# Work, Energy, and Simple Machines

## Mechanical Energy and Conservation of Energy

### Section Key Terms

### Mechanical Energy and Conservation of Energy

We saw earlier that mechanical energy can be either potential or kinetic. In this section we will see how energy is transformed from one of these forms to the other. We will also see that, in a closed system, the sum of these forms of energy remains constant.
Quite a bit of potential energy is gained by a roller coaster car and its passengers when they are raised to the top of the first hill. Remember that the potential part of the term means that energy has been stored and can be used at another time. You will see that this stored energy can either be used to do work or can be transformed into kinetic energy. For example, when an object that has gravitational potential energy falls, its energy is converted to kinetic energy. Remember that both work and energy are expressed in joules.
Refer back to . The amount of work required to raise the TV from point A to point B is equal to the amount of gravitational potential energy the TV gains from its height above the ground. This is generally true for any object raised above the ground. If all the work done on an object is used to raise the object above the ground, the amount work equals the object’s gain in gravitational potential energy. However, note that because of the work done by friction, these energy–work transformations are never perfect. Friction causes the loss of some useful energy. In the discussions to follow, we will use the approximation that transformations are frictionless.
Now, let’s look at the roller coaster in . Work was done on the roller coaster to get it to the top of the first rise; at this point, the roller coaster has gravitational potential energy. It is moving slowly, so it also has a small amount of kinetic energy. As the car descends the first slope, its PE is converted to KE. At the low point much of the original PE has been transformed to KE, and speed is at a maximum. As the car moves up the next slope, some of the KE is transformed back into PE and the car slows down.
On an actual roller coaster, there are many ups and downs, and each of these is accompanied by transitions between kinetic and potential energy. Assume that no energy is lost to friction. At any point in the ride, the total mechanical energy is the same, and it is equal to the energy the car had at the top of the first rise. This is a result of the law of conservation of energy, which says that, in a closed system, total energy is conservedthat is, it is constant. Using subscripts 1 and 2 to represent initial and final energy, this law is expressed as
Either side equals the total mechanical energy. The phrase in a  means we are assuming no energy is lost to the surroundings due to friction and air resistance. If we are making calculations on dense falling objects, this is a good assumption. For the roller coaster, this assumption introduces some inaccuracy to the calculation.

### Calculations involving Mechanical Energy and Conservation of Energy

### Practice Problems

### Check Your Understanding

### Section Summary

1. Mechanical energy may be either kinetic (energy of motion) or potential (stored energy).
2. Doing work on an object or system changes its energy.
3. Total energy in a closed, isolated system is constant.

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# Work, Energy, and Simple Machines

## Simple Machines

### Section Key Terms

### Simple Machines

Simple machines make work easier, but they do not decrease the amount of work you have to do. Why can’t simple machines change the amount of work that you do? Recall that in closed systems the total amount of energy is conserved. A machine cannot increase the amount of energy you put into it. So, why is a simple machine useful? Although it cannot change the amount of work you do, a simple machine can change the amount of force you must apply to an object, and the distance over which you apply the force. In most cases, a simple machine is used to reduce the amount of force you must exert to do work. The down side is that you must exert the force over a greater distance, because the product of force and distance, fd, (which equals work) does not change.
Let’s examine how this works in practice. In (a), the worker uses a type of lever to exert a small force over a large distance, while the pry bar pulls up on the nail with a large force over a small distance. (b) shows the how a lever works mathematically. The effort force, applied at F, lifts the load (the resistance force) which is pushing down at F. The triangular pivot is called the fulcrum; the part of the lever between the fulcrum and F is the effort arm, L; and the part to the left is the resistance arm, L. The mechanical advantage is a number that tells us how many times a simple machine multiplies the effort force. The ideal mechanical advantage, IMA, is the mechanical advantage of a perfect machine with no loss of useful work caused by friction between moving parts. The equation for IMA is shown in (b).
In general, the IMA = the resistance force, F, divided by the effort force, F. IMA also equals the distance over which the effort is applied, d, divided by the distance the load travels, d.
Getting back to conservation of energy, for any simple machine, the work put into the machine, W, equals the work the machine puts out, W. Combining this with the information in the paragraphs above, we can write
The equations show how a simple machine can output the same amount of work while reducing the amount of effort force by increasing the distance over which the effort force is applied.
Some levers exert a large force to a short effort arm. This results in a smaller force acting over a greater distance at the end of the resistance arm. Examples of this type of lever are baseball bats, hammers, and golf clubs. In another type of lever, the fulcrum is at the end of the lever and the load is in the middle, as in the design of a wheelbarrow.
The simple machine shown in  is called a wheel and axle. It is actually a form of lever. The difference is that the effort arm can rotate in a complete circle around the fulcrum, which is the center of the axle. Force applied to the outside of the wheel causes a greater force to be applied to the rope that is wrapped around the axle. As shown in the figure, the ideal mechanical advantage is calculated by dividing the radius of the wheel by the radius of the axle. Any crank-operated device is an example of a wheel and axle.
An inclined plane and a wedge are two forms of the same simple machine. A wedge is simply two inclined planes back to back.  shows the simple formulas for calculating the IMAs of these machines. All sloping, paved surfaces for walking or driving are inclined planes. Knives and axe heads are examples of wedges.
The screw shown in  is actually a lever attached to a circular inclined plane. Wood screws (of course) are also examples of screws. The lever part of these screws is a screw driver. In the formula for IMA, the distance between screw threads is called pitch and has the symbol P.
 shows three different pulley systems. Of all simple machines, mechanical advantage is easiest to calculate for pulleys. Simply count the number of ropes supporting the load. That is the IMA. Once again we have to exert force over a longer distance to multiply force. To raise a load 1 meter with a pulley system you have to pull N meters of rope. Pulley systems are often used to raise flags and window blinds and are part of the mechanism of construction cranes.
A complex machine is a combination of two or more simple machines. The wire cutters in  combine two levers and two wedges. Bicycles include wheel and axles, levers, screws, and pulleys. Cars and other vehicles are combinations of many machines.

