in hydrogen sulfide. This mineral deposit is composed of a mixture of substances.

Heterogeneous Mixtures

A heterogeneous mixture is a mixture of two or more chemical substances (elements or compounds), where the different components can be visually distinguished and easily separated by physical means. Examples include:

- mixtures of sand and water

- mixtures of sand and iron filings

- a conglomerate rock

- water and oil

- a salad

- trail mix

- mixtures of gold powder and silver powder

https://lab.concord.org/embeddable.html#interactives/sam/intermolecular-attractions/3-1-oil-and-water.jsonInteractive: Oil and Water: Explore the interactions that cause water and oil to separate from a mixture.

Homogenous Mixtures

A homogeneous mixture is a mixture of two or more chemical substances (elements or compounds), where the different components cannot be visually distinguished. Often separating the components of a homogeneous mixture is more challenging than separating the components of a heterogeneous mixture.

Distinguishing between homogeneous and heterogeneous mixtures is a matter of the scale of sampling. On a small enough scale, any mixture can be said to be heterogeneous, because a sample could be as small as a single molecule. In practical terms, if the property of interest is the same regardless of how much of the mixture is taken, the mixture is homogeneous.

A mixture’s physical properties, such as its melting point, may differ from those of its individual components. Some mixtures can be separated into their components by physical (mechanical or thermal) means.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=42

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=42

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=42

Classifying Matter (3 parts): Introduction to classifying matter as a substance or a mixture of substances. Mixtures are described as heterogeneous or homogeneous. Three common methods of separation are described.

Elements and Compounds

An element is a material that consists of a single type of atom, while a compound consists of two or more types of atoms.

Learning Objectives

Differentiate between elements and compounds and explore separation techniques

Key Takeaways

Key Points

- Elements are the simplest complete chemical substances. Each element corresponds to a single entry on the periodic table. An element is a material that consists of a single type of atom. Each atom type contains the same number of protons.

- Chemical bonds link elements together to form more complex molecules called compounds. A compound consists of two or more types of elements held together by covalent or ionic bonds.

- Elements cannot be divided into smaller units without large amounts of energy. Compounds, on the other hand, can have their bonds broken with practical amounts of energy, such as the heat from a fire.

- Matter can be broken down into two categories: pure substances and mixtures. Pure substances are further broken down into elements and compounds. Mixtures are physically combined structures that can be separated back into their original components.

Key Terms

- element: Any one of the simplest chemical substances that cannot be changed in a chemical reaction or by any chemical means. Made up of atoms that all have the same number of protons.

- chemical bond: Any of several attractive forces that serve to bind atoms together to form molecules.

- compound: A substance made from two or more elements. Consists of a fixed ratio of chemically bonded atoms. Has unique properties that are different from the properties of its individual elements.

Elements

A chemical element is a pure substance that consists of one type of atom. Each atom has an atomic number, which represents the number of protons that are in the nucleus of a single atom of that element. The periodic table of elements is ordered by ascending atomic number.

The chemical elements are divided into the metals, the metalloids, and the non-metals. Metals, typically found on the left side of the periodic table, are:

- often conductive to electricity

- malleable

- shiny

- sometimes magnetic.

Aluminum, iron, copper, gold, mercury and lead are metals.

In contrast, non-metals, found on the right side of the periodic table (to the right of the staircase), are:

- typically not conductive

- not malleable

- dull (not shiny)

- not magnetic.

Examples of elemental non-metals include carbon and oxygen.

Metalloids have some characteristics of metals and some characteristics of non-metals. Silicon and arsenic are metalloids.

As of November, 2011, 118 elements have been identified (the most recently identified was ununseptium, in 2010). Of these 118 known elements, only the first 98 are known to occur naturally on Earth. The elements that do not occur naturally on Earth are the synthetic products of man-made nuclear reactions. 80 of the 98 naturally-occurring elements are stable; the rest are radioactive, which means they decay into lighter elements over timescales ranging from fractions of a second to billions of years.

The periodic table: The periodic table shows 118 elements, including metals (blue), nonmetals (red), and metalloids (green).

Hydrogen and helium are by far the most abundant elements in the universe. However, iron is the most abundant element (by mass) in the composition of the Earth, and oxygen is the most common element in the layer that is the Earth’s crust.

Although all known chemical matter is composed of these elements, chemical matter itself constitutes only about 15% of the matter in the universe. The remainder is dark matter, a mysterious substance that is not composed of chemical elements. Dark matter lacks protons, neutrons, or electrons.

Compounds

Pure samples of isolated elements are uncommon in nature. While the 98 naturally occurring elements have all been identified in mineral samples from the Earth’s crust, only a small minority of them can be found as recognizable, relatively pure minerals. Among the more common of such “native elements” are copper, silver, gold, and sulfur. Carbon is also commonly found in the form of coal, graphite, and diamonds. The noble gases (e.g., neon) and noble metals (e.g., mercury) can also be found in their pure, non-bonded forms in nature. Still, most of these elements are found in mixtures.

When two distinct elements are chemically combined—i.e., chemical bonds form between their atoms—the result is called a chemical compound. Most elements on Earth bond with other elements to form chemical compounds, such as sodium (Na) and Chloride (Cl), which combine to form table salt (NaCl). Water is another example of a chemical compound. The two or more component elements of a compound can be separated through chemical reactions.

Chemical compounds have a unique and defined structure, which consists of a fixed ratio of atoms held together in a defined spatial arrangement by chemical bonds. Chemical compounds can be:

- molecular compounds held together by covalent bonds

- salts held together by ionic bonds

- intermetallic compounds held together by metallic bonds

- complexes held together by coordinate covalent bonds.

Pure chemical elements are not considered chemical compounds, even if they consist of diatomic or polyatomic molecules (molecules that contain only multiple atoms of a single element, such as H_2 or S_8).

9

Physical and Chemical Properties of Matter

Physical and Chemical Properties of Matter

Properties of matter can be classified as either extensive or intensive and as either physical or chemical.

Learning Objectives

Recognize the difference between physical and chemical, and intensive and extensive, properties

Key Takeaways

Key Points

- All properties of matter are either physical or chemical properties and physical properties are either intensive or extensive.

- Extensive properties, such as mass and volume, depend on the amount of matter being measured.

- Intensive properties, such as density and color, do not depend on the amount of the substance present.

- Physical properties can be measured without changing a substance’s chemical identity.

- Chemical properties can be measured only by changing a substance’s chemical identity.

Key Terms

- intensive property: Any characteristic of matter that does not depend on the amount of the substance present.

- extensive property: Any characteristic of matter that depends on the amount of matter being measured.

- physical property: Any characteristic that can be determined without changing the substance’s chemical identity.

- chemical property: Any characteristic that can be determined only by changing a substance’s molecular structure.

All properties of matter are either extensive or intensive and either physical or chemical. Extensive properties, such as mass and volume, depend on the amount of matter that is being measured. Intensive properties, such as density and color, do not depend on the amount of matter. Both extensive and intensive properties are physical properties, which means they can be measured without changing the substance’s chemical identity. For example, the freezing point of a substance is a physical property: when water freezes, it’s still water (H_2O)—it’s just in a different physical state.

Solid, liquids, and gases: Water can exist in several states, including ice (solid), water (liquid), and water vapor (gas).

A chemical property, meanwhile, is any of a material’s properties that becomes evident during a chemical reaction; that is, any quality that can be established only by changing a substance’s chemical identity. Chemical properties cannot be determined just by viewing or touching the substance; the substance’s internal structure must be affected for its chemical properties to be investigated.

Physical Properties

Physical properties are properties that can be measured or observed without changing the chemical nature of the substance. Some examples of physical properties are:

- color (intensive)

- density (intensive)

- volume (extensive)

- mass (extensive)

- boiling point (intensive): the temperature at which a substance boils

- melting point (intensive): the temperature at which a substance melts

Physical properties: Matter has mass and volume, as demonstrated by this concrete block. You can observe its mass by feeling how heavy it is when you try to pick it up; you can observe its volume by looking at it and noticing its size. Mass and volume are both examples of extensive physical properties.

Chemical Properties

Remember, the definition of a chemical property is that measuring that property must lead to a change in the substance’s chemical structure. Here are several examples of chemical properties:

- Heat of combustion is the energy released when a compound undergoes complete combustion (burning) with oxygen. The symbol for the heat of combustion is ΔHc.

- Chemical stability refers to whether a compound will react with water or air (chemically stable substances will not react). Hydrolysis and oxidation are two such reactions and are both chemical changes.

- Flammability refers to whether a compound will burn when exposed to flame. Again, burning is a chemical reaction—commonly a high-temperature reaction in the presence of oxygen.

- The preferred oxidation state is the lowest-energy oxidation state that a metal will undergo reactions in order to achieve (if another element is present to accept or donate electrons).

Physical and Chemical Changes to Matter

There are two types of change in matter: physical change and chemical change.

Learning Objectives

Identify the key features of physical and chemical changes

Key Takeaways

Key Points

- Physical changes only change the appearance of a substance, not its chemical composition.

- Chemical changes cause a substance to change into an entirely new substance with a new chemical formula.

- Chemical changes are also known as chemical reactions. The “ingredients” of a reaction are called reactants, and the end results are called products.

Key Terms

- chemical change: A process that causes a substance to change into a new substance with a new chemical formula.

- chemical reaction: A process involving the breaking or making of interatomic bonds and the transformation of a substance (or substances) into another.

- physical change: A process that does not cause a substance to become a fundamentally different substance.

There are two types of change in matter: physical change and chemical change. As the names suggest, a physical change affects a substance’s physical properties, and a chemical change affects its chemical properties. Many physical changes are reversible (such as heating and cooling), whereas chemical changes are often irreversible or only reversible with an additional chemical change.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=44

Physical & Chemical Changes: This video describes physical and chemical changes in matter.

Physical change: Blending a smoothie involves physical changes but no chemical changes.

Physical Changes

Another way to think about this is that a physical change does not cause a substance to become a fundamentally different substance but a chemical change causes a substance to change into something chemically new. Blending a smoothie, for example, involves two physical changes: the change in shape of each fruit and the mixing together of many different pieces of fruit. Because none of the chemicals in the smoothie components are changed during blending (the water and vitamins from the fruit are unchanged, for example), we know that no chemical changes are involved.

Cutting, tearing, shattering, grinding, and mixing are further types of physical changes because they change the form but not the composition of a material. For example, mixing salt and pepper creates a new substance without changing the chemical makeup of either component.

Phase changes are changes that occur when substances are melted, frozen, boiled, condensed, sublimated, or deposited. They are also physical changes because they do not change the nature of the substance.

Boiling water: Boiling water is an example of a physical change and not a chemical change because the water vapor still has the same molecular structure as liquid water (H_2O). If the bubbles were caused by the decomposition of a molecule into a gas (such as H_2O →H_2 and O_2), then boiling would be a chemical change.

Chemical Changes

Chemical changes are also known as chemical reactions. The “ingredients” of a reaction are called the reactants, and the end results are called the products. The change from reactants to products is signified by an arrow:

Reactants → Products

The formation of gas bubbles is often the result of a chemical change (except in the case of boiling, which is a physical change). A chemical change might also result in the formation of a precipitate, such as the appearance of a cloudy material when dissolved substances are mixed.

Rotting, burning, cooking, and rusting are all further types of chemical changes because they produce substances that are entirely new chemical compounds. For example, burned wood becomes ash, carbon dioxide, and water. When exposed to water, iron becomes a mixture of several hydrated iron oxides and hydroxides. Yeast carries out fermentation to produce alcohol from sugar.

An unexpected color change or release of odor also often indicates a chemical change. For example, the color of the element chromium is determined by its oxidation state; a single chromium compound will only change color if it undergoes an oxidation or reduction reaction. The heat from cooking an egg changes the interactions and shapes of the proteins in the egg white, thereby changing its molecular structure and converting the egg white from translucent to opaque.

The best way to be completely certain whether a change is physical or chemical is to perform chemical analyses, such as mass spectroscopy, on the substance to determine its composition before and after a reaction.

10

Units of Measurement

Standard Units (SI Units)

The International System of Units (abbreviated SI) is the metric system used in science, industry, and medicine.

Learning Objectives

Recognize SI units and their importance for measurment

Key Takeaways

Key Points

- Every field of science involves taking measurements, understanding them, and communicating them to others. In other words, we all have to speak the same basic language.

- The SI system, also called the metric system, is used around the world.

- There are seven basic units in the SI system: the meter (m), the kilogram (kg), the second (s), the kelvin (K), the ampere (A), the mole (mol), and the candela (cd).

Key Terms

- SI system: A series of units that is accepted and used throughout the scientific world.

The Need for a Common Language

Every field of science involves taking measurements, understanding them, and communicating them to others. In other words, we all have to speak the same basic language. Whether you are a chemist, a physicist, a biologist, an engineer, or even a medical doctor, you need a consistent way of communicating size, mass, shape, temperature, time, amount, energy, power, and speed.

Consider the screen on which you’re reading this text right now. It might be an LCD screen, which is made up of liquid crystals. The chemist developing a specific formulation for a liquid crystal has to meaningfully communicate information to an engineer so that the engineer knows how to manufacture it. The engineer, in turn has to be able to communicate with other engineers, physicists, and chemists to design the circuit boards, display screens, and electronic interfaces of the rest of the computer. If these people don’t all speak the same language, the enterprise will never get off the ground.

The International System of Units (abbreviated SI, from the French Système international d’unités) is the metric system used in science, industry, and medicine. Depending on your age and geographic location, you might be very familiar with the “imperial” system, which includes units such as gallons, feet, miles, and pounds. The imperial system is used for “everyday” measurements in a few places, such as the United States. But in most of the world (including Europe) and in all scientific circles, the SI system is in common use.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=46

Scientific Units the SI and Metric Units: Mr. Causey teaches scientific units of the SI system, the metric system, and the CGS system. Mr. Causey also shares the major prefixes and their meanings. Science measurements are based on the metric system, so it is important that you know the metric base units and prefixes.

Units of the SI System

There are seven base units in the SI system:

- the kilogram (kg), for mass

- the second (s), for time

- the kelvin (K), for temperature

- the ampere (A), for electric current

- the mole (mol), for the amount of a substance

- the candela (cd), for luminous intensity

- the meter (m), for distance

The Seven SI Units: This figure displays the fundamental SI units and the combinations that lead to more complex units of measurement.

It should be apparent that the move into modern times has greatly refined the conditions of measurement for each basic unit in the SI system, making the measurement of, for example, the luminous intensity of a light source a standard measurement in every laboratory in the world. A light source made to produce 20 cd will be the same regardless of whether it is made in the United States, in the UK, or anywhere else. The use of the SI system provides all scientists and engineers with a common language of measurement.

History of the SI System

The SI units of measurement have an interesting history. Over time they have been refined for clarity and simplicity.

- The meter (m), or metre, was originally defined as 1/10,000,000 of the distance from the Earth’s equator to the North Pole measured on the circumference through Paris. In modern terms, it is defined as the distance traveled by light in a vacuum over a time interval of 1/299,792,458 of a second.

- The kilogram (kg) was originally defined as the mass of a liter (i.e., of one thousandth of a cubic meter). It is currently defined as the mass of a platinum-iridium kilogram sample maintained by the Bureau International des Poids et Mesures in Sevres, France.

- The second (s) was originally based on a “standard day” of 24 hours, with each hour divided in 60 minutes and each minute divided in 60 seconds. However, we now know that a complete rotation of the Earth actually takes 23 hours, 56 minutes, and 4.1 seconds. Therefore, a second is now defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

- The ampere (A) is a measure of the amount of electric charge passing a point in an electric circuit per unit time. 6.241×10^18 electrons, or one coulomb, per second constitutes one ampere.

- The kelvin (K) is the unit of the thermodynamic temperature scale. This scale starts at 0 K. The incremental size of the kelvin is the same as that of the degree on the Celsius (also called centigrade) scale. The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (exactly 0.01 °C, or 32.018 °F).

- The mole (mol) is a number that relates molecular or atomic mass to a constant number of particles. It is defined as the amount of a substance that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12.

- The candela (cd) was so named to refer to “candlepower” back in the days when candles were the most common source of illumination (because so many people used candles, their properties were standardized). Now, with the prevalence of incandescent and fluorescent light sources, the candela is defined as the luminous intensity in a given direction of a source that emits monochromatic radiation of frequency Hertz and that has a radiant intensity in that direction of 1/683 watts per steradian.

SI Unit Prefixes

The basic SI units can be expressed as fractions and multiples of basic units by using a set of simple prefixes.

Learning Objectives

Convert between SI units

Key Takeaways

Key Points

- The set of prefixes is simple and easy to use.

- Prefixes cannot be combined.

- The set of prefixes is universal.

Key Terms

- prefix: One or more letters or syllables added to the beginning of a word to modify its meaning; for example, kilo can be added to gram to create kilogram

- fraction: a part of a whole, especially a comparatively small part

SI Unit Prefixes

Now that we know about the SI system and what it provides to the scientist and engineer, we can examine some aspects of actual measurement. The SI system utilizes a standard system of prefixes to the basic units that allow them to be more relevant to and descriptive of relative magnitude.

For example, when reading about chemical kinetics, you may encounter the terms “ms” or “ns,” meaning “millisecond” and “nanosecond” respectively. Once you’ve become accustomed to the practice of using the prefixes, you will immediately know that the millisecond is 1/1000 of one second and is 1 million times larger than a nanosecond, which is 1/1000000000 of one second, or 10-9 seconds.

Briefly review the basic SI units before you study the prefixes.

Unit name

Unit symbol

Quantity name

Quantity symbol

Dimension symbol

meter

m

length

l, x, r

L

kilogram[1]

kg

mass

m

M

second

s

time

t

T

ampere

A

electric current

I

I

kelvin

K

thermodynamic temperature

T

Θ

candela

cd

luminous intensity

Iv

J

mole

mol

amount of substance

n

N

There are 20 accepted prefixes. A prefix may be used to identify multiples of the original unit or fractions of the original unit. For example, kilo- denotes a multiple of a thousand, so there are one thousand meters in a kilometer. Milli- denotes a thousandth; therefore, there are one thousand millimeters in a meter.

Prefixes for SI units: The prefixes redefine the measurement as either a multiple or a fraction of the basic unit.

Keep in mind that prefixes should never be combined. Thus a millionth of a meter is a micrometer, not a millimillimeter, and a millionth of a kilogram is a milligram, not a microkilogram.

In older usage, a micron (a measurement often encountered in physics and engineering) is the same as a micrometer, 10-6 meters. Another older form of usage, the millimicron, is one thousandth of a micrometer, or 1 thousandth of 10-6 meters, or 10-9 meter, now called a nanometer. While these older terms are not in common usage, they are often encountered in older publications, and knowing their modern equivalents is an advantage.

Volume and Density

Density and volume are two common measurements in chemistry.

Learning Objectives

Describe the relationship between density and volume

Key Takeaways

Key Points

- The volume of a substance is related to the quantity of the substance present at a defined temperature and pressure.

- The volume of a substance can be measured in volumetric glassware, such as the volumetric flask and the graduated cylinder.

- Density indicates how much of a substance occupies a specific volume at a defined temperature and pressure. The density of a substance can be used to define the substance.

- Water is unusual because when water freezes, its solid form (ice) is less dense than liquid water, and thus floats on top of liquid water.

Key Terms

- volume: A unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric.

- density: A measure of the amount of matter contained in a given volume.

Volume and Density

The properties of a material may be described in many ways. Any amount of any substance will have a volume. If you have two containers of water that are different sizes, they each hold a different amount, or volume, of water. The unit for volume is a unit derived from the SI unit of length and is not a fundamental SI measurement.

If two water samples have different volumes, they still share a common measurement: the density. Density is another measurement derived from SI basic units. The density of a material is defined as its mass per unit volume. In this example, each volume of water is different and therefore has a specific and unique mass. The mass of water is expressed in grams (g) or kilograms (kg), and the volume is measured in liters (L), cubic centimeters (cm^3), or milliliters (mL). Density is calculated by the dividing the mass by the volume, so that density is measured as units of mass/volume, often g/mL. If both water samples are at the same temperature, their densities should be identical, regardless of the samples’ volume.

Measurement Tools

The measuring cup: The measuring cup is a common household utensil used for measuring the volumes of liquids.

If you have ever cooked in a kitchen, you have probably seen some sort of measuring cup, which allows the user to measure liquid volumes with reasonable accuracy. The measuring cup expresses liquid volume in the standard SI units of liters and milliliters. Most American measuring cups also measure liquid in the older system of cups and ounces.

Volumetric Glassware

Scientists who work in a laboratory must be familiar with typical laboratory glassware, often called volumetric glassware. These may include beakers, a volumetric flask, an Erlenmeyer flask, and a graduated cylinder. Each of these containers is used in a laboratory setting to measure liquid volumes for different purposes.

Laboratory volumetric glassware: Glassware, such as these beakers, is commonly used in a laboratory setting to conveniently measure and separate different volumes of liquids.

Density of Water

Different substances have different densities, so density is often used as a method to identify a material. Comparing the densities of two materials can also predict how substances will interact. Water is used as the common standard to substances, and it has a density of 1000 kg/m^3 at Standard Temperature and Pressure (called STP).

Using Water as a Density Comparison

When an object is placed in water, the object’s relative density determines whether it floats or sinks. If the object has a lower density than water, it will float to the top of the water. An object with a higher density will sink. For example, cork has a density of 240 kg/m^3, so it will float. Air has a density of approximately 1.2 kg/m^3, so it rises immediately to the top of a water column. The metals sodium (970 kg/m^3) and potassium (860 kg/m^3) will both float on water, while lead (11,340 kg/m^3) will sink.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=46

Density: A Story of Archimedes and the Gold Crown: Is the crown made of pure gold? An ancient Greek king needs to know if he’s been cheated by a goldsmith. He summons Archimedes, who decides to use density to determine the metal. But how can he determine the volume of the crown?

Liquids tend to form layers when added to water. The sugar alcohol glycerol (1,261 kg/m^3) will sink into the water and form a separate layer until it is thoroughly mixed (glycerol is soluble in water). Vegetable oil (approx. 900 kg/m^3) will float on water, and no matter how vigorously mixed, will always return as a layer on the water surface (oil is not soluble in water).

The Variable Density of Water

Water itself is a complicated and unique molecule. Even if the pressure is consistent, water’s density will change based on the temperature. Recall that the three basic forms of matter are solid, liquid and gas (ignore plasma for the time being). As a rule of thumb, almost all materials are more dense in their solid or crystalline form than in their liquid form; place the solid form of almost any material on the surface of its liquid form, and it will sink. Water, on the other hand, does something very special: ice (the solid form of water) floats on liquid water.

Look carefully at the relationship between water’s temperature and its density. Beginning at 100 °C, the density of water steadily increases, as far as 4 °C. At that point, the density trend reverses. At 0 °C, water freezes to ice and floats.

This table lists the densities of water at different temperatures and constant pressure.

The density of water at constant pressure

Temp (ºC)

Density (kg/m^3)

100

958.4

80

971.8

60

983.2

40

992.2

30

995.6502

25

997.0479

22

997.7735

20

998.2071

15

999.1026

10

999.7026

4

999.9720

0

999.8395

−10

998.117

−20

993.547

−30

983.854

The values below 0ºC refer to super cooled water

The implications of this simple fact are enormous: when a lake freezes, ice crusts at the surface and insulates the liquid below from freezing, while at the same time allowing the colder water (with a temp of approx. 4 °C and a high density) to sink to the bottom. If ice did not float, it would sink to the bottom, allowing more ice to form and sink, until the lake froze solid! Scuba divers and swimmers often encounter these water temperature gradients, and they might even encounter a water layer at the very bottom of a lake with a temperature of approximately 4 °C. That’s just about as cold as the lake will get at the bottom; as soon as the water gets colder, the liquid water becomes less dense and rises.

Layers of water in a winter lake: During the winter months of seasonal climates, the warmest water in most lakes and rivers is only 4°C. This 4°C water has the highest density and sinks to the bottom of the lake. As the water becomes colder (<4°C), it becomes less dense and rises to form ice on the surface of the lake. As a result, liquid water always exists in lakes and rivers during the winter months. This unique property of water enables animals and plants to survive under the frozen lake or winter, ensuring that all freshwater life does not go extinct each winter.

Temperature

The ability to measure temperature accurately was a major scientific advancement, putting absolute numbers on an observable phenomenon.

Learning Objectives

State the key developments in the history of temperature measurement

Key Takeaways

Key Points

- The measurement of temperature is accurate and reproducible.

- The measurement of temperature should be against accepted standards.

- Temperature can be calibrated to several scales including Celsius, Fahrenheit, and kelvin.

- Converting between different temperature scales is easy using the conversion equations.

- Kinetic energy arises from the motion of atoms and molecules. It is postulated that at absolute 0, there is no motion and therefore, no kinetic energy.

Key Terms

- temperature: A measure of cold or heat, often measurable with a thermometer.

- kelvin: A unit of measurement for temperature. It is one of the seven base units in the International System of Units (SI).

- Fahrenheit: An unit of measurement for temperature used most commonly in the United States.

- Celsius: A scale and unit of measurement for temperature where 0 °C is the freezing point of water. Also known as centigrade.

- Celsius: A scale and unit of measurement for temperature where 0 °C is the freezing point of water. Also known as centigrade.

How hot was it last summer? Will it be cold enough for skiing next week? Each of these questions forces quantification to a routine experience. Whether we’re talking about the weather, cooking food, or conducting a scientific experiment, we need to know how hot or how cold something is. To know this, one must be able to put some sort of accurate number on the concept. Although there is an entire field of study devoted to measuring temperature (thermometry), the focus of this section is on the fundamental measurements of temperature.

Monthly mean temperature: Temperature enables us to accurately measure and compare climates in different parts of the world.

The History of Measuring Temperature

To people in the 21st century, measuring temperature is a quick and easy thing to do. Thousands of years ago, however, things were different. Temperature-related phenomena were always being observed. Snow fell and collected in cold weather, and melted into liquid water when the air warmed in spring. Liquid water fell as rain when the air was warm. Ice melted when placed near a source of heat, and water completely boiled out of a pot on a hot stove. However, these are all qualitative observations. They do not generate a number: they do not tell us that water freezes at 0 °C, or that it boils at 100 °C. All we learn from observation is that heat and cold do something to water, or that water behaves differently when it is heated or cooled.

In the 16th and 17th centuries, scientists refined the observations and experiments of the Byzantines and Greeks to produce rudimentary devices relating the amount of “hotness” or “coldness” in the air. The devices they built were called thermoscopes. These basic measuring tools utilized the expansion and contraction of air and water when heated and cooled.

The concept was remarkable, but thermoscopes did not have a numeric scale. The thermoscope could not answer the question, “How hot is it today?” with a number, but it could give a relative measurement. The thermoscope was often a simple tube of gas over liquid. Thermoscopes also served as barometers (which measure pressure ). That made it difficult to use them as thermometers, but they reacted to both pressure and temperature. Even when early thermometers did have a numeric scale, the scales were not standardized.

The dawn of the 18th century saw great change in thermometers, thanks to the work of Isaac Newton, Anders Celsius, and Daniel Fahrenheit.

- Isaac Newton proposed a thermometer with a scale of 12 degrees between the freezing and boiling points of water.

- Fahrenheit was working with tubes filled with mercury, which has a very high coefficient of thermal expansion. This, combined with the quality and accuracy of Fahrenheit’s work, led to much greater sensitivity, and his thermometer was standardized against a brine solution and universally adopted, with the Fahrenheit scale being named in his honor.

- Anders Celsius proposed a 100 degree scale for the difference between freezing and boiling of water, and after a few minor adjustments, the Celsius, or centigrade, system was also widely adopted.

Thermometer calibrated with the Celsius scal: Celsius is a scale and unit of measurement for temperature where 0 °C is the freezing point of water. Our ability to accurately measure temperature enables us to measure the weather, cook food accurately, or conduct a scientific experiment.

Further advances led to faster-acting thermometers, which were useful in medicine and chemistry. Early thermometers did not record or hold the temperature they were measuring: if you removed the thermometer from the substance being measured, its reading would change. Scientists invented new thermometers that would maintain their reading, at least for a limited period of time, to reduce measurement errors and make it easier to record the temperature. Dial thermometers using bimetallic strips were also developed. The bimetallic strips are made from two dissimilar metals bonded together, with each metal having a different coefficient of thermal expansion. Upon heating or cooling, the two metals expand or contract at different rates, causing a bending or curvature to appear in the strip. This bending is useful as a transducer for the temperature reading; it can control a thermostatted circuit or drive a simple dial thermometer.

Absolute Zero

Through the development of temperature measurement, however, one question remained unanswered: “How cold can it really get? How cold is absolute 0?”

The trivial answer is “0 degrees,” but what exactly does that mean? Temperature itself is the measurement of the average kinetic energy of a substance. The kinetic energy arises from the motion of atoms and molecules, and it is postulated that at absolute 0, there is no motion and therefore, no kinetic energy. Therefore, the temperature must be “absolute 0.”

The question remains: how much colder is absolute 0 than 0 °C?

In 1848, Lord Kelvin (William Thomson) wrote a paper entitled “On An Absolute Thermometric Scale” about the need to seek out a thermodynamic zero temperature. Using the Celsius system for its measurement of degrees, Lord Kelvin calculated the ultimate cold temperature to be -273 °C. Today that is referred to as 0 K on the Kelvin thermodynamic temperature scale. Modern methods have refined the measurement to -273.16 °C.

Types of Temperature Scales

Temperature can be measured and represented in many different ways. The fundamental requirements of the practice involve accuracy, a standard, linearity, and reproducibility. The SI unit, chosen for its simplicity and relationship to thermodynamics, is the kelvin, named in honor of Lord Kelvin. While incrementally equal to the Celsius scale, the temperature in kelvins is a true representation of the kinetic energy in a thermodynamic sense. Chemistry and physics require many calculations involving temperature. Those calculations are always made in kelvins.

Comparison of temperature scales: Temperatures of some common events and substances in different units.

A comparison of temperature scales table illustrates a variety of temperature scales, some of which are no longer used. It is interesting to see the temperatures of commonly occurring events over these scales, and to imagine the great hurdles that were overcome in developing modern thermometry.

Conversion to and from kelvin: Use the equations in this table to calculate temperatures using the kelvin measurement system.

Although in most cases scientists are equipped with some sort of electronic calculator, there might be times when a conversion from one scale to another is required. Conversion tables can be used to convert a measurement to any scale from any other temperature scale, such as kelvin or Celsius.

Conversion to and from degrees Celsius: Use the equations in this table to convert temperatures to the Celsius measurement system.

1. Despite the prefix "kilo-," the kilogram is the base unit of mass. The kilogram, not the gram, is used in definitions of derived units. Nonetheless, units of mass are named as if the gram were the base unit. ↵

V

Acids and Bases

11

Acids and Bases

Nature of Acids and Bases

Acids and bases will neutralize one another to form liquid water and a salt.

Learning Objectives

Describe the general properties of acids and bases, comparing the three ways to define them

Key Takeaways

Key Points

- An acid is a substance that donates protons (in the Brønsted-Lowry definition) or accepts a pair of valence electrons to form a bond (in the Lewis definition).

- A base is a substance that can accept protons or donate a pair of valence electrons to form a bond.

- Bases can be thought of as the chemical opposite of acids. A reaction between an acid and base is called a neutralization reaction.

- The strength of an acid refers to its ability or tendency to lose a proton; a strong acid is one that completely dissociates in water.

Key Terms

- valence electron: Any of the electrons in the outermost shell of an atom; capable of forming bonds with other atoms.

- Lewis base: Any compound that can donate a pair of electrons and form a coordinate covalent bond.

- Lewis acid: Any compound that can accept a pair of electrons and form a coordinate covalent bond.

Acids

Acids have long been recognized as a distinctive class of compounds whose aqueous solutions exhibit the following properties:

- A characteristic sour taste.

- Changes the color of litmus from blue to red.

- Reacts with certain metals to produce gaseous H_2.

- Reacts with bases to form a salt and water.

Acidic solutions have a pH less than 7, with lower pH values corresponding to increasing acidity. Common examples of acids include acetic acid (in vinegar), sulfuric acid (used in car batteries), and tartaric acid (used in baking).

There are three common definitions for acids:

- Arrhenius acid: any substances that increases the concentration of hydronium ions (H_3O+) in solution.

- Brønsted-Lowry acid: any substance that can act as a proton donor.

- Lewis acid: any substance that can accept a pair of electrons.

Acid Strength and Strong Acids

The strength of an acid refers to how readily an acid will lose or donate a proton, oftentimes in solution. A stronger acid more readily ionizes, or dissociates, in a solution than a weaker acid. The six common strong acids are:

- hydrochloric acid (HCl)

- hydrobromic acid (HBr)

- hydroiodic acid (HI)

- sulfuric acid (H_2SO_4; only the first proton is considered strongly acidic)

- nitric acid (HNO_3)

- perchloric acid (HClO_4)

Each of these acids ionize essentially 100% in solution. By definition, a strong acid is one that completely dissociates in water; in other words, one mole of the generic strong acid, HA, will yield one mole of H+, one mole of the conjugate base, A−, with none of the unprotonated acid HA remaining in solution. By contrast, however, a weak acid, being less willing to donate its proton, will only partially dissociate in solution. At equilibrium, both the acid and the conjugate base will be present, along with a significant amount of the undissociated species, HA.

Factors Affecting Acid Strength

Two key factors contribute to overall strength of an acid:

- polarity of the molecule

- strength of the H-A bond

These two factors are actually related. The more polar the molecule, the more the electron density within the molecule will be drawn away from the proton. The greater the partial positive charge on the proton, the weaker the H-A bond will be, and the more readily the proton will dissociate in solution.

Acid strengths are also often discussed in terms of the stability of the conjugate base. Stronger acids have a larger Ka and a more negative pKa than weaker acids.

Metal and acid reaction: Zinc reacting with hydrochloric acid to form hydrogen gas.

Bases

There are three common definitions of bases:

- Arrhenius base: any compound that donates an hydroxide ion (OH–) in solution.

- Brønsted-Lowry base: any compound capable of accepting a proton.

- Lewis base: any compound capable of donating an electron pair.

In water, basic solutions will have a pH between 7-14.

Base Strength and Strong Bases

A strong base is the converse of a strong acid; whereas an acid is considered strong if it can readily donate protons, a base is considered strong if it can readily deprotonate (i.e, remove an H+ ion) from other compounds. As with acids, we often talk of basic aqueous solutions in water, and the species being deprotonated is often water itself. The general reaction looks like:

Thus, deprotonated water yields hydroxide ions, which is no surprise. The concentration of hydroxide ions increases as pH increases.

Most alkali metal and some alkaline earth metal hydroxides are strong bases in solution. These include:

- sodium hydroxide (NaOH)

- potassium hydroxide (KOH)

- lithium hydroxide (LiOH)

- rubidium hydroxide (RbOH)

- cesium hydroxide (CsOH)

- calcium hydroxide (Ca(OH)2)

- barium hydroxide (Ba(OH)2)

- strontium hydroxide (Sr(OH)2)

The alkali metal hydroxides dissociate completely in solution. The alkaline earth metal hydroxides are less soluble but are still considered to be strong bases.

Acid/Base Neutralization

Acids and bases react with one another to yield water and a salt. For instance:

This reaction is called a neutralization reaction.

Lewis bases and acids: A list of various Lewis bases (right) and Lewis acids (left).

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=62

Acids + Bases Made Easy! Part 1 – What the Heck is an Acid or Base? – Organic Chemistry – YouTube: Ever wondered what the heck an Acid or Base actually is? Were you ever super confused in high school or college chemistry? In this video I introduce to you guys what the heck an Acid and Base really is forgetting the Lewis or Bronstead/Lowry definitions and then we’ll go more in depth in parts 2,3, and 4.

The Arrhenius Definition

An Arrhenius acid dissociates in water to form hydrogen ions, while an Arrhenius base dissociates in water to form hydroxide ions.

Learning Objectives

Recall the Arrhenius acid definition and its limitations.

Key Takeaways

Key Points

- An Arrhenius acid increases the concentration of hydrogen (H+) ions in an aqueous solution, while an Arrhenius base increases the concentration of hydroxide (OH–) ions in an aqueous solution.

- The Arrhenius definitions of acidity and alkalinity are restricted to aqueous solutions and refer to the concentration of the solvent ions.

- The universal aqueous acid–base definition of the Arrhenius concept is described as the formation of a water molecule from a proton and hydroxide ion. Therefore, in Arrhenius acid–base reactions, the reaction between an acid and a base is a neutralization reaction.

Key Terms

- hydronium: The hydrated hydrogen ion ( $H_3O^+$ ).

- acidity: a measure of the overall concentration of hydrogen ions in solution

- alkalinity: a measure of the overall concentration of hydroxide ions in solution

The Arrhenius Definition

An acid-base reaction is a chemical reaction that occurs between an acid and a base. Several concepts exist that provide alternative definitions for the reaction mechanisms involved and their application in solving related problems. Despite several differences in definitions, their importance as different methods of analysis becomes apparent when they are applied to acid-base reactions for gaseous or liquid species, or when acid or base character may be somewhat less apparent.

The Arrhenius definition of acid-base reactions, which was devised by Svante Arrhenius, is a development of the hydrogen theory of acids. It was used to provide a modern definition of acids and bases, and followed from Arrhenius’s work with Friedrich Wilhelm Ostwald in establishing the presence of ions in aqueous solution in 1884. This led to Arrhenius receiving the Nobel Prize in Chemistry in 1903.

As defined by Arrhenius:

- An Arrhenius acid is a substance that dissociates in water to form hydrogen ions (H+). In other words, an acid increases the concentration of H+ ions in an aqueous solution. This protonation of water yields the hydronium ion (H_3O+); in modern times, H+ is used as a shorthand for H_3O+ because it is now known that a bare proton (H+) does not exist as a free species in aqueous solution.

- An Arrhenius base is a substance that dissociates in water to form hydroxide (OH–) ions. In other words, a base increases the concentration of OH– ions in an aqueous solution.

Limitations of the Arrhenius Definition

The Arrhenius definitions of acidity and alkalinity are restricted to aqueous solutions and refer to the concentration of the solvated ions. Under this definition, pure H_2SO_4 or HCl dissolved in toluene are not acidic, despite the fact that both of these acids will donate a proton to toluene. In addition, under the Arrhenius definition, a solution of sodium amide (NaNH_2) in liquid ammonia is not alkaline, despite the fact that the amide ion () will readily deprotonate ammonia. Thus, the Arrhenius definition can only describe acids and bases in an aqueous environment.

Arrhenius Acid-Base Reaction

An Arrhenius acid-base reaction is defined as the reaction of a proton and an hydroxide ion to form water:

Thus, an Arrhenius acid base reaction is simply a neutralization reaction.

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Chemistry 12.1 What are Acids and Bases? (Part 1 of 2) – YouTube: This introduction to acids and bases discusses their general properties and explains the Arrhenius definitions for acids and bases.

The Brønsted-Lowry Definition of Acids and Bases

A Brønsted-Lowry acid is any species capable of donating a proton; a Brønsted-Lowry base is any species capable of accepting a proton.

Learning Objectives

Differentiate Brønsted-Lowry and Arrhenius acids.

Key Takeaways

Key Points

- The formation of conjugate acids and bases is central to the Brønsted-Lowry definition of acids and bases. The conjugate base is the ion or molecule remaining after the acid has lost its proton, and the conjugate acid is the species created when the base accepts the proton.

- Interestingly, water is amphoteric and can act as both an acid and a base. Therefore, it can can play all four roles: conjugate acid, conjugate base, acid, and base.

- A Brønsted-Lowry acid -base reaction can be defined as: acid + base conjugate base + conjugate acid.

Key Terms

- amphoteric: Having the characteristics of both an acid and a base; capable of both donating and accepting a proton (amphiprotic).

- conjugate acid: The species created when a base accepts a proton.

- conjugate base: The species that is left over after an acid donates a proton.

Originally, acids and bases were defined by Svante Arrhenius. His original definition stated that acids were compounds that increased the concentration of hydrogen ions (H+) in solution, whereas bases were compounds that increased the concentration of hydroxide ions (OH–) in solutions. Problems arise with this conceptualization because Arrhenius’s definition is limited to aqueous solutions, referring to the solvation of aqueous ions, and is therefore not inclusive of acids dissolved in organic solvents. To solve this problem, Johannes Nicolaus Brønsted and Thomas Martin Lowry, in 1923, both independently proposed an alternative definition of acids and bases. In this newer system, Brønsted-Lowry acids were defined as any molecule or ion that is capable of donating a hydrogen cation (proton, H+), whereas a Brønsted-Lowry base is a species with the ability to gain, or accept, a hydrogen cation. A wide range of compounds can be classified in the Brønsted-Lowry framework: mineral acids and derivatives such as sulfonates, carboxylic acids, amines, carbon acids, and many more.

Brønsted-Lowry Acid/Base Reaction

Keep in mind that acids and bases must always react in pairs. This is because if a compound is to behave as an acid, donating its proton, then there must necessarily be a base present to accept that proton. The general scheme for a Brønsted-Lowry acid/base reaction can be visualized in the form:

acid + base conjugate base + conjugate acid

Here, a conjugate base is the species that is left over after the Brønsted acid donates its proton. The conjugate acid is the species that is formed when the Brønsted base accepts a proton from the Brønsted acid. Therefore, according to the Brønsted-Lowry definition, an acid-base reaction is one in which a conjugate base and a conjugate acid are formed (note how this is different from the Arrhenius definition of an acid-base reaction, which is limited to the reaction of H+ with OH– to produce water). Lastly, note that the reaction can proceed in either the forward or the backward direction; in each case, the acid donates a proton to the base.

Consider the reaction between acetic acid and water:

Here, acetic acid acts as a Brønsted-Lowry acid, donating a proton to water, which acts as the Brønsted-Lowry base. The products include the acetate ion, which is the conjugate base formed in the reaction, as well as hydronium ion, which is the conjugate acid formed.

Note that water is amphoteric; depending on the circumstances, it can act as either an acid or a base, either donating or accepting a proton. For instance, in the presence of ammonia, water will donate a proton and act as a Brønsted-Lowry acid:

Here, ammonia is the Brønsted-Lowry base. The conjugate acid formed in the reaction is the ammonium ion, and the conjugate base formed is hydroxide.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=62

Chemistry 12.1 What are Acids and Bases? (Part 2 of 2) – YouTube: This lesson continues to describe acids and bases according to their definition. We first look at the Brønsted-Lowry theory, and then describe Lewis acids and bases according to the Lewis Theory

Acid-Base Properties of Water

Water is capable of acting as either an acid or a base and can undergo self-ionization.

Learning Objectives

Explain the amphoteric properties of water.

Key Takeaways

Key Points

- The self- ionization of water can be expressed as: .

- The equilibrium constant for the self-ionization of water is known as KW; it has a value of .

- The value of KW leads to the convenient equation relating pH with pOH: pH + pOH = 14.

Key Terms

- ionization: Any process that leads to the dissociation of a neutral atom or molecule into charged particles (ions).

- autoprotolysis: The autoionization of water (or similar compounds) in which a proton (hydrogen ion) is transferred to form a cation and an anion.

- hydronium: The hydrated hydrogen ion ( $H_3O^+$ ).

Under standard conditions, water will self-ionize to a very small extent. The self-ionization of water refers to the reaction in which a water molecule donates one of its protons to a neighboring water molecule, either in pure water or in aqueous solution. The result is the formation of a hydroxide ion (OH–) and a hydronium ion (H_3O+). The reaction can be written as follows:

This is an example of autoprotolysis (meaning “self-protonating”) and it exemplifies the amphoteric nature of water (ability to act as both an acid and a base ).

Autoprotolysis of water: The self-ionization of water produces hydronium and hydroxide ions in solution.

The Water Ionization Constant, KW

Note that the self-ionization of water is an equilibrium reaction:

Like all equilibrium reactions, this reaction has an equilibrium constant. Because this is a special equilibrium constant, specific to the self-ionization of water, it is denoted KW; it has a value of 1.0 x 10−14. If we write out the actual equilibrium expression for KW, we get the following:

However, because H+ and OH– are formed in a 1:1 molar ratio, we have:

Now, note the definition of pH and pOH:

If we plug in the above value into our equation for pH, we find that:

Here we have the reason why neutral water has a pH of 7.0; it represents the condition at which the concentrations of H+ and OH– are exactly equal in solution.

pH, pOH, and pKW

We have already established that the equilibrium constant KW can be expressed as:

If we take the negative logarithm of both sides of this equation, we get the following:

However, because we know that pKW = 14, we can establish the following relationship:

This relationship always holds true for any aqueous solution, regardless of its level of acidity or alkalinity. Utilizing this equation is a convenient way to quickly determine pOH from pH and vice versa, as well as to determine hydroxide concentration given hydrogen concentration, or vice versa.

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Self-ionization of Water: Explanation of self-ionization of water and the formation of hydronium and hydroxide ions.

Acid Dissociation Constant (Ka)

The acid dissociation constant (Ka) is the measure of the strength of an acid in solution.

Learning Objectives

Compare and contrast acid strengths using Ka and pKa values.

Key Takeaways

Key Points

- An acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution.

- The dissociation constant is usually written as a quotient of the equilibrium concentrations (in mol/L): .

- Often times, the Ka value is expressed by using the pKa, which is equal to . The larger the value of pKa, the smaller the extent of dissociation.

- A weak acid has a pKa value in the approximate range of -2 to 12 in water. Acids with a pKa value of less than about -2 are said to be strong acids.

Key Terms

- dissociation: Referring to the process by which a compound breaks into its constituent ions in solution.

- equilibrium: The state of a reaction in which the rates of the forward and reverse reactions are equal.

The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution. Ka is the equilibrium constant for the following dissociation reaction of an acid in aqueous solution:

In the above reaction, HA (the generic acid), A– (the conjugate base of the acid), and H+ (the hydrogen ion or proton) are said to be in equilibrium when their concentrations do not change over time. As with all equilibrium constants, the value of Ka is determined by the concentrations (in mol/L) of each aqueous species at equilibrium. The Ka expression is as follows:

Acid dissociation constants are most often associated with weak acids, or acids that do not completely dissociate in solution. This is because strong acids are presumed to ionize completely in solution and therefore their Ka values are exceedingly large.

Ka and pKa

Due to the many orders of magnitude spanned by Ka values, a logarithmic measure of the acid dissociation constant is more commonly used in practice. The logarithmic constant (pKa) is equal to -log_10(Ka).

The larger the value of pKa, the smaller the extent of dissociation. A weak acid has a pKa value in the approximate range of -2 to 12 in water. Acids with a pKa value of less than about -2 are said to be strong acids. A strong acid is almost completely dissociated in aqueous solution; it is dissociated to the extent that the concentration of the undissociated acid becomes undetectable. pKa values for strong acids can be estimated by theoretical means or by extrapolating from measurements in non-aqueous solvents with a smaller dissociation constant, such as acetonitrile and dimethylsulfoxide.

Acetic acid dissociation: The acetic acid partially and reversibly dissociates into acetate and hydrogen ions.

Example

Acetic acid is a weak acid with an acid dissociation constant . What is the pKa for acetic acid?

pOH and Other p Scales

A p-scale is a negative logarithmic scale.

Learning Objectives

Convert between pH and pOH scales to solve acid-base equilibrium problems.

Key Takeaways

Key Points

- The p-scale is a negative logarithmic scale. It allows numbers with very small units of magnitude (for instance, the concentration of H+ in solution ) to be converted into more convenient numbers, often within the the range of -2 – 14.

- The most common p-scales are the pH and pOH scales, which measure the concentration of hydrogen and hydroxide ions. According to the water ion product, pH+pOH =14 for all aqueous solutions.

- Because of the convenience of the p-scale, it is used to also denote the small dissociation constants of acids and bases, which are given by the notation pKa and pKb.

Key Terms

- dissociation: the process by which compounds split into smaller constituent molecules, usually reversibly

- logarithm: for a number $x$, the power to which a given base number must be raised in order to obtain x; written logbx.; for example, log216 = 4 because 24 = 16

pH and pOH

Recall the reaction for the autoionization of water:

This reaction has a special equilibrium constant denoted KW, and it can be written as follows:

Because H+ and OH- dissociate in a one-to-one molar ratio,

If we take the negative logarithm of each concentration, we get:

Here we have the reason that neutral water has a pH of 7.0 -; this is the pH at which the concentrations of H+ and OH– are exactly equal.

Lastly, we should take note of the following relationship:

This relationship will always apply to aqueous solutions. It is a quick and convenient way to find pH from pOH, hydrogen ion concentration from hydroxide ion concentration, and more.

The pH and pOH Scale: Relation between p[OH] and p[H] (brighter red is more acidic, which is the lower numbers for the pH scale and higher numbers for the pOH scale; brighter blue is more basic, which is the higher numbers for the pH scale and lower numbers for the pOH scale).

pKa and pKb

Generically, this p-notation can be used for other scales. In acid -base chemistry, the amount by which an acid or base dissociates to form H+ or OH– ions in solution is often given in terms of their dissociation constants (Ka or Kb). However, because these values are often very small for weak acids and weak bases, the p-scale is used to simplify these numbers and make them more convenient to work with. Quite often we will see the notation pKa or pKb, which refers to the negative logarithms of Ka or Kb, respectively.

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Interactive: pH: Test the pH of things like coffee, spit, and soap to determine whether each is acidic, basic, or neutral. Visualize the relative number of hydroxide ions and hydronium ions in solution. Switch between logarithmic and linear scales. Investigate whether changing the volume or diluting with water affects the pH. Or you can design your own liquid!

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pH and pOH: This lesson introduces the pH scale and discusses the relationship between pH, [H+], [OH-] and pOH.

VI

Physics

12

Energy Conservation

Escape Speed

An object reaches escape speed when the sum of its kinetic energy and its gravitational potential energy is equal to zero.

Learning Objectives

Calculate the escape speed of an object given its kinetic energy and the gravitational potential energy

Key Takeaways

Key Points

- It is assumed that the velocity of the object at the ending point will be zero.

- The requisite escape speed of an object to escape a spherically symmetric body is given by: , where is the universal gravitational constant, is the mass of the body, and is the distance of the object from the body’s center of gravity.

- Escape speed is the required speed that an object has to have to go from a starting point in a gravitational potential field to an ending point that is infinitely far away.

- The speed at which a body rotates will affect the required velocity that an object must have relative to the surface of the body.

- Objects that have propulsion systems do not need to reach escape velocity.

Key Terms

- propulsion: Force causing movement.

- potential energy: The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)

- kinetic energy: The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.

Escape speed is the required starting speed required by an object to go from a starting point in a gravitational potential field to an ending point that is infinitely far away. It is assumed that the velocity of the object at the ending point will be zero.

Isaac Newton’s Analysis of Escape Speed: In this figure, Objects A and B don’t have the required escape speed and so they fall back to Earth after launch. Objects C and D don’t either, they achieve a circular and an elliptical orbit respectively. Object E is launched with sufficient escape velocity and escapes the Earth.

Imagine a situation in which a spaceship that does not have a propulsion system is launched straight away from a planet. (It is moot to discuss escape speed for objects with propulsion systems.) Let us assume that the only significant force that is acting on the spaceship is the force of gravity from the planet. The escape speed of the spaceship can calculated through a simple analysis of conservation of energy. The gravitational potential energy of the spaceship is:

Where is the universal gravitational constant (), is the mass of the planet, is the mass of the spaceship, and is the distance of the spaceship from the planet’s center of gravity.

At the ending point of the spaceship, goes to infinity. As goes to infinity, the value of the gravitational potential energy expression goes to 0.

The kinetic energy of the spaceship can be found from:

Where is the mass of the spaceship and is the velocity of the spaceship.

At the starting point of the spaceship, the velocity must have a magnitude equal to the escape speed (). The velocity of the spaceship is 0 at its ending point, and so consequently its kinetic energy is 0 in the end as well.

Summarizing the kinetic energy () and potential energy () of the spaceship at it’s initial () and final () states:

Due to conservation of energy, the initial energy must equal the final energy and so we can solve for :

Interestingly, if the spaceship were to fall to the planet from a point infinitely far away it would obtain a final speed of ses_e at the planet.

It should be noted that if an object is launched from a rotating body, such as the Earth, the speed at which the body rotates will affect the required velocity that an object must have relative to the surface of the body. If a rocket is launched tangentially from the Earth’s equator in the same direction that the Earth is turning, it will require a lower velocity relative to the Earth than if it were launched in the opposite direction to meet escape speed requirements.

Additionally, it is a misconception that powered vehicles (such as rockets) require escape speed to leave orbit and travel through outer-space. If the vehicle has a propulsion system to provide it with energy once it has left the surface of the planet, it is not necessary to initially meet escape speed requirements.

13

Introduction

Introduction to Work and Energy

Work is the energy associated with the action of a force.

Learning Objectives

Describe relationship between work, energy, and force

Key Takeaways

Key Points

- Work is the transfer of energy by a force acting on an object as it is displaced.

- The work done by a force is zero if the displacement is either zero or perpendicular to the force.

- The work done is positive if the force and displacement have the same direction, and the work done is negative if they have opposite direction.

Key Terms

- energy: A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.

The work done on a system by a constant force is the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as W=Fdcosθ, where W is work, F is the magnitude of the force on the system, d is the magnitude of the displacement of the system, and θ is the angle between the force vector F and the displacement vector d.

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Work, Power, and Energy: Biology is useful.

Take this example of work in action from: (A) The work done by the force F on this lawn mower is Fdcosθ. Note that Fcosθ is the component of the force in the direction of motion. (B) A person holding a briefcase does no work on it, because there is no motion. No energy is transferred to or from the briefcase. (C) The person moving the briefcase horizontally at a constant speed does no work on it, and transfers no energy to it. (D) Work is done on the briefcase by carrying it up stairs at constant speed, because there is necessarily a component of force F in the direction of the motion. Energy is transferred to the briefcase and could, in turn, be used to do work. (E) When the briefcase is lowered, energy is transferred out of the briefcase and into an electric generator. Here the work done on the briefcase by the generator is negative, removing energy from the briefcase, because F and d are in opposite directions.

Examples of Work: This is how work in progress and energy co-exist and operate. Work is the energy associated with the action of a force.

In physics, a force is said to do work when it acts on a body so that there is a displacement of the point of application, however small, in the direction of the force. Thus a force does work when there is movement under the action of the force. The work done by a constant force of magnitude F on a point that moves a distance d in the direction of the force is the product:

For example, if a force of 10 newton (F = 10 N) acts along point that travels two meters (d = 2 m), then it does the work W = (10 N)(2 m) = 20 N m = 20 J. This is approximately the work done lifting a 1 kg weight from the ground to over a person’s head against the force of gravity. Notice that the work is doubled either by lifting twice the weight in the same distance or by lifting the same weight twice the distance.

Work is closely related to energy. The conservation of energy states that the change in total internal energy of a system equals the added heat minus the work performed by the system (see the first law of thermodynamics, and ):

Baseball Pitcher: A baseball pitcher does work on a baseball by throwing the ball at some force, F, over some distance d, which for the average baseball field, is about 60 feet.

Also, from Newton’s second law for rigid bodies, it can be shown that work on an object is equal to the change in kinetic energy of that object:

The work of forces generated by a potential function is known as potential energy and the forces are said to be conservative. Therefore work on an object moving in a conservative force field is equal to minus the change of potential energy of the object:

This shows that work is the energy associated with the action of a force, and so has the physical dimensions and units of energy.

14

Work Done by a Constant Force

Force in the Direction of Displacement

The work done by a constant force is proportional to the force applied times the displacement of the object.

Learning Objectives

Contrast displacement and distance in constant force situations

Key Takeaways

Key Points

- Understanding work is quintessential to understanding systems in terms of their energy, which is necessary for higher level physics.

- Work is equivalent to the change in kinetic energy of a system.

- Distance is not the same as displacement. If a box is moved 3 meters forward and then 4 meters to the left, the total displacement is 5 meters, not 7 meters.

Key Terms

- work: A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.

Work Done by a Constant Force

When a force acts on an object over a distance, it is said to have done work on the object. Physically, the work done on an object is the change in kinetic energy that that object experiences. We will rigorously prove both of these claims.

The term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as “weight lifted through a height,” which is based on the use of early steam engines to lift buckets of water out of flooded ore mines. The SI unit of work is the newton-meter or joule (J).

Units

One way to validate if an expression is correct is to perform dimensional analysis. We have claimed that work is the change in kinetic energy of an object and that it is also equal to the force times the distance. The units of these two should agree. Kinetic energy – and all forms of energy – have units of joules (J). Likewise, force has units of newtons (N) and distance has units of meters (m). If the two statements are equivalent they should be equivalent to one another.

Displacement versus Distance

Often times we will be asked to calculate the work done by a force on an object. As we have shown, this is proportional to the force and the distance which the object is displaced, not moved. We will investigate two examples of a box being moved to illustrate this.

Example Problems

Here are a few example problems:

(1.a) Consider a constant force of two newtons (F = 2 N) acting on a box of mass three kilograms (M = 3 kg). Calculate the work done on the box if the box is displaced 5 meters.

(1.b) Since the box is displaced 5 meters and the force is 2 N, we multiply the two quantities together. The object’s mass will dictate how fast it is accelerating under the force, and thus the time it takes to move the object from point a to point b. Regardless of how long it takes, the object will have the same displacement and thus the same work done on it.

(2.a) Consider the same box (M = 3 kg) being pushed by a constant force of four newtons (F = 4 N). It begins at rest and is pushed for five meters (d = 5m). Assuming a frictionless surface, calculate the velocity of the box at 5 meters.

(2.b) We now understand that the work is proportional to the change in kinetic energy, from this we can calculate the final velocity. What do we know so far? We know that the block begins at rest, so the initial kinetic energy must be zero. From this we algebraically isolate and solve for the final velocity.

We see that the final velocity of the block is approximately 3.15 m/s.

Force at an Angle to Displacement

A force does not have to, and rarely does, act on an object parallel to the direction of motion.

Learning Objectives

Infer how to adjust one-dimensional motion for our three-dimensional world

Key Takeaways

Key Points

- Work done on an object along a given direction of motion is equal to the force times the displacement times the cosine of the angle.

- No work is done along a direction of motion if the force is perpendicular.

- When considering force parallel to the direction of motion, we omit the cosine term because it equals 1 which does not change the expression.

Key Terms

- dot product: A scalar product.

- work: A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.

The Fundamentals

Up until now, we have assumed that any force acting on an object has been parallel to the direction of motion. We have considered our motion to be one dimensional, only acting along the x or y axis. To best examine and understand how nature operators in our three-dimensional world, we will first discuss work in two dimensions in order to build our intuition.

A force does not have to, and rarely does, act on an object parallel to the direction of motion. In the past, we derived that W = F d; such that the work done on an object is the force acting on the object multiplied by the displacement. But this is not the whole story. This expression contains an assumed cosine term, which we do not consider for forces parallel to the direction of motion. “Why would we do such a thing? ” you may ask. We do this because the two are equivalent. If the angle of the force along the direction of motion is zero, such that the force is parallel to the direction of motion, then the cosine term equals one and does not change the expression. As we increase the force’s angle with respect to the direction of motion, less and less work is done along the direction that we are considering; and more and more work is being done in another, perpendicular, direction of motion. This process continues until we are perpendicular to our original direction of motion, such that the angle is 90, and the cosine term would equal zero; resulting in zero work being done along our original direction. Instead, we are doing work in another direction!

Angle: Recall that both the force and direction of motion are vectors. When the angle is 90 degrees, the cosine term goes to zero. When along the same direction, they equal one.

Let’s show this explicitly and then look at this phenomena in terms of a box moving along the x and y directions.

We have discussed that work is the integral of the force and the dot product respect to x. But in fact, dot product of force and a very small distance is equal to the two terms times cosine of the angle between the two. F * dx = Fdcos(theta). Explicitly,

A Box Being Pushed

Consider a coordinate system such that we have x as the abscissa and y as the ordinate. More so, consider a box being pushed along the x direction. What happens in the following three scenarios?

- The box is being pushed parallel to the x direction?

- The box is being pushed at an angle of 45 degrees to the x direction?

- The box is being pushed at an angle of 60 degrees to the x direction?

- The box is being pushed at an angle of 90 degrees to the x direction?

In the first scenario, we know that all of the force is acting on the box along the x-direction, which means that work will only be done along the x-direction. More so, a vertical perspective the box is not moving – it is unchanged in the y direction. Since the force is acting parallel to the direction of motion, the angle is equal to zero and our total work is simply the force times the displacement in the x-direction.

In the second scenario, the box is being pushed at an angle of 45 degrees to the x-direction; and thus also a 45 degree angle to the y-direction. When evaluated, the cosine of 45 degrees is equal to , or approximately 0.71. This means is that 71% of the force is contributing to the work along the x-direction. The other 29% is acting along the y-direction.

In the third scenario, we know that the force is acting at a 60 degree angle to the x-direction; and thus also a 30 degree angle to the y-direction. When evaluated, cosine of 60 degrees is equal to 1/2. This means that the force is equally acting in the x and y-direction! The work done is linear with respect to both x and y.

In the last scenario, the box is being pushed at an angle perpendicular to the x direction. In other words, we are pushing the box in the y-direction! Thus, the box’s position will be unchanged and experience no displacement along the x-axis. The work done in the x direction will be zero.

15

Power

What is Power?

In physics, power is the rate of doing work—the amount of energy consumed per unit time.

Learning Objectives

Relate power to the transfer, use, and transformation of different types of energy

Key Takeaways

Key Points

- Power implies that energy is transferred, perhaps changing form.

- Energy transfer can be used to do work, so power is also the rate at which this work is performed.

- The unit of power is the joule per second (J/s), known as the watt.

Key Terms

- power: A measure of the rate of doing work or transferring energy.

- watt: In the International System of Units, the derived unit of power; the power of a system in which one joule of energy is transferred per second.

In physics, power is the rate of doing work. It is the amount of energy consumed per unit time. The unit of power is the joule per second (J/s), known as the watt (in honor of James Watt, the eighteenth-century developer of the steam engine). For example, the rate at which a lightbulb transforms electrical energy into heat and light is measured in watts (W)—the more wattage, the more power, or equivalently the more electrical energy is used per unit time.

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Power: A brief overview of power in an algebra-based physics course.

Energy transfer can be used to do work, so power is also the rate at which this work is performed. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is expended during the running because the work is done in a shorter amount of time. The output power of an electric motor is the product of the torque the motor generates and the angular velocity of its output shaft. The power expended to move a vehicle is the product of the traction force of the wheels and the velocity of the vehicle.

Examples of power are limited only by the imagination, because there are as many types as there are forms of work and energy. Sunlight reaching Earth’s surface carries a maximum power of about 1.3 kilowatts per square meter (kW/m^2). A tiny fraction of this is retained by Earth over the long term. Our consumption rate of fossil fuels is far greater than the rate at which they are stored, so it is inevitable that they will be depleted. Power implies that energy is transferred, perhaps changing form. It is never possible to change one form completely into another without losing some of it as thermal energy. For example, a 60-W incandescent bulb converts only 5 W of electrical power to light, with 55 W dissipating into thermal energy. Furthermore, the typical electric power plant converts only 35 to 40 percent of its fuel into electricity. The remainder becomes a huge amount of thermal energy that must be dispersed as heat transfer, as rapidly as it is created. A coal-fired power plant may produce 1,000 megawatts; 1 megawatt (MW) is 106 W of electric power. But the power plant consumes chemical energy at a rate of about 2,500 MW, creating heat transfer to the surroundings at a rate of 1,500 MW.

Coal-fired Power Plant: Tremendous amounts of electric power are generated by coal-fired power plants such as this one in China, but an even larger amount of power goes into heat transfer to the surroundings. The large cooling towers here are needed to transfer heat as rapidly as it is produced. The transfer of heat is not unique to coal plants but is an unavoidable consequence of generating electric power from any fuel—nuclear, coal, oil, natural gas, or the like.

Humans: Work, Energy, and Power

The human body converts energy stored in food into work, thermal energy, and/or chemical energy that is stored in fatty tissue.

Learning Objectives

Identify what factors play a role in basal metabolic rate (BMR)

Key Takeaways

Key Points

- The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate, and the corresponding rate when at rest is called the basal metabolic rate (BMR).

- The energy included in the basal metabolic rate is divided among various systems in the body, with the largest fraction going to the liver and spleen, and the brain coming next.

- About 75% of food calories are used to sustain basic body functions included in the basal metabolic rate.

- Work done by a person is sometimes called useful work, which is work done on the outside world, such as lifting weights.

- The energy consumption of people during various activities can be determined by measuring their oxygen use, because the digestive process is basically one of oxidizing food.

Key Terms

- basal metabolic rate: The amount of energy expended while at rest in a neutrally temperate environment, in the post-absorptive state.

Humans: Work, Energy, and Power

Our own bodies, like all living organisms, are energy conversion machines. Conservation of energy implies that the chemical energy stored in food is converted into work, thermal energy, or stored as chemical energy in fatty tissue, as shown in. Energy consumed by humans is converted to work, thermal energy, and stored fat. By far the largest fraction goes to thermal energy, although the fraction varies depending on the type of physical activity. The fraction going into each form depends both on how much we eat and on our level of physical activity. If we eat more than is needed to do work and stay warm, the remainder goes into body fat.

Energy Conversion in Humans: Energy consumed by humans is converted to work, thermal energy, and stored fat. By far the largest fraction goes to thermal energy, although the fraction varies depending on the type of physical activity.

Functions that Require Energy

All bodily functions, from thinking to lifting weights, require energy. The many small muscle actions accompanying all quiet activity, from sleeping to head scratching, ultimately become thermal energy, as do less visible muscle actions by the heart, lungs, and digestive tract. Shivering, in fact, is an involuntary response to low body temperature that pits muscles against one another to produce thermal energy in the body (and do no work). The kidneys and liver consume a surprising amount of energy, but the biggest surprise of all it that a full 25% of all energy consumed by the body is used to maintain electrical potentials in all living cells. (Nerve cells use this electrical potential in nerve impulses. ) This bioelectrical energy ultimately becomes mostly thermal energy, but some is utilized to power chemical processes such as in the kidneys and liver, and in fat production.

Basal Metabolic Rate

The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate. The total energy conversion rate of a person at rest is called the basal metabolic rate (BMR) and is divided among various systems in the body. The largest fraction goes to the liver and spleen, with the brain coming next. Of course, during vigorous exercise, the energy consumption of the skeletal muscles and heart increase markedly. About 75% of the calories burned in a day go into these basic functions. The BMR is a function of age, gender, total body weight, and amount of muscle mass (which burns more calories than body fat). Athletes have a greater BMR due to this last factor.

Useful Work

Work done by a person is sometimes called useful work, which is work done on the outside world, such as lifting weights. Useful work requires a force exerted through a distance on the outside world, and so it excludes internal work, such as that done by the heart when pumping blood. Useful work does include that done in climbing stairs or accelerating to a full run, because these are accomplished by exerting forces on the outside world. Forces exerted by the body are nonconservative, so that they can change the mechanical energy (KE+PE) of the system worked upon, and this is often the goal.

For example, what is the power output for a 60.0-kg woman who runs up a 3.00 m high flight of stairs in 3.50 s, starting from rest but having a final speed of 2.00 m/s?.

Woman Running Up Stairs: When this woman runs upstairs starting from rest, she converts the chemical energy originally from food into kinetic energy and gravitational potential energy. Her power output depends on how fast she does this.

Her power output depends on how fast she does this. The work going into mechanical energy is W= KE + PE. At the bottom of the stairs, we take both KE and PEg as initially zero; thus,

where h is the vertical height of the stairs. Because all terms are given, we can calculate W and then divide it by time to get power. Substituting the expression for W into the definition of power given in the previous equation, P=W/t yields

.

Entering known values yields

The woman does 1764 J of work to move up the stairs compared with only 120 J to increase her kinetic energy; thus, most of her power output is required for climbing rather than accelerating.

Energy consumption is directly proportional to oxygen consumption because the digestive process is basically one of oxidizing food. We can measure the energy people use during various activities by measuring their oxygen use. Approximately 20 kJ of energy are produced for each liter of oxygen consumed, independent of the type of food.

16

Temperature and Temperature Scales

Celsius Scale

Celsius, or centigrade, is a scale and unit of measurement for temperature. It is one of the most commonly used temperature units.

Learning Objectives

Explain how the Celsius scale is defined

Key Takeaways

Key Points

- The degree Celsius (°C) can refer to a specific temperature on the Celsius scale as well as a unit to indicate a temperature interval, a difference between two temperatures or an uncertainty.

- The Celsius scale is currently defined by two different temperatures: absolute zero and the triple point of Vienna Standard Mean Ocean Water (VSMOW; specially purified water).

- Based on this, the relationship between Celsius and Kelvin is as follows: .

Key Terms

- kelvin: in the International System of Units, the base unit of thermodynamic temperature; 1/273.16 of the thermodynamic temperature of the triple point of water; symbolized as K

- absolute zero: The coldest possible temperature: zero on the Kelvin scale and approximately -273.15°C and -459.67°F. The total absence of heat; the temperature at which motion of all molecules would cease.

- standard atmosphere: an international reference pressure defined as 101.325 kPa and formerly used as a unit of pressure

Celsius, also known as centigrade, is a scale to measure temperature. The unit of measurement is the degree Celsius (°C). It is one of the most commonly used temperature units in the world. The unit system is named after the Swedish astronomer Anders Celsius (1701-1744), who developed a similar temperature scale.

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Temperature Scales: A brief introduction to temperature and temperature scales for students studying thermal physics or thermodynamics.

Thermometer: A thermometer calibrated in degrees Celsius

From 1743 until 1954, 0°C was defined as the freezing point of water, and 100°C was defined as the boiling point of water, both at a pressure of one standard atmosphere, with mercury as the working material. Although these defining correlations are commonly taught in schools today, by international agreement the unit “degree Celsius” and the Celsius scale are currently defined by two different temperatures: absolute zero and the triple point of Vienna Standard Mean Ocean Water (VSMOW; specially purified water). This definition also precisely relates the Celsius scale to the Kelvin scale, which defines the SI base unit of thermodynamic temperature and which uses the symbol K. Absolute zero, the lowest temperature possible (the temperature at which matter reaches minimum entropy), is defined as being precisely 0K and -273.15°C. The temperature of the triple point of water is defined as precisely 273.16K and 0.01°C. Based on this, the relationship between degree Celsius and Kelvin is as follows:

Phase Diagram of Water: In this typical phase diagram of water, the green lines mark the freezing point, and the blue line marks the boiling point, showing how they vary with pressure. The dotted line illustrates the anomalous behavior of water. Note that water changes states based on the pressure and temperature.

.

Besides expressing specific temperatures along its scale (e.g., “Gallium melts at 29.7646°C” and “The temperature outside is 23 degrees Celsius”), the degree Celsius is also suitable for expressing temperature intervals — differences between temperatures, or their uncertainties (e.g. “The output of the heat exchanger is hotter by 40 degrees Celsius” and “Our standard uncertainty is ±3°C”). Because of this dual usage, one must not rely upon the unit name or its symbol to denote that a quantity is a temperature interval; it must be clear through context or explicit statement that the quantity is an interval.

Farenheit Scale

In the Fahrenheit scale, the freezing of water is defined at 32 degrees, while the boiling point of water is defined to be 212 degrees.

Learning Objectives

Explain how the Fahrenheit scale is defined and convert between it and Celsius

Key Takeaways

Key Points

- The Fahrenheit system puts the boiling and freezing points of water exactly 180 degrees apart. Therefore, a degree on the Fahrenheit scale is 1⁄180 of the interval between the freezing point and the boiling point.

- To convert °F to °C, you can use the following formula: . The Fahrenheit and Celsius scales intersect at −40°.

- The Fahrenheit scale was replaced by the Celsius scale in most countries during the mid to late 20th century. Fahrenheit remains the official scale of the United States, Cayman Islands, Palau, Bahamas and Belize.

Key Terms

- brine: a solution of salt (usually sodium chloride) in water

- frigorific mixture: A mixture of two or more chemicals that reaches an equilibrium temperature independent of the temperature of any of its constituent chemicals. The temperature is also relatively independent of the quantities of mixtures as long as a significant amount of each original chemical is present in its pure form

The Fahrenheit scale measures temperature. It is based on a scale proposed in 1724 by physicist Daniel Gabriel Fahrenheit (1686-1736). The unit of this scale is the degree Fahrenheit (°F). On this scale, water’s freezing point is defined to be 32 degrees, while water’s boiling point is defined to be 212 degrees.

Historically, the zero point of the Fahrenheit scale was determined by evaluating a thermometer placed in brine. Fahrenheit himself used a mixture of ice, water, and ammonium chloride (a salt) at a 1:1:1 ratio. This is a frigorific mixture, which stabilizes its temperature automatically; the stable temperature of this mixture was defined as 0 °F (-17.78 °C). The second determining point, 32 degrees, was a mixture of just ice and water at a 1:1 ratio. The third determining point, 96 degrees, was approximately the temperature of the human body, then called “blood-heat. ”

The Fahrenheit system puts the boiling and freezing points of water exactly 180 degrees apart. Therefore, a degree on the Fahrenheit scale is 1/180 of the interval between the freezing point and the boiling point. On the Celsius scale, the freezing and boiling points of water are 100 degrees apart. A temperature interval of 1 °F is equal to an interval of 5/9 degrees Celsius (°C). To convert °F to °C, you can use the following formula:

The Fahrenheit and Celsius scales intersect at -40° (-40 °F and -40 °C represent the same temperature). Absolute zero (-273.15 °C, or 0K) is defined as -459.67 °F.

The Fahrenheit scale was replaced by the Celsius scale in most countries in the mid- to late-20th century, though Canada retains it as a supplementary scale that can be used alongside the Celsius scale. The Fahrenheit scale remains the official scale of the United States, the Cayman Islands, Palau, the Bahamas, and Belize.

Fig 2: Comparison of Celsius vs Fahrenheit scales.

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Temperature Scales: A brief introduction to temperature and temperature scales for students studying thermal physics or thermodynamics.

Absolute Zero

Absolute zero is the coldest possible temperature; formally, it is the temperature at which entropy reaches its minimum value.

Learning Objectives

Explain why absolute zero is a natural choice as the null point for a temperature unit system

Key Takeaways

Key Points

- Absolute zero is universal in the sense that all matter is in ground state at this temperature. Therefore, it is a natural choice as the null point for a temperature unit system.

- K system at absolute zero still possesses quantum mechanical zero-point energy, the energy of its ground state. However, in the interpretation of classical thermodynamics, kinetic energy can be zero, and the thermal energy of matter vanishes.

- The lowest temperature that has been achieved in the laboratory is in the 100 pK range, where pK (pico- Kelvin ) is equivalent to 10-12 K. The lowest natural temperature ever recorded is approximately 1K, observed in the rapid expansion of gases leaving the Boomerang Nebula.

Key Terms

- entropy: A measure of how evenly energy (or some analogous property) is distributed in a system.

- thermodynamics: a branch of natural science concerned with heat and its relation to energy and work

Absolute zerois the coldest possible temperature. Formally, it is the temperature at which entropy reaches its minimum value. More simply put, absolute zero refers to a state in which all the energy of a system is extracted (by definition, the lowest energy state the system can have). Absolute zero is universal in the sense that all matteris in ground state at this temperature. Therefore, it is a natural choice as the null point for a temperature unit system.

Graph of Pressure Versus Temperature: Graph of pressure versus temperature for various gases kept at a constant volume. Note that all of the graphs extrapolate to zero pressure at the same temperature

To be precise, a system at absolute zero still possesses quantum mechanical zero-point energy, the energy of its ground state. The uncertainty principle states that the position of a particle cannot be determined with absolute precision; therefore a particle is in motion even if it is at absolute zero, and a ground state still carries a minimal amount of kinetic energy. However, in the interpretation of classical thermodynamics, kinetic energy can be zero, and the thermal energy of matter vanishes.

The zero point of a thermodynamic temperature scale, such as the Kelvin scale, is set at absolute zero. By international agreement, absolute zero is defined as 0K on the Kelvin scale and as -273.15° on the Celsius scale (equivalent to -459.67° on the Fahrenheit scale). Scientists have brought systems to temperatures very close to absolute zero, at which point matter exhibits quantum effects such as superconductivity and superfluidity. The lowest temperature that has been achieved in the laboratory is in the 100 pK range, where pK (pico-Kelvin) is equivalent to 10-12 K. The lowest natural temperature ever recorded is approximately 1K, observed in the rapid expansion of gases leaving the Boomerang Nebula, shown below.

Boomerang Nebula: The rapid expansion of gases resulting in the Boomerang Nebula causes the lowest observed temperature outside a laboratory.

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Temperature Scales: A brief introduction to temperature and temperature scales for students studying thermal physics or thermodynamics.

Kelvin Scale

The kelvin is a unit of measurement for temperature; the null point of the Kelvin scale is absolute zero, the lowest possible temperature.

Learning Objectives

Explain how the Kelvin scale is defined

Key Takeaways

Key Points

- 0K ( absolute zero ) is universal because all thermal motions of all matter are maximally suppressed at this temperature. Absolute zero is therefore the natural choice as the null point of the Kelvin scale.

- The Kelvin scale is used extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature.

- To convert kelvin to degree Celsius, we use the following formula:.

Key Terms

- absolute zero: The coldest possible temperature: zero on the Kelvin scale and approximately -273.15°C and -459.67°F. The total absence of heat; the temperature at which motion of all molecules would cease.

- Triple point: The unique temperature and pressure at which the solid, liquid and gas phases of a substance are all in equilibrium.

- ideal gas: A hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and with the walls of the container.

The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units (SI) and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using absolute zero as its null point. In the classical description of thermodynamics, absolute zero is the temperature at which all thermal motion ceases.

The choice of absolute zero as null point for the Kelvin scale is logical. Different types of matter boil or freeze at different temperatures, but at 0K (absolute zero), all thermal motions of any matter are maximally suppressed. The Kelvin scale is used extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature.

The Kelvin scale is named after Glasgow University engineer and physicist William Thomson, 1st Baron Kelvin (1824-1907), who wrote of the need for an “absolute thermometric scale. ” Unlike the degree Fahrenheit and the degree Celsius, the kelvin is not referred to or typeset as a degree. The kelvin is the primary unit of measurement in the physical sciences, but it is often used in conjunction with the degree Celsius, which has the same magnitude. The kelvin is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (exactly 0.01°C, or 32.018°F). To convert kelvin to degrees Celsius, we use the following formula:

Subtracting 273.16K from the temperature of the triple point of water, 0.01°C, makes absolute zero (0K) equivalent to -273.15°C and -460°F.

Calculating U

Relationships Between the Temperature Scales: Relationships between the Fahrenheit, Celsius, and Kelvin temperature scales, rounded to the nearest degree. The relative sizes of the scales are also shown

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Temperature Scales: A brief introduction to temperature and temperature scales for students studying thermal physics or thermodynamics.

17

Phase Changes

Phase Changes and Energy Conservation

During a phase transition, certain properties of the medium change, often discontinuously, as a result of some external condition.

Learning Objectives

Describe behavior of the medium during a phase transition

Key Takeaways

Key Points

- The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter and, in rare cases, plasma.

- Once water reaches the boiling point, extra energy is used to change the state of matter and increase the potential energy instead of the kinetic energy.

- Plots of pressure versus temperatures, an example of a phase diagram, provide considerable insight into thermal properties of substances.

Key Terms

- intermolecular: from one molecule to another; between molecules

- plasma: a state of matter consisting of partially ionized gas

- thermodynamic: Relating to the conversion of heat into other forms of energy.

A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature or pressure. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter and, in rare cases, plasma.

As an example, if you boil water, it never goes above 100 degrees Celsius. Only after it has completely evaporated will it get any hotter. This is because once water reaches the boiling point, extra energy is used to change the state of matter and increase the potential energy instead of the kinetic energy. The opposite happens when water freezes. To boil or melt one mole of a substance, a certain amount of energy is required. These amounts of energy are the molar heat of vaporization and molar heat of fusion. If that amount of energy is added to a mole of that substance at boiling or freezing point, all of it will melt or boil, but the temperature won’t change.

Temperature increases linearly with heat, until the melting point. But the heat added does not change the temperature; that heat energy is instead used to break intermolecular bonds and convert ice into water. At this point, there is a mixture of both ice and water. Once all ice has been melted, the temperature again rises linearly with heat added. At the boiling point, temperature no longer rises with heat added because the energy is once again being used to break intermolecular bonds. Once all water has been boiled to steam, the temperature will continue to rise linearly as heat is added.

Temperature vs. Heat: This graph shows the temperature of ice as heat is added.

The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so PT graphs are called phase diagrams. Using the graph, if you know the pressure and temperature you can determine the phase of water. The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist (that is, they exist together in ratios, depending on pressure and temperature). For example, the boiling point of water is 100º C at 1.00 atm. As the pressure increases, the boiling temperature rises steadily to 374º C at a pressure of 218 atm. A pressure cooker (or even a covered pot) will cook food faster because the water can exist as a liquid at temperatures greater than 100º C without all boiling away. The curve ends at a point called the critical point, because at higher temperatures the liquid phase does not exist at any pressure. The critical temperature for oxygen is -118ºC, so oxygen cannot be liquefied above this temperature.