### Calculating Mechanical Advantage and Efficiency of Simple Machines

In general, the IMA = the resistance force, Fr, divided by the effort force, Fe. IMA also equals the distance over which the effort is applied, d, divided by the distance the load travels, d.
Refer back to the discussions of each simple machine for the specific equations for the IMA for each type of machine.
No simple or complex machines have the actual mechanical advantages calculated by the IMA equations. In real life, some of the applied work always ends up as wasted heat due to friction between moving parts. Both the input work (W) and output work (W) are the result of a force, F, acting over a distance, d.
The efficiency output of a machine is simply the output work divided by the input work, and is usually multiplied by 100 so that it is expressed as a percent.
Look back at the pictures of the simple machines and think about which would have the highest efficiency. Efficiency is related to friction, and friction depends on the smoothness of surfaces and on the area of the surfaces in contact. How would lubrication affect the efficiency of a simple machine?

### Practice Problems

### Check Your Understanding

### Section Summary

1. The six types of simple machines make work easier by changing the fd term so that force is reduced at the expense of increased distance.
2. The ratio of output force to input force is a machine’s mechanical advantage.
3. Combinations of two or more simple machines are called complex machines.
4. The ratio of output work to input work is a machine’s efficiency.

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# Special Relativity

## Introduction

Have you ever dreamed of traveling to other planets in faraway star systems? The trip might seem possible by traveling fast enough, but you will read in this chapter why it is not. In 1905, Albert Einstein developed the theory of special relativity. Einstein developed the theory to help explain inconsistencies between the equations describing electromagnetism and Newtonian mechanics, and to explain why the ether did not exist. This theory explains the limit on an object’s speed among other implications.
Relativity is the study of how different observers moving with respect to one another measure the same events. Galileo and Newton developed the first correct version of classical relativity. Einstein developed the modern theory of relativity. Modern relativity is divided into two parts. Special relativity deals with observers moving at constant velocity. General relativity deals with observers moving at constant acceleration. Einstein’s theories of relativity made revolutionary predictions. Most importantly, his predictions have been verified by experiments.
In this chapter, you learn how experiments and puzzling contradictions in existing theories led to the development of the theory of special relativity. You will also learn the simple postulates on which the theory was based; a postulate is a statement that is assumed to be true for the purposes of reasoning in a scientific or mathematic argument.