Phase Diagram of Water: In this typical phase diagram of water, the green lines mark the freezing point, and the blue line marks the boiling point, showing how they vary with pressure. The dotted line illustrates the anomalous behavior of water. Note that water changes states based on the pressure and temperature.

Humidity, Evaporation, and Boiling

The amount of water vapor in air is a result of evaporation or boiling, until an equilibrium is reached.

Learning Objectives

Explain why water boils at 100 °C

Key Takeaways

Key Points

- Relative humidity is the fraction of water vapor in a gas compared to the saturation value.

- Since the kinetic energy of a molecule is proportional to its temperature, evaporation proceeds more quickly at higher temperatures.

- Vapor pressure increases with temperature because molecular speeds are higher as temperature increases.

- Water boils at 100 °C because the vapor pressure exceeds atmospheric pressure at this temperature.

Key Terms

- equilibrium: The state of a body at rest or in uniform motion, the resultant of all forces on which is zero.

- vapor pressure: The pressure that a vapor exerts, or the partial pressure if it is mixed with other gases.

- humidity: The amount of water vapor in the air.

Overview

The term relative humidity refers to how much water vapor is in the air compared with the maximum possible. At its maximum, denoted as saturation, the relative humidity is 100%, and evaporation is inhibited. The amount of water vapor the air can hold depends on its temperature. For example, relative humidity rises in the evening, as air temperature declines, sometimes reaching the dew point. At the dew point temperature, relative humidity is 100%, and fog may result from the condensation of water droplets if they are small enough to stay in suspension. Conversely, if one wished to dry something, it is more effective to blow hot air over it rather than cold air, because, among other things, hot air can hold more water vapor.

Evaporation

The capacity of air to hold water vapor is based on vapor pressure of water. The liquid and solid phases are continuously giving off vapor because some of the molecules have high enough speeds to enter the gas phase, a process called evaporation; see (a). For the molecules to evaporate, they must be located near the surface, be moving in the proper direction, and have sufficient kinetic energy to overcome liquid-phase intermolecular forces. When only a small proportion of the molecules meet these criteria, the rate of evaporation is low. Since the kinetic energy of a molecule is proportional to its temperature, evaporation proceeds more quickly at higher temperatures.

If a lid is placed over the container, as in (b), evaporation continues, increasing the pressure, until sufficient vapor has built up for condensation to balance evaporation. Then equilibrium has been achieved, and the vapor pressure is equal to the partial pressure of water in the container. Vapor pressure increases with temperature because molecular speeds are higher as temperature increases.

As the faster-moving molecules escape, the remaining molecules have lower average kinetic energy, and the temperature of the liquid decreases. This phenomenon is also called evaporative cooling. This is why evaporating sweat cools the human body. Evaporation also tends to proceed more quickly with higher flow rates between the gaseous and liquid phase and in liquids with higher vapor pressure. For example, laundry on a clothes line will dry (by evaporation) more rapidly on a windy day than on a still day.

Application for Boiling Water

Why does water boil at 100ºC? The vapor pressure of water at 100ºC is 1.01×10^5 Pa, or 1.00 atm. Thus, it can evaporate without limit at this temperature and pressure. But why does it form bubbles when it boils? This is because water ordinarily contains significant amounts of dissolved air and other impurities, which are observed as small bubbles of air in a glass of water. If a bubble starts out at the bottom of the container at 20ºC, it contains water vapor (about 2.30%). The pressure inside the bubble is fixed at 1.00 atm (we ignore the slight pressure exerted by the water around it). As the temperature rises, the amount of air in the bubble stays the same, but the water vapor increases; the bubble expands to keep the pressure at 1.00 atm. At 100ºC, water vapor enters the bubble continuously since the partial pressure of water is equal to 1.00 atm in equilibrium. It cannot reach this pressure, however, since the bubble also contains air and total pressure is 1.00 atm. The bubble grows in size and thereby increases the buoyant force. The bubble breaks away and rises rapidly to the surface, resulting in boiling. (See. )

Close-up of the Boiling Process: (a) An air bubble in water starts out saturated with water vapor at 20ºC. (b) As the temperature rises, water vapor enters the bubble because its vapor pressure increases. The bubble expands to keep its pressure at 1.00 atm. (c) At 100ºC, water vapor enters the bubble continuously because water’s vapor pressure exceeds its partial pressure in the bubble, which must be less than 1.00 atm. The bubble grows and rises to the surface.

18

Waves

Waves

Wave motion transfers energy from one point to another, usually without permanent displacement of the particles of the medium.

Learning Objectives

Describe process of energy and mass transfer during wave motion

Key Takeaways

Key Points

- A wave can be thought of as a disturbance or oscillation that travels through space-time, accompanied by a transfer of energy.

- The direction a wave propagates is perpendicular to the direction it oscillates for transverse waves.

- A wave does not move mass in the direction of propagation; it transfers energy.

Key Terms

- medium: The material or empty space through which signals, waves or forces pass.

- direction of propagation: The axis along which the wave travels.

- wave: A moving disturbance in the energy level of a field.

Vibrations and waves are extremely important phenomena in physics. In nature, oscillations are found everywhere. From the jiggling of atoms to the large oscillations of sea waves, we find examples of vibrations in almost every physical system. In physics a wave can be thought of as a disturbance or oscillation that travels through space-time, accompanied by a transfer of energy. Wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium —that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations.

The emphasis of the last point highlights an important misconception of waves. Waves transfer energy not mass. An easy way to see this is to imagine a floating ball a few yards out to sea. As the waves propagate (i.e., travel) towards the shore, the ball will not come towards the shore. It may come to shore eventually due to the tides, current or wind, but the waves themselves will not carry the ball with them. A wave only moves mass perpendicular to the direction of propagation—in this case up and down, as illustrated in the figure below:

Wave motion: The point along the axis is analogous to the floating ball at sea. We notice that while it moves up and down it does not move in the direction of the wave’s propagation.

A wave can be transverse or longitudinal depending on the direction of its oscillation. Transverse waves occur when a disturbance causes oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations are parallel to the direction of propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse. Sound, for example, is a longitudinal wave.

The description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic (light) wave transfer caused by vibration. Therefore, concepts such as mass, momentum, inertia or elasticity become crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved. In this chapter we will closely examine the difference between longitudinal and transverse waves along with some of the properties they possess. We will also learn how waves are fundamental in describing motion of many applicable physical systems.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=112

The Wave Equation: A brief introduction to the wave equation, discussing wave velocity, frequency, wavelength, and period.

Transverse Waves

Transverse waves propagate through media with a speed orthogonally to the direction of energy transfer.

Learning Objectives

Describe properties of the transverse wave

Key Takeaways

Key Points

- Transverse waves oscillate in the z-y plane but travel along the x axis.

- A transverse wave has a speed of propagation given by the equation v = fλ.

- The direction of energy transfer is perpendicular to the motion of the wave.

Key Terms

- wavelength: The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.

- trough: A long, narrow depression between waves or ridges.

- speed of propagation: The speed at which a wave moves through a medium.

- crest: The ridge or top of a wave.

- transverse wave: Any wave in which the direction of disturbance is perpendicular to the direction of travel.

- direction of propagation: The axis along which the wave travels.

A transverse wave is a moving wave that consists of oscillations occurring perpendicular (or right angled) to the direction of energy transfer. If a transverse wave is moving in the positive x-direction, its oscillations are in up and down directions that lie in the y–z plane. Light is an example of a transverse wave. For transverse waves in matter, the displacement of the medium is perpendicular to the direction of propagation of the wave. A ripple on a pond and a wave on a string are easily visualized transverse waves.

Transverse waves are waves that are oscillating perpendicularly to the direction of propagation. If you anchor one end of a ribbon or string and hold the other end in your hand, you can create transverse waves by moving your hand up and down. Notice though, that you can also launch waves by moving your hand side-to-side. This is an important point. There are two independent directions in which wave motion can occur. In this case, these are the y and z directions mentioned above. depicts the motion of a transverse wave. Here we observe that the wave is moving in t and oscillating in the x-y plane. A wave can be thought as comprising many particles (as seen in the figure) which oscillate up and down. In the figure we observe this motion to be in x-y plane (denoted by the red line in the figure). As time passes the oscillations are separated by units of time. The result of this separation is the sine curve we expect when we plot position versus time.

Sine Wave: The direction of propagation of this wave is along the t axis.

When a wave travels through a medium–i.e., air, water, etc., or the standard reference medium (vacuum)–it does so at a given speed: this is called the speed of propagation. The speed at which the wave propagates is denoted and can be found using the following formula:

where v is the speed of the wave, f is the frequency , and is the wavelength. The wavelength spans crest to crest while the amplitude is 1/2 the total distance from crest to trough. Transverse waves have their applications in many areas of physics. Examples of transverse waves include seismic S (secondary) waves, and the motion of the electric (E) and magnetic (M) fields in an electromagnetic plane waves, which both oscillate perpendicularly to each other as well as to the direction of energy transfer. Therefore an electromagnetic wave consists of two transverse waves, visible light being an example of an electromagnetic wave.

Wavelength and Amplitude: The wavelength is the distance between adjacent crests. The amplitude is the 1/2 the distance from crest to trough.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=112

Two Types of Waves: Longitudinal vs. Transverse: Even ocean waves!

Longitudinal Waves

Longitudinal waves, sometimes called compression waves, oscillate in the direction of propagation.

Learning Objectives

Give properties and provide examples of the longitudinal wave

Key Takeaways

Key Points

- While longitudinal waves oscillate in the direction of propagation, they do not displace mass since the oscillations are small and involve an equilibrium position.

- The longitudinal ‘waves’ can be conceptualized as pulses that transfer energy along the axis of propagation.

- Longitudinal waves can be conceptualized as pressure waves characterized by compression and rarefaction.

Key Terms

- rarefaction: a reduction in the density of a material, especially that of a fluid

- Longitudinal: Running in the direction of the long axis of a body.

- compression: to increase in density; the act of compressing, or the state of being compressed; compaction

Longitudinal Waves

Longitudinal waves have the same direction of vibration as their direction of travel. This means that the movement of the medium is in the same direction as the motion of the wave. Some longitudinal waves are also called compressional waves or compression waves. An easy experiment for observing longitudinal waves involves taking a Slinky and holding both ends. After compressing and releasing one end of the Slinky (while still holding onto the end), a pulse of more concentrated coils will travel to the end of the Slinky.

Longitudinal Waves: A compressed Slinky is an example of a longitudinal wave. The wave propagates in the same direction of oscillation.

Like transverse waves, longitudinal waves do not displace mass. The difference is that each particle which makes up the medium through which a longitudinal wave propagates oscillates along the axis of propagation. In the example of the Slinky, each coil will oscillate at a point but will not travel the length of the Slinky. It is important to remember that energy, in this case in the form of a pulse, is being transmitted and not the displaced mass.

Longitudinal waves can sometimes also be conceptualized as pressure waves. The most common pressure wave is the sound wave. Sound waves are created by the compression of a medium, usually air. Longitudinal sound waves are waves of alternating pressure deviations from the equilibrium pressure, causing local regions of compression and rarefaction. Matter in the medium is periodically displaced by a sound wave, and thus oscillates. When people make a sound, whether it is through speaking or hitting something, they are compressing the air particles to some significant amount. By doing so, they create transverse waves. When people hear sounds, their ears are sensitive to the pressure differences and interpret the waves as different tones.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=112

Two Types of Waves: Longitudinal vs. Transverse: Even ocean waves!

Water Waves

Water waves can be commonly observed in daily life, and comprise both transverse and longitudinal wave motion.

Learning Objectives

Describe particle movement in water waves

Key Takeaways

Key Points

- The particles which make up a water wave move in circular paths.

- If the waves move slower than the wind above them, energy is transfered from the wind to the waves.

- The oscillations are greatest on the surface of the wave and become weaker deeper in the fluid.

Key Terms

- phase velocity: The velocity of propagation of a pure sine wave of infinite extent and infinitesimal amplitude.

- group velocity: The propagation velocity of the envelope of a modulated travelling wave, which is considered as the propagation velocity of information or energy contained in it.

- plane wave: A constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector.

Water waves, which can be commonly observed in our daily lives, are of specific interest to physicists. Describing detailed fluid dynamics in water waves is beyond the scope of introductory physics courses. Although we often observe water wave propagating in 2D, in this atom we will limit our discussion to 1D propagation.

Water waves: Surface waves in water

The uniqueness of water waves is found in the observation that they comprise both transverse and longitudinal wave motion. As a result, the particles composing the wave move in clockwise circular motion, as seen in. Oscillatory motion is highest at the surface and diminishes exponentially with depth. Waves are generated by wind passing over the surface of the sea. As long as the waves propagate slower than the wind speed just above the waves, there is an energy transfer from the wind to the waves. Both air pressure differences between the upwind and the lee side of a wave crest, as well as friction on the water surface by the wind (making the water to go into the shear stress), contribute to the growth of the waves.

In the case of monochromatic linear plane waves in deep water, particles near the surface move in circular paths, creating a combination of longitudinal (back and forth) and transverse (up and down) wave motions. When waves propagate in shallow water (where the depth is less than half the wavelength ), the particle trajectories are compressed into ellipses. As the wave amplitude (height) increases, the particle paths no longer form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as Stokes drift.

Plane wave: We see a wave propagating in the direction of the phase velocity. The wave can be thought to be made up of planes orthogonal to the direction of the phase velocity.

Since water waves transport energy, attempts to generate power from them have been made by utilizing the physical motion of such waves. Although larger waves are more powerful, wave power is also determined by wave speed, wavelength, and water density. Deep water corresponds with a water depth larger than half the wavelength, as is a common case in the sea and ocean. In deep water, longer-period waves propagate faster and transport their energy faster. The deep-water group velocity is half the phase velocity. In shallow water for wavelengths larger than about twenty times the water depth (as often found near the coast), the group velocity is equal to the phase velocity. These methods have proven viable in some cases but do not provide a fully sustainable form of renewable energy to date.

Water waves: The motion water waves causes particles to follow clockwise circular motion. This is a result of the wave having both transverse and longitudinal properties.

Wavelength, Freqency in Relation to Speed

Waves are defined by its frequency, wavelength, and amplitude among others. They also have two kinds of velocity: phase and group velocity.

Learning Objectives

Identify major characteristic properties of waves

Key Takeaways

Key Points

- The wavelength is the spatial period of the wave.

- The frequency of a wave refers to the number of cycles per unit time and is not to be confused with angular frequency.

- The phase velocity can be expressed as the product of wavelength and frequency.

Key Terms

- wave speed: The absolute value of the velocity at which the phase of any one frequency component of the wave travels.

- wavelength: The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.

- frequency: The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.

Characteristics of Waves

Waves have certain characteristic properties which are observable at first notice. The first property to note is the amplitude. The amplitude is half of the distance measured from crest to trough. We also observe the wavelength, which is the spatial period of the wave (e.g. from crest to crest or trough to trough). We denote the wavelength by the Greek letter .

The frequency of a wave is the number of cycles per unit time — one can think of it as the number of crests which pass a fixed point per unit time. Mathematically, we make the observation that,

Frequencies of different sine waves.: The red wave has a low frequency sine there is very little repetition of cycles. Conversely we say that the purple wave has a high frequency. Note that time increases along the horizontal.

where T is the period of oscillation. Frequency and wavelength can also be related-* with respects to a “speed” of a wave. In fact,

where v is called the wave speed, or more commonly,the phase velocity, the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity.

Finally, the group velocity of a wave is the velocity with which the overall shape of the waves’ amplitudes — known as the modulation or envelope of the wave — propagates through space. In, one may see that the overall shape (or “envelope”) propagates to the right, while the phase velocity is negative.

Fig 2: This shows a wave with the group velocity and phase velocity going in different directions. (The group velocity is positive and the phase velocity is negative. )

Energy Transportation

Waves transfer energy which can be used to do work.

Learning Objectives

Relate direction of energy and wave transportation

Key Takeaways

Key Points

- Waves which are more massive transfer more energy.

- Waves with greater velocities transfer more energy.

- Energy of a wave is transported in the direction of the waves transportation.

Key Terms

- energy: A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.

- power: A measure of the rate of doing work or transferring energy.

- work: A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.

Energy transportion is essential to waves. It is a common misconception that waves move mass. Waves carry energy along an axis defined to be the direction of propagation. One easy example is to imagine that you are standing in the surf and you are hit by a significantly large wave, and once you are hit you are displaced (unless you hold firmly to your ground!). In this sense the wave has done work (it applied a force over a distance). Since work is done over time, the energy carried by a wave can be used to generate power.

Water Wave: Waves that are more massive or have a greater velocity transport more energy.

Similarly we find that electromagnetic waves carry energy. Electromagnetic radiation (EMR) carries energy—sometimes called radiant energy—through space continuously away from the source (this is not true of the near-field part of the EM field). Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. EMR also carries both momentum and angular momentum. These properties may all be imparted to matter with which it interacts (through work). EMR is produced from other types of energy when created, and it is converted to other types of energy when it is destroyed. The photon is the quantum of the electromagnetic interaction, and is the basic “unit” or constituent of all forms of EMR. The quantum nature of light becomes more apparent at high frequencies (or high photon energy). Such photons behave more like particles than lower-frequency photons do.

Electromagnetic Wave: Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D diagram shows a plane linearly polarized wave propagating from left to right.

In general, there is a relation of waves which states that the velocity () of a wave is proportional to the frequency () times the wavelength ():

We also know that classical momentum is given by which relates to force via Newton’s second law:

EM waves with higher frequencies carry more energy. This is a direct result of the equations above. Since we find that higher frequencies imply greater velocity. If velocity is increased then we have greater momentum which implies a greater force (it gets a little bit tricky when we talk about particles moving close to the speed of light, but this observation holds in the classical sense). Since energy is the ability of an object to do work, we find that for a greater force correlates to more energy transfer. Again, this is an easy phenomenon to experience empirically; just stand in front of a faster wave and feel the difference!

19

Sound Intensity and Level

Intensity

Sound Intensity is the power per unit area carried by a wave. Power is the rate that energy is transferred by a wave.

Learning Objectives

Calculate sound intensity from power

Key Takeaways

Key Points

- Sound intensity can be found from the following equation: Δ p – change in pressure, or amplitude ρ – density of the material the sound is traveling through vw – speed of observed sound.

- The larger your sound wave oscillation, the more intense your sound will be.

- Although the units for sound intensity are technically watts per meter squared, it is much more common for it to be referred to as decibels, dB.

Key Terms

- decibel: A common measure of sound intensity that is one tenth of a bel on the logarithmic intensity scale. It is defined as , where and are the relative powers of the sound.

- amplitude: The maximum absolute value of some quantity that varies.

Overview of Intensity

Sound Intensity is the power per unit area carried by a wave. Power is the rate that energy is transferred by a wave.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=114

Sound Intensity and Decibels

The equation used to calculate this intensity, I, is:Where P is the power going through the area, A. The SI unit for intensity is watts per meter squared or,. This is the general intensity formula, but lets look at it from a sound perspective.

Sound Intensity

Sound intensity can be found from the following equation:Δp – change in pressure, or amplitudeρ – density of the material the sound is traveling throughvw – speed of observed sound.Now we have a way to calculate the sound intensity, so lets talk about observed intensity. The pressure variation, amplitude, is proportional to the intensity, So it is safe to say that the larger your sound wave oscillation, the more intense your sound will be. This figure shows this concept.

Sound Intensity: Graphs of the gauge pressures in two sound waves of different intensities. The more intense sound is produced by a source that has larger-amplitude oscillations and has greater pressure maxima and minima. Because pressures are higher in the greater-intensity sound, it can exert larger forces on the objects it encounters

Although the units for sound intensity are technically watts per meter squared, it is much more common for it to be referred to as decibels, dB. A decibel is a ratio of the observed amplitude, or intensity level to a reference, which is 0 dB. The equation for this is:β – decibel levelI – Observed intensityI_0– Reference intensity.For more on decibels, please refer to the Decibel Atom.

For a reference point on intensity levels, below are a list of a few different intensities:

- 0 dB, I = 1×10-12 –> Threshold of human hearing

- 10 dB, I = 1×10-11 –> Rustle of leaves

- 60 dB, I = 1×10-6 –> Normal conversation

- 100 dB, I = 1×10-2 –> Loud siren

- 160 dB, I = 1×10^4–> You just burst your eardrums

Human Perception of Sound

The study of human perception of sound is called psychoacoustics.

Learning Objectives

Explain how frequency is perceived by humans

Key Takeaways

Key Points

- Frequency is perceived by humans as pitch. The sound intensity is what humans can hear, and is generally only a specific range of sound, usually from 20 Hz to 20,000 Hz. The factors that go into a sound are its intensity, frequency and overtones (which are like interference or background noises).

- Your ear is made up of three major sections: the inner, middle and outer ear.

- Your cochlea, which is in your inner ear, not only transmits sound waves to your brain, but also contains a liquid that helps humans maintain their balance.

Key Terms

- eardrum: A thin membrane that separates the outer ear from the middle ear and transmits sound from the air to the malleus.

- cochlea: The complex, spirally coiled, tapered cavity of the inner ear in which sound vibrations are converted into nerve impulses.

The study of the human perception of sound is called psychoacoustics. Many factors go into hearing, including wave properties, sensory and brain processes. First, the wave has to be made, and it has a specific wavelength and frequency. Then the sound wave reaches the human ear, and is processed through many areas. Finally, the sound wave makes it through the ear and to the human brain, where even more action happens. You might think that when something makes a noise that you hear it instantaneously but, in reality, it goes through many steps first.

Wave Properties

We are not going to go into too much detail about the wave’s physical properties, since it is out of the scope of this atom, but remember:

- Frequency is perceived by humans as pitch;

- The sound intensity is the amplitude;

- Humans can only hear a specific range of sound, usually from 20 Hz to 20,000 Hz;

- The factors that go into a sound are its intensity, frequency and overtones (which are like interference, or background noises).

The Human Ear

The human ear is made up of three main sections, as shown in:

The Human Ear: A detailed diagram of the human ear.

1. The outer ear

2. The middle ear

3. The inner ear

We are going to start where a sound wave would start and follow it on its journey from outside your ear all the way to your brain. When you look at someone’s ear, you are really only seeing the pinna, the outer most portion of the ear. It collects and focuses the sound wave. The wave then goes through your ear canal to the eardrum. The sound waves cause the eardrum to vibrate. Then we are in the middle ear, which has three very, very small bones: the malleus, incus and stapes. These can also be referred to as the hammer, anvil and stirrup, respectively. These three bones transmit the signal to the elliptical window. This is the beginning of the inner ear. The sound waves are then transmitted from the elliptical window through the inner ear’s semicircular canals, the cochlea, and the audio nerve, which is filled with fluid. This fluid is what allows the body to detect movements and maintain balance. Your cochlea is shaped like a snail, and is full of teeny tiny hairs. These hairs vibrate differently depending on the frequencies. These vibrations release electrical impulses to the auditory nerve and are then sent to your brain, where they are understood as sound. So while this seems to happen very quickly, sound waves have to travel a long way before you ever hear anything!

Decibels

The decibel is a logarithmic unit used to quantify sound levels, by comparing a physical quantity to a reference level.

Learning Objectives

Identify how decibel is quantified in acoustics

Key Takeaways

Key Points

- In acoustics, the decibel is quantified relative to a reference which has been set at a sound pressure level of 20 micropascals, and is called a 0 dB.

- The following equation is used to calculate the sound pressure level, or amplitude: so is the reference pressure which is 20 micropascals or 0 dB, and s is the observed sound pressure.

- The human ear has a standard sound threshold of 120 dB, which expressed logarithmically is around 1012. This is a standard threshold, but it also depends on frequency. Loudness is a measure of sound intensity taking frequency into account, and is called a A-weighted decibel, dB(A), or a phon.

Key Terms

- decibel: A common measure of sound intensity that is one tenth of a bel on the logarithmic intensity scale. It is defined as dB = 10 * log10(P 1/P 2), where P1 and P2 are the relative powers of the sound.

- phon: A unit of apparent loudness, equal in number to the intensity in decibels of a 1,000-hertz tone judged to be as loud as the sound being measured.

The decibel, dB, is commonly used to quantify sound levels, although it is not a unit of sound, but a unit of pressure. The decibel is a logarithmic unit that indicates the ratio if a physical quantity to a reference level. It is one tenth of a Bel, which was named after the inventor of the telephone, Alexander Graham Bell. The word decibell comes from the prefix, deci, that is 1/10 of the word it precedes For more information on how to convert units, refer to the unit conversion atom. Although the decibel can be used to talk about a number of different subjects, in this atom we are going to cover its use in acoustics and sound level.

In acoustics, the decibel is quantified relative to a reference which has been set at a sound pressure level of 20 micropascals, and is called a 0 dB. This reference level is a typical threshold of human hearing perception. The following equation is used to calculate the sound pressure level, or amplitude:so is the reference pressure which is 20 micropascals or 0 dB, and s is the observed sound pressure.The human ear has a standard sound threshold of 120 dB, which expressed logarithmically is around 10^12. This is a standard threshold, but it also depends on frequency. Loudness is a measure of sound intensity taking frequency into account, and is called a A-weighted decibel, dB(A), or a phon. This figure shows The Fletcher Munson Chart, which demonstrates the different sound frequencies and decibels that the human ear perceives as the same.

The Fletcher Munson Chart: The Fletcher-Munson equal-loudness contours. Phons are labelled in blue

20

Overview

Electric Charge in the Atom

Atoms contain negatively charged electrons and positively charged protons; the number of each determines the atom’s net charge.

Learning Objectives

Identify factors that determine the atom’s net charge

Key Takeaways

Key Points

- A proton is a positively charged particle located in the nucleus of an atom. An electron has times the mass of a proton, but an equal and opposite negative charge.

- An elementary charge — that of a proton or electron — is approximately equal to 1.6×10-19Coulombs.

- Unlike protons, electrons can move from atom to atom. If an atom has an equal number of protons and electrons, its net charge is 0. If it gains an extra electron, it becomes negatively charged and is known as an anion. If it loses an electron, it becomes positively charged and is known as a cation.

Key Terms

- nucleus: the massive, positively charged central part of an atom, made up of protons and neutrons

Overview of Atomic Electrical Charges

Atoms, the fundamental building blocks of all molecules, consist of three types of particles: protons, neutrons, and electrons. Of these three subatomic particle types, two (protons and electrons) carry a net electric charge, while neutrons are neutral and have no net charge.

Both protons and electrons have charge that is quantized. That is, the magnitude of their respective charges, which are equal each other, is 1. This standard value is equal to approximately 1.6×10-19 Coulombs.

Protons

Protons are found in the center of the atom; they, with neutrons, make up the nucleus. Protons have a charge of +1 and a mass of 1 atomic mass unit, which is approximately equal to 1.66×10-24 grams. The number of protons in an atom defines the identity of the element (an atom with 1 proton is hydrogen, for example, and an atom with two protons is helium). As such, protons are relatively stable; their number rarely changes, only in the instance of radioactive decay.

Electrons

Electrons are found in the periphery of the atom and have a charge of -1. They are much smaller than protons; their mass is amu. Typically in modeling atoms, protons and neutrons are regarded as stationary, while electrons move about in the space outside the nucleus like a cloud. The negatively charged electronic cloud indicates the regions of the space where electrons are likely to be found. The electrons cloud patterns are extremely complex and is of no importance to the discussion of electric charge in the atom. More important is the fact that electrons are labile; that is, they can be transferred from one atom to the next. It is through electronic transfer that atoms become charged.

Ions

In the ground state, an atom will have an equal number of protons and electrons, and thus will have a net charge of 0. However, because electrons can be transferred from one atom to another, it is possible for atoms to become charged. Atoms in such a state are known as ions.

If a neutral atom gains an electron, it becomes negative. This kind of ion is called an anion.

If a neutral atom loses an electron, it becomes positive. This kind of ion is called a cation.

The steady flow of electrons is called current. Current is what flows through electrical wires and powers electronics items, from light bulbs to televisions.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=116

Electric Charge: A brief overview of atoms, ions, and electrical charge.

Planetary Model of an Atom: Small electrons orbit the large and relatively fixed nucleus of protons and neutrons.

Properties of Electric Charges

Electric charge is a fundamental physical property of matter that has many parallels to mass.

Learning Objectives

Describe properties of electric charge, such as its relativistic invariance and its conservation in closed systems

Key Takeaways

Key Points

- Charge is measured in Coulombs (C), which represent 6.242×10^18 e, where e is the charge of a proton. Charges can be positive or negative, and as such a singular proton has a charge of 1.602×10−19 C, while an electron has a charge of -1.602×10−19 C.

- Electric charge, like mass, is conserved. The force generated by two charges is of the same form as that generated by two masses and, like gravity, force from an electrical field is both conservative and central.

- Electric charge is a relativistic invariant. That is, charge (unlike mass) is independent of speed. Whereas the mass of a particle will exponentially rise as its speed approaches that of light, charge will remain constant.

Key Terms

- coulomb: In the International System of Units, the derived unit of electric charge; the amount of electric charge carried by a current of 1 ampere flowing for 1 second. Symbol: C

- gravity: Resultant force on Earth’s surface, of the attraction by the Earth’s masses, and the centrifugal pseudo-force caused by the Earth’s rotation.

- electric field: A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.

Properties of Electric Charge

Electric charge, like mass and volume, is a physical property of matter. Its SI unit is known as the Coulomb (C), which represents 6.242×10^18e, where e is the charge of a proton. Charges can be positive or negative; a singular proton has a charge of 1.602×10−19 C, while an electron has a charge of -1.602×10−19 C.

Invariance

Like mass, electric charge in a closed system is conserved. As long as a system is impermeable, the amount of charge inside it will neither increase nor decrease; it can only be transferred. However, electric charge differs from other properties—like mass—in that it is a relativistic invariant. That is, charge is independent of speed. The mass of a particle will rise exponentially as its speed approaches that of light, its charge, however, will remain constant.

The independence of electric charge from speed was proven through an experiment in which one fast-moving helium nucleus (two protons and two neutrons bound together) was proven to have the same charge as two separate, slow-moving deuterium nuclei (one proton and one neutron bound together in each nucleus).

Attraction and Repulsion

Electric charge is a property that produces forces that can attract or repel matter. Mass is similar, although it can only attract matter, not repel it. Still, the formula describing the interactions between charges is remarkably similar to that which characterizes the interactions between masses. For electric fields, the force (F) is related to the charges (q_1, q_2) and the distance (r) between them as:

where π and  are constants. This is known as Coulomb’s Law.

Coulomb’s Law: The forces (F_1 and F_2) sum to produce the total force, which is calculated by Coulomb’s Law and is proportional to the product of the charges q_1 and q_2, and inversely proportional to the square of the distance (r_21) between them.

The formula for gravitational force has exactly the same form as Coulomb’s Law, but relates the product of two masses (rather than the charges) and uses a different constant. Both act in a vacuum and are central (depend only on distance between the forces) and conservative (independent of path taken). However, it should be noted that when comparing similar terms, charge-based interaction is substantially greater than that based on mass. For example, the electric repulsion between two electrons is about 10^42 times stronger than their gravitational attraction.

Charge Separation

Charge separation, often referred to as static electricity, is the building of space between particles of opposite charges.

Learning Objectives

Identify factors that can create charge separation

Key Takeaways

Key Points

- Because electrons are labile (i.e., they can be transferred from atom to atom), it is possible for “charge separation” to occur. This phenomenon is often commonly referred to as static electricity.

- Charge separation can be created by friction, pressure, heat, and other charges.

- Charge separation can reach a critical level, whereat it is discharged. Lightning is a common example.

Key Terms

- discharge: the act of releasing an accumulated charge

- static electricity: an electric charge that has built up on an insulated body, often due to friction

- nucleus: the massive, positively charged central part of an atom, made up of protons and neutrons

All matter is composed of atoms made up of negatively-charged electrons and positively-charged protons. In the ground state, each atom is of neutral charge—its protons and electrons are equal in number, and it exists with no permanent dipole. Because electrons are labile (i.e., they can be transferred from atom to atom) it is possible for the phenomenon of “charge separation” (often referred to as static electricity) to occur.

Static Electricity: Due to friction between her hair and the plastic slide, the girl on the left has created charge separation, resulting in her hair being attracted to the slide.

In chemistry, this charge separation is illustrated simply by the transfer of an electron from one atom to another as an ionic bond is formed. In physics, there are many other instances of charge separation that cannot be written as formal chemical reactions. Consider, for example, rubbing a balloon on your hair. Once you pull the balloon away, your hair will stand on end and “reach” towards the balloon. This is because electrons from one have transferred to the other, causing one to be positive and the other to be negative. Thus, the opposite charges attract. A similar example can be seen in playground slides (as shown in ).

Charge separation can be created not only by friction, but by pressure, heat, and other charges. Both pressure and heat increase the energy of a material and can cause electrons to break free and separate from their nuclei. Charge, meanwhile, can attract electrons to or repel them from a nucleus. For example, a nearby negative charge can “push” electrons away from the nucleus around which they typically orbit. Charge separation occurs often in the natural world. It can have an extreme effect if it reaches a critical level, whereat it becomes discharged. Lightning is a common example.

Polarization

Dielectric polarization is the phenomenon that arises when positive and negative charges in a material are separated.

Learning Objectives

Identify two ways polarization can occur on the molecular level

Key Takeaways

Key Points

- Dielectrics are insulators that are capable of being polarized by an electric field. That is, their charges cannot flow freely, but can still be induced to redistribute unevenly.

- Electric fields applied to atoms will push electrons away from the field. In the case of polar molecules, the negative ends thereof will align themselves away from the field while the positive ends will be towards the field.

- An instantaneous polarization occurs when ions, through natural, random vibrations, become distributed asymmetrically such that one area is more dense with one type of ion than another.

Key Terms

- dipole moment: The vector product of the charge on either pole of a dipole and the distance separating them.

- dielectric: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).

- insulator: A substance that does not transmit heat (thermal insulator), sound (acoustic insulator) or electricity (electrical insulator).

The concept of polarity is very broad and can be applied to molecules, light, and electric fields. For the purposes of this atom, we focus on its meaning in the context of what is known as dielectric polarization—the separation of charges in materials.

Dielectrics

A dielectric is an insulator that can be polarized by an electric field, meaning that it is a material in which charge does not flow freely, but in the presence of an electric field it can shift its charge distribution. Positive charge in a dielectric will migrate towards the applied field, while negative charges will shift away. This creates a weak local field within the material that opposes the applied field.

Different materials will react differently to an induced field, depending on their dielectric constant. This constant is the degree of their polarizability (the extent to which they become polarized).

Atomic Model

The most basic view of dielectrics involves considering their charged components: protons and electrons. If an electric field is applied to an atom, the electrons in the atom will migrate away from the applied field. The protons, however, remain relatively exposed to the field. This separation creates a dipole moment, as shown in.

Reaction of an Atom to an Applied Electric Field: When an electric field (E) is applied, electrons drift away from the field. Their average location is displaced from the average location of the protons (which hasn’t moved) by a distance of d. The atom’s dipole moment is represented by M.

Dipole Polarization

On the molecular level, polarization can occur with both dipoles and ions. In polar bonds, electrons are more attracted to one nucleus than to the other. One example of a dipole molecule is water, (H_2O), which has a bent shape (the H-O-H angle is 104.45°) and in which the oxygen pulls electron density away from the H atoms, leaving the H relatively positive and the O relatively negative, as shown in.

Water Molecule: Water is an example of a dipole molecule, which has a bent shape (the H-O-H angle is 104.45°) and in which the oxygen pulls electron density away from the H atoms, leaving the H relatively positive and the O relatively negative.

When a dipolar molecule is exposed to an electric field, the molecule will align itself with the field, with the positive end towards the electric field and the negative end away from it.

Ionic Polarization

Ionic compounds are those that are formed from permanently charge-separated ions. For example, table salt (NaCl) is formed from Na+ and Cl– ions that are not formally bound to one another through a chemical bond, but interact very strongly due to their opposite charges.

Ions are still free from one another and will naturally move at random. If they happen to move in a way that is asymmetrical, and results in a greater concentration of positive ions in one area and a greater concentration of negative ions in another, the sample of ionic compound will be polarized—a phenomenon is known as ionic polarization.

Static Electricity, Charge, and the Conservation of Charge

Electric charge is a physical property that is perpetually conserved in amount; it can build up in matter, which creates static electricity.

Learning Objectives

Formulate rules that apply to the creation and the destruction of electric charge

Key Takeaways

Key Points

- Electric charge is a physical property of matter created by an imbalance in the number of protons and electrons in a substance.

- Charge can be created or destroyed. However, any creation or elimination of charge occurs at a ratio of 1:1 between positive and negative charges.

- Static electricity is when an excess of electric charge collects on an object’s surface.

Key Terms

- electric charge: A quantum number that determines the electromagnetic interactions of some subatomic particles; by convention, the electron has an electric charge of -1 and the proton +1, and quarks have fractional charge.

- discharge: the act of releasing an accumulated charge

- static electricity: an electric charge that has built up on an insulated body, often due to friction

Electric charge is a physical property of matter. It is created by an imbalance in a substance’s number of protons and electrons. The matter is positively charged if it contains more protons than electrons, and it is negatively charged if it contains more electrons than protons. In both instances, charged particles will experience a force when in the presence of other charged matter.

Charges of like sign (positive and positive, or negative and negative) will repel each other, whereas charges of opposite sign (positive and negative) will attract each another, as shown in.

Charge Repulsion and Attraction: Charges of like sign (positive and positive, or negative and negative) will repel each other, whereas charges of opposite sign (positive and negative) will attract each other.

The SI unit for charge is the Coulomb (C), which is approximately equal to elementary charges. (An elementary charge is the magnitude of charge of a proton or electron. )

Conservation of Charge

Charge, like matter, is essentially constant throughout the universe and over time. In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved.

For any finite volume, the law of conservation of charge (Q) can be written as a continuity equation:

where Q(t_1) is the charge in the system at a given time, Q(t_2) is the charge in the same system at a later time, Qin is the charge that has entered the system between the two times, and Qout is the amount of charge that has left the system between the two times.

This does not mean that individual positive and negative charges cannot be created or destroyed. Electric charge is carried by subatomic particles such as electrons and protons, which can be created and destroyed. For example, when particles are destroyed, equal numbers of positive and negative charges are destroyed, keeping the net amount of charge unchanged.

Static Electricity

Static electricity is when an excess of electric charge collects on an object’s surface. It can be created through contact between materials, a buildup of pressure or heat, or the presence of a charge. Static electricity can also be created through friction between a balloon (or another object) and human hair (see ). It can be observed in storm clouds as a result of pressure buildup; lightning (see ) is the discharge that occurs after the charge exceeds a critical concentration.

Static Electricity: Due to friction between her hair and the plastic slide, the girl on the left has created charge separation, resulting in her hair being attracted to the slide.

Lightning: Lightning is a dramatic natural example of static discharge.

Conductors and Insulators

Based on the ability to conduct current, materials are divided into conductors and insulators.

Learning Objectives

Identify conductors and insulators among common materials

Key Takeaways

Key Points

- Resistivity, a physical property that measures the ability of a material to carry current, is the main factor in determining whether a substance is a conductor or an insulator.

- Conductors contain electric charges that, when exposed to a potential difference, move towards one pole or the other. This flow of charge is electric current.

- Insulators are materials in which the internal charge cannot flow freely, and thus cannot conduct electric current to an appreciable degree when exposed to an electric field.

Key Terms

- conductor: A material which contains movable electric charges.

- insulator: A substance that does not transmit heat (thermal insulator), sound (acoustic insulator) or electricity (electrical insulator).

- resistivity: In general, the resistance to electric current of a material; in particular, the degree to which a material resists the flow of electricity.

Overview

All materials can be categorized as either insulators or conductors based on a physical property known as resistivity.

An insulator is a material in which, when exposed to an electric field, the electric charges do not flow freely—it has a high resistivity. Conversely, a conductor is a material that permits the flow of electric charges in one or more directions—its resistivity is low.

Conductors

All conductors contain electric charges that, when exposed to a potential difference, move towards one pole or the other. The positive charges in a conductor will migrate towards the negative end of the potential difference; the negative charges in the material will move towards the positive end of the potential difference. This flow of charge is electric current.

Ionic substances and solutions can conduct electricity, but the most common and effective conductors are metals. Copper is commonly used in wires due to its high conductivity and relatively inexpensive price. However, gold-plated wires are sometimes used in instances in which especially high conductivity is necessary.

Every conductor has a limit to its ampacity, or amount of current it can carry. This usually is the current at which the heat released due to resistance melts the material.

Insulators

Insulators are materials in which the internal charge cannot flow freely, and thus cannot conduct electric current to an appreciable degree when exposed to an electric field.

While there is no perfect insulator with infinite resistivity, materials like glass, paper and Teflon have very high resistivity and can effectively serve as insulators in most instances.

Just as conductors are used to carry electrical current through wires, insulators are commonly used as coating for the wires.

Insulators, like conductors, have their physical limits. When exposed to enough voltage, an insulator will experience what is known as electrical breakdown, in which current suddenly spikes through the material as it becomes a conductor.

Conductor and Insulator in a Wire: This wire consists of a core of copper (a conductor) and a coating of polyethylene (an insulator). The copper allows current to flow through the wire, while the polyethylene ensures that the current does not escape.

The Millikan Oil-Drop Experiment

In 1911, using charged droplets of oil, Robert Millikan was able to determine the charge of an electron.

Learning Objectives

Explain the difference in value of a real electron’s charge and the charge measured by Robert Millikan

Key Takeaways

Key Points

- The Oil-Drop Experiment involved ionizing droplets of oil as they fell through the air, and balancing the force of gravity with the force of an electric field applied by electrodes above and below the droplet.

- Millikan could not directly count the number of electrons on each oil droplet, but found that the common denominator between all measured charges was equal to 1.5924(17)×10−19 C, and thus concluded that this value was the charge of an electron.

- The measured value of an electron’s charge, 1.5924(17)×10−19 C, differs from the accepted value of 1.602176487(40)×10−19 C by less than one percent.

Key Terms

- voltage: The amount of electrostatic potential between two points in space.

- terminal velocity: The speed at which an object in free-fall and not in a vacuum ceases to accelerate downwards because the force of gravity is equal and opposite to the drag force acting against it.

- electric field: A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.

The Oil-Drop Experiment

The Oil-Drop Experiment, otherwise known as the Millikan Oil-Drop Experiment, is one of the most influential studies in the history of physical science.

Performed by Robert Millikan and Harvey Fletcher in 1911, the experiment was designed to determine the charge of a single electron, otherwise known as the elementary electric charge.

Millikan designed his experiment to measure the force on oil droplets between two electrodes.

He used an atomizer to spray a mist of tiny oil droplets into a chamber, which included a hole. Some droplets would fall through this hole and into a chamber, where he measured their terminal velocity and calculated their mass.

Millikan then exposed the droplets to X-rays, which ionized molecules in the air and caused electrons to attach to the oil droplets, thus making them charged. The top and bottom of the chamber were attached to a battery, and the potential difference between the top and bottom produced an electric field that acted on the charged oil drops.

Adjusting the voltage perfectly, Millikan was able to balance the force of gravity (which was exerted downward) with the force of the electric field on the charged particles (which was exerted upward), causing the oil droplets to be suspended in mid-air.

Simplified scheme of Millikan’s oil-drop experiment: This apparatus has a parallel pair of horizontal metal plates. A uniform electric field is created between them. The ring has three holes for illumination and one for viewing through a microscope. Special oil for vacuum apparatus is sprayed into the chamber, where drops become electrically charged. The droplets enter the space between the plates and can be controlled by changing the voltage across the plates.

Millikan then calculated the charge on particles suspended in mid-air. His assumptions were that the force of gravity, which is the product of mass (m) and gravitational acceleration (g), was equal to the force of the electric field (the product of the charge (q) and the electric field (E)):

Since he already knew the mass of the oil droplets and the acceleration due to gravity (9.81 m/s^2), as well as the energy of the x-rays he was using, he was able to calculate the charge.

Although the charge of each droplet was unknown, Millikan adjusted the strength of the X-rays ionizing the air and measured many values of (q) from many different oil droplets. In each instance, the charge measured was a multiple of 1.5924(17)×10−19 C. Thus, it was concluded that the elementary electric charge was 1.5924(17)×10−19 C.

The results were very accurate. The calculated value from the Oil-Drop Experiment differs by less than one percent of the current accepted value of 1.602176487(40)×10−19 C.

The Oil-Drop Experiment was tremendously influential at the time, not only for determining the charge of an electron, but for helping prove the existence of particles smaller than atoms. At the time, it was not fully accepted that protons, neutrons, and electrons existed.

21

Coulomb’s Law

Superposition of Forces

The superposition principle (superposition property) states that for all linear forces the total force is a vector sum of individual forces.

Learning Objectives

Apply the superposition principle to determine the net response caused by two or more stimuli

Key Takeaways

Key Points

- The superposition principle suggests that the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

- Total Coulomb force on a test charge due to a group of charges is equal to the vector sum of all the Coulomb forces between the test charge and other individual charges.

- The superposition of forces is not limited to Coulomb forces. It applies to any types (or combinations) of forces.

Key Terms

- Lorentz force: The force exerted on a charged particle in an electromagnetic field.

- unit vector: A vector with length 1.

- electrostatic force: The electrostatic interaction between electrically charged particles; the amount and direction of attraction or repulsion between two charged bodies.

The superposition principle (also known as superposition property) states that: for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. For Coulomb’s law, the stimuli are forces. Therefore, the principle suggests that total force is a vector sum of individual forces.

Coulomb Force

The scalar form of Coulomb’s Law relates the magnitude and sign of the electrostatic force F, acting simultaneously on two point charges q_1 and q_2:

,

Lorentz Force on a Moving Particle: Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.

where r is the separation distance and ε_0 is electric permittivity. If the product q_1q_2 is positive, the force between them is repulsive; if q_1q_2 is negative, the force between them is attractive. The principle of linear superposition allows the extension of Coulomb’s law to include any number of point charges—in order to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting force vector happens to be parallel to the electric field vector at that point, with that point charge removed.

To calculate the force on a small test charge q at position , due to a system of N discrete charges:

,

where qi and ri are the magnitude and position vector of the i-th charge, respectively, and is a unit vector in the direction of (a vector pointing from charges qi to q. )

Of course, our discussion of superposition of forces applies to any types (or combinations) of forces. For example, when a charge is moving in the presence of a magnetic field as well as an electric field, the charge will feel both electrostatic and magnetic forces. Total force, affecting the motion of the charge, will be the vector sum of the two forces. (In this particular example of the moving charge, the force due to the presence of electromagnetic field is collectively called Lorentz force (see ).

Spherical Distribution of Charge

The charge distribution around a molecule is spherical in nature, and creates a sort of electrostatic “cloud” around the molecule.

Learning Objectives

Describe shape of a Coulomb force from a spherical distribution of charge

Key Takeaways

Key Points

- The force between two objects is inversely proportional to the square of the distance between two objects.

- The attraction or repulsion forces within the spherical distribution of charge is stronger closer to the molecule and becomes weaker as the distance from the molecule increases.

- This law also accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids.

Key Terms

- electrostatic force: The electrostatic interaction between electrically charged particles; the amount and direction of attraction or repulsion between two charged bodies.

Through the work of scientists in the late 18th century, the main features of the electrostatic force —the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb ‘s law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.

Charge distribution in a water molecule: Schematic representation of the outer electron cloud of a neutral water molecule. The electrons spend more time near the oxygen than the hydrogens, giving a permanent charge separation as shown. Water is thus a polar molecule. It is more easily affected by electrostatic forces than molecules with uniform charge distributions.

Modern experiments have verified Coulomb’s law to great precision. For example, it has been shown that the force is inversely proportional to distance between two objects squared (F∝1/r^2) to an accuracy of 1 part in 1016. No exceptions have ever been found, even at the small distances within the atom.

Coulomb’s law holds even within the atoms, correctly describing the force between the positively charged nucleus and each of the negatively charged electrons. This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids.

Generally, as the distance between ions increases, the energy of attraction approaches zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.

An electric field is a vector field which associates to each point of the space the Coulomb force that will experience a test unity charge. Given the electric field, the strength and direction of a force F on a quantity charge q in an electric field E is determined by the electric field. For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is towards for a negative charge.

This distribution around a charged molecule is spherical in nature, and creates a sort of electrostatic “cloud” around the molecule. The attraction or repulsion forces within the spherical distribution of charge is stronger closer to the molecule, and becomes weaker as the distance from the molecule increases.

This image shows the outer electron cloud of a neutral water molecule. The charge distribution of the oxygen molecule is negative, and attracts the two positive hydrogen molecules. The attraction between the two opposing charges forms a neutral water molecule. It is a polar molecule because there is still a permanent charge separation because the electrons spend more time near the oxygen than the hydrogens.

Solving Problems with Vectors and Coulomb’s Law

Coulomb’s Law, which calculates the electric force between charged particles, can be written in vector notation as r+.

Learning Objectives

Explain when the vector notation of Coulomb’s Law can be used

Key Takeaways

Key Points

- The vector notation of Coulomb ‘s Law can be used in the simple example of two point charges where only one of which is a source of charge.

- The total force on the field charge for multiple point source charges is the sum of these individual forces.

- Coulomb’s Law can be further simplified and applied to a fixed number of charge points.

Key Terms

- coulomb’s law: the mathematical equation calculating the electrostatic force vector between two charged particles

- electrostatic force: The electrostatic interaction between electrically charged particles; the amount and direction of attraction or repulsion between two charged bodies.

Solving Problems with Vectors and Coulomb’s Law

Electric Force Between Two Point Charges

To address the electrostatic forces among electrically charged particles, first consider two particles with electric charges q and Q, separated in empty space by a distance r. Suppose that we want to find the electric force vector on charge q. (The electric force vector has both a magnitude and a direction. ) We can express the location of charge q as rq, and the location of charge Q as rQ. In this way we can know both how strong the electric force is on a charge, but also what direction that force is directed in. Coulomb’s Law using vectors can be written as:

In this equation, k is equal to ,where is the permittivity of free space and is the relative permittivity of the material in which the charges are immersed. The variables , , and are in bold because they are vectors. Thus, we need to find by performing standard vector subtraction. This means that we need to subtract the corresponding components of vector from vector .

This vector notation can be used in the simple example of two point charges where only one of which is a source of charge.

Application of Coulomb’s Law: In a simple example, the vector notation of Coulomb’s Law can be used when there are two point charges and only one of which is a source charge.

Electric Force on a Field Charge Due to Fixed Source Charges

Suppose there is more than one point source charges providing forces on a field charge. diagrams a fairly simple example with three source charges (shown in green and indexed by subscripts) and one field charge (in red, designated q). We assume that the source charges are fixed in space, and the field charge q is subject to forces from the source charges.

Multiple point charges: Coulomb’s Law applied to more than one point source charges providing forces on a field charge.

Note the coordinate system that has been chosen. All of the charges lie on the corners of a square, and the origin is chosen to collocate with the lower right source charge, and aligned with the square. Since we can have only one origin of coordinates, no more than one of the source points can lie at the origin, and the displacements from different source points to the field point differ. The total force on the field charge q is due to applications of the force described in the vector notation of Coulomb’s Law from each of the source charges. The total force is therefore the sum of these individual forces.

Displacements of field charge: The displacements of the field charge from each source charge are shown as light blue arrows.

Applying Coulomb’s Law three times and summing the results gives us:

.

This equation can further be simplified and applied to a fixed number of charge points.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=118

Coulomb’s Law: In this video I continue with my series of tutorial videos on Electrostatics. It’s pitched at undergraduate level and while it is mainly aimed at physics majors, it should be useful to anybody taking a first course in electricity and magnetism such as engineers etc..

22

Electric Current

The Battery

A battery is a device that converts chemical energy directly to electrical energy.

Learning Objectives

Describe the functions and identify the major components of a battery

Key Takeaways

Key Points

- A battery stores electrical potential from the chemical reaction. When it is connected to a circuit, that electric potential is converted to kinetic energy as the electrons travel through the circuit.

- The voltage or potential difference between two points is defined to be the change in potential energy of a charge q moved from point 1 to point 2, divided by the charge.

- The voltage of a battery is synonymous with its electromotive force, or emf. This force is responsible for the flow of charge through the circuit, known as the electric current.

Key Terms

- battery: A device that produces electricity by a chemical reaction between two substances.

- current: The time rate of flow of electric charge.

- voltage: The amount of electrostatic potential between two points in space.

Symbol of a Battery in a Circuit Diagram: This is the symbol for a battery in a circuit diagram. It originated as a schematic drawing of the earliest type of battery, a voltaic pile. Notice the positive cathode and negative anode. This orientation is important when drawing circuit diagrams to depict the correct flow of electrons.

A battery is a device that converts chemical energy directly to electrical energy. It consists of a number of voltaic cells connected in series by a conductive electrolyte containing anions and cations. One half-cell includes electrolyte and the anode, or negative electrode; the other half-cell includes electrolyte and the cathode, or positive electrode. In the redox (reduction-oxidation) reaction that powers the battery, cations are reduced (electrons are added) at the cathode, while anions are oxidized (electrons are removed) at the anode. The electrodes do not touch each other but are electrically connected by the electrolyte. Some cells use two half-cells with different electrolytes. A separator between half-cells allows ions to flow, but prevents mixing of the electrolytes.

Each half-cell has an electromotive force (or emf), determined by its ability to drive electric current from the interior to the exterior of the cell. The net emf of the cell is the difference between the emfs of its half-cells, or the difference between the reduction potentials of the half-reactions.

The electrical driving force across the terminals of a cell is known as the terminal voltage (difference) and is measured in volts. When a battery is connected to a circuit, the electrons from the anode travel through the circuit toward the cathode in a direct circuit. The voltage of a battery is synonymous with its electromotive force, or emf. This force is responsible for the flow of charge through the circuit, known as the electric current.

A battery stores electrical potential from the chemical reaction. When it is connected to a circuit, that electric potential is converted to kinetic energy as the electrons travel through the circuit. Electric potential is defined as the potential energy per unit charge (q). The voltage, or potential difference, between two points is defined to be the change in potential energy of a charge q moved from point 1 to point 2, divided by the charge. Rearranged, this mathematical relationship can be described as:

Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other. The car battery can move more charge than the motorcycle battery, although both are 12V batteries.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=120

Ideal and Real Batteries: A brief introduction to ideal and real batteries for students studying circuits.

Current and Voltage Measurements in Circuits

The electrical current is directly proportional to the voltage applied and inversely related to the resistance in a circuit.

Learning Objectives

Describe the relationship between the electrical current, voltage, and resistance in a circuit

Key Takeaways

Key Points

- A simple circuit consists of a voltage source and a resistor.

- Ohm ‘s law gives the relationship between current I, voltage V, and resistance R in a simple circuit: I = V/R.

- The SI unit for measuring the rate of flow of electric charge is the ampere, which is equal to a charge flowing through some surface at the rate of one coulomb per second.

Key Terms

- electrical current: the movement of charge through a circuit

- ohm: in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω

- ampere: A unit of electrical current; the standard base unit in the International System of Units. Abbreviation: amp. Symbol: A.

To understand how to measure current and voltage in a circuit, you must also have a general understanding of how a circuit works and how its electrical measurements are related.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=120

What is Voltage?: This video helps with a conceptual understanding of voltage.

An electrical circuit is a type of network that has a closed loop, which provides a return path for the current. A simple circuit consists of a voltage source and a resistor and can be schematically represented as in.

A Simple Circuit: A simple electric circuit made up of a voltage source and a resistor

According to Ohm’s law, The electrical current I, or movement of charge, that flows through most substances is directly proportional to the voltage V applied to it. The electric property that impedes current (crudely similar to friction and air resistance) is called resistance R. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is inversely proportional to current. Ohm’s law can therefore be written as follows:

where I is the current through the conductor in amperes, V is the potential difference measured across the conductor in volts, and R is the resistance of the conductor in ohms (Ω). More specifically, Ohm’s law states that R in this relation is constant, independent of the current. Using this equation, we can calculate the current, voltage, or resistance in a given circuit.

For example, if we had a 1.5V battery that was connected in a closed circuit to a lightbulb with a resistance of 5Ω, what is the current flowing through the circuit? To solve this problem, we would just substitute the given values into Ohm’s law: I = 1.5V/5Ω; I = 0.3 amperes. If we know the current and the resistance, we can rearrange the Ohm’s law equation and solve for voltage V:

A Microscopic View: Drift Speed

The drift velocity is the average velocity that a particle achieves due to an electric field.

Learning Objectives

Relate the drift velocity with the velocity of free charges in conductors

Key Takeaways

Key Points

- There is an electric field in conductors that causes electrons to drift in the direction opposite to the field. The drift velocity is the average velocity of these free charges.

- The expression for the relationship between the current and drift velocity can be obtained by considering the number of free charges in a segment of wire.

- I = qnAv relates the drift velocity to the current, where I is the current through a wire of cross-sectional area A made of a material with a free charge density n. The carriers of the current each have a charge q and move with a drift velocity of magnitude v.

Key Terms

- drift velocity: The average velocity of the free charges in a conductor.

Drift Speed

Electrical signals are known to move very rapidly. Telephone conversations carried by currents in wires cover large distances without noticeable delays. Lights come on as soon as a switch is flicked. Most electrical signals carried by currents travel at speeds on the order of 10^8m/s, a significant fraction of the speed of light. Interestingly, the individual charges that make up the current move much more slowly on average, typically drifting at speeds on the order of 10−4m/s.

The high speed of electrical signals results from the fact that the force between charges acts rapidly at a distance. Thus, when a free charge is forced into a wire, the incoming charge pushes other charges ahead of it, which in turn push on charges farther down the line. The resulting electrical shock wave moves through the system at nearly the speed of light. To be precise, this rapidly moving signal or shock wave is a rapidly propagating change in the electric field.

Electrons Moving Through a Conductor: When charged particles are forced into this volume of a conductor, an equal number are quickly forced to leave. The repulsion between like charges makes it difficult to increase the number of charges in a volume. Thus, as one charge enters, another leaves almost immediately, carrying the signal rapidly forward.

Drift Velocity

Good conductors have large numbers of free charges in them. In metals, the free charges are free electrons. The distance that an individual electron can move between collisions with atoms or other electrons is quite small. The electron paths thus appear nearly random, like the motion of atoms in a gas. However, there is an electric field in the conductor that causes the electrons to drift in the direction shown (opposite to the field, since they are negative). The drift velocity vdis the average velocity of the free charges after applying the field. The drift velocity is quite small, since there are so many free charges. Given an estimate of the density of free electrons in a conductor (the number of electrons per unit volume), it is possible to calculate the drift velocity for a given current. The larger the density, the lower the velocity required for a given current.

Drift Speed: Free electrons moving in a conductor make many collisions with other electrons and atoms. The path of one electron is shown. The average velocity of the free charges is called the drift velocity and is in the direction opposite to the electric field for electrons. The collisions normally transfer energy to the conductor, requiring a constant supply of energy to maintain a steady current.

It is possible to obtain an expression for the relationship between the current and drift velocity by considering the number of free charges in a segment of wire. The number of free charges per unit volume is given the symbol n and depends on the material. Ax is the volume of a segment, so that the number of free charges in it is nAx. The charge ΔQ in this segment is thus qnAx, where q is the amount of charge on each carrier. (Recall that for electrons, q is 1.60×10−19C. ) The current is the charge moved per unit time. Thus, if all the original charges move out of this segment in time t, the current is:

Notably, x/Δt is the magnitude of the drift velocity vd, since the charges move an average distance x in a time t. Rearranging terms gives: I = qnAvd, where I is the current through a wire of cross-sectional area A made of a material with a free charge density n. The carriers of the current each have charges q and move with a drift velocity of magnitude vd.

Current density is the electric current per unit area of cross-section. It has units of Amperes per square meter.

23

Resistance and Resistors

Ohm’s Law

Ohm’s Law states that current is proportional to voltage; circuits are ohmic if they obey the relation V=IR.

Learning Objectives

Contrast shape of current-voltage plots for ohmic and non-ohmic circuits

Key Takeaways

Key Points

- Voltage drives current while resistance impedes it.

- Ohm ‘s Law refers to the proportion relation between voltage and current. It also applies to the specific equation V=IR, which is valid when considering circuits that contain simple resistors (whose resistance is independent of voltage and current).

- Circuits or components that obey the relation V=IR are known as ohmic and have current-voltage plots that are linear and pass through the origin.

- There are components and circuits that are non-ohmic; their I-V plots are not linear and/or don’t pass through the origin.

Key Terms

- simple circuit: A circuit with a single voltage source and a single resistor.

- ohmic: That which obeys Ohm’s law.

Ohm’s Law

What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference V that creates an electric field. The electric field, in turn, exerts force on charges, causing current. The current that flows through most substances is directly proportional to the voltage V applied to it. The German physicist Georg Simon Ohm (1787-1854) was the first to experimentally demonstrate that the current in a metal wire is directly proportional to the voltage applied: .

This important relationship is known as Ohm’s law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn’t always occur. Recall that while voltage drives current, resistance impedes it. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. The current is therefore inversely proportional to the resistance: .

Simple Circuit: A simple electric circuit in which a closed path for current to flow is supplied by conductors (usually metal wires) connecting a load to the terminals of a battery, represented by the red parallel lines. The zigzag symbol represents the single resistor and includes any resistance in the connections to the voltage source.

The unit for resistance is the ohm where 1Ω = 1 V/A. We can combine the two relations above to obtain I = V/R. This relationship is also called Ohm’s law. In this form Ohm’s law really defines resistance for certain materials. Ohm’s law (like Hooke’s law) is not universally valid. The many substances for which Ohm’s law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance R that is independent of voltage V and current I. An object that has simple resistance is called a resistor, even if its resistance is small.

Voltage Drop: The voltage drop across a resistor in a simple circuit equals the voltage output of the battery.

Additional insight is gained by solving I=V/R for V, yielding V=IR. This expression for V can be interpreted as the voltage drop across a resistor produced by the flow of current I. The phrase IR drop is often used for this voltage. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since E=qΔV, and the same q flows through each. Thus, the energy supplied by the voltage source and the energy converted by the resistor are equal.

In a true ohmic device, the same value of resistance will be calculated from R = V/I regardless of the value of the applied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curve is linear (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I will equal R. Or if the current is forced to some value I, then the measured voltage V divided by that current I is also R. We visualize the plot of I versus V as a straight line. There are, however, components of electrical circuits which do not obey Ohm’s law; that is, their relationship between current and voltage (their I–V curve) is nonlinear (or non-ohmic). An example is the p-n junction diode.

Current-Voltage Curves: The I–V curves of four devices: two resistors, a diode, and a battery. The two resistors follow Ohm’s law: The plot is a straight line through the origin. The other two devices do not follow Ohm’s law.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=122

Ohm’s Law: A brief overview of Ohm’s Law.

Temperature and Superconductivity

Superconductivity is a phenomenon of zero electrical resistance and expulsion of magnetic fields in certain materials below a critical temp.

Learning Objectives

Describe behaviors of a superconductor below a critical temperature and in a weak external magnetic field

Key Takeaways

Key Points

- Superconductivity is a Superconductivity is a thermodynamic phase and possesses certain distinguishing properties which are largely independent of microscopic details.

- In superconducting materials, the characteristics of superconductivity appear when the temperature is lowered below a critical temperature. The onset of superconductivity is accompanied by abrupt changes in various physical properties.

- When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected.

- Superconductors are able to maintain a current with no applied voltage.

Key Terms

- high-temperature superconductors: Materials that behave as superconductors at unusually high temperatures (above about 30 K).

- critical temperature: In superconducting materials, the characteristics of superconductivity appears at (and continues below) this temperature.

- superconductivity: The property of a material whereby it has no resistance to the flow of an electric current.

Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature. It was discovered by Heike Kamerlingh Onnes (shown in ) on April 8, 1911 in Leiden.

Heike Kamerlingh Onnes: Heike Kamerlingh Onnes (1853-1926).

Most of the physical properties of superconductors vary from material to material, such as the heat capacity and the critical temperature, critical field, and critical current density at which superconductivity is destroyed. On the other hand, there is a class of properties independent of the underlying material. For instance, all superconductors have exactly zero resistivity to low applied currents when there is no magnetic field present or if the applied field does not exceed a critical value. The existence of these “universal” properties implies that superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties that are largely independent of microscopic details.

In superconducting materials, the characteristics of superconductivity appear when the temperature T is lowered below a critical temperature Tc. The onset of superconductivity is accompanied by abrupt changes in various physical properties—the hallmark of a phase transition. For example, the electronic heat capacity is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear, as illustrated in.

When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected. Rather, the field penetrates the superconductor to a very small distance (characterized by a parameter λ), called the London penetration depth. It decays exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm.

Superconducting phase transition: Behavior of heat capacity (cv, blue) and resistivity (ρ, green) at the superconducting phase transition.

Superconductors are also able to maintain a current with no applied voltage whatsoever—a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a current lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature.

The value of this critical temperature varies from material to material. Usually, conventional superconductors have critical temperatures ranging from around 20 K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. As of 2009, the highest critical temperature found for a conventional superconductor is 39 K for magnesium diboride (MgB_2), although this material’s exotic properties cause some doubt about accurately classifying it as a “conventional” superconductor. High-temperature superconductors can have much higher critical temperatures. For example, YBa_2Cu_3O_7, one of the first cuprate superconductors to be discovered, has a critical temperature of 92 K; mercury-based cuprates have been found with critical temperatures in excess of 130 K. It is of note that the chemical composition and crystal structure of superconducting materials can be quite complex, as seen in.

Unit Cell of YBaCuO superconductor: Unit Cell of YBaCuO superconductor. Atoms are indicated with different colors.

Resistance and Resistivity

Resistance and resistivity describe the extent to which an object or material impedes the flow of electric current.

Learning Objectives

Identify properties of the material that are described by the resistance and resistivity

Key Takeaways

Key Points

- The resistance of an object (i.e., a resistor ) depends on its shape and the material of which it is composed.

- Resistivity ρ is an intrinsic property of a material and directly proportional to the total resistance R, an extrinsic quantity that depends on the length and cross-sectional area of a resistor.

- The resistivity of different materials varies by an enormous amount. Likewise, resistors range over many orders of magnitude.

- Resistors are arranged in series or parallel configurations. The equivalent resistance of a network of resistors in series is the sum of all the resistance. The inverse of the equivalent resistance of a network of resistors in parallel is the sum of the inverse of the resistance of each resistor.

Key Terms

- series equivalent resistance: The resistance of a network of resistors arranged such that the voltage across the network is the sum of the voltage across each resistor. In this case, the equivalent resistance is the sum of the resistance of all the resistors in the network.

- parallel equivalent resistance: the resistance of a network such that each resistor is subject to the same potential difference (voltage), so that the currents through them add. In this case the inverse of the equivalent resistance is equal to the sum of the inverse resistance of all the resistors in the network.

- resistivity: In general, the resistance to electric current of a material; in particular, the degree to which a material resists the flow of electricity.

Resistance and Resistivity

Resistance is the electric property that impedes a current. A current flowing through a wire (or resistor) is like water flowing through a pipe, and the voltage drop across the wire is like the pressure drop which pushes water through the pipe. Resistance is proportional to how much pressure is required to achieve a given flow, while conductance is proportional to how much flow occurs for a given pressure. Conductance and resistance are reciprocals. The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor is easy to analyze, and by so doing we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder’s electric resistance R is directly proportional to its length L, similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again, similar to the flow of fluid through a pipe). In fact, R is inversely proportional to the cylinder’s cross-sectional area A.

Cylindrical Resistor: A uniform cylinder of length L and cross-sectional area A. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area A, the smaller its resistance.

As mentioned, for a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity ρ of a substance so that theresistanceR of an object is directly proportional to ρ. Resistivity ρ is an intrinsic property of a material, independent of its shape or size. In contrast, the resistance R is an extrinsic property that does depend on the size an shape of the resistor. (A similar intrinsic/extrinsic relation exists between heat capacity C and the specific heat c). Recall that an object whose resistance is proportional to the voltage and current is known as a resistor.

Typical Resistor: A typical axial-lead resistor.

What determines resistivity? The resistivity of different materials varies by an enormous amount. For example, the conductivity of teflon is about 1030 times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal has large numbers of “delocalized” electrons that are not stuck in any one place, but free to move across large distances, whereas in an insulator (like teflon), each electron is tightly bound to a single atom, and a great force is required to pull it away. Likewise, resistors range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of 10^12 Ω or more. A dry person may have a hand-to-foot resistance of 10^5 Ω, whereas the resistance of the human heart is about 10^3 Ω. A meter-long piece of large-diameter copper wire may have a resistance of 10−5 Ω, and superconductors have no resistance at all (they are non-ohmic). The potential difference (voltage) seen across the network is the sum of those voltages, thus the total resistance (the series equivalent resistance) can be found as the sum of those resistances:

.

As a special case, the resistance of N resistors connected in series, each of the same resistance R, is given by NR.Resistors in a parallel configuration are each subject to the same potential difference (voltage), however the currents through them add. Thus the equivalent resistance (Req) of the network can be computed:

.

The parallel equivalent resistance can be represented in equations by two vertical lines “||” (as in geometry) as a simplified notation. Occasionally two slashes “//” are used instead of “||”, in case the keyboard or font lacks the vertical line symbol. For the case of two resistors in parallel, this can be calculated using:

.

As a special case, the resistance of N resistors connected in parallel, each of the same resistance R, is given by R/N. A resistor network that is a combination of parallel and series connections can be broken up into smaller parts that are either one or the other, such as is shown in.

Resistor Network: In this combination circuit, the circuit can be broken up into a series component and a parallel component.

However, some complex networks of resistors cannot be resolved in this manner. These require a more sophisticated circuit analysis. One practical application of these relationships is that a non-standard value of resistance can generally be synthesized by connecting a number of standard values in series or parallel. This can also be used to obtain a resistance with a higher power rating than that of the individual resistors used. In the special case of N identical resistors all connected in series or all connected in parallel, the power rating of the individual resistors is thereby multiplied by N.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=122

Resistance, Resistors, and Resistivity: A brief overview of resistance, resistors, and resistivity.

Dependence of Resistance on Temperature

Resistivity and resistance depend on temperature with the dependence being linear for small temperature changes and nonlinear for large.

Learning Objectives

Compare temperature dependence of resistivity and resistance for large and small temperature changes

Key Takeaways

Key Points

- Over temperature changes of 100ºC or less, resistivity (ρ) varies with temperature change ΔT as: where ρ_0 is the original resistivity and α is the temperature coefficient of resistivity.

- For large temperature changes, nonlinear variation of resistivity with temperature is observed.

- The resistance of an object demonstrates similar temperature dependence as resistivity since resistance is directly proportional to resistivity.

Key Terms

- resistivity: In general, the resistance to electric current of a material; in particular, the degree to which a material resists the flow of electricity.

- temperature coefficient of resistivity: An empirical quantity, denoted by α, which describes the change in resistance or resistivity of a material with temperature.

- semiconductor: A substance with electrical properties intermediate between a good conductor and a good insulator.

The resistivity of all materials depends on temperature. Some materials can become superconductors (zero resistivity) at very low temperatures (see ). Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal, for example, create more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about 100ºC or less), resistivity ρ varies with temperature change ΔT as expressed in the following equation:

Resistance of a sample of mercury: The resistance of a sample of mercury is zero at very low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.

where ρ_0 is the original resistivity and α is the temperature coefficient of resistivity. For larger temperature changes, α may vary, or a nonlinear equation may be needed to find ρ. For this reason it is usual to specify a suffix for the temperature at which the substance was measured (such as α_15) and the relationship only holds in a range of temperatures around the reference. Note that α is positive for metals, meaning their resistivity increases with temperature. The temperature coefficient is typically +3×10−3 K−1 to +6×10−3 K−1 for metals near room temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (made of copper, manganese and nickel), for example, has α close to zero, so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

Note also that α is negative for semiconductors, meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing ρ with temperature is also related to the type and amount of impurities present in the semiconductors.

The resistance of an object also depends on temperature, since R_0 is directly proportional to ρ. For a cylinder we know R=ρL/A, so if L and A do not change greatly with temperature, R will have the same temperature dependence as ρ. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on L and A is about two orders of magnitude less than on ρ. ) Thus,

is the temperature dependence of the resistance of an object, where R_0 is the original resistance and R is the resistance after a temperature change T. Numerous thermometers are based on the effect of temperature on resistance (see ). One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small so it quickly comes into thermal equilibrium with the part of a person it touches.

Thermometers: These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance.

24

Electric Power and Energy

Energy Usage

The energy used is the time integral of the electric power.

Learning Objectives

Formulate the relationship between the energy usage and the electric power

Key Takeaways

Key Points

- Recall that power is the rate work is done, or the rate at which energy is consumed or produced. In terms of current and voltage it is P=IV.

- The energy used is the amount of charge q moved through voltage V in a time interval t. It is equal to the integral of power over time.

- A common unit used to describe energy usage is the kilowatt-hour, the energy of 1000 W acting over one hour.

Key Terms

- kilowatt-hour: a unit of electrical energy equal to that done by one kilowatt acting for one hour; equal to 3.6 mega-joules. Symbol: kWh.

In many cases it is necessary to calculate the energy usage by an electric device or a collection of devices, such as in a home. For example, we (or the electric power utility) may want to calculate the amount of money owed for electricity consumed. In another case, we might need to determine the energy necessary to power a component or apparatus for a given period of time. The last distinction is crucial – the energy used by a circuit or component is the time integral of the electric power.

Power

Recall that power is the rate at which work is done – or the rate at which energy is consumed or produced – and is measured in watts (W). The electric power in watts produced by an electric current I consisting of a charge of Q coulombs every t seconds passing through an electric potential (voltage) difference of V is , where Q is electric charge in coulombs, t is time in seconds, I is electric current in amperes, and V is electric potential or voltage in volts.

In resistive circuits where Ohm ‘s Law applies, the power can be expressed as , where R is the electrical resistance. Power isn’t necessarily constant; it may vary over time. The general expression for electric power is then

where the current I and voltage V may be time variable.

Energy

In any given time interval, the energy consumed (or provided, depending on your perspective) is given by , where E is the electric energy, V is the voltage, and q is the amount of charge moved in the time interval under consideration. We can relate the total energy consumed to the power by integrating over time:. Positive energy corresponds to consumed energy and negative energy corresponds to energy production. Note that a circuit element having a power profile that is both positive and negative over some time interval could consume or produce energy according to the sign of the integral of power. If the power is constant over the time interval then the energy can be expressed simply as:

.

Units of Energy Usage

We are of course very familiar with the SI unit of energy, the joule. However, typically, residential energy bills state household energy consumption in kilowatt-hours (kWh). Additionally, this unit is often seen elsewhere when the energy usage of power consuming devices, structures, or jurisdictions is under consideration. We can parse out the conversion from kilowatt-hours to joules in this way: 1 W = 1 J/s and a kilowatt is 1000 W while one hour is 3,600 seconds, so 1 kWh is (1000 J/s)(3600 s)=3,600,000 joules. This is the scale of American home energy usage, which is on the order of hundreds of kilowatt-hours per month.

Reducing Energy Usage

The electrical energy (E) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This will not only reduce the cost, but it will also result in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20% of a home’s use of energy goes to lighting, while the number for commercial establishments is closer to 40%. Fluorescent lights are about four times more efficient than incandescent lights—this is true for both the long tubes and the compact fluorescent lights (CFL). Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color, flicker, shape, and high initial investment for CFLs have been addressed in recent years. ) The heat transfer from these CFLs is less, and they last up to 10 times longer.

Compact Fluorescent Light (CFL): CFLs are much more efficient than incandescent bulbs and so consume much less energy for the intensity light produces.

25

Resistors in Series and Parallel

Resisitors in Series

The total resistance in the circuit with resistors connected in series is equal to the sum of the individual resistances.

Learning Objectives

Calculate the total resistance in the circuit with resistors connected in series

Key Takeaways

Key Points

- The same current flows through each resistor in series.

- Individual resistors in series do not get the total source voltage, but divide it.

- The total resistance in a series circuit is equal to the sum of the individual resistances: .

Key Terms

- series: A number of things that follow on one after the other or are connected one after the other.

- resistance: The opposition to the passage of an electric current through that element.

Overview

Most circuits have more than one component, called a resistor, that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections. The total resistance of a combination of resistors depends on both their individual values and how they are connected.

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Series Circuits: A brief introduction to series circuit and series circuit analysis, including Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).

Resistors in Series

Resistors are in series whenever the flow of charge, or the current, must flow through components sequentially.

Resistors in Series: These four resistors are connected in series because if a current was applied at one end, it would flow through each resistor sequentially to the end.

shows resistors in series connected to a voltage source. The total resistance in the circuit is equal to the sum of the individual resistances, since the current has to pass through each resistor in sequence through the circuit.

Resistors connected in a series circuit: Three resistors connected in series to a battery (left) and the equivalent single or series resistance (right).

Using Ohm ‘s Law to Calculate Voltage Changes in Resistors in Series

According to Ohm’s law, the voltage drop, V, across a resistor when a current flows through it is calculated by using the equation V=IR, where I is current in amps (A) and R is the resistance in ohms (Ω).

So the voltage drop across R_1 is V_1=IR_1, across R_2 is V_2=IR_2, and across R_3 is V_3=IR_3. The sum of the voltages would equal: V=V_1+V_2+V_3, based on the conservation of energy and charge. If we substitute the values for individual voltages, we get:

or

This implies that the total resistance in a series is equal to the sum of the individual resistances. Therefore, for every circuit with N number of resistors connected in series:

Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

Since voltage and resistance have an inverse relationship, individual resistors in series do not get the total source voltage, but divide it. This is indicated in an example of when two light bulbs are connected together in a series circuit with a battery. In a simple circuit consisting of one 1.5V battery and one light bulb, the light bulb would have a voltage drop of 1.5V across it. If two lightbulbs were connected in series with the same battery, however, they would each have 1.5V/2, or 0.75V drop across them. This would be evident in the brightness of the lights: each of the two light bulbs connected in series would be half as dim as the single light bulb. Therefore, resistors connected in series use up the same amount of energy as a single resistor, but that energy is divided up between the resistors depending on their resistances.

Resistors in Parallel

The total resistance in a parallel circuit is equal to the sum of the inverse of each individual resistances.

Learning Objectives

Calculate the total resistance in the circuit with resistors connected in parallel

Key Takeaways

Key Points

- The total resistance in a parallel circuit is less than the smallest of the individual resistances.

- Each resistor in parallel has the same voltage of the source applied to it (voltage is constant in a parallel circuit).

- Parallel resistors do not each get the total current; they divide it (current is dependent on the value of each resistor and the number of total resistors in a circuit).

Key Terms

- resistance: The opposition to the passage of an electric current through that element.

- parallel: An arrangement of electrical components such that a current flows along two or more paths.

Overview

Resistors in a circuit can be connected in series or in parallel. The total resistance of a combination of resistors depends on both their individual values and how they are connected.

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Parallel Circuits: A brief overview of parallel circuit analysis using VIRP tables for high school physics students.

Resistors in Parallel

Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.

Resistors in Parallel: A parallel connection of resistors.

Each resistor draws the same current it would if it were the only resistor connected to the voltage source. This is true of the circuitry in a house or apartment. Each outlet that is connected to a appliance (the “resistor”) can operate independently, and the current does not have to pass through each appliance sequentially.

Ohm ‘s Law and Parallel Resistors

Each resistor in the circuit has the full voltage. According to Ohm’s law, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current is the sum of these currents:

Parallel resistors: Three resistors connected in parallel to a battery and the equivalent single or parallel resistance.

Substituting the expressions for individual currents gives:

or

This implies that the total resistance in a parallel circuit is equal to the sum of the inverse of each individual resistances. Therefore, for every circuit with number or resistors connected in parallel,

This relationship results in a total resistance that is less than the smallest of the individual resistances. When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, so the total resistance is lower.

Each resistor in parallel has the same full voltage of the source applied to it, but divide the total current amongst them. This is exemplified by connecting two light bulbs in a parallel circuit with a 1.5V battery. In a series circuit, the two light bulbs would be half as dim when connected to a single battery source. However, if the two light bulbs were connected in parallel, they would be equally as bright as if they were connected individually to the battery. Because the same full voltage is being applied to both light bulbs, the battery would also die more quickly, since it is essentially supplying full energy to both light bulbs. In a series circuit, the battery would last just as long as it would with a single light bulb, only the brightness is then divided amongst the bulbs.

Combination Circuits

A combination circuit can be broken up into similar parts that are either series or parallel.

Learning Objectives

Describe arrangement of resistors in a combination circuit and its practical implications

Key Takeaways

Key Points

- More complex connections of resistors are sometimes just combinations of series and parallel.

- Various parts of a combination circuit can be identified as series or parallel, reduced to their equivalents, and then further reduced until a single resistance is left.

- Resistance in wires reduces the current and power delivered to a resistor. If the resistance in wires is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant and affect power output into appliances.