# Special Relativity

## Postulates of Special Relativity

### Section Key Terms

### Scientific Experiments and Problems

Relativity is not new. Way back around the year 1600, Galileo explained that motion is relative. Wherever you happen to be, it seems like you are at a fixed point and that everything moves with respect to you. Everyone else feels the same way. Motion is always measured with respect to a fixed point. This is called establishing a frame of reference. But the choice of the point is arbitrary, and all frames of reference are equally valid. A passenger in a moving car is not moving with respect to the driver, but they are both moving from the point of view of a person on the sidewalk waiting for a bus. They are moving even faster as seen by a person in a car coming toward them. It is all relative.
Light is involved in the discussion of relativity because theories related to electromagnetism are inconsistent with Galileo’s and Newton’s explanation of relativity. The true nature of light was a hot topic of discussion and controversy in the late 19th century. At the time, it was not generally believed that light could travel across empty space. It was known to travel as waves, and all other types of energy that propagated as waves needed to travel though a material medium. It was believed that space was filled with an invisible medium that light waves traveled through. This imaginary (as it turned out) material was called the ether (also spelled aether). It was thought that everything moved through this mysterious fluid. In other words, ether was the one fixed frame of reference. The Michelson–Morley experiment proved it was not.
In 1887, Albert Michelson and Edward Morley designed the interferometer shown in  to measure the speed of Earth through the ether. A light beam is split into two perpendicular paths and then recombined. Recombining the waves produces an inference pattern, with a bright fringe at the locations where the two waves arrive in phase; that is, with the crests of both waves arriving together and the troughs arriving together. A dark fringe appears where the crest of one wave coincides with a trough of the other, so that the two cancel. If Earth is traveling through the ether as it orbits the sun, the peaks in one arm would take longer than in the other to reach the same location. The places where the two waves arrive in phase would change, and the interference pattern would shift. But, using the interferometer, there was no shift seen! This result led to two conclusions: that there is no ether and that the speed of light is the same regardless of the relative motion of source and observer. The Michelson–Morley investigation has been called the most famous failed experiment in history.
To see what Michelson and Morley expected to find when they measured the speed of light in two directions, watch this animation. In the video, two people swimming in a lake are represented as an analogy to light beams leaving Earth as it moves through the ether (if there were any ether). The swimmers swim away from and back to a platform that is moving through the water. The swimmers swim in different directions with respect to the motion of the platform. Even though they swim equal distances at the same speed, the motion of the platform causes them to arrive at different times.

### Einstein’s Postulates

The results described above left physicists with some puzzling and unsettling questions such as, why doesn’t light emitted by a fast-moving object travel faster than light from a street lamp? A radical new theory was needed, and Albert Einstein, shown in , was about to become everyone’s favorite genius. Einstein began with two simple postulates based on the two things we have discussed so far in this chapter.
1. The laws of physics are the same in all inertial reference frames.
2. The speed of light is the same in all inertial reference frames and is not affected by the speed of its source.
The speed of light is given the symbol c and is equal to exactly 299,792,458 m/s. This is the speed of light in vacuum; that is, in the absence of air. For most purposes, we round this number off to
        The term inertial reference frame simply refers to a frame of reference where all objects follow Newton’s first law of motion: Objects at rest remain at rest, and objects in motion remain in motion at a constant velocity in a straight line, unless acted upon by an external force. The inside of a car moving along a road at constant velocity and the inside of a stationary house are inertial reference frames.
Einstein’s postulates were carefully chosen, and they both seemed very likely to be true. Einstein proceeded despite realizing that these two ideas taken together and applied to extreme conditions led to results that contradict Newtonian mechanics. He just took the ball and ran with it.
In the traditional view, velocities are additive. If you are running at 3 m/s and you throw a ball forward at a speed of 10 m/s, the ball should have a net speed of 13 m/s. However, according to relativity theory, the speed of a moving light source is not added to the speed of the emitted light.
In addition, Einstein’s theory shows that if you were moving forward relative to Earth at nearly c (the speed of light) and could throw a ball forward at c, an observer at rest on the earth would not see the ball moving at nearly twice the speed of light. The observer would see it moving at a speed that is still less than c. This result conforms to both of Einstein’s postulates: The speed of light has a fixed maximum and neither reference frame is privileged.
Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turn green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.
Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps on a moving train. (See )
A woman (observer A) is seated in the center of a rail car, with two flash lamps at opposite sides equidistant from her. Multiple light rays that are emitted from the flash lamps move towards observer A, as shown with arrows. A velocity vector arrow for the rail car is shown towards the right. A man (observer B) standing on the platform is facing the woman and also observes the flashes of light.
Observer A moves with the lamps on the rail car as the rail car moves towards the right of observer B. Observer B receives the light flashes simultaneously, and sees the bulbs as both having flashed at the same time. However, he sees observer A receive the flash from the right first. Because the pulse from the right reaches her first, in her frame of reference she sees the bulbs as not having flashed simultaneously. Here, a relative velocity between observers affects whether two events at well-separated locations are observed to be simultaneous. Simultaneity, or whether different events occur at the same instant, depends on the frame of reference of the observer. Remember that velocity equals distance divided by time, so t = d/v. If velocity appears to be different, then duration of time appears to be different.
This illustrates the power of clear thinking. We might have guessed incorrectly that, if light is emitted simultaneously, then two observers halfway between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of thought experiment (in German, Gedankenexperiment). He very carefully considered how an observation is made and disregarded what might seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his theory of relativity. No experiments after that of Michelson and Morley were able to detect any ether medium. We will describe later how experiments also confirmed other predictions of special relativity, such as the distance between two objects and the time interval of two events being different for two observers moving with respect to each other.
In summary: Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events). Two events are not necessarily simultaneous to all observers.
The discrepancies between Newtonian mechanics and relativity theory illustrate an important point about how science advances. Einstein’s theory did not replace Newton’s but rather extended it. It is not unusual that a new theory must be developed to account for new information. In most cases, the new theory is built on the foundation of older theory. It is rare that old theories are completely replaced.
In this chapter, you will learn about the theory of special relativity, but, as mentioned in the introduction, Einstein developed two relativity theories: special and general.  summarizes the differences between the two theories.