Key Terms

- series: A number of things that follow on one after the other or are connected one after the other.

- parallel: An arrangement of electrical components such that a current flows along two or more paths.

- combination circuit: An electrical circuit containing multiple resistors that are connected in a combination of both series and parallel connections.

Combination Circuits

More complex connections of resistors are sometimes just combinations of series and parallel. This is commonly encountered, especially when wire resistances is considered. In that case, wire resistance is in series with other resistances that are in parallel.

A combination circuit can be broken up into similar parts that are either series or parallel, as diagrammed in. In the figure, the total resistance can be calculated by relating the three resistors to each other as in series or in parallel. R_1 and R_2 are connected in parallel in relation to each other, so we know that for that subset, the inverse of resistance would be equal to:

Resistor Network: In this combination circuit, the circuit can be broken up into a series component and a parallel component.

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Combination Circuits: Two parallel resistors in series with one resistor.

or

R_3 is connected in series to both R_1 and R_2, so the resistance would be calculated as:

Complex Combination Circuits

For more complicated combination circuits, various parts can be identified as series or parallel, reduced to their equivalents, and then further reduced until a single resistance is left, as shown in. In this figure, the combination of seven resistors was identified by being either in series or in parallel. In the initial image, the two circled sections show resistors that are in parallel.

Reducing a combination circuit: This combination of seven resistors has both series and parallel parts. Each is identified and reduced to an equivalent resistance, and these are further reduced until a single equivalent resistance is reached.

Reducing those parallel resistors into a single R value allows us to visualize the circuit in a more simplified manner. In the top right image, we can see that the circled portion contains two resistors in series. We can further reduce that to another R value by adding them. The next step shows that the circled two resistors are in parallel. Reducing those highlights that the last two are in series, and thus can be reduced to a single resistance value for the entire circuit.

One practical implication of a combination circuit is that resistance in wires reduces the current and power delivered to a resistor. Combination circuit can be transformed into a series circuit, based on an understanding of the equivalent resistance of parallel branches to a combination circuit. A series circuit can be used to determine the total resistance of the circuit. Essentially, wire resistance is a series with the resistor. It thus increases the total resistance and decreases the current. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant. If a large current is drawn, the IR drop in the wires can also be significant.

Charging a Battery: EMFs in Series and Parallel

When voltage sources are connected in series, their emfs and internal resistances are additive; in parallel, they stay the same.

Learning Objectives

Compare the resistances and electromotive forces for the voltage sources connected in the same and opposite polarity, and in series and in parallel

Key Takeaways

Key Points

- Emfs connected in the same polarity in series are additive and result in a higher total emf.

- Two emfs connected in the opposite polarity in series have a total emf equal to the difference between them, and can be used to charge the lower voltage source.

- Two voltage sources with identical emfs connected in parallel have a net emf equivalent to one emf source, however, the net internal resistance is less, and therefore produces a higher current.

Key Terms

- parallel: An arrangement of electrical components such that a current flows along two or more paths.

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- series: A number of things that follow on one after the other or are connected one after the other.

When more than one voltage source is used, they can be connected either in series or in parallel, similar to resistors in a circuit. When voltage sources are in series facing the same direction, their internal resistances add and their electromotive force, or emf, add algebraically. These types of voltage sources are common in flashlights, toys, and other appliances. Usually, the cells are in series in order to produce a larger total emf.

Flashlight and Bulb: A series connection of two voltage sources in the same direction. This schematic represents a flashlight with two cells (voltage sources) and a single bulb (load resistance) in series.

A battery is a multiple connection of voltaic cells. The disadvantage of series connections of cells in this manner, though, is that their internal resistances add. This can sometimes be problematic. For example, if you placed two 6v batteries in your car instead of the typical 12v single battery, you would be adding both the emfs and the internal resistances of each battery. You would therefore end up with the same 12v emf, though the internal resistance would then be doubled, causing issues for you when you want to start your engine.

But, if the cells oppose one another—such as when one is put into an appliance backwards—the total emf is less, since it is the algebraic sum of the individual emfs. When it is reversed, it produces an emf that opposes the other, and results in a difference between the two voltage sources.

Battery Charger: This represents two voltage sources connected in series with their emfs in opposition. Current flows in the direction of the greater emf and is limited by the sum of the internal resistances. (Note that each emf is represented by script E in the figure. ) A battery charger connected to a battery is an example of such a connection. The charger must have a larger emf than the battery to reverse current through it.

When two voltage sources with identical emfs are connected in parallel and also connected to a load resistance, the total emf is the same as the individual emfs. But the total internal resistance is reduced, since the internal resistances are in parallel. Thus, the parallel connection can produce a larger current.

Two Identical EMFs: Two voltage sources with identical emfs (each labeled by script E) connected in parallel produce the same emf but have a smaller total internal resistance than the individual sources. Parallel combinations are often used to deliver more current.

EMF and Terminal Voltage

The output, or terminal voltage of a voltage source such as a battery, depends on its electromotive force and its internal resistance.

Learning Objectives

Express the relationship between the electromotive force and terminal voltage in a form of equation

Key Takeaways

Key Points

- The electromotive force (emf) is the potential difference of a source when no current is flowing.

- Terminal voltage is the voltage output of a device is measured across its terminals.

- Terminal voltage is calculated by V = emf – Ir.

Key Terms

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- terminal voltage: The voltage output of a device measured across its terminals.

- potential difference: The difference in potential energy between two points in an electric field; the difference in charge between two points in an electrical circuit; voltage.

When you forget to turn off your car lights, they slowly dim as the battery runs down. Why don’t they simply blink off when the battery’s energy is gone? Their gradual dimming implies that battery output voltage decreases as the battery is depleted. The reason for the decrease in output voltage for depleted or overloaded batteries is that all voltage sources have two fundamental parts—a source of electrical energy and an internal resistance.

Electromotive Force

All voltage sources create a potential difference and can supply current if connected to a resistance. On a small scale, the potential difference creates an electric field that exerts force on charges, causing current. We call this potential difference the electromotive force (abbreviated emf). Emf is not a force at all; it is a special type of potential difference of a source when no current is flowing. Units of emf are volts.

Electromotive force is directly related to the source of potential difference, such as the particular combination of chemicals in a battery. However, emf differs from the voltage output of the device when current flows. The voltage across the terminals of a battery, for example, is less than the emf when the battery supplies current, and it declines further as the battery is depleted or loaded down. However, if the device’s output voltage can be measured without drawing current, then output voltage will equal emf (even for a very depleted battery).

Terminal Voltage

presents a schematic representation of a voltage source. The voltage output of a device is measured across its terminals and is called its terminal voltage V. Terminal voltage is given by the equation:

Schematic Representation of a Voltage Source: Any voltage source (in this case, a carbon-zinc dry cell) has an emf related to its source of potential difference, and an internal resistance r related to its construction. (Note that the script E stands for emf. ) Also shown are the output terminals across which the terminal voltage V is measured. Since V=emf−Ir, terminal voltage equals emf only if there is no current flowing.

,

where r is the internal resistance and I is the current flowing at the time of the measurement.

I is positive if current flows away from the positive terminal. The larger the current, the smaller the terminal voltage. Likewise, it is true that the larger the internal resistance, the smaller the terminal voltage.

26

Magnetism and Magnetic Fields

Electric Currents and Magnetic Fields

An electric current will produce a magnetic field, which can be visualized as a series of circular field lines around a wire segment.

Learning Objectives

Describe shape of a magnetic field produced by an electric current flowing through a wire

Key Takeaways

Key Points

- A wire carrying electric current will produce a magnetic field with closed field lines surrounding the wire.

- Another version of the right hand rules can be used to determine the magnetic field direction from a current—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it. See.

- The Biot-Savart Law can be used to determine the magnetic field strength from a current segment. For the simple case of an infinite straight current-carrying wire it is reduced to the form .

- A more fundamental law than the Biot-Savart law is Ampere ‘s Law, which relates magnetic field and current in a general way. It is written in integral form as , where Ienc is the enclosed current and μ_0 is a constant.

- A current-carrying wire feels a force in the presence of an external magnetic field. It is found to be , where ℓ is the length of the wire, i is the current, and θ is the angle between the current direction and the magnetic field.

Key Terms

- Biot-Savart Law: An equation that describes the magnetic field generated by an electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The law is valid in the magnetostatic approximation, and is consistent with both Ampère’s circuital law and Gauss’s law for magnetism.

- Ampere’s Law: An equation that relates magnetic fields to electric currents that produce them. Using Ampere’s law, one can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present.

Electric Current and Magnetic Fields

Electric current produces a magnetic field. This magnetic field can be visualized as a pattern of circular field lines surrounding a wire. One way to explore the direction of a magnetic field is with a compass, as shown by a long straight current-carrying wire in. Hall probes can determine the magnitude of the field. Another version of the right hand rule emerges from this exploration and is valid for any current segment—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.

Magnetic Field Generated by Current: (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.

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Magnets and Magnetic Fields: A brief introduction to magnetism for introductory physics students.

Magnitude of Magnetic Field from Current

The equation for the magnetic field strength (magnitude) produced by a long straight current-carrying wire is:

For a long straight wire where I is the current, r is the shortest distance to the wire, and the constant 0=4π10−7 T⋅m/A is the permeability of free space. (μ_0 is one of the basic constants in nature, related to the speed of light. ) Since the wire is very long, the magnitude of the field depends only on distance from the wire r, not on position along the wire. This is one of the simplest cases to calculate the magnetic field strenght from a current.

The magnetic field of a long straight wire has more implications than one might first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vectorsum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Integral calculus is needed to sum the field for an arbitrary shape current. The Biot-Savart law is written in its complete form as:

where the integral sums over the wire length where vector dℓ is the direction of the current; r is the distance between the location of dℓ, and the location at which the magnetic field is being calculated; and r̂ is a unit vector in the direction of r. The reader may apply the simplifications in calculating the magnetic field from an infinite straight wire as above and see that the Biot-Savart law reduces to the first, simpler equation.

Ampere’s Law

A more fundamental law than the Biot-Savart law is Ampere’s Law, which relates magnetic field and current in a general way. In SI units, the integral form of the original Ampere’s circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S through which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. You can think of the “surface” as the cross-sectional area of a wire carrying current.

The mathematical statement of the law states that the total magnetic field around some path is directly proportional to the current which passes through that enclosed path. It can be written in a number of forms, one of which is given below.

where the magnetic field is integrated over a curve (circumfrence of a wire), equivalent to integrating the current density (in amperes per square meter, Am-2) over the cross section area of the wire (which is equal to the permeability constant times the enclosed current Ienc). Ampere’s law is always valid for steady currents and can be used to calculate the B-field for certain highly symmetric situations such as an infinite wire or an infinite solenoid. Ampere’s Law is also a component of Maxwell’s Equations.

Force on a Current-Carrying Wire

The force on a current carrying wire (as in ) is similar to that of a moving charge as expected since a charge carrying wire is a collection of moving charges. A current-carrying wire feels a force in the presence of a magnetic field. Consider a conductor (wire) of length ℓ, cross section A, and charge q which is due to electric current i. If this conductor is placed in a magnetic field of magnitude B which makes an angle with the velocity of charges (current) in the conductor, the force exerted on a single charge q is

Force on a Current-Carrying Wire: The right hand rule can be used to determine the direction of the force on a current-carrying wire placed in an external magnetic field.

So, for N charges where

the force exerted on the conductor is

where i = nqvA. The right hand rule can give you the direction of the force on the wire, as seen in the above figure. Note that the B-field in this case is the external field.

Permanent Magnets

Permanent magnets are objects made from ferromagnetic material that produce a persistent magnetic field.

Learning Objectives

Give examples and counterexamples of permanent magnets

Key Takeaways

Key Points

- Permanent magnets are objects made from magnetized material and produce continual magnetic fields. Everyday examples include refrigerator magnets used to hold notes on a refrigerator door.

- Materials that can be magnetized, which are also the ones that are strongly attracted to a magnet, are called ferromagnetic. Examples of these materials include iron, nickel, and cobalt.

- The counterexample to a permanent magnet is an electromagnet, which only becomes magnetized when an electric current flows through it.

- Magnets always have a north pole and a south pole, so if one were to split a permanent magnet in half, two smaller magnets would be created, each with a north pole and south pole.

- Permanent magnets are made from ferromagnetic materials that are exposed to a strong external magnetic field and heated to align their internal microcrystalline structure, making them very hard to demagnetize.

Key Terms

- permanent magnet: A material, or piece of such material, which retains its magnetism even when not subjected to any external magnetic fields.

- ferromagnetic: Of a material, such as iron or nickel, that is easily magnetized.

- electromagnet: A magnet which attracts metals only when electrically activated.

Permanent Magnets

Overview

Recall that a magnet is a material or object that generates a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.

Types of Magnets

A permanent magnet is an object made from a material that is magnetized and creates its own persistent magnetic field. An everyday example is a refrigerator magnet used to hold notes on a refrigerator door. Materials that can be magnetized, which are also the ones that are strongly attracted to a magnet, are called ferromagnetic. These include iron, nickel, cobalt, some alloys of rare earth metals, and some naturally occurring minerals such as lodestone. Although ferromagnetic materials are the only ones attracted to a magnet strongly enough to be commonly considered magnetic, all other substances respond weakly to a magnetic field, by one of several other types of magnetism. The counterexample to a permanent magnet is an electromagnet, which only becomes magnetized when an electric current flows through it.

Example of a Permanent Magnet: An example of a permanent magnet: a “horseshoe magnet” made of alnico, an iron alloy. The magnet is made in the shape of a horseshoe to bring the two magnetic poles close to each other, to create a strong magnetic field there that can pick up heavy pieces of iron.

Polarity

All magnets have two poles, one called the north pole and one called the south pole. Like poles repel and unlike polls attract (in analogy to positive and negative charges in electrostatics). North and south poles always exist in pairs (there are no magnetic monopoles in nature), so if one were to split a permanent magnet in half, two smaller magnets would be created, each with a north pole and south pole.

North and South Poles Always Come in Pairs: North and south poles always occur in pairs. Attempts to separate them result in more pairs of poles. If we continue to split the magnet, we will eventually get down to an iron atom with a north pole and a south pole—these, too, cannot be separated.

Manufacturing Permanent Magnets

Ferromagnetic materials can be divided into magnetically “soft” materials like annealed iron, which can be magnetized but do not tend to stay magnetized, and magnetically “hard” materials, which do. Permanent magnets are made from “hard” ferromagnetic materials such as alcino and ferrite that are subjected to special processing in a powerful magnetic field during manufacture, to align their internal microcrystalline structure, making them very hard to demagnetize.

When a magnet is brought near a previously unmagnetized ferromagnetic material, it causes local magnetization of the material with unlike poles closest. (This results in the attraction of the previously unmagnetized material to the magnet. ) On the microscopic scale, there are regions in the unmagnetized ferromagnetic material that act like small bar magnets. In each region the poles of individual atoms are aligned. However, before magnetization these regions are small and randomly oriented throughout the unmagnetized ferromagnetic objects, so there is no net magnetic field. In response to an external magnetic field like the one applied in the above figure, these regions grow and become aligned. This arrangement can become permanent when the ferromagnetic material is heated and then cooled.

Making a Ferromagnet: An unmagnetized piece of iron is placed between two magnets, heated, and then cooled, or simply tapped when cold. The iron becomes a permanent magnet with the poles aligned as shown: its south pole is adjacent to the north pole of the original magnet, and its north pole is adjacent to the south pole of the original magnet. Note that there are attractive forces between the magnets.

Magnetic Field Lines

Magnetic field lines are useful for visually representing the strength and direction of the magnetic field.

Learning Objectives

Relate the strength of the magnetic field with the density of the magnetic field lines

Key Takeaways

Key Points

- The magnetic field direction is the same direction a compass needle points, which is tangent to the magnetic field line at any given point.

- The strength of the B-field is inversely proportional to the distance between field lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines.

- A magnetic field line can never cross another field line. The magnetic field is unique at every point in space.

- Magnetic field lines are continuous and unbroken, forming closed loops. Magnetic field lines are defined to begin on the north pole of a magnet and terminate on the south pole.

Key Terms

- B-field: A synonym for the magnetic field.

- magnetic field lines: A graphical representation of the magnitude and the direction of a magnetic field.

Magnetic Field Lines

Einstein is said to have been fascinated by a compass as a child, perhaps musing on how the needle felt a force without direct physical contact. His ability to think deeply and clearly about action at a distance, particularly for gravitational, electric, and magnetic forces, later enabled him to create his revolutionary theory of relativity. Since magnetic forces act at a distance, we define a magnetic field to represent magnetic forces. A pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. The direction of magnetic field lines is defined to be the direction in which the north end of a compass needle points. The magnetic field is traditionally called the B-field.

Visualizing Magnetic Field Lines: Magnetic field lines are defined to have the direction that a small compass points when placed at a location. (A) If small compasses are used to map the magnetic field around a bar magnet, they will point in the directions shown: away from the north pole of the magnet, toward the south pole of the magnet (recall that Earth’s north magnetic pole is really a south pole in terms of definitions of poles on a bar magnet. ) (B) Connecting the arrows gives continuous magnetic field lines. The strength of the field is proportional to the closeness (or density) of the lines. (C) If the interior of the magnet could be probed, the field lines would be found to form continuous closed loops.

Mapping the magnetic field of an object is simple in principle. First, measure the strength and direction of the magnetic field at a large number of locations (or at every point in space). Then, mark each location with an arrow (called a vector ) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field (producing a vector field). You can “connect” the arrows to form magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength.

Magnetic field lines are like the contour lines (constant altitude) on a topographic map in that they represent something continuous, and a different mapping scale would show more or fewer lines. An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the “number” of field lines through a surface. These concepts can be quickly translated to their mathematical form. For example, the number of field lines through a given surface is the surface integral of the magnetic field.

Bar Magnet and Magnetic Field Lines: The direction of magnetic field lines represented by the alignment of iron filings sprinkled on paper placed above a bar magnet.

Various phenomena have the effect of “displaying” magnetic field lines as though the field lines are physical phenomena. For example, iron filings placed in a magnetic field line up to form lines that correspond to “field lines. ” Magnetic fields’ lines are also visually displayed in polar auroras, in which plasma particle dipole interactions create visible streaks of light that line up with the local direction of Earth’s magnetic field.

Small compasses used to test a magnetic field will not disturb it. (This is analogous to the way we tested electric fields with a small test charge. In both cases, the fields represent only the object creating them and not the probe testing them. ) Figure 15051 shows how the magnetic field appears for a current loop and a long straight wire, as could be explored with small compasses. A small compass placed in these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction of B. Note the symbols used for field into and out of the paper. We’ll explore the consequences of these various sources of magnetic fields in further sections.

Mapping Magnetic Field Lines: Small compasses could be used to map the fields shown here. (A) The magnetic field of a circular current loop is similar to that of a bar magnet. (B) A long and straight wire creates a field with magnetic field lines forming circular loops. (C) When the wire is in the plane of the paper, the field is perpendicular to the paper. Note that the symbols used for the field pointing inward (like the tail of an arrow) and the field pointing outward (like the tip of an arrow).

Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. We use magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of themselves). The properties of magnetic field lines can be summarized by these rules:

1. The direction of the magnetic field is tangent to the field line at any point in space. A small compass will point in the direction of the field line.

2. The strength of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines (called the areal density).

3. Magnetic field lines can never cross, meaning that the field is unique at any point in space.

4. Magnetic field lines are continuous, forming closed loops without beginning or end. They go from the north pole to the south pole.

The last property is related to the fact that the north and south poles cannot be separated. It is a distinct difference from electric field lines, which begin and end on the positive and negative charges. If magnetic monopoles existed, then magnetic field lines would begin and end on them.

Geomagnetism

Earth’s magnetic field is caused by electric currents in the molten outer core and varies with time.

Learning Objectives

Explain the origin of the Earth’s magnetic field and its importance for the life on Earth

Key Takeaways

Key Points

- Earth is largely protected from the solar wind, a stream of energetic charged particles emanating from the sun, by its magnetic field. These particles would strip away the ozone layer, which protects Earth from harmful ultraviolet rays.

- Earth’s magnetic field is generated by a feedback loop in the liquid outer core: Current loops generate magnetic fields; a changing magnetic field generates an electric field; and the electric and magnetic fields exert a force on the charges that are flowing in currents (the Lorentz force).

- The geomagnetic field varies with time. Currents in the ionosphere and magnetosphere cause changes over short time scales, while dramatic geomagnetic reversald (where the North and South poles switch locations) occur at apparently random intervals ranging from 0.1 to 50 million years.

Key Terms

- dynamo: A mechanism by which a celestial body such as Earth or a star generates a magnetic field over astronomical timescales via a rotating, convecting, and electrically conducting fluid.

Geomagnetism

The Structure of Earth’s Magnetic Field

Earth is largely protected from the solar wind, a stream of energetic charged particles emanating from the sun, by its magnetic field, which deflects most of the charged particles. These particles would strip away the ozone layer, which protects Earth from harmful ultraviolet rays. The region above the ionosphere, and extending several tens of thousands of kilometers into space, is called the magnetosphere. This region protects Earth from cosmic rays that would strip away the upper atmosphere, including the ozone layer that protects our planet from harmful ultraviolet radiation. The magnetic field strength ranges from approximately 25 to 65 microteslas (0.25 to 0.65 G; by comparison, a strong refrigerator magnet has a field of about 100 G). The intensity of the field is greatest near the poles and weaker near the equator. An isodynamic chart of Earth’s magnetic field, shows a minimum intensity over South America while there are maxima over northern Canada, Siberia, and the coast of Antarctica south of Australia. Near the surface of Earth, its magnetic field can be closely approximated by the field of a magnetic dipole positioned at the center of Earth and tilted at an angle of about 10° with respect to the rotational axis of Earth.

Physical Origin

Earth’s magnetic field is mostly caused by electric currents in the liquid outer core, which is composed of highly conductive molten iron. A magnetic field is generated by a feedback loop: Current loops generate magnetic fields (Ampère’s law); a changing magnetic field generates an electric field (Faraday’s law); and the electric and magnetic fields exert a force on the charges that are flowing in currents (the Lorentz force). These effects can be combined into a partial differential equation called the magnetic induction equation:

In this equation u is the velocity of the fluid, B is the magnetic field, and eta is the magnetic diffusivity. The first term on the right hand side of the induction equation is a diffusion term. If Earth’s dynamo shut off, the dipole part would disappear in a few tens of thousands of years. The motion of the molten outer iron core is sustained by convection, or motion driven by buoyancy. The temperature increases toward the center of Earth, and the higher temperature of the fluid lower down makes it buoyant. The Coriolis effect, caused by the overall planetary rotation, tends to organize the flow into rolls aligned along the north-south polar axis.

Origin of Earth’s Magnetic Field: A schematic illustrating the relationship between motion of conducting fluid, organized into rolls by the Coriolis force, and the magnetic field the motion generates.

Electric currents induced in the ionosphere generate magnetic fields (ionospheric dynamo region). Such a field is always generated near where the atmosphere is closest to the sun, causing daily alterations that can deflect surface magnetic fields by as much as one degree. Typical daily variations of field strength are about 25 nanoteslas (nT), with variations over a few seconds of typically around 1 nT.

Time Variations

The geomagnetic field changes on time scales from milliseconds to millions of years. Shorter time scales mostly arise from currents in the ionosphere (ionospheric dynamo region) and magnetosphere, and some changes can be traced to geomagnetic storms or daily variations in currents. Changes over time scales of a year or more mostly reflect changes in Earth’s interior, particularly the iron-rich core. Frequently, Earth’s magnetosphere is hit by solar flares causing geomagnetic storms, provoking displays of aurorae. At present, the overall geomagnetic field is becoming weaker; the present strong deterioration corresponds to a 10 to 15 percent decline over the last 150 years and has accelerated in the past several years. Geomagnetic intensity has declined almost continuously from a maximum 35 percent above the modern value achieved approximately 2,000 years ago. Earth’s magnetic North Pole is drifting from northern Canada toward Siberia with a presently accelerating rate—10 km per year at the beginning of the 20th century, up to 40 km per year in 2003, and since then has only accelerated.

Although Earth’s field is generally well approximated by a magnetic dipole with its axis near the rotational axis, there are occasional dramatic events where the North and South geomagnetic poles trade places. These events are called geomagnetic reversals. Evidence for these events can be found worldwide in basalts, sediment cores taken from the ocean floors, and seafloor magnetic anomalies. Reversals occur at apparently random intervals ranging from less than 0.1 million years to as much as 50 million years. The most recent such event, called the Brunhes-Matuyama reversal, occurred about 780,000 years ago.

27

Magnetic Flux, Induction, and Faraday’s Law

Induced EMF and Magnetic Flux

Faraday’s law of induction states that an electromotive force is induced by a change in the magnetic flux.

Learning Objectives

Explain the relationship between the magnetic field and the electromotive force

Key Takeaways

Key Points

- It is a change in the magnetic field flux that results in an electromotive force (or voltage).

- The magnetic flux (often denoted Φ or ΦB) through a surface is the component of the magnetic field passing through that surface.

- In the most general form, magnetic flux is defined as . It is the integral (sum) of all of the magnetic field passing through infinitesimal area elements dA.

Key Terms

- vector area: A vector whose magnitude is the area under consideration, and whose direction is perpendicular to the surface area.

- galvanometer: An analog measuring device, denoted by G, that measures current flow using a needle deflection caused by a magnetic field force acting upon a current-carrying wire.

Induced EMF

The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in the following figure. When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted (or guided) to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in a separate coil on the bottom.

Faraday’s Apparatus: This is Faraday’s apparatus for demonstrating that a magnetic field can produce a current. A change in the field produced by the top coil induces an EMF and, hence, a current in the bottom coil. When the switch is opened and closed, the galvanometer registers currents in opposite directions. No current flows through the galvanometer when the switch remains closed or open.

It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. Closing and opening the switch induces the current. It is the change in magnetic field that creates the current. More basic than the current that flows is the electromotive force (EMF) that causes it. The current is a result of an EMF induced by a changing magnetic field, whether or not there is a path for current to flow.

Magnetic Flux

The magnetic flux (often denoted Φ or ΦB) through a surface is the component of the magnetic field passing through that surface. The magnetic flux through some surface is proportional to the number of field lines passing through that surface. The magnetic flux passing through a surface of vector area A is

,

where B is the magnitude of the magnetic field (having the unit of Tesla, T), A is the area of the surface, and θ is the angle between the magnetic field lines and the normal (perpendicular) to A.

For a varying magnetic field, we first consider the magnetic flux through an infinitesimal area element dA, where we may consider the field to be constant:

Varying Magnetic Field: Each point on a surface is associated with a direction, called the surface normal; the magnetic flux through a point is then the component of the magnetic field along this normal direction.

A generic surface, A, can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the surface integral

.

Faraday’s Law of Induction and Lenz’ Law

Faraday’s law of induction states that the EMF induced by a change in magnetic flux is , when flux changes by Δ in a time Δt.

Learning Objectives

Express the Faraday’s law of induction in a form of equation

Key Takeaways

Key Points

- The minus in the Faraday’s law means that the EMF creates a current I and magnetic field B that oppose the change in flux Δthis is known as Lenz’ law.

- Faraday’s law of induction is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators, and solenoids.

- Faraday’s law states that the EMF induced by a change in magnetic flux depends on the change in flux Δ, time Δt, and number of turns of coils.

Key Terms

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- solenoid: A coil of wire that acts as a magnet when an electric current flows through it.

- flux: The rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux or magnetic flux.

Faraday’s Law of Induction

Faraday’s law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators, and solenoids.

Faraday’s experiments showed that the EMF induced by a change in magnetic flux depends on only a few factors. First, EMF is directly proportional to the change in flux Δ. Second, EMF is greatest when the change in time Δt is smallest—that is, EMF is inversely proportional to Δt. Finally, if a coil has N turns, an EMF will be produced that is N times greater than for a single coil, so that EMF is directly proportional to N. The equation for the EMF induced by a change in magnetic flux is

.

This relationship is known as Faraday’s law of induction. The units for EMF are volts, as is usual.

Lenz’ Law

The minus sign in Faraday’s law of induction is very important. The minus means that the EMF creates a current I and magnetic field B that oppose the change in flux Δthis is known as Lenz’ law. The direction (given by the minus sign) of the EMF is so important that it is called Lenz’ law after the Russian Heinrich Lenz (1804–1865), who, like Faraday and Henry, independently investigated aspects of induction. Faraday was aware of the direction, but Lenz stated it, so he is credited for its discovery.

Lenz’ Law: (a) When this bar magnet is thrust into the coil, the strength of the magnetic field increases in the coil. The current induced in the coil creates another field, in the opposite direction of the bar magnet’s to oppose the increase. This is one aspect of Lenz’s law—induction opposes any change in flux. (b) and (c) are two other situations. Verify for yourself that the direction of the induced Bcoil shown indeed opposes the change in flux and that the current direction shown is consistent with the right hand rule.

Energy Conservation

Lenz’ law is a manifestation of the conservation of energy. The induced EMF produces a current that opposes the change in flux, because a change in flux means a change in energy. Energy can enter or leave, but not instantaneously. Lenz’ law is a consequence. As the change begins, the law says induction opposes and, thus, slows the change. In fact, if the induced EMF were in the same direction as the change in flux, there would be a positive feedback that would give us free energy from no apparent source—conservation of energy would be violated.

Motional EMF

Motion in a magnetic field that is stationary relative to the Earth induces motional EMF (electromotive force).

Learning Objectives

Identify process that induces motional electromotive force

Key Takeaways

Key Points

- Faraday’s law of induction can be used to calculate the motional EMF when a change in magnetic flux is caused by a moving element in a system.

- That a moving magnetic field produces an electric field (and conversely that a moving electric field produces a magnetic field) is part of the reason electric and magnetic forces are now considered as different manifestations of the same force.

- Any change in magnetic flux induces an electromotive force (EMF) opposing that change—a process known as induction. Motion is one of the major causes of induction.

Key Terms

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- magnetic flux: A measure of the strength of a magnetic field in a given area.

- induction: The generation of an electric current by a varying magnetic field.

As seen in previous Atoms, any change in magnetic flux induces an electromotive force (EMF) opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an EMF, and a coil moved toward a magnet produces a similar EMF. In this Atom, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional EMF.

Motional EMF

Consider the situation shown in. A rod is moved at a speed v along a pair of conducting rails separated by a distance ℓ in a uniform magnetic field B. The rails are stationary relative to B, and are connected to a stationary resistor R (the resistor could be anything from a light bulb to a voltmeter). Consider the area enclosed by the moving rod, rails and resistor. B is perpendicular to this area, and the area is increasing as the rod moves. Thus the magnetic flux enclosed by the rails, rod and resistor is increasing. When flux changes, an EMF is induced according to Faraday’s law of induction.

Motional EMF: (a) A motional emf=Bℓv is induced between the rails when this rod moves to the right in the uniform magnetic field. The magnetic field B is into the page, perpendicular to the moving rod and rails and, hence, to the area enclosed by them. (b) Lenz’s law gives the directions of the induced field and current, and the polarity of the induced emf. Since the flux is increasing, the induced field is in the opposite direction, or out of the page. Right hand rule gives the current direction shown, and the polarity of the rod will drive such a current.

To find the magnitude of EMF induced along the moving rod, we use Faraday’s law of induction without the sign:

.

In this equation, N=1 and the flux Φ=BAcosθ. We have θ=0º and cosθ=1, since B is perpendicular to A. Now Δ=Δ(BA)=BΔA, since B is uniform. Note that the area swept out by the rod is ΔA=ℓx. Entering these quantities into the expression for EMF yields:

.

To find the direction of the induced field, the direction of the current, and the polarity of the induced EMF we apply Lenz’ law, as explained in Faraday’s Law of Induction: Lenz’ Law. As seen in Fig 1 (b), F lux is increasing, since the area enclosed is increasing. Thus the induced field must oppose the existing one and be out of the page. (The right hand rule requires that I be counterclockwise, which in turn means the top of the rod is positive, as shown. )

Electric Field vs. Magnetic Field

There are many connections between the electric force and the magnetic force. That a moving magnetic field produces an electric field (and conversely that a moving electric field produces a magnetic field) is part of the reason electric and magnetic forces are now considered as different manifestations of the same force (first noticed by Albert Einstein). This classic unification of electric and magnetic forces into what is called the electromagnetic force is the inspiration for contemporary efforts to unify other basic forces.

Back EMF, Eddy Currents, and Magnetic Damping

Back EMF, eddy currents, and magnetic damping are all due to induced EMF and can be explained by Faraday’s law of induction.

Learning Objectives

Explain the relationship between the motional electromotive force, eddy currents, and magnetic damping

Key Takeaways

Key Points

- Input EMF that powers a motor can be opposed by the motor’s self-generated EMF, called the back EMF of the motor.

- If motional EMF can cause a current loop in the conductor, the current is called an eddy current.

- Eddy currents can produce significant drag, called magnetic damping, on the motion involved.

Key Terms

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- Faraday’s law of induction: A basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

Back EMF

Motors and generators are very similar. (Read our Atoms on “Electric Generators” and “Electric Motors. “) Generators convert mechanical energy into electrical energy, whereas motors convert electrical energy into mechanical energy. Furthermore, motors and generators have the same construction. When the coil of a motor is turned, magnetic flux changes, and an electromotive force (EMF), consistent with Faraday’s law of induction, is induced. The motor thus acts as a generator whenever its coil rotates. This will happen whether the shaft is turned by an external input, like a belt drive, or by the action of the motor itself. That is, when a motor is doing work and its shaft is turning, an EMF is generated. Lenz’ law tells us the induced EMF opposes any change, so that the input EMF that powers the motor will be opposed by the motor’s self-generated EMF, called the back EMF of the motor.

Eddy Current

As discussed in “Motional EMF,” motional EMF is induced when a conductor moves in a magnetic field or when a magnetic field moves relative to a conductor. If motional EMF can cause a current loop in the conductor, we refer to that current as an eddy current. Eddy currents can produce significant drag, called magnetic damping, on the motion involved.

Consider the apparatus shown in, which swings a pendulum bob between the poles of a strong magnet. If the bob is metal, there is significant drag on the bob as it enters and leaves the field, quickly damping the motion. If, however, the bob is a slotted metal plate, as shown in (b), there is a much smaller effect due to the magnet. There is no discernible effect on a bob made of an insulator.

Device for Exploring Eddy Currents and Magnetic Damping: A common physics demonstration device for exploring eddy currents and magnetic damping. (a) The motion of a metal pendulum bob swinging between the poles of a magnet is quickly damped by the action of eddy currents. (b) There is little effect on the motion of a slotted metal bob, implying that eddy currents are made less effective. (c) There is also no magnetic damping on a nonconducting bob, since the eddy currents are extremely small.

shows what happens to the metal plate as it enters and leaves the magnetic field. In both cases, it experiences a force opposing its motion. As it enters from the left, flux increases, and so an eddy current is set up (Faraday’s law) in the counterclockwise direction (Lenz’ law), as shown. Only the right-hand side of the current loop is in the field, so that there is an unopposed force on it to the left (right hand rule). When the metal plate is completely inside the field, there is no eddy current if the field is uniform, since the flux remains constant in this region. But when the plate leaves the field on the right, flux decreases, causing an eddy current in the clockwise direction that, again, experiences a force to the left, further slowing the motion. A similar analysis of what happens when the plate swings from the right toward the left shows that its motion is also damped when entering and leaving the field.

Conducting Plate Passing Between the Poles of a Magnet: A more detailed look at the conducting plate passing between the poles of a magnet. As it enters and leaves the field, the change in flux produces an eddy current. Magnetic force on the current loop opposes the motion. There is no current and no magnetic drag when the plate is completely inside the uniform field.

When a slotted metal plate enters the field, as shown in, an EMF is induced by the change in flux, but it is less effective because the slots limit the size of the current loops. Moreover, adjacent loops have currents in opposite directions, and their effects cancel. When an insulating material is used, the eddy current is extremely small, and so magnetic damping on insulators is negligible. If eddy currents are to be avoided in conductors, then they can be slotted or constructed of thin layers of conducting material separated by insulating sheets.

Eddy Currents Induced in a Slotted Metal Plate: Eddy currents induced in a slotted metal plate entering a magnetic field form small loops, and the forces on them tend to cancel, thereby making magnetic drag almost zero.

Changing Magnetic Flux Produces an Electric Field

Faraday’s law of induction states that changing magnetic field produces an electric field: .

Learning Objectives

Describe the relationship between the changing magnetic field and an electric field

Key Takeaways

Key Points

- Faraday’s law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force.

- An alternative, differential form of Faraday’s law of induction is express in the equation .

- Faraday’s law of induction is one of the four equations in Maxwell’s equations, governing all electromagnetic phenomena.

Key Terms

- vector area: A vector whose magnitude is the area under consideration and whose direction is perpendicular to the plane.

- Maxwell’s equations: A set of equations describing how electric and magnetic fields are generated and altered by each other and by charges and currents.

- Stokes’ theorem: a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

We have studied Faraday’s law of induction in previous atoms. We learned the relationship between induced electromotive force (EMF) and magnetic flux. In a nutshell, the law states that changing magnetic field produces an electric field , Faraday’s law of induction is expressed as , where is induced EMF and is magnetic flux. (“N” is dropped from our previous expression. The number of turns of coil is included can be incorporated in the magnetic flux, so the factor is optional. ) Faraday’s law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). In this Atom, we will learn about an alternative mathematical expression of the law.

Faraday’s Experiment: Faraday’s experiment showing induction between coils of wire: The liquid battery (right) provides a current which flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).

Differential form of Faraday’s law

The magnetic flux is , where is a vector area over a closed surface S. A device that can maintain a potential difference, despite the flow of current is a source of electromotive force. (EMF) The definition is mathematically , where the integral is evaluated over a closed loop C.

Faraday’s law now can be rewritten . Using the Stokes’ theorem in vector calculus, the left hand side is. Also, note that in the right hand side. Therefore, we get an alternative form of the Faraday’s law of induction: .This is also called a differential form of the Faraday’s law. It is one of the four equations in Maxwell’s equations, governing all electromagnetic phenomena.

Electric Generators

Electric generators convert mechanical energy to electrical energy; they induce an EMF by rotating a coil in a magnetic field.

Learning Objectives

Explain how an electromotive force is induced in electric generators

Key Takeaways

Key Points

- An electric generator rotates a coil in a magnetic field, inducing an EMF given as a function of time by ε=NABw sinωt.

- Generators supply almost all of the power for the electric power grids which provide most of the world’s electric power.

- A motor becomes a generator when its shaft rotates.

Key Terms

- electromotive force: (EMF)—The voltage generated by a battery or by the magnetic force according to Faraday’s Law. It is measured in units of volts, not newtons, and thus, is not actually a force.

- turbine: Any of various rotary machines that use the kinetic energy of a continuous stream of fluid (a liquid or a gas) to turn a shaft.

Electric generators are devices that convert mechanical energy to electrical energy. They induce an electromotive force (EMF) by rotating a coil in a magnetic field. It is a device that converts mechanical energy to electrical energy. A generator forces electric charge (usually carried by electrons) to flow through an external electrical circuit. Possible sources of mechanical energy include: a reciprocating or turbine steam engine, water falling through a turbine or waterwheel, an internal combustion engine, a wind turbine, a hand crank, compressed air, or any other source of mechanical energy. Generators supply almost all of the power for the electric power grids which provide most of the world’s electric power.

Steam Turbine Generator: A modern steam turbine generator.

Basic Setup

Consider the setup shown in. Charges in the wires of the loop experience the magnetic force because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. However, those in the top and bottom segments feel a force perpendicular to the wire; this force does not cause a current. We can thus find the induced EMF by considering only the side wires. Motional EMF is given to be EMF=Bℓv, where the velocity v is perpendicular to the magnetic field B (see our Atom on “Motional EMF”). Here, the velocity is at an angle θ with B, so that its component perpendicular to B is vsinθ.

Diagram of an Electric Generator: A generator with a single rectangular coil rotated at constant angular velocity in a uniform magnetic field produces an emf that varies sinusoidally in time. Note the generator is similar to a motor, except the shaft is rotated to produce a current rather than the other way around.

Thus in this case the EMF induced on each side is EMF=Bℓvsinθ, and they are in the same direction. The total EMF around the loop is then:

.

This expression is valid, but it does not give EMF as a function of time. To find the time dependence of EMF, we assume the coil rotates at a constant angular velocity ω. The angle θ is related to angular velocity by θ=ωt, so that:

.

Now, linear velocity v is related to angular velocity by v=rω. Here r=w/2, so that v=(w/2)ω, and:

.

Noting that the area of the loop is A=ℓw, and allowing for N loops, we find that:

is the EMF induced in a generator coil of N turns and area A rotating at a constant angular velocity in a uniform magnetic field B.

Generators illustrated in this Atom look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates.

Electric Motors

An electric motor is a device that converts electrical energy into mechanical energy.

Learning Objectives

Explain how force is generated into electric motors

Key Takeaways

Key Points

- Most electric motors use the interaction of magnetic fields and current -carrying conductors to generate force.

- Current in a conductor consists of moving charges. Therefore, a current-carrying coil in a magnetic field will also feel the Lorentz force.

- In a motor, a current-carrying coil in a magnetic field experiences a force on both sides of the coil, which creates a twisting force (called a torque) that makes it turn.

Key Terms

- Lorentz force: The force exerted on a charged particle in an electromagnetic field.

- torque: A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)

The basic principles of operation for a motor are the same as those for a generator, except that a motor converts electrical energy into mechanical energy (motion). (Read our atom on electric generators first. ) Most electric motors use the interaction of magnetic fields and current-carrying conductors to generate force. Electric motors are found in applications as diverse as industrial fans, blowers and pumps, machine tools, household appliances, power tools, and disk drives.

Lorentz Force

If you were to place a moving charged particle in a magnetic field, it would experience a force called the Lorentz force:

Right-Hand Rule: Right-hand rule showing the direction of the Lorentz force

where v is the speed of the moving charge, q is the charge, and B is the magnetic field. Current in a conductor consists of moving charges. Therefore, a current-carrying coil in a magnetic field will also feel the Lorentz force. For a straight current carrying wire that is not moving, the Lorentz force is:

where F is the force (in newtons, N), I is the current in the wire (in amperes, A), L is the length of the wire that is in the magnetic field (in m), and B is the magnetic field strength (in teslas, T). The direction of the Lorentz force is perpendicular to both the direction of the flow of current and the magnetic field and can be found using the right-hand rule, shown in. Using your right hand, point your thumb in the direction of the current, and point your first finger in the direction of the magnetic field. Your third finger will now be pointing in the direction of the force.

Torque: The force on opposite sides of the coil will be in opposite directions because the charges are moving in opposite directions. This means the coil will rotate.

Mechanics of a Motor

Both motors and generators can be explained in terms of a coil that rotates in a magnetic field. In a generator the coil is attached to an external circuit that is then turned. This results in a changing flux, which induces an electromagnetic field. In a motor, a current-carrying coil in a magnetic field experiences a force on both sides of the coil, which creates a twisting force (called a torque) that makes it turn. Any coil carrying current can feel a force in a magnetic field. This force is the Lorentz force on the moving charges in the conductor. The force on opposite sides of the coil will be in opposite directions because the charges are moving in opposite directions. This means the coil will rotate.

Inductance

Inductance is the property of a device that tells how effectively it induces an emf in another device or on itself.

Learning Objectives

Describe properties of an inductor, distinguishing mutual inductance and self-inductance

Key Takeaways

Key Points

- Mutual inductance is the effect of two devices in inducing emfs in each other. A change in current ΔI_1/Δt in one induces an emf emf2 in the seccond: EMF_2= −M ΔI_1/Δt, where M is defined to be the mutual inductance between the two devices.

- Self-inductance is the effect of the device inducing emf in itself.

- A device that exhibits significant self-inductance is called an inductor, and the EMF induced in it by a change in current through it is EMF = −L ΔI/Δt.

Key Terms

- Faraday’s law of induction: A basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

- transformer: A static device that transfers electric energy from one circuit to another by magnetic coupling. Their main use is to transfer energy between different voltage levels, which allows choosing most appropriate voltage for power generation, transmission and distribution separately.

Induction is the process in which an emf is induced by changing magnetic flux. Transformers, for example, are designed to be particularly effective at inducing a desired voltage and current with very little loss of energy to other forms (see our Atom on “Transformers. “) Is there a useful physical quantity related to how “effective” a given device is? The answer is yes, and that physical quantity is called inductance.

Mutual Inductance

Mutual inductance is the effect of Faraday’s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See, where simple coils induce emfs in one another.

Mutual Inductance in Coils: These coils can induce emfs in one another like an inefficient transformer. Their mutual inductance M indicates the effectiveness of the coupling between them. Here a change in current in coil 1 is seen to induce an emf in coil 2. (Note that “E2 induced” represents the induced emf in coil 2. )

In the many cases where the geometry of the devices is fixed, flux is changed by varying current. We therefore concentrate on the rate of change of current, ΔI/Δt, as the cause of induction. A change in the current I_1 in one device, coil 1, induces an EMF_2 in the other. We express this in equation form as

,

where M is defined to be the mutual inductance between the two devices. The minus sign is an expression of Lenz’s law. The larger the mutual inductance M, the more effective the coupling.

Nature is symmetric here. If we change the current I2 in coil 2, we induce an emf1 in coil 1, which is given by

,

where M is the same as for the reverse process. Transformers run backward with the same effectiveness, or mutual inductance M.

Self-Inductance

Self-inductance, the effect of Faraday’s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also increase, inducing a counter emf, as required by Lenz’s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a fixed geometry, and so the change in flux is due entirely to the change in current ΔI through the device. The induced emf is related to the physical geometry of the device and the rate of change of current. It is given by

,

where L is the self-inductance of the device. A device that exhibits significant self-inductance is called an inductor. Again, the minus sign is an expression of Lenz’s law, indicating that emf opposes the change in current.

A Quantitative Interpretation of Motional EMF

A a motional EMF is an electromotive force (EMF) induced by motion relative to a magnetic field B.

Learning Objectives

Formulate two views that are applied to calculate the electromotive force

Key Takeaways

Key Points

- Motional and induced EMF are the same phenomenon, just observed in different reference frames. Equivalence of the two phenomena is what triggered Einstein to work on special relativity.

- The EMF produced due to the relative motion of the loop and magnet is given as (Eq. 1), where L is the length of the object moving at speed v relative to the magnet.

- The EMF can be calculated from two different points of view: 1) in terms of the magnetic force on moving electrons in a magnetic field, and 2) in terms of the rate of change in magnetic flux. Both yield the same result.

Key Terms

- special relativity: A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.

- magnetic field: A condition in the space around a magnet or electric current in which there is a detectable magnetic force, and where two magnetic poles are present.

- frame of reference: A coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it.

An electromotive force (EMF) induced by motion relative to a magnetic field B is called a motional EMF. You might have noticed that motional EMF is very similar to the induced EMF caused by a changing magnetic field. In this Atom we see that they are indeed the same phenomenon, shown in different frame of reference.

Motional EMF

In the case where a conductor loop is moving into magnet shown in (a), magnetic force on a moving charge in the loop is given by (Lorentz force, e: electron charge).

Conductor Loop Moving Into a Magnet: (a) Motional EMF. The current loop is moving into a stationary magnet. The direction of the magnetic field is into the screen. (b) Induced EMF. Current loop is stationary, and the magnet is moving.

Due to the force, electrons will keep building up on one side (bottom end in the figure) until enough of an electric field opposing the motion of electrons is established across the rod, which is . Equating the two forces, we get .

Therefore, the motional EMF over the length L of the side of the loop is given by (Eq. 1), where L is the length of the object moving at speed v relative to the magnet.

Induced EMF

Since the rate of change of the magnetic flux passing through the loop is (A: area of the loop that magnetic field pass through), the induced EMF (Eq. 2).

Equivalence of the Motional and Induced EMF

From Eq. 1 and Eq. 2 we can confirm that motional and induced EMF yield the same result. In fact, the equivalence of the two phenomena is what triggered Albert Einstein to examine special relativity. In his seminal paper on special relativity published in 1905, Einstein begins by mentioning the equivalence of the two phenomena:

“…… for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy , producing a current  at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case. “

Mechanical Work and Electrical Energy

Mechanical work done by an external force to produce motional EMF is converted to heat energy; energy is conserved in the process.

Learning Objectives

Apply the law of conservation of energy to describe the production motional electromotive force with mechanical work

Key Takeaways

Key Points

- Motional EMF produced by a moving conductor in a uniform field is given as follows .

- To keep the rod moving at a constant speed v, we have to apply an external force Fext constantly on the rod along its motion.

- Lenz’ law guarantees that the motion of the rod is opposed, and therefore the law of energy conservation is not violated.

Key Terms

- motional EMF: An EMF (electromotive force) induced by motion relative to a magnetic field.

- Faraday’s law of induction: A basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

We learned about motional EMF previously (see our Atom on “Motional EMF”). For the simple setup shown below, motional EMF produced by a moving conductor (in a uniform field) is given as follows:

where B is the magnetic field, l is the length of the conducting rod, and v is the (constant) speed of its motion. (B, l, and v are all perpendicular to each other as shown in the image below.)

Motional EMF: (a) A motional emf=Bℓv is induced between the rails when this rod moves to the right in the uniform magnetic field. The magnetic field B is into the page, perpendicular to the moving rod and rails and, hence, to the area enclosed by them. (b) Lenz’s law gives the directions of the induced field and current, and the polarity of the induced emf. Since the flux is increasing, the induced field is in the opposite direction, or out of the page. Right hand rule gives the current direction shown, and the polarity of the rod will drive such a current.

Conservation of Energy

In this atom, we will consider the system from the energy perspective. As the rod moves and carries current i, it will feel the Lorentz force

.

To keep the rod moving at a constant speed v, we must constantly apply an external force Fext(equal to magnitude of FL and opposite in its direction) to the rod along its motion. Since the rod is moving at v, the power P delivered by the external force would be:

.

In the final step, we used the first equation we talked about. Note that this is exactly the power dissipated in the loop (= current voltage). Therefore, we conclude that the mechanical work done by an external force to keep the rod moving at a constant speed is converted to heat energy in the loop. More generally, mechanical work done by an external force to produce motional EMF is converted to heat energy. Energy is conserved in the process.

Lenz’ Law

We learned in the Atom “Faraday’s Law of Induction and Lenz’ Law” that Lenz’ law is a manifestation of the conservation of energy. As we see in the example in this Atom, Lenz’ law guarantees that the motion of the rod is opposed because of nature’s tendency to oppose a change in magnetic field. If the induced EMF were in the same direction as the change in flux, there would be a positive feedback causing the rod to fly away from the slightest perturbation.

Energy in a Magnetic Field

Magnetic field stores energy. The energy density is given as .

Learning Objectives

Express the energy density of a magnetic field in a form of equation

Key Takeaways

Key Points

- Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field.

- For linear, non-dispersive, materials (such that B = μH where μ, called the permeability, is frequency-independent), the energy density is: .

- The energy stored by an inductor is .

Key Terms

- permeability: A quantitative measure of the degree of magnetization of a material in the presence of an applied magnetic field (measured in newtons per ampere squared in SI units).

- inductor: A passive device that introduces inductance into an electrical circuit.

- ferromagnet: Materials that show a permanent magnetic property.

Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials this same energy is released when the magnetic field is destroyed. Therefore, this energy can be modeled as being “stored” in the magnetic field.

Magnetic Field Created By A Solenoid: Magnetic field created by a solenoid (cross-sectional view) described using field lines. Energy is “stored” in the magnetic field.

Energy Stored in a Magnetic Field

For linear, non-dispersive, materials (such that B = μH where μ, called the permeability, is frequency-independent), the energy density is:

.

Energy density is the amount of energy stored in a given system or region of space per unit volume. If there are no magnetic materials around, μ can be replaced by μ_0. The above equation cannot be used for nonlinear materials, though; a more general expression (given below) must be used.

In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δB is:

.

Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state. For hysteretic materials such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.

Energy Stored in the Field of a Solenoid

The energy stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. This is given by:

.

Proof: Power that should be supplied to an inductor with inductance L to run current I through it it given as

.

Therefore

.

Transformers

Transformers transform voltages from one value to another; its function is governed by the transformer equation.

Learning Objectives

Apply the transformer equation to compare the secondary and primary voltages

Key Takeaways

Key Points

- Transformers are often used at several points in the power distribution systems and also in many household power adapters.

- Transformer equation states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils: .

- Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. This leads us to another useful eqaution: . If voltage increases, current decreases. Conversely, if voltage decreases, current increases.

Key Terms

- magnetic flux: A measure of the strength of a magnetic field in a given area.

- Faraday’s law of induction: A basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

Transformers change voltages from one value to another. For example, devices such as cell phones, laptops, video games, power tools and small appliances have a transformer (built into their plug-in unit) that changes 120 V into the proper voltage for the device. Transformers are also used at several points in power distribution systems, as shown in. Power is sent long distances at high voltages, as less current is required for a given amount of power (this means less line loss). Because high voltages pose greater hazards, transformers are employed to produce lower voltage at the user’s location.

Transformer Setup: Transformers change voltages at several points in a power distribution system. Electric power is usually generated at greater than 10 kV, and transmitted long distances at voltages over 200 kV—sometimes as great as 700 kV—to limit energy losses. Local power distribution to neighborhoods or industries goes through a substation and is sent short distances at voltages ranging from 5 to 13 kV. This is reduced to 120, 240, or 480 V for safety at the individual user site.

The type of transformer considered here is based on Faraday’s law of induction, and is very similar in construction to the apparatus Faraday used to demonstrate that magnetic fields can create currents (illustrated in ). The two coils are called the primary and secondary coils. In normal use, the input voltage is placed on the primary, and the secondary produces the transformed output voltage. Not only does the iron core trap the magnetic field created by the primary coil, its magnetization increases the field strength. Since the input voltage is AC, a time-varying magnetic flux is sent to the secondary, inducing its AC output voltage.

Simple Transformer: A typical construction of a simple transformer has two coils wound on a ferromagnetic core that is laminated to minimize eddy currents. The magnetic field created by the primary is mostly confined to and increased by the core, which transmits it to the secondary coil. Any change in current in the primary induces a current in the secondary.The figure shows a simple transformer with two coils wound on either sides of a laminated ferromagnetic core. The set of coil on left side of the core is marked as the primary and there number is given as N p. The voltage across the primary is given by V p. The set of coil on right side of the core is marked as the secondary and there number is represented as N s. The voltage across the secondary is given by V s. A symbol of the transformer is also shown below the diagram. It consists of two inductor coils separated by two equal parallel lines representing the core.

Transformer Equation

For the simple transformer shown in, the output voltage Vs depends almost entirely on the input voltage Vp and the ratio of the number of loops in the primary and secondary coils. Faraday’s law of induction for the secondary coil gives its induced output voltage Vs as:

,

where Ns is the number of loops in the secondary coil and Δ/Δt is the rate of change of magnetic flux. Note that the output voltage equals the induced EMF (Vs=EMFs), provided coil resistance is small. The cross-sectional area of the coils is the same on either side, as is the magnetic field strength, so /Δt is the same on either side. The input primary voltage Vp is also related to changing flux by:

.

Taking the ratio of these last two equations yields a useful relationship:

.

This is known as the transformer equation, which simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils. The output voltage of a transformer can be less than, greater than or equal to the input voltage, depending on the ratio of the number of loops in their coils. Some transformers even provide a variable output by allowing connection to be made at different points on the secondary coil. A step-up transformer is one that increases voltage, whereas a step-down transformer decreases voltage.

Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. Equating the power input and output,

.

Combining this results with the transformer equation, we find:

.

So if voltage increases, current decreases. Conversely, if voltage decreases, current increases.

28

The Electromagnetic Spectrum

Radio Waves

Radio waves are EM (Electromagnetic)waves that have wavelengths between 1 millimeter and 100 kilometers (or 300 GHz and 3 kHz in frequency).

Learning Objectives

Compare properties of AM and FM radio waves

Key Takeaways

Key Points

- The lowest frequency portion of the electromagnetic spectrum is designated as “radio,” generally considered to have wavelengths within 1 millimeter to 100 kilometers or frequencies within 300 GHz to 3 kHz.

- There is a wide range of subcategories contained within radio including AM and FM radio. Radio waves can be generated by natural sources such as lightning or astronomical phenomena; or by artificial sources such as broadcast radio towers, cell phones, satellites and radar.

- AM radio waves are used to carry commercial radio signals in the frequency range from 540 to 1600 kHz. The abbreviation AM stands for amplitude modulation—the method for placing information on these waves. AM waves have constant frequency, but a varying amplitude.

- FM radio waves are also used for commercial radio transmission in the frequency range of 88 to 108 MHz. FM stands for frequency modulation, which produces a wave of constant amplitude but varying frequency.

Key Terms

- AM radio waves: Waves used to carry commercial radio signals between 540 and 1600 kHz. Information is carried by amplitude variation, while the frequency remains constant.

- FM radio waves: Waves used to carry commercial radio signals between 88 and 108 MHz. Information is carried by frequency modulation, while the signal amplitude remains constant.

- radio waves: Designates a portion of the electromagnetic spectrum having frequencies ranging from 300 GHz to 3 kHz, or equivalently, wavelengths from 1 millimeter to 100 kilometers.

Radio Waves

Radio waves are a type of electromagnetic (EM) radiation with wavelengths in the electromagnetic spectrum longer than infrared light. They have have frequencies from 300 GHz to as low as 3 kHz, and corresponding wavelengths from 1 millimeter to 100 kilometers. Like all other electromagnetic waves, radio waves travel at the speed of light. Naturally occurring radio waves are made by lightning or by astronomical objects. Artificially generated radio waves are used for fixed and mobile radio communication, broadcasting, radar and other navigation systems, communications satellites, computer networks and innumerable other applications. Different frequencies of radio waves have different propagation characteristics in the Earth’s atmosphere—long waves may cover a part of the Earth very consistently, shorter waves can reflect off the ionosphere and travel around the world, and much shorter wavelengths bend or reflect very little and travel on a line of sight.

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves encompass the high frequency portion of the radio section of the EM spectrum.

Types of Radio Waves and Applications

Radio waves have many uses—the category is divided into many subcategories, including microwaves and electromagnetic waves used for AM and FM radio, cellular telephones and TV.

The lowest commonly encountered radio frequencies are produced by high-voltage AC power transmission lines at frequencies of 50 or 60 Hz. These extremely long wavelength electromagnetic waves (about 6000 km) are one means of energy loss in long-distance power transmission.

Extremely low frequency (ELF) radio waves of about 1 kHz are used to communicate with submerged submarines. The ability of radio waves to penetrate salt water is related to their wavelength (much like ultrasound penetrating tissue)—the longer the wavelength, the farther they penetrate. Since salt water is a good conductor, radio waves are strongly absorbed by it; very long wavelengths are needed to reach a submarine under the surface.

AM Radio Waves

AM radio waves are used to carry commercial radio signals in the frequency range from 540 to 1600 kHz. The abbreviation AM stands for amplitude modulation—the method for placing information on these waves. A carrier wave having the basic frequency of the radio station (for instance, 1530 kHz) is varied or modulated in amplitude by an audio signal. The resulting wave has a constant frequency, but a varying amplitude.

AM Radio: Amplitude modulation for AM radio. (a) A carrier wave at the station’s basic frequency. (b) An audio signal at much lower audible frequencies. (c) The amplitude of the carrier is modulated by the audio signal without changing its basic frequency.

FM Radio Waves

FM radio waves are also used for commercial radio transmission, but in the frequency range of 88 to 108 MHz. FM stands for frequency modulation, another method of carrying information. In this case, a carrier wave having the basic frequency of the radio station (perhaps 105.1 MHz) is modulated in frequency by the audio signal, producing a wave of constant amplitude but varying frequency.

FM Radio: Frequency modulation for FM radio. (a) A carrier wave at the station’s basic frequency. (b) An audio signal at much lower audible frequencies. (c) The frequency of the carrier is modulated by the audio signal without changing its amplitude.

Since audible frequencies range up to 20 kHz (or 0.020 MHz) at most, the frequency of the FM radio wave can vary from the carrier by as much as 0.020 MHz. For this reason, the carrier frequencies of two different radio stations cannot be closer than 0.020 MHz. An FM receiver is tuned to resonate at the carrier frequency and has circuitry that responds to variations in frequency, reproducing the audio information.

FM radio is inherently less subject to noise from stray radio sources than AM radio because amplitudes of waves add noise. Thus, an AM receiver would interpret noise added onto the amplitude of its carrier wave as part of the information. An FM receiver can be fashioned to reject amplitudes other than that of the basic carrier wave and only look for variations in frequency. Thus, since noise produces a variation in amplitude, it is easier to reject noise from FM.

TV

Electromagnetic waves also broadcast television transmission. However, as the waves must carry a great deal of visual as well as audio information, each channel requires a larger range of frequencies than simple radio transmission. TV channels utilize frequencies in the range of 54 to 88 MHz and 174 to 222 MHz (the entire FM radio band lies between channels 88 MHz and 174 MHz). These TV channels are called VHF (very high frequency). Other channels called UHF (ultra high frequency) utilize an even higher frequency range of 470 to 1000 MHz.

The TV video signal is AM, while the TV audio is FM. Note that these frequencies are those of free transmission with the user utilizing an old-fashioned roof antenna. Satellite dishes and cable transmission of TV occurs at significantly higher frequencies, and is rapidly evolving with the use of the high-definition or HD format.

Microwaves

Microwaves are electromagnetic waves with wavelengths ranging from one meter to one millimeter (frequencies between 300 MHz and 300 GHz).

Learning Objectives

Distinguish three ranges of the microwave portion of the electromagnetic spectrum

Key Takeaways

Key Points

- The microwave region of the electromagnetic (EM) spectrum is generally considered to overlap with the highest frequency (shortest wavelength ) radio waves.

- The prefix “micro-” in “microwave” is not meant to suggest a wavelength in the micrometer range. It indicates that microwaves are “small” compared to waves used in typical radio broadcasting in that they have shorter wavelengths.

- The microwave portion of the electromagnetic spectrum can be subdivided into three ranges listed below from high to low frequencies: extremely high frequency (30 to 300 GHz), super high frequency (3 to 30 GHz), and ultra-high frequency (300 MHz to 3 GHz).

- Microwave sources include artificial devices such as circuits, transmission towers, radar, masers, and microwave ovens, as well as natural sources such as the Sun and the Cosmic Microwave Background.

- Microwaves can also be produced by atoms and molecules. They are, for example, a component of electromagnetic radiation generated by thermal agitation. The thermal motion of atoms and molecules in any object at a temperature above absolute zero causes them to emit and absorb radiation.

Key Terms

- terahertz radiation: Electromagnetic waves with frequencies around one terahertz.

- thermal agitation: The thermal motion of atoms and molecules in any object at a temperature above absolute zero, causing them to emit and absorb radiation.

- radar: A method of detecting distant objects and determining their position, velocity, or other characteristics by analysis of sent radio waves (usually microwaves) reflected from their surfaces.

Microwaves

Microwaves are electromagnetic waves with wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently with frequencies between 300 MHz (0.3 GHz) and 300 GHz. The microwave region of the electromagnetic (EM) spectrum is generally considered to overlap with the highest frequency (shortest wavelength) radio waves. As is the case for all EM waves, microwaves travel in a vacuum at the speed of light. The prefix “micro-” in “microwave” is not meant to suggest a wavelength in the micrometer range. It indicates that microwaves are “small” because have shorter wavelengths as compared to waves used in typical radio broadcasting. The boundaries between far infrared light, terahertz radiation, microwaves, and ultra-high-frequency radio waves are fairly arbitrary. They are used variously between different fields of study (see figure).

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves overlap with the high frequency portion of the radio section of the EM spectrum.

Subcategories of Microwaves

The microwave portion of the radio spectrum can be subdivided into three ranges, listed below from high to low frequencies.

- Extremely high frequency (EHF) is the highest microwave frequency band. EHF runs the range of frequencies from 30 to 300 gigahertz, above which electromagnetic radiation is considered as far infrared light, also referred to as terahertz radiation. This frequency range corresponds to a wavelength range of 10 to 1 millimeter, so it is sometimes called the millimeter band. This band is commonly used in radio astronomy and remote sensing.

- Super high frequency (SHF) is the designation for electromagnetic wave frequencies in the range of 3 GHz to 30 GHz. This band of frequencies is known also as the centimeter band because the wavelengths range from ten to one centimeters. This frequency range is used for most radar transmitters, microwave ovens, wireless LANs, cell phones, satellite communication, microwave radio relay links, and numerous short range terrestrial data links.

- Ultra-high frequency (UHF) designates the microwave frequency range of electromagnetic waves between 300 MHz and 3 GHz, also known as the decimeter band because the wavelengths range from one to ten decimeters, or 10 centimeters to 1 meter. They are used for television broadcasting, cordless phones, walkie-talkies, satellite communication, and numerous other applications.

Sources of Microwaves

Microwaves are the highest-frequency electromagnetic waves that can be produced by currents in macroscopic circuits and devices. Microwaves can also be produced by atoms and molecules—e.g., they are a component of electromagnetic radiation generated by thermal agitation. The thermal motion of atoms and molecules in any object at a temperature above absolute zero causes them to emit and absorb radiation.

Since it is possible to carry more information per unit time on high frequencies, microwaves are quite suitable for communications devices. Most satellite-transmitted information is carried on microwaves, as are land-based long-distance transmissions. A clear line of sight between transmitter and receiver is needed because of the short wavelengths involved.

The sun also emits microwave radiation, although most of it is blocked by Earth’s atmosphere. The Cosmic Microwave Background Radiation (CMBR) is microwave radiation that permeates all of space, and its discovery supports the Big Bang theory of the origin of the universe.

Cosmic Microwave Background: Cosmic background radiation of the Big Bang mapped with increasing resolution.

Devices Employing Microwaves

High-power microwave sources use specialized vacuum tubes to generate microwaves. These devices operate on different principles from low-frequency vacuum tubes, using the ballistic motion of electrons in a vacuum under the influence of controlling electric or magnetic fields, and include the magnetron (used in microwave ovens), klystron, traveling-wave tube (TWT), and gyrotron.

Cavity Magnetron: Cutaway view inside a cavity magnetron as used in a microwave oven.

Microwaves are used by microwave ovens to heat food. Microwaves at a frequency of 2.45 GHz are produced by accelerating electrons. The microwaves then induce an alternating electric field in the oven. Water and some other constituents of food have a slightly negative charge at one end and a slightly positive charge at one end (called polar molecules). The range of microwave frequencies is specially selected so that the polar molecules, in trying to maintain their orientation with the electric field, absorb these energies and increase their temperatures—a process called dielectricheating.

Radar, first developed in World War II, is a common application of microwaves. By detecting and timing microwave echoes, radar systems can determine the distance to objects as diverse as clouds and aircraft. A Doppler shift in the radar echo can determine the speed of a car or the intensity of a rainstorm. Sophisticated radar systems can map the Earth and other planets, with a resolution limited by wavelength. The shorter the wavelength of any probe, the smaller the detail it is possible to observe.

A maser is a device similar to a laser, which amplifies light energy by stimulating photons. The maser, rather than amplifying visible light energy, amplifies the lower-frequency, longer-wavelength microwaves and radio frequency emissions.

Infrared Waves

Infrared (IR) light is EM radiation with wavelengths longer than those of visible light from 0.74 µm to 1 mm (300 GHz to 1 THz).

Learning Objectives

Distinguish three ranges of the infrared portion of the spectrum, and describe processes of absorption and emission of infrared light by molecules

Key Takeaways

Key Points

- Infrared light includes most of the thermal radiation emitted by objects near room temperature. Infrared light is emitted or absorbed by molecules when they change their rotational-vibrational movements.

- The infrared portion of the spectrum can be divided into three regions in wavelength: far-infrared, from 300 GHz (1 mm) to 30 THz (10 μm); mid-infrared, from 30 to 120 THz (10 to 2.5 μm); and near-infrared, from 120 to 400 THz (2,500 to 750 nm).

- Infrared radiation is popularly known as ” heat radiation,” but light and electromagnetic waves of any frequency will heat surfaces that absorb them.

- The concept of emissivity is important in understanding the infrared emissions of objects. This is a property of a surface which describes how its thermal emissions deviate from the ideal of a black body.

- Infrared radiation can be used to remotely determine the temperature of objects (if the emissivity is known). This is termed thermography, mainly used in military and industrial applications.

Key Terms

- emissivity: The energy-emitting propensity of a surface, usually measured at a specific wavelength.

- thermography: Any of several techniques for the remote measurement of the temperature variations of a body, especially by creating images produced by infrared radiation.

- thermal radiation: The electromagnetic radiation emitted from a body as a consequence of its temperature; increasing the temperature of the body increases the amount of radiation produced, and shifts it to shorter wavelengths (higher frequencies) in a manner explained only by quantum mechanics.

Infrared Waves

Infrared (IR) light is electromagnetic radiation with longer wavelengths than those of visible light, extending from the nominal red edge of the visible spectrum at 0.74 micrometers (µm) to 1 mm. This range of wavelengths corresponds to a frequency range of approximately 300 GHz to 400 THz, and includes most of the thermal radiation emitted by objects near room temperature. Infrared light is emitted or absorbed by molecules when they change their rotational-vibrational movements.

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves encompass the high frequency portion of the radio section of the EM spectrum.

Subcategories of IR Waves

The infrared part of the electromagnetic spectrum covers the range from roughly 300 GHz (1 mm) to 400 THz (750 nm). It can be divided into three parts: It can be divided into three parts:

Atmospheric Transmittance: This is a plot of Earth’s atmospheric transmittance (or opacity) to various wavelengths of electromagnetic radiation. Most UV wavelengths are absorbed by oxygen and ozone in Earth’s atmosphere. Observations of astronomical UV sources must be done from space.

- Far-infrared, from 300 GHz (1 mm) to 30 THz (10 μm) – The lower part of this range may also be called microwaves. This radiation is typically absorbed by so-called rotational modes in gas-phase molecules, by molecular motions in liquids, and by phonons in solids. The water in Earth’s atmosphere absorbs so strongly in this range that it renders the atmosphere in effect opaque. However, there are certain wavelength ranges (“windows”) within the opaque range that allow partial transmission, and can be used for astronomy. The wavelength range from approximately 200 μm up to a few mm is often referred to as “sub-millimeter” in astronomy, reserving far infrared for wavelengths below 200 μm.

- Mid-infrared, from 30 to 120 THz (10 to 2.5 μm) – Hot objects (black-body radiators) can radiate strongly in this range, and human skin at normal body temperature radiates strongly at the lower end of this region. This radiation is absorbed by molecular vibrations, where the different atoms in a molecule vibrate around their equilibrium positions. This range is sometimes called the fingerprint region, since the mid-infrared absorption spectrum of a compound is very specific for that compound.

- Near-infrared, from 120 to 400 THz (2,500 to 750 nm) – Physical processes that are relevant for this range are similar to those for visible light. The highest frequences in this region can be detected directly by some types of photographic film, and by many types of solid state image sensors for infrared photography and videography.

Note that in some fields the boundaries of these categories differ slightly; for example, in astronomy “near-infrared” is considered to extend to 5 μm rather than 2.5 μm.

Heat and Thermal Radiation

Infrared radiation is popularly known as “heat radiation,” but light and electromagnetic waves of any frequency will heat surfaces that absorb them. Infrared light from the Sun only accounts for 49% of the heating of the Earth, with the rest being caused by visible light that is absorbed then re-radiated at longer wavelengths. Visible light or ultraviolet-emitting lasers can char paper and incandescently hot objects emit visible radiation. Objects at room temperature will emit radiation mostly concentrated in the 8 to 25 µm band, but this is not distinct from the emission of visible light by incandescent objects and ultraviolet by even hotter objects (see sections on black body radiation and Wien’s displacement law).

Heat is energy in transient form that flows due to temperature difference. Unlike heat transmitted by thermal conduction or thermal convection, radiation can propagate through a vacuum.

The concept of emissivity is important in understanding the infrared emissions of objects. This is a property of a surface which describes how its thermal emissions deviate from the ideal of a black body. To further explain, two objects at the same physical temperature will not “appear” the same temperature in an infrared image if they have differing emissivities.

Sources of IR Waves

As stated above, while infrared radiation is commonly referred to as heat radiation, only objects emitting with a certain range of temperatures and emissivities will produce most of their electromagnetic emission in the infrared part of the spectrum. However, this is the case for most objects and environments humans encounter in our daily lives. Humans, their surroundings, and the Earth itself emit most of their thermal radiation at wavelengths near 10 microns, the boundary between mid and far infrared according to the delineation above. The range of wavelengths most relevant to thermally emitting objects on earth is often called the thermal infrared. Many astronomical objects emit detectable amounts of IR radiation at non-thermal wavelengths.

Infrared radiation can be used to remotely determine the temperature of objects (if the emissivity is known). This is termed thermography, mainly used in military and industrial applications but the technology is reaching the public market in the form of infrared cameras on cars due to the massively reduced production costs.

Thermography: A thermographic image of a dog

Applications of IR waves extend to heating, communication, meteorology, spectroscopy, astronomy, biological and medical science, and even the analysis of works of art.

Visible Light

Visible light is the portion of the electromagnetic spectrum that is visible to the human eye, ranging from roughly 390 to 750 nm.

Learning Objectives

Distinguish six ranges of the visible spectrum

Key Takeaways

Key Points

- Visible light is produced by vibrations and rotations of atoms and molecules, as well as by electronic transitions within atoms and molecules. We say the atoms and molecules are excited when they absorb and relax when they emit through electronic transitions.

- This figure shows the visible part of the spectrum, together with the colors associated with particular pure wavelengths. Red light has the lowest frequencies and longest wavelengths, while violet has the highest frequencies and shortest wavelengths.

- Colors that can be produced by visible light of a narrow band of wavelengths are called pure spectral colors. They can be delineated roughly in wavelength as: violet (380-450 nm), blue (450-495 nm), green (495-570 nm), yellow (570-590 nm), orange (590-620 nm), and red (620 to 750 nm).

- Visible wavelengths pass through the optical window, the Earth’s atmosphere allows this region of the electromagnetic spectrum to pass through largely unattenuated (see opacity plot in.

- The portion of the EM spectrum used by photosynthesic organisms is called the photosynthetically active region (PAR) and corresponds to solar radiation between 400 and 700 nm, substantially overlapping with the range of human vision.

Key Terms

- spectral color: a color that is evoked by a single wavelength of light in the visible spectrum, or by a relatively narrow band of wavelengths. Every wavelength of light is perceived as a spectral color, in a continuous spectrum; the colors of sufficiently close wavelengths are indistinguishable.

- optical window: the optical portion of the electromagnetic spectrum that passes through the atmosphere all the way to the ground. The window runs from around 300 nanometers (ultraviolet-C) at the short end up into the range the eye can use, roughly 400-700 nm and continues up through the visual infrared to around 1100 nm, which is thermal infrared.

- visible light: the part of the electromagnetic spectrum, between infrared and ultraviolet, that is visible to the human eye

Visible Light

Visible light, as called the visible spectrum, is the portion of the electromagnetic spectrum that is visible to (can be detected by) the human eye. Electromagnetic radiation in this range of wavelengths is often simply referred to as “light”. A typical human eye will respond to wavelengths from about 390 to 750 nm (0.39 to 0.75 µm). In terms of frequency, this corresponds to a band in the vicinity of 400–790 THz. A light-adapted eye generally has its maximum sensitivity at around 555 nm (540 THz), in the green region of the optical spectrum. The spectrum does not, however, contain all the colors that the human eyes and brain can distinguish. Unsaturated colors such as pink, or purple variations such as magenta, are absent, for example, because they can be made only by a mix of multiple wavelengths.

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves encompass the high frequency portion of the radio section of the EM spectrum.

Visible light is produced by vibrations and rotations of atoms and molecules, as well as by electronic transitions within atoms and molecules. The receivers or detectors of light largely utilize electronic transitions. We say the atoms and molecules are excited when they absorb and relax when they emit through electronic transitions.

Visible Spectrum: A small part of the electromagnetic spectrum that includes its visible components. The divisions between infrared, visible, and ultraviolet are not perfectly distinct, nor are those between the seven rainbow colors.

The figure above shows this part of the spectrum, together with the colors associated with particular pure wavelengths. Red light has the lowest frequencies and longest wavelengths, while violet has the highest frequencies and shortest wavelengths. Blackbody radiation from the Sun peaks in the visible part of the spectrum but is more intense in the red than in the violet, making the Sun yellowish in appearance.

Colors that can be produced by visible light of a narrow band of wavelengths (monochromaticlight) are called pure spectral colors. Quantitatively, the regions of the visible spectrum encompassing each spectral color can be delineated roughly as:

- red – 620 to 750 nm (400-484 THz)

Note that each color can come in many shades, since the spectrum is continuous. The human eye is insensitive to electromagnetic radiation outside this range. By definition any images presented with data recorded from wavelengths other than those in the visible part of the spectrum (such as IR images of humans or animals or astronomical X-ray images) are necessarily in false color.

Visible Light and Earth’s Atmosphere

Visible wavelengths pass through the “optical window”, the region of the electromagnetic spectrum which allows wavelengths to pass largely unattenuated through the Earth’s atmosphere (see opacity plot in. An example of this phenomenon is that clean air scatters blue light more than red wavelengths, and so the midday sky appears blue.

Atmospheric Transmittance: This is a plot of Earth’s atmospheric transmittance (or opacity) to various wavelengths of electromagnetic radiation. Most UV wavelengths are absorbed by oxygen and ozone in Earth’s atmosphere. Observations of astronomical UV sources must be done from space.

The optical window is also called the visible window because it overlaps the human visible response spectrum. This is not coincidental as humanity’s ancestors evolved vision that could make use of the most plentiful wavelengths of light.The near infrared (NIR) window lies just out of the human vision, as well as the Medium Wavelength IR (MWIR) window and the Long Wavelength or Far Infrared (LWIR or FIR) window though other animals may experience them.

A consequence of the existence of the optical window in Earth’s atmosphere is the relatively balmy temperature conditions on Earth’s surface. The Sun’s luminosity function peaks in the visible range and light in that range is able to travel to the surface of the planet unattenuated due to the optical window. This allows visible light to heat the surface. The surface of the planet then emits energy primarily in infrared wavelengths, which has much greater difficulty escaping (and thus causing the planet to cool) due to the opacity of the atmosphere in the infrared. Earth’s surface would be much cooler without this effect.

Photosynthesis

Plants, like animals, have evolved to utilize and respond to parts of the electromagnetic spectrum they are embedded in. Plants (and many bacteria) convert the light energy captured from the Sun into chemical energy that can be used to fuel the organism’s activities. In plants, algae, and cyanobacteria, photosynthesis uses carbon dioxide and water, releasing oxygen as a waste product. Photosynthesis is vital for all aerobic life on Earth (such as humans and animals). The portion of the EM spectrum used by photosynthesic organisms is called the photosynthetically active region (PAR) and corresponds to solar radiation between 400 and 700 nm, substantially overlapping with the range of human vision. This is again not coincidental; the light in this range is the most plentiful to organisms on the surface of Earth because the Sun emits about half of its luminosity in this wavelength range and it is allowed to pass freely through the optical windows in Earth’s atmosphere.

Ultraviolet Light

Ultraviolet (UV) light is electromagnetic radiation with a wavelength shorter than that of visible light in the range 10 nm to 400 nm.

Learning Objectives

Identify wavelength range characteristic for ultraviolet light and its biological effects

Key Takeaways

Key Points

- Ultraviolet light gets its name because the spectrum consists of electromagnetic waves with frequencies higher than those that humans identify as the color violet.

- Most UV is non- ionizing radiation, though UV with higher energies (10-120 nm) is ionizing. All UV can have harmful effects on biological matter (such as causing cancers) with the highest energies causing the most damage.

- The danger posed by lower energy UV radiation is derived from the ultraviolet photon ‘s power to alter chemical bonds in molecules, even without having enough energy to ionize atoms.

- Solar UV radiation is commonly subdivided into three regions: UV-A (320–400 nm), UV-B (290–320 nm), and UV-C (220–290 nm), ranked from long to shorter wavelengths (from smaller to larger energies).

- Most UV-B and all UV-C is absorbed by ozone (O_3) molecules in the upper atmosphere. Consequently, 99% of the solar UV radiation reaching the Earth’s surface is UV-A.

Key Terms

- ozone layer: A region of the stratosphere, between 15 and 30 kilometres in altitude, containing a relatively high concentration of ozone; it absorbs most solar ultraviolet radiation.

- ionizing radiation: high-energy radiation that is capable of causing ionization in substances through which it passes; also includes high-energy particles

- non-ionizing radiation: Radiation that does not cause atmospheric ionization; electrically neutral radiation.

Ultraviolet Light

Ultraviolet (UV) light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, that is, in the range 10 nm to 400 nm, corresponding to photon energies from 3 eV to 124 eV (1 eV = 1.6e-19 J; EM radiation with frequencies higher than those of visible light are often expressed in terms of energy rather than frequency). It is so-named because the spectrum consists of electromagnetic waves with frequencies higher than those that humans identify as the color violet. These frequencies are invisible to humans, but visible to a number of insects and birds.