### Practice Problems

### Check Your Understanding

### Section Summary

1. One postulate of special relativity theory is that the laws of physics are the same in all inertial frames of reference.
2. The other postulate is that the speed of light in a vacuum is the same in all inertial frames.
3. Einstein showed that simultaneity, or lack of it, depends on the frame of reference of the observer.

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# Special Relativity

## Consequences of Special Relativity

### Section Key Terms

### Relativistic Effects on Time, Distance, and Momentum

Consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect. Time dilation is the phenomenon of time passing more slowly for an observer who is moving relative to another observer.
For example, suppose an astronaut measures the time it takes for light to travel from the light source, cross her ship, bounce off a mirror, and return. (See .) How does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on the earth? Asking this question (another thought experiment) produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the earth bound observer. The passage of time is different for the two observers because the distance the light travels in the astronaut’s frame is smaller than in the earth bound frame. Light travels at the same speed in each frame, and so it will take longer to travel the greater distance in the earth bound frame.
The relationship between Δt and Δto is given by
where
       is the relativistic factor given by
and v and c are the speeds of the moving observer and light, respectively.
Notice that when the velocity v is small compared to the speed of light c, then v/c becomes small, and
       becomes close to 1. When this happens, time measurements are the same in both frames of reference. Relativistic effects, meaning those that have to do with special relativity, usually become significant when speeds become comparable to the speed of light. This is seen to be the case for time dilation.
You may have seen science fiction movies in which space travelers return to Earth after a long trip to find that the planet and everyone on it has aged much more than they have. This type of scenario is a based on a thought experiment, known as the twin paradox, which imagines a pair of twins, one of whom goes on a trip into space while the other stays home. When the space traveler returns, she finds her twin has aged much more than she. This happens because the traveling twin has been in two frames of reference, one leaving Earth and one returning.
Time dilation has been confirmed by comparing the time recorded by an atomic clock sent into orbit to the time recorded by a clock that remained on Earth. GPS satellites must also be adjusted to compensate for time dilation in order to give accurate positioning.
Have you ever driven on a road, like that shown in , that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it is about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers moving with respect to one other.
One thing all observers agree upon is their relative speed. When one observer is traveling away from another, they both see the other receding at the same speed, regardless of whose frame of reference is chosen. Remember that speed equals distance divided by time: v = d/t. If the observers experience a difference in elapsed time, they must also observe a difference in distance traversed. This is because the ratio d/t must be the same for both observers.
The shortening of distance experienced by an observer moving with respect to the points whose distance apart is measured is called length contraction. Proper length, L0, is the distance between two points measured in the reference frame where the observer and the points are at rest. The observer in motion with respect to the points measures L. These two lengths are related by the equation
Because
       is the same expression used in the time dilation equation above, the equation becomes
To see how length contraction is seen by a moving observer, go to this simulation. Here you can also see that simultaneity, time dilation, and length contraction are interrelated phenomena.
This link is to a simulation that illustrates the relativity of simultaneous events.
In classical physics, momentum is a simple product of mass and velocity. When special relativity is taken into account, objects that have mass have a speed limit. What effect do you think mass and velocity have on the momentum of objects moving at relativistic speeds; i.e., speeds close to the speed of light?
Momentum is one of the most important concepts in physics. The broadest form of Newton’s second law is stated in terms of momentum. Momentum is conserved in classical mechanics whenever the net external force on a system is zero. This makes momentum conservation a fundamental tool for analyzing collisions. We will see that momentum has the same importance in modern physics. Relativistic momentum is conserved, and much of what we know about subatomic structure comes from the analysis of collisions of accelerator-produced relativistic particles.
One of the postulates of special relativity states that the laws of physics are the same in all inertial frames. Does the law of conservation of momentum survive this requirement at high velocities? The answer is yes, provided that the momentum is defined as follows.
Relativistic momentum, p, is classical momentum multiplied by the relativistic factor

where
       is the rest mass of the object (that is, the mass measured at rest, without any
       factor involved),
       is its velocity relative to an observer, and

 as before, is the relativistic factor. We use the mass of the object as measured at rest because we cannot determine its mass while it is moving.
Note that we use
       for velocity here to distinguish it from relative velocity
       between observers. Only one observer is being considered here. With
       defined in this way,
       is conserved whenever the net external force is zero, just as in classical physics. Again we see that the relativistic quantity becomes virtually the same as the classical at low velocities. That is, relativistic momentum
       becomes the classical
       at low velocities, because
       is very nearly equal to 1 at low velocities.
Relativistic momentum has the same intuitive feel as classical momentum. It is greatest for large masses moving at high velocities. Because of the factor
       however, relativistic momentum behaves differently from classical momentum by approaching infinity as
       approaches
        (See .) This is another indication that an object with mass cannot reach the speed of light. If it did, its momentum would become infinite, which is an unreasonable value.
Relativistic momentum is defined in such a way that the conservation of momentum will hold in all inertial frames. Whenever the net external force on a system is zero, relativistic momentum is conserved, just as is the case for classical momentum. This has been verified in numerous experiments.

### Mass-Energy Equivalence

Let us summarize the calculation of relativistic effects on objects moving at speeds near the speed of light. In each case we will need to calculate the relativistic factor, given by
where v and  c are as defined earlier. We use u as the velocity of a particle or an object in one frame of reference, and v for the velocity of one frame of reference with respect to another.

### Time Dilation

Elapsed time on a moving object,
        as seen by a stationary observer is given by
       where
       is the time observed on the moving object when it is taken to be the frame or reference.

### Length Contraction

Length measured by a person at rest with respect to a moving object, L, is given by
where L0 is the length measured on the moving object.

### Relativistic Momentum

Momentum, p, of an object of mass, m, traveling at relativistic speeds is given by
        where u is velocity of a moving object as seen by a stationary observer.

### Relativistic Energy

The original source of all the energy we use is the conversion of mass into energy. Most of this energy is generated by nuclear reactions in the sun and radiated to Earth in the form of electromagnetic radiation, where it is then transformed into all the forms with which we are familiar. The remaining energy from nuclear reactions is produced in nuclear power plants and in Earth’s interior. In each of these cases, the source of the energy is the conversion of a small amount of mass into a large amount of energy. These sources are shown in .
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor. The result of his analysis is that a particle or object of mass m moving at velocity u has relativistic energy given by
This is the expression for the total energy of an object of mass m at any speed u and includes both kinetic and potential energy. Look back at the equation for
       and you will see that it is equal to 1 when u is 0; that is, when an object is at rest. Then the rest energy, E0, is simply
This is the correct form of Einstein’s famous equation.
This equation is very useful to nuclear physicists because it can be used to calculate the energy released by a nuclear reaction. This is done simply by subtracting the mass of the products of such a reaction from the mass of the reactants. The difference is the m in
        Here is a simple example:
A positron is a type of antimatter that is just like an electron, except that it has a positive charge. When a positron and an electron collide, their masses are completely annihilated and converted to energy in the form of gamma rays. Because both particles have a rest mass of 9.11 × 10–31 kg, we multiply the mc2 term by 2. So the energy of the gamma rays is
where we have the expression for the joule (J) in terms of its SI base units of kg, m, and s. In general, the nuclei of stable isotopes have less mass then their constituent subatomic particles. The energy equivalent of this difference is called the binding energy of the nucleus. This energy is released during the formation of the isotope from its constituent particles because the product is more stable than the reactants. Expressed as mass, it is called the mass defect. For example, a helium nucleus is made of two neutrons and two protons and has a mass of 4.0003 atomic mass units (u). The sum of the masses of two protons and two neutrons is 4.0330 u. The mass defect then is 0.0327 u. Converted to kg, the mass defect is 5.0442 × 10–30 kg. Multiplying this mass times c2 gives a binding energy of 4.540 × 10–12 J. This does not sound like much because it is only one atom. If you were to make one gram of helium out of neutrons and protons, it would release 683,000,000,000 J. By comparison, burning one gram of coal releases about 24 J.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Time dilates, length contracts, and momentum increases as an object approaches the speed of light.
2. Energy and mass are interchangeable, according to the relationship E = mc2. The laws of conservation of mass and energy are combined into the law of conservation of mass-energy.