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves encompass the high frequency portion of the radio section of the EM spectrum.

UV light is found in sunlight (where it constitutes about 10% of the energy in vacuum) and is emitted by electric arcs and specialized lights such as black lights. It can cause chemical reactions, and causes many substances to glow or fluoresce. Most ultraviolet is classified as non-ionizing radiation. The higher energies of the ultraviolet spectrum from wavelengths about 10 nm to 120 nm (‘extreme’ ultraviolet) are ionizing, but this type of ultraviolet in sunlight is blocked by normal molecular oxygen (O_2) in air, and does not reach the ground. However, the entire spectrum of ultraviolet radiation has some of the biological features of ionizing radiation, in doing far more damage to many molecules in biological systems than is accounted for by simple heating effects (an example is sunburn). These properties derive from the ultraviolet photon’s power to alter chemical bonds in molecules, even without having enough energy to ionize atoms.

Although ultraviolet radiation is invisible to the human eye, most people are aware of the effects of UV on the skin, called suntan and sunburn. In addition to short wave UV blocked by oxygen, a great deal (>97%) of mid-range ultraviolet (almost all UV above 280 nm and most up to 315 nm) is blocked by the ozone layer, and like ionizing short wave UV, would cause much damage to living organisms if it penetrated the atmosphere. After atmospheric filtering, only about 3% of the total energy of sunlight at the zenith is ultraviolet, and this fraction decreases at other sun angles. Much of it is near-ultraviolet that does not cause sunburn, but is still capable of causing long term skin damage and cancer. An even smaller fraction of ultraviolet that reaches the ground is responsible for sunburn and also the formation of vitamin D (peak production occurring between 295 and 297 nm) in all organisms that make this vitamin (including humans). The UV spectrum thus has many effects, both beneficial and damaging, to human health.

Atmospheric Transmittance: This is a plot of Earth’s atmospheric opacity (opposite of transmittance) to various wavelengths of electromagnetic radiation, including visible light. Visible light passes relatively unimpeded through the atmosphere in the “optical window.” Most UV wavelengths are absorbed by oxygen and ozone in Earth’s atmosphere. Observations of astronomical UV sources must be done from space.

Subcategories of UV Light

Solar UV radiation is commonly subdivided into three regions: UV-A (320–400 nm), UV-B (290–320 nm), and UV-C (220–290 nm), ranked from long to shorter wavelengths (from smaller to larger energies). Most UV-B and all UV-C is absorbed by ozone (O_3) molecules in the upper atmosphere. Consequently, 99% of the solar UV radiation reaching the Earth’s surface is UV-A.

There are other schemes for dividing UV into different categories, another common one is: near-ultraviolet (NUV – 300-400 nm), middle ultraviolet (MUV – 200-300 nm), far ultraviolet (FUV – 200-122 nm), and extreme ultraviolet (EUV- 121-10 nm).

Harmful Effects

An overexposure to UVB radiation can cause sunburn and some forms of skin cancer. In humans, prolonged exposure to solar UV radiation may result in acute and chronic health effects on the skin, eye, and immune system. Moreover, UVC can cause adverse effects that can variously be mutagenic or carcinogenic.

DNA UV Mutation: Ultraviolet photons harm the DNA molecules of living organisms in different ways. In one common damage event, adjacent thymine bases bond with each other, instead of across the “ladder. ” This “thymine dimer” makes a bulge, and the distorted DNA molecule does not function properly.

The International Agency for Research on Cancer of the World Health Organization has classified all categories and wavelengths of ultraviolet radiation as a Group 1 carcinogen. This is the highest level designation for carcinogens and means that “there is enough evidence to conclude that it can cause cancer in humans. ”

Beneficial Effects

UVB exposure induces the production of vitamin D in the skin. The majority of positive health effects are related to this vitamin. It has regulatory roles in calcium metabolism (which is vital for normal functioning of the nervous system, as well as for bone growth and maintenance of bone density), immunity, cell proliferation, insulin secretion, and blood pressure.

X-Rays

X-rays are electromagnetic waves with wavelengths in the range of 0.01 to 10 nanometers and energies in the range of 100 eV to 100 keV.

Learning Objectives

Distinguish two categories of X-rays and their biological effects

Key Takeaways

Key Points

- X-rays have shorter wavelengths (higher energy ) than UV waves and, generally, longer wavelengths (lower energy) than gamma rays. Sometimes X-rays are called Röntgen radiation, after Wilhelm Röntgen, who is usually credited as their discoverer.

- Because X-rays have very high energy they are known as ionizing radiation and can harm living tissue. A very high radiation dose over a short amount of time causes radiation sickness, while lower doses can give an increased risk of radiation-induced cancer.

- Lower doses of X-ray radiation can be very effectively used in medical radiography and X-ray spectroscopy. In the case of medical radiography, the benefits of using X-rays for examination far outweighs the risk.

- X-rays are broken up into broad two categories: hard X-rays with energies above 5-10 keV (below 0.2-0.1 nm wavelength) and soft X-rays with energies 100 eV – 5 keV (10 – 0.1 nm wavelength). Hard X-rays are more useful for radiography because they pass through tissue.

- The distinction between X-rays and gamma rays is somewhat arbitrary and there is substantial overlap at the high energy boundary. However, in general they are distinguished by their source, with gamma rays originating from the nucleus and X-rays from the electrons in the atom.

Key Terms

- X-ray spectroscopy: The use of an X-ray spectrometer for chemical analysis.

- x-ray crystallography: A technique in which the patterns formed by the diffraction of X-rays on passing through a crystalline substance yield information on the lattice structure of the crystal, and the molecular structure of the substance.

- radiograph: An image, often a photographic negative, produced by radiation other than normal light; especially an X-ray photograph.

X-Rays

X-rays are electromagnetic waves with wavelengths in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz (3×10^16 Hz to 3×10^19 Hz) and energies in the range 100 eV to 100 keV. They are shorter in wavelength than UV rays and longer than gamma rays. In many languages, X-radiation is called Röntgen radiation, after Wilhelm Röntgen, who is usually credited as its discoverer, and who had named it X-radiation to signify an unknown type of radiation.

Electromagnetic Spectrum: The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Microwaves encompass the high frequency portion of the radio section of the EM spectrum.

Properties and Applications

X-ray photons carry enough energy to ionize atoms and disrupt molecular bonds. This makes it a type of ionizing radiation and thereby harmful to living tissue. A very high radiation dose over a short amount of time causes radiation sickness, while lower doses can give an increased risk of radiation-induced cancer. In medical imaging this increased cancer risk is generally greatly outweighed by the benefits of the examination. The ionizing capability of X-rays can be utilized in cancer treatment to kill malignant cells using radiation therapy. It is also used for material characterization using X-ray spectroscopy.

X-Ray Spectrum and Applications: X-rays are part of the electromagnetic spectrum, with wavelengths shorter than those of visible light. Different applications use different parts of the X-ray spectrum.

X-rays with photon energies above 5 to 10 keV (below 0.2-0.1 nm wavelength), are called hard X-rays, while those with lower energy are called soft X-rays. Due to their penetrating ability, hard X-rays are widely used to image the inside of objects (e.g., in medical radiography and airport security). As a result, the term X-ray is metonymically used to refer to a radiographic image produced using this method, in addition to the method itself. Since the wavelength of hard X-rays are similar to the size of atoms, they are also useful for determining crystal structures by X-ray crystallography. By contrast, soft X-rays are easily absorbed in air and the attenuation length of 600 eV (~2 nm) X-rays in water is less than 1 micrometer.

In medical diagnostic applications, the low energy (soft) X-rays are unwanted, since they are totally absorbed by the body, increasing the radiation dose without contributing to the image. Hence, a thin metal sheet, often of aluminum, called an X-ray filter, is usually placed over the window of the X-ray tube, absorbing the low energy part in the spectrum. This is called hardening the beam since it shifts the center of the spectrum towards higher energy (or harder) X-rays.

Distinction Between X-Rays and Gamma Rays

The distinction between X-rays and gamma rays is somewhat arbitrary. The most frequent method of distinguishing between X- and gamma radiation is the basis of wavelength, with radiation shorter than some arbitrary wavelength, such as 10−11 m, defined as gamma rays. The electromagnetic radiation emitted by X-ray tubes generally has a longer wavelength than the radiation emitted by radioactive nuclei. Historically, therefore, an alternative means of distinguishing between the two types of radiation has been by their origin: X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus. There is overlap between the wavelength bands of photons emitted by electrons outside the nucleus, and photons emitted by the nucleus. Like all electromagnetic radiation, the properties of X-rays (or gamma rays) depend only on their wavelength and polarization.

Gamma Rays

Gamma rays are very high frequency electromagnetic waves usually emitted from radioactive decay with frequencies greater than 10^19 Hz.

Learning Objectives

Identify wavelength range characteristic for gamma rays, noting their biological effects and distinguishing them from gamma rays

Key Takeaways

Key Points

- Gamma rays are the highest energy EM radiation and typically have energies greater than 100 keV, frequencies greater than 10^19 Hz, and wavelengths less than 10 picometers.

- Gamma rays from radioactive decay are defined as gamma rays no matter what their energy, so that there is no lower limit to gamma energy derived from radioactive decay. Gamma decay commonly produces energies of a few hundred keV, and almost always less than 10 MeV.

- Gamma rays have characteristics identical to X-rays of the same frequency—they differ only in source. Gamma rays are usually distinguished by their origin: X-rays are emitted by definition by electrons outside the nucleus, while gamma rays are emitted by the nucleus.

- Natural sources of gamma rays include gamma decay from naturally occurring radioisotopes such as potassium-40, and also as a secondary radiation from atmospheric interactions with cosmic ray particles. Exotic astrophysical processes will also produce gamma rays.

- Gamma rays are ionizing radiation and are thus biologically hazardous. The most biological damaging forms of gamma radiation occur at energies between 3 and 10 MeV.

Key Terms

- gamma ray: A very high frequency (and therefore very high energy) electromagnetic radiation emitted as a consequence of radioactivity.

- ionizing radiation: high-energy radiation that is capable of causing ionization in substances through which it passes; also includes high-energy particles

- gamma decay: A nuclear reaction with the emission of a gamma ray.

Gamma Rays

Gamma radiation, also known as gamma rays or hyphenated as gamma-rays and denoted as γ, is electromagnetic radiation of high frequency and therefore high energy. Gamma rays typically have frequencies above 10 exahertz (or >10^19 Hz), and therefore have energies above 100 keV and wavelengths less than 10 picometers (less than the diameter of an atom). However, this is not a hard and fast definition, but rather only a rule-of-thumb description for natural processes. Gamma rays from radioactive decay are defined as gamma rays no matter what their energy, so that there is no lower limit to gamma energy derived from radioactive decay. Gamma decay commonly produces energies of a few hundred keV, and almost always less than 10 MeV.

Gamma Decay: Illustration of an emission of a gamma ray (γ) from an atomic nucleus

Gamma rays are ionizing radiation and are thus biologically hazardous. They are classically produced by the decay from high energy states of atomic nuclei, a process called gamma decay, but are also created by other processes. Paul Villard, a French chemist and physicist, discovered gamma radiation in 1900, while studying radiation emitted from radium during its gamma decay. Villard’s radiation was named “gamma rays” by Ernest Rutherford in 1903.

Gamma Ray Sources

Natural sources of gamma rays on Earth include gamma decay from naturally occurring radioisotopes such as potassium-40, and also as a secondary radiation from various atmospheric interactions with cosmic ray particles. Some rare terrestrial natural sources that produce gamma rays that are not of a nuclear origin, are lightning strikes and terrestrial gamma-ray flashes, which produce high energy emissions from natural high-energy voltages. Gamma rays are produced by a number of astronomical processes in which very high-energy electrons are produced. Such electrons produce secondary gamma rays by the mechanisms of bremsstrahlung, inverse Compton scattering and synchrotron radiation. A large fraction of such astronomical gamma rays are screened by Earth’s atmosphere and must be detected by spacecraft. Notable artificial sources of gamma rays include fission such as occurs in nuclear reactors, and high energy physics experiments, such as neutral pion decay and nuclear fusion.

Gamma Rays vs. X-Rays

Gamma rays have characteristics identical to X-rays of the same frequency—they differ only in source. At higher frequencies, γ rays are more penetrating and more damaging to living tissue. They have many of the same uses as X-rays, including cancer therapy. Gamma radiation from radioactive materials is used in nuclear medicine.

The distinction between X-rays and gamma rays has changed in recent decades. Originally, the electromagnetic radiation emitted by X-ray tubes almost invariably had a longer wavelength than the radiation (gamma rays) emitted by radioactive nuclei. Older literature distinguished between X- and gamma radiation on the basis of wavelength, with radiation shorter than some arbitrary wavelength, such as 10−11 m, defined as gamma rays. However, with artificial sources now able to duplicate any electromagnetic radiation that originates in the nucleus, as well as far higher energies, the wavelengths characteristic of radioactive gamma ray sources vs. other types, now completely overlap. Thus, gamma rays are now usually distinguished by their origin: X-rays are emitted by definition by electrons outside the nucleus, while gamma rays are emitted by the nucleus.

Exceptions to this convention occur in astronomy, where gamma decay is seen in the afterglow of certain supernovas, but other high energy processes known to involve other than radioactive decay are still classed as sources of gamma radiation. A notable example is extremely powerful bursts of high-energy radiation normally referred to as long duration gamma-ray bursts, which produce gamma rays by a mechanism not compatible with radioactive decay. These bursts of gamma rays, thought to be due to the collapse of stars called hypernovas, are the most powerful events so far discovered in the cosmos. Astrophysical processes are the only sources for very high energy gamma rays (~100 MeV).

Gamma Ray Sky Map: This is an image of the entire sky in 100 MeV or greater gamma rays as seen by the EGRET instrument aboard the CGRO spacecraft. Bright spots within the galactic plane are pulsars (spinning neutron stars with strong magnetic fields), while those above and below the plane are thought to be quasars (galaxies with supermassive black holes actively accreting matter).

Health Effects

All ionizing radiation causes similar damage at a cellular level, but because rays of alpha particles and beta particles are relatively non-penetrating, external exposure to them causes only localized damage (e.g., radiation burns to the skin). Gamma rays and neutrons are more penetrating, causing diffuse damage throughout the body (e.g., radiation sickness, cell’s DNA damage, cell death due to damaged DNA, increasing incidence of cancer) rather than burns. External radiation exposure should also be distinguished from internal exposure, due to ingested or inhaled radioactive substances, which, depending on the substance’s chemical nature, can produce both diffuse and localized internal damage. The most biological damaging forms of gamma radiation occur at energies between 3 and 10 MeV.

29

Reflection, Refraction, and Dispersion

The Law of Reflection and Its Consequences

The law of reflection states that the angle of reflection equals the angle of incidence.

Learning Objectives

Formulate the relationship between the angle of reflection and the angle of incidence

Key Takeaways

Key Points

- Light strikes different parts of a rough surface at different angles and is reflected, or diffused, in many different directions.

- A mirror has a smooth surface (compared with the wavelength of light) and so reflects light at specific angles.

- We see the light reflected off a mirror coming from a direction determined by the law of reflection.

Key Terms

- reflection: the property of a propagated wave being thrown back from a surface (such as a mirror)

Whenever you look into a mirror or squint at sunlight glinting off a lake, you are seeing a reflection. When you look at the text in a book, you are actually seeing the light that is reflected from it. Large telescopes use reflections to form images of stars and other astronomical objects. In fact, the only way we can see an object that does not itself emit light is if that object reflects light.

The law of reflection is illustrated in, which also shows how the angles are measured relative to the perpendicular to the surface at the point where the light ray strikes. The law of reflection is very simple: The angle of reflection equals the angle of incidence. When we see our reflection in a mirror, it appears that our image is actually behind the mirror — we see the light coming from a direction determined by the law of reflection. The angles are such that our image appears exactly the same distance behind the mirror as we stand away from the mirror.

Mirror Reflection: An image in a mirror appears as though it is behind the mirror. The two rays shown are those that strike the mirror at just the correct angles to be reflected into the eyes of the viewer. The image appears to come from the direction the rays are coming from when they enter the viewer’s eyes.

Law of Reflection: The law of reflection states that the angle of reflection equals the angle of incidence: θr = θi. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.

We expect to see reflections off a smooth surface. However, light strikes different parts of a rough surface at different angles, and it is reflected in many different directions (“diffused”). Diffused light is what allows us to see a sheet of paper from any angle. Many objects, such as people, clothing, leaves, and walls, have rough surfaces and can be seen from all sides. A mirror, on the other hand, has a smooth surface (compared with the wavelength of light) and reflects light at specific angles. When the moon reflects off the surface of a lake, a combination of these effects takes place.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=134

Reflection: A brief overview of reflection and the law of reflection.

The Law of Refraction: Snell’s Law and the Index of Refraction

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes.

Learning Objectives

Formulate the relationship between the index of refraction and the speed of light

Key Takeaways

Key Points

- The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction.

- The index of refraction is n=c/v, where v is the speed of light in the material, c is the speed of light in vacuum, and n is the index of refraction.

- Snell’s law, the law of refraction, is stated in equation form as .

Key Terms

- refraction: Changing of a light ray’s direction when it passes through variations in matter.

- index of refraction: For a material, the ratio of the speed of light in vacuum to that in the material.

It is easy to notice some odd things when looking into a fish tank. For example, you may see the same fish appearing to be in two different places. This is because light coming from the fish to us changes direction when it leaves the tank, and in this case, it can travel two different paths to get to our eyes. The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction. Refraction is responsible for a tremendous range of optical phenomena, from the action of lenses to voice transmission through optical fibers.

Law of Refraction: Looking at the fish tank as shown, we can see the same fish in two different locations, because light changes directions when it passes from water to air. In this case, the light can reach the observer by two different paths, and so the fish seems to be in two different places. This bending of light is called refraction and is responsible for many optical phenomena.

Refraction: The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction.

Speed of Light

The speed of light c not only affects refraction, it is one of the central concepts of Einstein’s theory of relativity. The speed of light varies in a precise manner with the material it traverses. It makes connections between space and time and alters our expectations that all observers measure the same time for the same event, for example. The speed of light is so important that its value in a vacuum is one of the most fundamental constants in nature as well as being one of the four fundamental SI units.

Why does light change direction when passing from one material ( medium ) to another? It is because light changes speed when going from one material to another.

Law of Refraction

A ray of light changes direction when it passes from one medium to another. As before, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In mediums that have a greater index of refraction the speed of light is less. Imagine moving your hand through the air and then moving it through a body of water. It is more difficult to move your hand through the water, and thus your hand slows down if you are applying the same amount of force. Similarly, light travels slower when moving through mediums that have higher indices of refraction.

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the law of refraction, or “Snell’s Law,” which is stated in equation form as:

n_1sinθ_1 = n_2sinθ_2

Here n_1 and n_2 are the indices of refraction for medium 1 and 2, and θ_1 and θ_2 are the angles between the rays and the perpendicular in medium 1 and 2. The incoming ray is called the incident ray and the outgoing ray the refracted ray, and the associated angles the incident angle and the refracted angle. The law of refraction is also called Snell’s law after the Dutch mathematician Willebrord Snell, who discovered it in 1621. Snell’s experiments showed that the law of refraction was obeyed and that a characteristic index of refraction n could be assigned to a given medium.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=134

Understanding Snell’s Law with the Index of Refraction: This video introduces refraction with Snell’s Law and the index of refraction.The second video discusses total internal reflection (TIR) in detail. http://www.youtube.com/watch?v=fvrvqm3Erzk

Total Internal Reflection and Fiber Optics

Total internal reflection happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle.

Learning Objectives

Formulate conditions required for the total internal reflection

Key Takeaways

Key Points

- The critical angle is the angle of incidence above which total internal reflection occurs and given as .

- The critical angle is only defined when n2/n1 is less than 1.

- If light is incident on an optical fiber with an angle of incidence greater than the critical angle then the light will remain trapped inside the glass strand. Light can travel over a very long distance without a significant loss.

Key Terms

- Snell’s law: A formula used to describe the relationship between the angles of incidence and refraction.

- cladding: One or more layers of materials of lower refractive index, in intimate contact with a core material of higher refractive index.

Total internal reflection is a phenomenon that happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflectance occurs.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=134

What is Total Internal Reflection?: Describes the concept of total internal reflection, derives the equation for the critical angle and shows one example.

Critical angle

The critical angle is the angle of incidence above which total internal reflection occurs. The angle of incidence is measured with respect to the normal at the refractive boundary (see diagram illustrating Snell’s law ). Consider a light ray passing from glass into air. The light emanating from the interface is bent towards the glass. When the incident angle is increased sufficiently, the transmitted angle (in air) reaches 90 degrees. It is at this point no light is transmitted into air. The critical angle is given by Snell’s law, . Here, n_1 and n_2 are refractive indices of the media, and and are angles of incidence and refraction, respectively. To find the critical angle, we find the value for when = 90° and thus . The resulting value of is equal to the critical angle . So the critical angle is only defined when n_2/n_1 is less than 1.

Fig 1: Refraction of light at the interface between two media, including total internal reflection.

Optical Fiber

Total internal reflection is a powerful tool since it can be used to confine light. One of the most common applications of total internal reflection is in fibre optics. An optical fibre is a thin, transparent fibre, usually made of glass or plastic, for transmitting light. The construction of a single optical fibre is shown in.

Fig 2: Fibers in bundles are clad by a material that has a lower index of refraction than the core to ensure total internal reflection, even when fibers are in contact with one another. This shows a single fiber with its cladding.

The basic functional structure of an optical fiber consists of an outer protective cladding and an inner core through which light pulses travel. The overall diameter of the fiber is about 125 μm and that of the core is just about 50 μm. The difference in refractive index of the cladding and the core allows total internal reflection in the same way as happens at an air-water surface show in. If light is incident on a cable end with an angle of incidence greater than the critical angle then the light will remain trapped inside the glass strand. In this way, light travels very quickly down the length of the cable over a very long distance (tens of kilometers). Optical fibers are commonly used in telecommunications, because information can be transported over long distances, with minimal loss of data. Another common use can be found in medicine in endoscopes. The field of applied science and engineering concerned with the design and application of optical fibers are called fiber optics.

Total Polarization

Brewster’s angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a surface.

Learning Objectives

Calculate the Brewster’s angle from the indices of refraction and discuss its physical mechanism

Key Takeaways

Key Points

- When light hits a surface at a Brewster angle, reflected beam is linearly polarized.

- The physical mechanism for the Brewster’s angle can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light.

- Brewsters’ angle is given as .

Key Terms

- dipole: A separation of positive and negative charges.

- dielectric: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).

- polarizer: An optical filter that passes light of a specific polarization and blocks waves of other polarizations.

Brewster’s angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light (whose electric field is polarized in the same plane as the incident ray and the surface normal). One can imagine that light incident on the surface is absorbed, and then re-radiated by oscillating electric dipoles at the interface between the two media. The refracted light is emitted perpendicular to the direction of the dipole moment; no energy can be radiated in the direction of the dipole moment. Thus, if the angle of reflection θ_1 (angle of reflection) is equal to the alignment of the dipoles (90 – θ_2), where θ_2 is angle of refraction, no light is reflected.

Fig 1: An illustration of the polarization of light that is incident on an interface at Brewster’s angle.

This geometric condition can be expressed as , where θ_1 is the angle of incidence and θ_2 is the angle of refraction. Using Snell’s law (n_1sinθ1 = n_2sinθ_2), one can calculate the incident angle θ_1 = B at which no light is reflected: Solving for θB gives

When light hits a surface at a Brewster angle, reflected beam is linearly polarized. shows an example, where the reflected beam was nearly perfectly polarized and hence, blocked by a polarizer on the right picture. Polarized sunglasses use the same principle to reduce glare from the sun reflecting off horizontal surfaces such as water or road.

Fig 2: Photograph taken of a window with a camera polarizer filter rotated to two different angles. In the picture at left, the polarizer is aligned with the polarization angle of the window reflection. In the picture at right, the polarizer has been rotated 90° eliminating the heavily polarized reflected sunlight.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=134

Polarization Experience: A polarizing filter allows light of a particular plane of polarization to pass, but scatters the rest of the light. When two polarizing filters are crossed, almost no light gets through. Some materials have molecules that rotate the plane of polarization of light. When one of these materials is placed between crossed polarizing filters, more light is allowed to pass through.

Dispersion: Rainbows and Prisims

Dispersion is defined as the spreading of white light into its full spectrum of wavelengths.

Learning Objectives

Describe production of rainbows by a combination of refraction and reflection processes

Key Takeaways

Key Points

- Dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion can occur for any type of wave and always involves wavelength-dependent processes.

- For a given medium, n increases as wavelength decreases and is greatest for violet light. Thus violet light is bent more than red light, as can be seen with a prism.

- In a rainbow, light enters a drop of water and is reflected from the back of the drop. The light is refracted both as it enters and as it leaves the drop.

Key Terms

- refraction: Changing of a light ray’s direction when it passes through variations in matter.

We see about six colors in a rainbow—red, orange, yellow, green, blue, and violet; sometimes indigo is listed, too. These colors are associated with different wavelengths of light. White light, in particular, is a fairly uniform mixture of all visible wavelengths. Sunlight, considered to be white, actually appears to be a bit yellow because of its mixture of wavelengths, but it does contain all visible wavelengths. The sequence of colors in rainbows is the same sequence as the colors plotted versus wavelength. What this implies is that white light is spread out according to wavelength in a rainbow. Dispersion is defined as the spreading of white light into its full spectrum of wavelengths. More technically, dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion, as a general phenomenon, can occur for any type of wave and always involves wavelength-dependent processes.

Colors of a Rainbow: Even though rainbows are associated with seven colors, the rainbow is a continuous distribution of colors according to wavelengths.

Refraction is responsible for dispersion in rainbows and many other situations. The angle of refraction depends on the index of refraction, as we saw in the Law of Refraction. We know that the index of refraction n depends on the medium. But for a given medium, n also depends on wavelength. Note that, for a given medium, n increases as wavelength decreases and is greatest for violet light. Thus violet light is bent more than red light and the light is dispersed into the same sequence of wavelengths.

Pure Light and Light Dispersion: (a) A pure wavelength of light falls onto a prism and is refracted at both surfaces. (b) White light is dispersed by the prism (shown exaggerated). Since the index of refraction varies with wavelength, the angles of refraction vary with wavelength. A sequence of red to violet is produced, because the index of refraction increases steadily with decreasing wavelength.

Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you look away from the sun. Light enters a drop of water and is reflected from the back of the drop. The light is refracted both as it enters and as it leaves the drop. Since the index of refraction of water varies with wavelength, the light is dispersed, and a rainbow is observed. (There is no dispersion caused by reflection at the back surface, since the law of reflection does not depend on wavelength. ) The actual rainbow of colors seen by an observer depends on the myriad of rays being refracted and reflected toward the observer’s eyes from numerous drops of water. The arc of a rainbow comes from the need to be looking at a specific angle relative to the direction of the sun.

Light Reflecting on Water Droplet: Part of the light falling on this water drop enters and is reflected from the back of the drop. This light is refracted and dispersed both as it enters and as it leaves the drop.

30

Lenses

Thin Lenses and Ray Tracing

Ray tracing is the technique of determining the paths light rays take; often thin lenses (the light ray bending only once) are assumed.

Learning Objectives

Describe properties of a thin lens and the purpose of ray tracing

Key Takeaways

Key Points

- When light interacts with objects several times as large as its wavelength, it travels in straight lines and acts like a ray. A ray is simply a straight line that originates at a point.

- Ray tracing is the method for determining the paths light takes through matter, such as optical systems that include lenses.

- A thin lens is defined as one with a thickness that allows rays to refract, as illustrated in, but that does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces but the lens is thin enough to assume that light rays bend only once.

- There are five basic rules for tracing rays through a lens.

Key Terms

- focal point: A focus—a point at which rays of light or other radiation converge.

- ray tracing: A technique used in optics for analysis of optical systems.

- thin lens: A thin lens is defined to be one whose thickness allows rays to refract but does not allow properties such as dispersion and aberrations.

Thin Lenses and Ray Tracing

Ray tracing is the technique of determining or following (tracing) the paths that light rays take. Experiments, as well as our own experiences, show that when light interacts with objects several times as large as its wavelength, it travels in straight lines and acts like a ray. (A ray is simply a straight line that originates at a point. ) Its wave characteristics are not pronounced in such situations. Since the wavelength of light is less than a micron (a thousandth of a millimeter), it acts like a ray in the many common situations in which it encounters objects larger than a micron, such as lenses.

For rays passing through matter, the law of refraction is used to trace the paths. Here we use ray tracing to help us understand the action of lenses in situations ranging from forming images on film to magnifying small print to correcting nearsightedness. While ray tracing for complicated lenses, such as those found in sophisticated cameras, may require computer techniques, there is a set of simple rules for tracing rays through thin lenses. A thin lens is defined to be one whose thickness allows rays to refract, as illustrated in, but does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces but the lens is thin enough toassume that light rays bend only once. Another way of saying this is that the lens thickness is much much smaller than the focal length of the lens. A thin symmetrical lens has two focal points, one on either side and both at the same distance from the lens. (See. ) Another important characteristic of a thin lens is that light rays through its center are deflected by a negligible amount, as seen in the center rays in the first two figures. The treatment of a lens as a thin lens is known as the “thin lens approximation. ”

Convex Lens: Rays of light entering a converging lens parallel to its axis converge at its focal point F. (Ray 2 lies on the axis of the lens. ) The distance from the center of the lens to the focal point is the lens’s focal length f. An expanded view of the path taken by ray 1 shows the perpendiculars and the angles of incidence and refraction at both surfaces.

Thin Lens: Thin lenses have the same focal length on either side. (a) Parallel light rays entering a converging lens from the right cross at its focal point on the left. (b) Parallel light rays entering a diverging lens from the right seem to come from the focal point on the right.

Rules for Ray Tracing

Using paper, pencil, and a straight edge, ray tracing can accurately describe the operation of a lens. The rules for ray tracing for thin lenses are based on the illustrations included in this section:

1. A ray entering a converging lens parallel to its axis passes through the focal point F of the lens on the other side. ( See rays 1 and 3 in. )

2. A ray entering a diverging lens parallel to its axis seems to come from the focal point F. (See rays 1 and 3 in. )

3. A ray passing through the center of either a converging or a diverging lens does not change direction. (See ray 2 in and. )

4. A ray entering a converging lens through its focal point exits parallel to its axis. (The reverse of rays 1 and 3 in. )

5. A ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis. (The reverse of rays 1 and 3 in ).

Diverging Lens: Rays of light entering a diverging lens parallel to its axis are diverged, and all appear to originate at its focal point F. The dashed lines are not rays—they indicate the directions from which the rays appear to come. The focal length f of a diverging lens is negative. An expanded view of the path taken by ray 1 shows the perpendiculars and the angles of incidence and refraction at both surfaces.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=136

Ray Diagrams, Concave Lens and Convex Mirror: Shows how to draw the ray diagrams for locating the image produced by a concave lens and a convex mirror.

The Thin Lens Equation and Magnification

The thin lens equation relates the object distance do, image distance di, and focal length f.

Learning Objectives

Formulate five basic rules of ray tracing

Key Takeaways

Key Points

- Ray tracing can be used to construct an image from the light rays originating from an object that pass through a lens. The image is located at the point where the rays cross. By choosing several points from an object the entire image can be constructed.

- We define do to be the object distance, the distance of an object from the center of a lens. Image distance di is defined to be the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols ho and hi, respectively.

- The thin lens equation quickly provides the relation between di, do, and the focal length f. It can be derived from a geometric analysis of ray tracing for thin lenses and is given by .

- The magnification m of an image is the ratio between the image and object height (hi/ho). The magnification is related to do, di, ho, and hi by the following relation: .

Key Terms

- thin lens equation: Relates object distance do, image distance di, and focal length f:

- image distance: The distance of the image from the center of the lens.

- magnification: The apparent enlargement of an object in an image.

The Thin Lens Equation and Magnification

Image Formation by Thin Lenses

How does a lens form an image of an object? We can use the technique of ray tracing to illustrate how lenses form images. We can also develop equations to describe the images quantitatively. Recall the five basic rules of ray tracing:

1. A ray entering a converging lens parallel to its axis passes through the focal point F of the lens on the other side.

2. A ray entering a diverging lens parallel to its axis seems to come from the focal point F.

3. A ray passing through the center of either a converging or a diverging lens does not change direction.

4. A ray entering a converging lens through its focal point exits parallel to its axis.

5. A ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis.

Consider an object some distance away from a converging lens, as shown in. To find the location and size of the image formed, we trace the paths of selected light rays originating from one point on the object (in this case the top of the person’s head). The figure shows three rays from the top of the object that can be traced using the five ray tracing rules. Rays leave this point going in many directions, but we concentrate on only a few with paths that are easy to trace. The first ray is one that enters the lens parallel to its axis and passes through the focal point on the other side (rule 1). The second ray passes through the center of the lens without changing direction (rule 3). The third ray passes through the nearer focal point on its way into the lens and leaves the lens parallel to its axis (rule 4). The three rays cross at the same point on the other side of the lens. The image of the top of the person’s head is located at this point. All rays that come from the same point on the top of the person’s head are refracted in such a way as to cross at the point shown. Rays from another point on the object, such as her belt buckle, will also cross at another common point, forming a complete image, as shown. Although three rays are traced in, only two are necessary to locate the image. It is best to trace rays for which there are simple ray tracing rules. Before applying ray tracing to other situations, let us consider the example shown in in more detail.

Image Formation with a Thin Lens: Ray tracing is used to locate the image formed by a lens. Rays originating from the same point on the object are traced—the three chosen rays each follow one of the rules for ray tracing, so that their paths are easy to determine. The image is located at the point where the rays cross. In this case, a real image—one that can be projected on a screen—is formed.

Several important distances appear in. We define do as the object distance—the distance of an object from the center of a lens. Image distance diis defined as the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols ho and hi, respectively. Images that appear upright relative to the object have heights that are positive and those that are inverted have negative heights. Using the rules of ray tracing and making a scale drawing with paper and pencil, like that in, we can accurately describe the location and size of an image. But the real benefit of ray tracing is in visualizing how images are formed in a variety of situations. To obtain numerical information, we use a pair of equations that can be derived from a geometric analysis of ray tracing for thin lenses. The thin lens equation is:

We define the ratio of image height to object height (hi/ho) as the magnification m. The magnification is related to do, di, ho, and hi by the following relation:

In many cases both of these equations are referred to together as the thin lens equations. The thin lens equations are broadly applicable to all situations involving thin lenses (and “thin” mirrors).

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=136

Thin Lens Equations for a Convex Lens: Shows how to use the thin lens equation to calculate the image distance, image height and image orientation for convex lenses when the object distance is greater the the focal length (f).

Combinations of Lenses

A compound lens is an array of simple lenses with a common axis.

Learning Objectives

Calculate focal length for a compound lens from the focal lengths of simple lenses

Key Takeaways

Key Points

- The use of multiple elements allows for the correction of more optical aberrations, such as the chromatic abberation caused by the wavelength -dependent index of refraction in glass, than is possible using a single lens.

- If the lenses of focal lengths f_1 and f_2are “thin”, the combined focal length f of the lenses is given by while if the lenses are separated by some distance d then the combined focal length is given by .

- If the separation distance is equal to the sum of the focal lengths (d = f_1+f_2), the combined focal length is infinite. This corresponds to a pair of lenses that transform a collimated beam into another collimated beam. This type of system is called an afocal system (a simple optical telescope).

- An achromatic doublet is a kind of compound lens designed to bring two wavelengths (typically red and blue/violet) into focus in the same plane. This (partially) corrects for the chromatic aberration found in a single simple lens. See.

Key Terms

- aberration: The convergence to different foci, by a lens or mirror, of rays of light emanating from one and the same point, or the deviation of such rays from a single focus; a defect in a focusing mechanism that prevents the intended focal point.

- afocal system: An optical system that produces no net convergence or divergence of the beam, i.e. has an infinite effective focal length. This type of system can be created with a pair of optical elements where the distance between the elements is equal to the sum of each element’s focal length (d = f1+f2).

- achromatic doublet: A type of lens made up of two simple lenses paired together designed so that the chromatic aberration of each lens partially offsets the other; in this way light in a range of wavelengths may be brought to the same focus.

COMPOUND LENSES

In contrast to a simple lens, which consists of only one optical element, a compound lens is an array of simple lenses (elements) with a common axis. The use of multiple elements allows for the correction of more optical aberrations, such as the chromatic aberration caused by the wavelength-dependent index of refraction in glass, than is possible using a single lens. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations.

The simplest case is where lenses are placed in contact: if the lenses of focal lengths f_1 and f_2 are “thin”, the combined focal length f of the lenses is given by

.

Since 1/f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.

If two thin lenses are separated in air by some distance d (where d is smaller than the focal length of the first lens), the focal length for the combined system is given by

.

BACK FOCAL LENGTH

The distance from the second lens to the focal point of the combined lenses is called the back focal length (BFL).

.

As d tends to zero, the value of the BFL tends to the value of f given for thin lenses in contact.

If the separation distance is equal to the sum of the focal lengths (d = f_1+f_2), the combined focal length and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam (see ). This type of system is called an afocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by

Keplerian Telescope: All refracting telescopes use the same principles. The combination of an objective lens 1 and some type of eyepiece 2 is used to gather more light than the human eye could collect on its own, focus it 5, and present the viewer with a brighter, clearer, and magnified virtual image 6. The magnification can be found by dividing the focal length of the objective lens by the focal length of the eyepiece.

which is the ratio of the input beam width to the output beam width. Note the sign convention: a telescope with two convex lenses (f_1 > 0, f_2 > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (f_1 > 0 >f_2) produces a positive magnification and the image is upright.

ACHROMATS

An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue/violet) into focus in the same plane.

The most common type of achromat is the achromatic doublet, which is composed of two individual lenses made from glasses with different amounts of dispersion Typically, one element is a negative (concave) element made out of flint, which has relatively high dispersion, and the other is a positive (convex) element made of crown glass, which has lower dispersion. The lens elements are mounted next to each other, often cemented together, and shaped so that the chromatic aberration of one is counterbalanced by that of the other.

In the most common type (shown in ), the positive power of the crown lens element is not quite equaled by the negative power of the flint lens element. Together they form a weak positive lens that will bring two different wavelengths of light to a common focus. Negative doublets, in which the negative-power element predominates, are also made.

Achromatic Doublet: (a) Chromatic aberration is caused by the dependence of a lens’s index of refraction on color (wavelength). The lens is more powerful for violet (V) than for red (R), producing images with different locations and magnifications. (b) Multiple-lens systems, such as this achromatic doublet, can partially correct chromatic aberrations, but they may require lenses of different materials and add to the expense of optical systems such as cameras.

The Lensmaker’s Equation

The lensmaker’s formula is used to relate the radii of curvature, the thickness, the refractive index, and the focal length of a thick lens.

Learning Objectives

Compare idealized thin lenses with real lenses

Key Takeaways

Key Points

- If a lens is biconvex, a beam of light travelling parallel to the lens axis and passing through the lens will be focused to a spot on the axis, at a certain distance behind the lens (i.e. the focal length ). In this case, the lens is called a positive or converging lens. See.

- If a lens is biconcave, a beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. See.

- The focal length of a thick lens in air can be calculated from the lensmaker’s equation: .

- The signs of the lens’ radii of curvature indicate whether the corresponding surfaces are convex or concave. The signs are reversed for the back surface of the lens: if R_2 is positive the surface is concave, and if R_2 is negative the surface is convex.

- The lensmaker’s equation can be greatly simplified if the lens thickness d is very small compared to R_1 and R_2. In this case, the thin lens approximation can then be made and the lensmaker’s equation can be approximated as .

Key Terms

- thick lens: Lenses whose thicknesses are not negligible (i.e., one cannot make the simple assumption that a light ray is refracted only once in the lens).

- surface vertices: The points where each surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the other cardinal points must be known with respect to the vertices to describe the physical system.

The Lensmaker’s Equation

Thick Lenses

Unlike idealized thin lenses, real lenses have a finite thickness between their two surfaces of curvature. An ideal thin lens with two surfaces of equal curvature would have zero optical power, meaning that it would neither converge nor diverge light. A lens whose thickness is not negligible is called a thick lens. In this case, we can not simply assume that a light ray is only refracted once while traveling through the lens. Instead the extent of the refraction must be dependent on the thickness of the lens.

Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or just convex) if both surfaces are convex. If the lens is biconvex, a beam of light travelling parallel to the lens axis and passing through the lens will be converged (or focused) to a spot on the axis, at a certain distance behind the lens (i.e. the focal length). In this case, the lens is called a positive or converging lens. See for a diagram of a positive (converging) lens.

Thick Converging Lens: Diagram of a positive (converging) lens. The lensmaker’s formula relates the radii of curvature, the index of refraction of the lens, the thickness of the lens, and the focal length.

If the lens is biconcave, a beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam after passing through the lens appears to be emanating from a particular point on the axis in front of the lens; the distance from this point to the lens is also known as the focal length, although it is negative with respect to the focal length of a converging lens. See for a diagram of a negative (diverging) lens.

Negative Diverging Lens: Diagram of a negative (diverging) lens. The lensmaker’s formula relates the radii of curvature, the index of refraction of the lens, the thickness of the lens, and the focal length.

The focal length of a thick lens in air can be calculated from the lensmaker’s equation:

where

- P is the power of the lens,

- f is the focal length of the lens,

- n is the refractive index of the lens material,

- R_1is the radius of curvature of the lens surface closest to the light source,

- R_2 is the radius of curvature of the lens surface farthest from the light source, d and is the thickness of the lens (the distance along the lens axis between the two surface vertices).

Sign convention of Radii R_1 and R_2

The signs of the lens’ radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies, but for our treatment if R_1 is positive the first surface is convex, and if R_1 is negative the surface is concave. The signs are reversed for the back surface of the lens: if R_2 is positive the surface is concave, and if R_2 is negative the surface is convex. If either radius is infinite, the corresponding surface is flat. With this convention the signs are determined by the shapes of the lens surfaces, and are independent of the direction in which light travels through the lens.

Thin Lens Approximation

The above equation can be greatly simplified if the lens thickness d is very small compared to R_1 and R_2. In this case, the thin lens approximation can then be made and the lensmaker’s equation can be approximated as

The focal length f is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, 1/f, is the optical of the lens. If the focal length is in meters, this gives the optical power in diopters (inverse meters).

Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back, although other properties of the lens, such as the aberrations are not necessarily the same in both directions.

Refraction Through Lenses

Because the index of refraction of a lens is greater than air, a ray moves towards the perpendicular as it enters and away as it leaves.

Learning Objectives

Compare the effect of a convex lens and a concave lens on the light rays

Key Takeaways

Key Points

- Recall that the a ray will bend as it enters a medium with a different refractive index. Since the refractive index of a lens is greater than air, a light ray will move towards the perpendicular as it enters and away as it leaves.

- A convex lens has been shaped so that all light rays that enter it parallel to its axis cross one another at a single point on the opposite side of the lens (the focal point ). Such a lens is called a converging (or convex) lens for the converging effect it has on light rays. See.

- A concave lens is a diverging lens, because it causes the light rays to bend away (diverge) from its axis. shows the effect it has on rays of light that enter it parallel to its axis (the path taken by ray 2 in the figure is the axis of the lens).

- The greater effect a lens has on light rays, the more powerful it is said to be. A powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length than a weak lens. The power of a lens is given by the equation .

Key Terms

- convex lens: A lens having at least one convex surface, such that light passing through it, may be brought to a focus.

- concave lens: A lens having at least one concave surface, such that light rays passing through it bend away from its optical axis.

- focal point: A focus—a point at which rays of light or other radiation converge.

Refraction Through Lenses

Lenses are found in a huge array of optical instruments, ranging from the simple magnifying glass to a camera lens to the lens of the human eye. The word lens derives from the Latin word for lentil bean—the shape of which is similar to that of the convex lens (as shown in ). The convex lens is shaped so that all light rays that enter it parallel to its axis cross one another at a single point on the opposite side of the lens. The axis is defined as a line normal to the lens at its center (as shown in ). Such a lens is called a converging (or convex) lens for the corresponding effect it has on light rays. The expanded view of the path of one ray through the lens illustrates how the ray changes direction both as it enters and as it leaves the lens.

Since the index of refraction of the lens is greater than that of air, the ray moves towards the perpendicular as it enters, and away from the perpendicular as it leaves (this is in accordance with the law of refraction). Due to the lens’s shape, light is thus bent toward the axis at both surfaces. The point at which the rays cross is defined as the focal point F of the lens. The distance from the center of the lens to its focal point is defined as the focal length f of the lens. shows how a converging lens, such as that in a magnifying glass, can concentrate (converge) the nearly parallel light rays from the sun towards a small spot.

Magnifying Glass: Sunlight focused by a converging magnifying glass can burn paper. Light rays from the sun are nearly parallel and cross at the focal point of the lens. The more powerful the lens, the closer to the lens the rays will cross.

The greater effect a lens has on light rays, the more powerful it is said to be. For example, a powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length than a weak lens. The light will also focus into a smaller, more intense spot for a more powerful lens. The power P of a lens is defined as the inverse of its focal length. In equation form:

shows the effect of a concave lens on rays of light entering it parallel to its axis (the path taken by ray 2 in the figure is the axis of the lens). The concave lens is a diverging lens, because it causes the light rays to bend away (diverge) from its axis. In this case, the lens is shaped so that all light rays entering it parallel to its axis appear to originate from the same point F, defined as the focal point of a diverging lens. The distance from the center of the lens to the focal point is again called the focal length f of the lens. Note that the focal length and power of a diverging lens are defined as negative. For example, if the distance to F in is 5.00 cm, then the focal length is f=–5.00 cm and the power of the lens is P=–20 D. The expanded view of the path of one ray through the lens illustrates how the shape of the lens (given the law of refraction) causes the ray to follow its particular path and be diverged.

In subsequent sections we will examine the technique of ray tracing to describe the formation of images by lenses. Additionally, we will explore how image locations and characteristics can be quantified with the help of a set of geometric optics equations.

31

Mirrors

Image Reflection by a Plane Mirror

A mirror is a reflective surface that bounces off light, thus producing a real or virtual image.

Learning Objectives

Describe interaction of the light with a mirror surface

Key Takeaways

Key Points

- Reflected images can be either real or virtual. In a plane mirror, the images are virtual.

- The virtual images in a plane mirror have a left-right inversion.

- Drawing a ray diagram is a way to predict what a reflected image will look like.

Key Terms

- virtual image: A virtual image occurs when light rays do not actually meet at the image

Plane Mirrors and Reflection

A mirror is a reflective surface that does not allow the passage of light and instead bounces it off, thus producing an image. The most common mirrors are flat and called plane mirrors. These mirrors are made by putting a thin layer of silver nitrate or aluminium behind a flat piece of glass.

When you place an object in front of a mirror, you see an image of the same object in the mirror. The object is the source of the incident rays, and the image is formed by the reflected rays. An image formed by reflection may be real or virtual. A “real” image occurs when light rays actually intersect at the image, and become inverted, or turned upside down. A “virtual” image occurs when light rays do not actually meet at the image. Instead, you “see” the image because your eye projects light rays backward. You are fooled into seeing an image! A virtual image is right side up (upright).

In flat, or plane mirrors, the image is a virtual image, and is the same distance behind the mirror as the object is in front of the mirror. The image is also the same size as the object. These images are also parity inverted, which means they have a left-right inversion.

Ray Diagrams

The way that we can predict how a reflection will look is by drawing a ray diagram. These diagrams can be used to find the position and size of the image and whether that image is real or virtual. These are the steps you follow to draw a ray diagram:

1. Draw the plane mirror as a straight line on a principal axis. The principal axis is an imaginary line that is drawn perpendicular to the mirror.

2. Draw the object as an arrow in front of the mirror.

3. Draw the image of the object, by using the principle that the image is placed at the same distance behind the mirror that the object is in front of the mirror. The image size is also the same as the object size. shows these first three steps.

4. Place a dot at the point the eye is located.

5. Pick one point on the image and draw the reflected ray that travels to the eye as it sees this point. Remember to add an arrowhead.

6. Draw the incident ray for light traveling from the corresponding point on the object to the mirror, such that the law of reflection is obeyed.

7. Continue for other extreme points on the object (i.e. the tip and base of the arrow). A completed ray diagram is shown in

The angle in which a light ray hits the mirror is the same angle in which it will be reflected back. If, for example, a light ray leaves the top of an object travelling parallel to the principal axis, it will hit the mirror at a 0 degree angle, and be reflected back at 0 degrees. When this happens, we say the ray hit the mirror normally. If the light ray hit the object at a 30 degree angle, it will be reflected back at a 30 degree angle.

Image Formation by Spherical Mirrors: Reflection and Sign Conventions

A mirror is a reflective surface that light does not pass through, made by a layer of silver nitrate or aluminium behind piece of glass.

Learning Objectives

Distinguish properties of the concave and the convex mirrors

Key Takeaways

Key Points

- Images in mirrors can be either real or virtual.

- A summary of the properties of the concave mirrors are shown below: converging real image inverted image in front of mirror.

- A summary of the properties of the convex mirrors are shown below: diverging virtual image upright image behind mirror.

Key Terms

- concave: curved like the inner surface of a sphere or bowl

- convex: curved or bowed outward like the outside of a bowl or sphere or circle

Overview

A mirror is a reflective surface that light does not pass through, but bounces off of and this produces an image. Mirrors are made by putting a thin layer of silver nitrate or aluminium behind a flat piece of glass.

When you place an object in front of a mirror, you see the same object in the mirror. This image that appears to be behind the mirror is called the image. The object is the source of the incident rays, and the image is formed by the reflected rays. An image formed by reflection may be real or virtual. A real image occurs when light rays actually intersect at the image, and is inverted, or upside down. A virtual image occurs when light rays do not actually meet at the image. Instead, you “see” the image because your eye projects light rays backward. A virtual image is right side up (upright).

This section will cover spherical mirrors. Spherical mirrors can be either concave or convex. The center of curvature is the point at the center of the sphere and describes how big the sphere is. These concepts are shown in.

Spherical Mirrors: This figure shows the difference between a concave and convex mirror.

Concave Mirrors

In a concave mirror, the principal axis is a line that is perpendicular to the center of the mirror. The easiest way to visualize what a image will look like in this type of mirror is a ray diagram. Before that can be done, the focal point must first be defined. This point is half way between the mirror and the center of curvature on the principal axis. The distance to the focal point from the mirror is called the focal length. We can see from the figure that this focal length is also equal to half of the radius of the curvature. shows the ray diagram of a concave mirror.

Concave Ray Diagram: This is a ray diagram of a concave mirror. The steps taken to draw are the same as those in a plane mirror.

A summary of the properties of concave mirrors is shown below:

- converging

- real image

- inverted

- image in front of mirror

Convex Mirrors

In convex mirrors, the principal axis is the same as in a plane or concave mirror, perpendicular to the center of the mirror. In this case, the focal point is behind the mirror. A convex mirror has a negative focal length because of this. The focal point is the same distance from the mirror as in a concave mirror. This is shown in.

Convex Mirror Ray Diagram: A convex mirror with three rays drawn to locate the image. Each incident ray is reflected according to the Law of Reflection. The reflected rays diverge. If the reflected rays are extended behind the mirror, then their intersection gives the location of the image behind the mirror. For a convex mirror, the image is virtual and upright.

A summary of the properties of convex mirrors is shown below:

- diverging

- virtual image

- upright

- image behind mirror

VII

Heat and Heat Transfer

32

Introduction

Overview of Heat

Heat is a measurable form of energy that can be transferred from one body to another; it is not a substance.

Learning Objectives

Distinguish three modes of heat transfer

Key Takeaways

Key Points

- Heat is a crucial concept that touches every aspect of our lives. James Clerk Maxwell set down important principles that couple into the definition of heat.

- The quantity of heat transfer can be directly measured or estimated indirectly through the science of calorimetry.

- There are three modes of heat transfer: conduction, convection, and radiation.

Key Terms

- heat transfer: The transmission of thermal energy via conduction, convection, or radiation.

- calorimetry: The science of measuring the heat absorbed or evolved during the course of a chemical reaction or change of state.

Introduction to Heat and Heat Transfer

Energy can exist in many forms and heat is one of the most intriguing. Heat is often hidden, as it only exists when in transit, and is transferred by a number of distinctly different methods. Heat transfer touches every aspect of our lives and helps us understand how the universe functions. It explains the chill we feel on a clear breezy night, or why Earth’s core has yet to cool. This module defines and explores heat transfer, its effects, and the methods by which heat is transferred. These topics are fundamental, as well as practical, and will often be referred to in the modules ahead.

Examples of Heat Transfer: (a) The chilling effect of a clear breezy night is produced by the wind and by radiative heat transfer to cold outer space. (b) There was once great controversy about the Earth’s age, but it is now generally accepted to be about 4.5 billion years old. Much of the debate is centered on the Earth’s molten interior. According to our understanding of heat transfer, if the Earth is really that old, its center should have cooled off long ago. The discovery of radioactivity in rocks revealed the source of energy that keeps the Earth’s interior molten, despite heat transfer to the surface, and from there to cold outer space.

Definitions

Scottish physicist James Clerk Maxwell, in his 1871 classic Theoryof Heat, was one of many who began to build on the already established idea that heat was something to do with matter in motion. This was the same idea put forwards by Sir Benjamin Thompson in 1798, who said he was only following on from the work of many others. One of Maxwell’s recommended books was Heat as a Mode of Motion by John Tyndall. Maxwell outlined four stipulations for the definition of heat:

- It is something which may be transferred from one body to another.

- It is a measurable quantity, and thus treated mathematically.

- It cannot be treated as a substance, because it may be transformed into something that is not a substance, such as mechanical work.

- Heat is one of the forms of energy.

In the following sections, we will define heat more rigorously, paying particular attention to how it can be measured and quantified.

Estimation of Quantity of Heat

The quantity of heat transferred by some process can either be directly measured, or determined indirectly through calculations based on other quantities. Direct measurement is by calorimetry and is the primary empirical basis of the idea of quantity of heat transferred in a process. The transferred heat is measured by changes in a body of known properties, for example, temperature rise, change in volume or length, or phase change, such as melting of ice. Indirect estimations of quantity of heat transferred rely on the law of conservation of energy, and, in particular cases, on the first law of thermodynamics (explored in the following sections). Indirect estimation is the primary approach of many theoretical studies of quantity of heat transferred.

Heat Transfer Methods

After defining and quantifying heat transfer and its effects on physical systems, we will discuss the methods by which heat is transferred. So many processes involve heat transfer, so that it is hard to imagine a situation where no heat transfer occurs. Yet every process involving heat transfer takes place by only three methods:

1. Conduction is heat transfer through stationary matter by physical contact. ( The matter is stationary on a macroscopic scale—we know there is thermal motion of the atoms and molecules at any temperature above absolute zero. ) Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction.

2. Convection is the heat transfer by the macroscopic movement of a fluid. This type of transfer takes place in a forced-air furnace and in weather systems, for example.

3. Heat transfer by radiation occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed. An obvious example is the warming of the Earth by the Sun. A less obvious example is thermal radiation from the human body.

Heat as Energy Transfer

Heat is the spontaneous transfer of energy due to a temperature difference.

Learning Objectives

Identify SI and common units of heat

Key Takeaways

Key Points

- If two objects at different temperature are brought together, energy will transfer from the hotter object to the cooler one until both are at the same temperature. This transfer of energy is known as heat.

- Heat should not be confused with temperature. Temperature describes the internal state of an object, while heat refers to the energy transferred to or from the object.

- Since heat is a form of energy, its SI unit is the joule. Other common units of heat energy include the calorie and kilocalorie, equal to 4.186 and 4,186 joules, respectively.

- Because heat and work both involve the transfer of energy, they can each produce the same effects. The concept of the mechanical equivalent of heat was instrumental in establishing the principle of conservation of energy.

Key Terms

- kilocalorie: A non-SI unit of energy equal to 1,000 calories or 4,186 joules; equal to the “calorie” or “Calorie” used in nutritional labeling. Symbol: kcal.

- thermal equilibrium: Two systems are in thermal equilibrium if they could transfer heat between each other, but don’t.

- mechanical equivalent of heat: The work needed to produce the same effects as heat transfer.

Heat as Energy Transfer

Consider two objects at different temperatures that are brought together. Energy is transferred from the hotter object to the cooler one, until both objects reach thermal equilibrium (i.e., both become the same temperature). How is this energy transferred? No work is done by either object, because no force acts through a distance. The transfer of energy is caused by the temperature difference, and ceases once the temperatures are equal. This observation leads to the following definition of heat: Heat is the spontaneous transfer of energy due to a temperature difference.

Heat Transfer and Equilibrium: (a) The soft drink and the ice have different temperatures, T1 and T2, and are not in thermal equilibrium. (b) When the soft drink and ice are allowed to interact, energy is transferred until they reach the same temperature T, achieving equilibrium. Heat transfer occurs due to the difference in temperatures. In fact, since the soft drink and ice are both in contact with the surrounding air and bench, the equilibrium temperature will be the same for both.

Where Is the Most Heat Lost?: Use movable thermometers to discover where a house has poor insulation.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=108

Heat Transfer: A brief introduction to heat transfer for students.

Heat is often confused with temperature. For example, we may say the heat was unbearable, when we actually mean that the temperature was high. Heat is a form of energy, whereas temperature is not. The misconception arises because we are sensitive to the flow of heat, rather than the temperature.

Units

Owing to the fact that heat is a form of energy, it has the SI unit of joule (J). The calorie (cal) is a common unit of energy, defined as the energy needed to change the temperature of 1.00 g of water by 1.00ºC —specifically, between 14.5ºC and 15.5ºC, since there is a slight temperature dependence. Another common unit of heat is the kilocalorie (kcal), which is the energy needed to change the temperature of 1.00 kg of water by 1.00ºC. Since mass is often specified in kilograms, kilocalorie is commonly used. Food calories (given the notation Cal, and sometimes called “big calorie”) are actually kilocalories (1kilocalorie=1000 calories), a fact not easily determined from package labeling in the United States, but more common in Europe and elsewhere. In some engineering fields, the British Thermal Unit (BTU), equal to about 1.055 kilo-joules, is widely used.

Figure 1 Equivalence of Heat and Work: Schematic depiction of Joule’s experiment that established the equivalence of heat and work

The total amount of energy transferred as heat is conventionally written as Q for algebraic purposes. Heat released by a system into its surroundings is by convention a negative quantity (Q < 0); when a system absorbs heat from its surroundings, it is positive (Q > 0).

Mechanical Equivalent of Heat

It is also possible to change the temperature of a substance by doing work. Work can transfer energy into or out of a system. This realization helped establish the fact that heat is a form of energy. James Prescott Joule (1818–1889) performed many experiments to establish the mechanical equivalent of heat—the work needed to produce the same effects as heat transfer. In terms of the units used for these two terms, the best modern value for this equivalence is 1.000 kcal = 4186 J. We consider this equation as the conversion between two different units of energy.

Figure 1 shows one of Joule’s most famous experimental setups for demonstrating the mechanical equivalent of heat. It demonstrated that work and heat can produce the same effects, and helped establish the principle of conservation of energy. Gravitational potential energy (PE) (work done by the gravitational force) is converted into kinetic energy (KE), and then randomized by viscosity and turbulence into increased average kinetic energy of atoms and molecules in the system, producing a temperature increase. His contributions to the field of thermodynamics were so significant that the SI unit of energy was named after him.

Heat added or removed from a system changes its internal energy (a concept we will discuss in the following section) and thus its temperature. Such a temperature increase is observed while cooking. However, adding heat does not necessarily increase the temperature. An example is melting of ice; that is, when a substance changes from one phase to another. Work done on the system or by the system can also change the internal energy of the system. Joule demonstrated that the temperature of a system can be increased by stirring. If an ice cube is rubbed against a rough surface, work is done by the frictional force. A system has a well-defined internal energy, but we cannot say that it has a certain “heat content” or “work content.” We use the phrase “heat transfer” to emphasize its nature.

Internal Energy

The internal energy of a system is the sum of all kinetic and potential energy in a system.

Learning Objectives

Express the internal energy in terms of kinetic and potential energy

Key Takeaways

Key Points

- While a system does not contain ‘ heat,’ it does contain a total amount of energy called internal energy.

- The internal energy is the energy necessary to create a system, minus the energy necessary to displace its surroundings.

- Most of the time, we are interested in the change in internal energy rather than the total internal energy.

- The first law of thermodynamics, , describes small changes in internal energy.

Key Terms

- internal energy: The sum of all energy present in the system, including kinetic and potential energy; equivalently, the energy needed to create a system, excluding the energy necessary to displace its surroundings.

- isolated system: A system that does not interact with its surroundings, that is, its total energy and mass stay constant.

Internal Energy

James Joule showed that both heat and work can produce the same change in the internal energy of a substance, establishing the principle of the mechanical equivalence of heat. Heat is emphatically a quantity that solely describes energy being transferred. It makes no sense to speak of the total ‘heat’ an object or system contains. However, a system does contain a quantifiable amount of energy called the internal energy of a system. The internal energy of a system is the quantity that changes with the addition or subtraction of work or heat. It is closely related to temperature.

Definition

The internal energy is the energy required to create a system, excluding the energy necessary to displace its surroundings. Internal energy has two components: kinetic energy and potential energy. The kinetic energy consists of all the energy involving the motions of the particles constituting the system, including translation, vibration, and rotation. The potential energy is associated with the static constituents of matter, static electric energy of atoms within molecules or crystals, and the energy from chemical bonds. The equation describing the total internal energy of a system is then:

.

We can also think of the internal energy as the sum of all the energy states of each component in the system:

.

At any finite temperature, kinetic and potential energies are constantly converted into each other, but the total energy remains constant in an isolated system. The kinetic energy portion of internal energy gives rise to the temperature of the system. We can use statistical mechanics to relate the (somewhat) random motions of particles in a system to the mean kinetic energy of the ensemble of particles, and thus the empirically measurable quantity expressed as temperature.

We can see that internal energy is an extensive property: it depends on the size of the system or on the amount of substance it contains.

In most cases, we are not concerned with the total amount of internal energy in the system, as it is rarely convenient or necessary to consider all energies belonging to the system. Rather, we are far more interested in the change in internal energy, given some transfer of work or heat. This can be expressed as:

.

Q is heat added to a system and Wmech is the mechanical work performed by the surroundings due to pressure or volume changes in the system. All other perturbations and energies added by other processes, such as an electric current introduced into an electronic circuit, is summarized as the term Wextra.

We can calculate a small change in internal energy of the system by considering the infinitesimal amount of heat δQ added to the system minus the infinitesimal amount of work δW done by the system:

.

This expression is the first law of thermodynamics.

The Sun and Internal Energy: Nuclear fusion in the sun converts nuclear potential energy into available internal energy and keeps the temperature of the Sun very high.

33

Specific Heat

Heat Capacity

The heat capacity measures the amount of heat necessary to raise the temperature of an object or system by one degree Celsius.

Learning Objectives

Explain the enthalpy in a system with constant volume and pressure

Key Takeaways

Key Points

- Heat capacity is the measurable physical quantity that characterizes the amount of heat required to change a substance’s temperature by a given amount. It is measured in joules per Kelvin and given by.

- The heat capacity is an extensive property, scaling with the size of the system.

- The heat capacity of most systems is not constant (though it can often be treated as such). It depends on the temperature, pressure, and volume of the system under consideration.

Key Terms

- heat capacity: The amount of heat energy needed to raise the temperature of an object or unit of matter by one degree Celsius; in units of joules per kelvin (J/K).

- enthalpy: the total amount of energy in a system, including both the internal energy and the energy needed to displace its environment

Heat Capacity

Heat capacity (usually denoted by a capital C, often with subscripts), or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance’s temperature by a given amount. In SI units, heat capacity is expressed in units of joules per kelvin (J/K).

An object’s heat capacity (symbol C) is defined as the ratio of the amount of heat energy transferred to an object to the resulting increase in temperature of the object.

Heat capacity is an extensive property, so it scales with the size of the system. A sample containing twice the amount of substance as another sample requires the transfer of twice as much heat (Q) to achieve the same change in temperature (ΔT). For example, if it takes 1,000 J to heat a block of iron, it would take 2,000 J to heat a second block of iron with twice the mass as the first.

The Measurement of Heat Capacity

The heat capacity of most systems is not a constant. Rather, it depends on the state variables of the thermodynamic system under study. In particular, it is dependent on temperature itself, as well as on the pressure and the volume of the system, and the ways in which pressures and volumes have been allowed to change while the system has passed from one temperature to another. The reason for this is that pressure-volume work done to the system raises its temperature by a mechanism other than heating, while pressure-volume work done by the system absorbs heat without raising the system’s temperature. (The temperature dependence is why the definition a calorie is formally the energy needed to heat 1 g of water from 14.5 to 15.5 °C instead of generally by 1 °C. )

Different measurements of heat capacity can therefore be performed, most commonly at constant pressure and constant volume. The values thus measured are usually subscripted (by p and V, respectively) to indicate the definition. Gases and liquids are typically also measured at constant volume. Measurements under constant pressure produce larger values than those at constant volume because the constant pressure values also include heat energy that is used to do work to expand the substance against the constant pressure as its temperature increases. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.

Thermodynamic Relations and Definition of Heat Capacity

The internal energy of a closed system changes either by adding heat to the system or by the system performing work. Recalling the first law of thermodynamics,

.

For work as a result of an increase of the system volume we may write,

.

If the heat is added at constant volume, then the second term of this relation vanishes and one readily obtains

.

This defines the heat capacity at constant volume, CV. Another useful quantity is the heat capacity at constant pressure, CP. With the enthalpy of the system given by

,

our equation for dU changes to

,

and therefore, at constant pressure, we have

.

Specific Heat

The specific heat is an intensive property that describes how much heat must be added to a particular substance to raise its temperature.

Learning Objectives

Summarize the quantitative relationship between heat transfer and temperature change

Key Takeaways

Key Points

- Unlike the total heat capacity, the specific heat capacity is independent of mass or volume. It describes how much heat must be added to a unit of mass of a given substance to raise its temperature by one degree Celsius. The units of specific heat capacity are J/(kg °C) or equivalently J/(kg K).

- The heat capacity and the specific heat are related by C=cm or c=C/m.

- The mass m, specific heat c, change in temperature ΔT, and heat added (or subtracted) Q are related by the equation: Q=mcΔT.

- Values of specific heat are dependent on the properties and phase of a given substance. Since they cannot be calculated easily, they are empirically measured and available for reference in tables.

Key Terms

- specific heat capacity: The amount of heat that must be added (or removed) from a unit mass of a substance to change its temperature by one degree Celsius. It is an intensive property.

Specific Heat

The heat capacity is an extensive property that describes how much heat energy it takes to raise the temperature of a given system. However, it would be pretty inconvenient to measure the heat capacity of every unit of matter. What we want is an intensive property that depends only on the type and phase of a substance and can be applied to systems of arbitrary size. This quantity is known as the specific heat capacity (or simply, the specific heat), which is the heat capacity per unit mass of a material. Experiments show that the transferred heat depends on three factors: (1) The change in temperature, (2) the mass of the system, and (3) the substance and phase of the substance. The last two factors are encapsulated in the value of the specific heat.

Heat Transfer and Specific Heat Capacity: The heat Q transferred to cause a temperature change depends on the magnitude of the temperature change, the mass of the system, and the substance and phase involved. (a) The amount of heat transferred is directly proportional to the temperature change. To double the temperature change of a mass m, you need to add twice the heat. (b) The amount of heat transferred is also directly proportional to the mass. To cause an equivalent temperature change in a doubled mass, you need to add twice the heat. (c) The amount of heat transferred depends on the substance and its phase. If it takes an amount Q of heat to cause a temperature change ΔT in a given mass of copper, it will take 10.8 times that amount of heat to cause the equivalent temperature change in the same mass of water assuming no phase change in either substance.

A YouTube element has been excluded from this version of the text. You can view it online here: https://open.ocolearnok.org/physicalscience/?p=110

Specific Heat Capacity: This lesson relates heat to a change in temperature. We discuss how the amount of heat needed for a temperature change is dependent on mass and the substance involved, and that relationship is represented by the specific heat capactiy of the substance, C.

The dependence on temperature change and mass are easily understood. Because the (average) kinetic energy of an atom or molecule is proportional to the absolute temperature, the internal energy of a system is proportional to the absolute temperature and the number of atoms or molecules. Since the transferred heat is equal to the change in the internal energy, the heat is proportional to the mass of the substance and the temperature change. The transferred heat also depends on the substance so that, for example, the heat necessary to raise the temperature is less for alcohol than for water. For the same substance, the transferred heat also depends on the phase (gas, liquid, or solid).

The quantitative relationship between heat transfer and temperature change contains all three factors:

,

where Q is the symbol for heat transfer, m is the mass of the substance, and ΔT is the change in temperature. The symbol c stands for specific heat and depends on the material and phase.

The specific heat is the amount of heat necessary to change the temperature of 1.00 kg of mass by 1.00ºC. The specific heat c is a property of the substance; its SI unit is J/(kg⋅K) or J/(kg⋅C). Recall that the temperature change (ΔT) is the same in units of kelvin and degrees Celsius. Note that the total heat capacity C is simply the product of the specific heat capacity c and the mass of the substance m, i.e.,

or ,

where ϱ is the density of the substance and V is its volume.

Values of specific heat must generally be looked up in tables, because there is no simple way to calculate them. Instead, they are measured empirically. In general, the specific heat also depends on the temperature. The table below lists representative values of specific heat for various substances. Except for gases, the temperature and volume dependence of the specific heat of most substances is weak. The specific heat of water is five times that of glass and ten times that of iron, which means that it takes five times as much heat to raise the temperature of water the same amount as for glass and ten times as much heat to raise the temperature of water as for iron. In fact, water has one of the largest specific heats of any material, which is important for sustaining life on Earth.

Specific Heats: Listed are the specific heats of various substances. These values are identical in units of cal/(g⋅C).3. cv at constant volume and at 20.0ºC, except as noted, and at 1.00 atm average pressure. Values in parentheses are cp at a constant pressure of 1.00 atm.

Calorimetry

Calorimetry is the measurement of the heat of chemical reactions or physical changes.

Learning Objectives

Analyze the relationship between the gas constant for an ideal gas yield and volume

Key Takeaways

Key Points

- A calorimeter is used to measure the heat generated (or absorbed) by a physical change or chemical reaction. The science of measuring these changes is known as calorimetry.

- In order to do calorimetry, it is crucial to know the specific heats of the substances being measured.

- Calorimetry can be performed under constant volume or constant pressure. The type of calculation done depends on the conditions of the experiment.

Key Terms

- constant-pressure calorimeter: An instrument used to measure the heat generated during changes that do not involve changes in pressure.

- calorimeter: An apparatus for measuring the heat generated or absorbed by either a chemical reaction, change of phase or some other physical change.

- constant-volume calorimeter: An instrument used to measure the heat generated during changes that do not involve changes in volume.

Calorimetry

Overview

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. Calorimetry is performed with a calorimeter. A simple calorimeter just consists of a thermometer attached to a metal container full of water suspended above a combustion chamber. The word calorimetry is derived from the Latin word calor, meaning heat. Scottish physician and scientist Joseph Black, who was the first to recognize the distinction between heat and temperature, is said to be the founder of calorimetry.

Calorimetry requires that the material being heated have known thermal properties, i.e. specific heat capacities. The classical rule, recognized by Clausius and by Kelvin, is that the pressure exerted by the calorimetric material is fully and rapidly determined solely by its temperature and volume; this rule is for changes that do not involve phase change, such as melting of ice. There are many materials that do not comply with this rule, and for them, more complex equations are required than those below.

Ice Calorimeter: The world’s first ice-calorimeter, used in the winter of 1782-83, by Antoine Lavoisier and Pierre-Simon Laplace, to determine the heat evolved in variouschemical changes; calculations which were based on Joseph Black’s prior discovery of latent heat. These experiments mark the foundation of thermochemistry.

Basic Calorimetry at Constant Value

Constant-volume calorimetry is calorimetry performed at a constant volume. This involves the use of a constant-volume calorimeter (one type is called a Bomb calorimeter). For constant-volume calorimetry:

where δQ is the increment of heat gained by the sample, CV is the heat capacity at constant volume, cv is the specific heat at constant volume, and ΔT is the change in temperature.

Measuring Enthalpy Change

To find the enthalpy change per mass (or per mole) of a substance A in a reaction between two substances A and B, the substances are added to a calorimeter and the initial and final temperatures (before the reaction started and after it has finished) are noted. Multiplying the temperature change by the mass and specific heat capacities of the substances gives a value for the energy given off or absorbed during the reaction:

Dividing the energy change by how many grams (or moles) of A were present gives its enthalpy change of reaction. This method is used primarily in academic teaching as it describes the theory of calorimetry. It does not account for the heat loss through the container or the heat capacity of the thermometer and container itself. In addition, the object placed inside the calorimeter shows that the objects transferred their heat to the calorimeter and into the liquid, and the heat absorbed by the calorimeter and the liquid is equal to the heat given off by the metals.

Constant-Pressure Calorimetry

A constant-pressure calorimeter measures the change in enthalpy of a reaction occurring in solution during which the atmospheric pressure remains constant. An example is a coffee-cup calorimeter, which is constructed from two nested Styrofoam cups and a lid with two holes, allowing insertion of a thermometer and a stirring rod. The inner cup holds a known amount of a solute, usually water, that absorbs the heat from the reaction. When the reaction occurs, the outer cup provides insulation. Then

where Cp is the specific heat at constant pressure, ΔH is the enthalpy of the solution, ΔT is the change in temperature, W is the mass of the solute, and M is the molecular mass of the solute. The measurement of heat using a simple calorimeter, like the coffee cup calorimeter, is an example of constant-pressure calorimetry, since the pressure (atmospheric pressure) remains constant during the process. Constant-pressure calorimetry is used in determining the changes in enthalpy occurring in solution. Under these conditions the change in enthalpy equals the heat (Q=ΔH).

Specific Heat for an Ideal Gas at Constant Pressure and Volume

An ideal gas has different specific heat capacities under constant volume or constant pressure conditions.

Learning Objectives

Explain how to derive the adiabatic index

Key Takeaways

Key Points

- The specific heat at constant volume for a gas is given as .

- The specific heat at constant pressure for an ideal gas is given as .

- The heat capacity ratio (or adiabatic index ) is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.

Key Terms

- Fundamental Thermodynamic Relation: In thermodynamics, the fundamental thermodynamic relation expresses an infinitesimal change in internal energy in terms of infinitesimal changes in entropy, and volume for a closed system in thermal equilibrium in the following way: dU=TdS-PdV. Here, U is internal energy, T is absolute temperature, S is entropy, P is pressure and V is volume.

- adiabatic index: The ratio of the heat capacity at constant pressure to heat capacity at constant volume.

- specific heat: The ratio of the amount of heat needed to raise the temperature of a unit mass of substance by a unit degree to the amount of heat needed to raise that of the same mass of water by the same amount.

Specific Heat for an Ideal Gas at Constant Pressure and Volume

The heat capacity at constant volume of nR = 1 J·K−1 of any gas, including an ideal gas is:

This represents the dimensionless heat capacity at constant volume; it is generally a function of temperature due to intermolecular forces. For moderate temperatures, the constant for a monoatomic gas is cv=3/2 while for a diatomic gas it is cv=5/2 (see ). Macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules.

Molecular internal vibrations: When a gas is heated, translational kientic energy of molecules in the gas will increase. In addition, molecules in the gas may pick up many characteristic internal vibrations. Potential energy stored in these internal degrees of freedom contributes to specific heat of the gas.

The heat capacity at constant pressure of 1 J·K−1 ideal gas is:

where H=U+pV is the enthalpy of the gas.

Measuring the heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume (this implies the containing vessel must be nearly rigid or at least very strong). It is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws.

Utilizing the Fundamental Thermodynamic Relation we can show:

where the partial derivatives are taken at: constant volume and constant number of particles, and at constant pressure and constant number of particles, respectively.

The heat capacity ratio or adiabatic index is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor:

For an ideal gas, evaluating the partial derivatives above according to the equation of state, where R is the gas constant for an ideal gas yields:

substituting:

This equation reduces simply to what is known as Mayer’s relation:

Julius Robert Mayer: Julius Robert von Mayer (November 25, 1814 – March 20, 1878), a German physician and physicist, was one of the founders of thermodynamics. He is best known for his 1841 enunciation of one of the original statements of the conservation of energy (or what is now known as one of the first versions of the first law of thermodynamics): “Energy can be neither created nor destroyed. ” In 1842, Mayer described the vital chemical process now referred to as oxidation as the primary source of energy for any living creature. His achievements were overlooked and credit for the discovery of the mechanical equivalent of heat was attributed to James Joule in the following year. von Mayer also proposed that plants convert light into chemical energy.

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It is a simple equation relating the heat capacities under constant temperature and under constant pressure.

Solving Problems with Calorimetry

Calorimetry is used to measure the amount of heat produced or consumed in a chemical reaction.

Learning Objectives

Explain a bomb calorimeter is used to measure heat evolved in a combustion reaction

Key Takeaways

Key Points

- Calorimetry is used to measure amounts of heat transferred to or from a substance.

- A calorimeter  is a device used to measure the amount of heat involved in a chemical or physical process.

- This means that the amount of heat produced or consumed in the reaction equals the amount of heat absorbed or lost by the solution.

Key Terms

- heat of reaction: The enthalpy change in a chemical reaction; the amount of heat that a systems gives up to its surroundings so it can return to its initial temperature.

- combustion: A process where two chemicals are combined to produce heat.

Calorimeters are designed to minimize energy exchange between the system being studied and its surroundings. They range from simple coffee cup calorimeters used by introductory chemistry students to sophisticated bomb calorimeters used to determine the energy content of food.

Calorimetry is used to measure amounts of heat transferred to or from a substance. To do so, the heat is exchanged with a calibrated object (calorimeter). The change in temperature of the measuring part of the calorimeter is converted into the amount of heat (since the previous calibration was used to establish its heat capacity ). The measurement of heat transfer using this approach requires the definition of a system (the substance or substances undergoing the chemical or physical change) and its surroundings (the other components of the measurement apparatus that serve to either provide heat to the system or absorb heat from the system). Knowledge of the heat capacity of the surroundings, and careful measurements of the masses of the system and surroundings and their temperatures before and after the process allows one to calculate the heat transferred as described in this section.

A calorimeter is a device used to measure the amount of heat involved in a chemical or physical process. For example, when an exothermic reaction occurs in solution in a calorimeter, the heat produced by the reaction is absorbed by the solution, which increases its temperature. When an endothermic reaction occurs, the heat required is absorbed from the thermal energy of the solution, which decreases its temperature. The temperature change, along with the specific heat and mass of the solution, can then be used to calculate the amount of heat involved in either case.

Coffee-Cup Calorimeters

General chemistry students often use simple calorimeters constructed from polystyrene cups. These easy-to-use “coffee cup” calorimeters allow more heat exchange with their surroundings, and therefore produce less accurate energy values.

Structure of the Constant Volume (or “Bomb”) Calorimeter

Bomb Calorimeter: This is the picture of a typical setup of bomb calorimeter.

A different type of calorimeter that operates at constant volume, colloquially known as a bomb calorimeter, is used to measure the energy produced by reactions that yield large amounts of heat and gaseous products, such as combustion reactions. (The term “bomb” comes from the observation that these reactions can be vigorous enough to resemble explosions that would damage other calorimeters.) This type of calorimeter consists of a robust steel container (the “bomb”) that contains the reactants and is itself submerged in water. The sample is placed in the bomb, which is then filled with oxygen at high pressure. A small electrical spark is used to ignite the sample. The energy produced by the reaction is trapped in the steel bomb and the surrounding water. The temperature increase is measured and, along with the known heat capacity of the calorimeter, is used to calculate the energy produced by the reaction. Bomb calorimeters require calibration to determine the heat capacity of the calorimeter and ensure accurate results. The calibration is accomplished using a reaction with a known q, such as a measured quantity of benzoic acid ignited by a spark from a nickel fuse wire that is weighed before and after the reaction. The temperature change produced by the known reaction is used to determine the heat capacity of the calorimeter. The calibration is generally performed each time before the calorimeter is used to gather research data.

Example: Identifying a Metal by Measuring Specific Heat

A 59.7 g piece of metal that had been submerged in boiling water was quickly transferred into 60.0 mL of water initially at 22.0 °C. The final temperature is 28.5 °C. Use these data to determine the specific heat of the metal. Use this result to identify the metal.

Solution

Assuming perfect heat transfer, the heat given off by metal is the negative of the heat taken in by water, or:

In expanded form, this is:

Noting that since the metal was submerged in boiling water, its initial temperature was 100.0 °C; and that for water, 60.0 mL = 60.0 g; we have:

Solving this:

Our experimental specific heat is closest to the value for copper (0.39 J/g °C), so we identify the metal as copper.

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