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# Thermal Energy, Heat, and Work

## Introduction

Heat is something familiar to all of us. We feel the warmth of the summer sun, the hot vapor rising up out of a cup of hot cocoa, and the cooling effect of our sweat. When we feel warmth, it means that heat is transferring energy to our bodies; when we feel cold, that means heat is transferring energy away from our bodies. Heat transfer is the movement of thermal energy from one place or material to another, and is caused by temperature differences. For example, much of our weather is caused by Earth evening out the temperature across the planet through wind and violent storms, which are driven by heat transferring energy away from the equator towards the cold poles. In this chapter, we’ll explore the precise meaning of heat, how it relates to temperature as well as to other forms of energy, and its connection to work.


# Thermal Energy, Heat, and Work

## Temperature and Thermal Energy

### Section Key Terms

### Temperature

What is temperature? It’s one of those concepts so ingrained in our everyday lives that, although we know what it means intuitively, it can be hard to define. It is tempting to say that temperature measures heat, but this is not strictly true. Heat is the transfer of energy due to a temperature difference. Temperature is defined in terms of the instrument we use to tell us how hot or cold an object is, based on a mechanism and scale invented by people. Temperature is literally defined as what we measure on a thermometer.
Heat is often confused with temperature. For example, we may say that the heat was unbearable, when we actually mean that the temperature was high. This is because we are sensitive to the flow of energy by heat, rather than the temperature. Since heat, like work, transfers energy, it has the SI unit of joule (J).
Atoms and molecules are constantly in motion, bouncing off one another in random directions. Recall that kinetic energy is the energy of motion, and that it increases in proportion to velocity squared. Without going into mathematical detail, we can say that thermal energy—the energy associated with heat—is the average kinetic energy of the particles (molecules or atoms) in a substance. Faster moving molecules have greater kinetic energies, and so the substance has greater thermal energy, and thus a higher temperature. The total internal energy of a system is the sum of the kinetic and potential energies of its atoms and molecules. Thermal energy is one of the subcategories of internal energy, as is chemical energy.
To measure temperature, some scale must be used as a standard of measurement. The three most commonly used temperature scales are the Fahrenheit, Celsius, and Kelvin scales. Both the Fahrenheit scale and Celsius scale are relative temperature scales, meaning that they are made around a reference point. For example, the Celsius scale uses the freezing point of water as its reference point; all measurements are either lower than the freezing point of water by a given number of degrees (and have a negative sign), or higher than the freezing point of water by a given number of degrees (and have a positive sign). The boiling point of water is 100
   for the Celsius scale, and its unit is the degree Celsius  .
On the Fahrenheit scale, the freezing point of water is at 32
 , and the boiling point is at 212
  . The unit of temperature on this scale is the degree Fahrenheit  . Note that the difference in degrees between the freezing and boiling points is greater for the Fahrenheit scale than for the Celsius scale. Therefore, a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Since 100 Celsius degrees span the same range as 180 Fahrenheit degrees, one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale (because
  ). This relationship can be used to convert between temperatures in Fahrenheit and Celsius (see ).
The Kelvin scale is the temperature scale that is commonly used in science because it is an absolute temperature scale. This means that the theoretically lowest-possible temperature is assigned the value of zero. Zero degrees on the Kelvin scale is known as absolute zero; it is theoretically the point at which there is no molecular motion to produce thermal energy. On the original Kelvin scale first created by Lord Kelvin, all temperatures have positive values, making it useful for scientific work. The official temperature unit on this scale is the kelvin, which is abbreviated as K. The freezing point of water is 273.15 K, and the boiling point of water is 373.15 K.
Although absolute zero is possible in theory, it cannot be reached in practice. The lowest temperature ever created and measured during a laboratory experiment was
   K, at Helsinki University of Technology in Finland. In comparison, the coldest recorded temperature for a place on Earth’s surface was 183 K (–89 	°C
), at Vostok, Antarctica, and the coldest known place (outside the lab) in the universe is the Boomerang Nebula, with a temperature of 1 K. Luckily, most of us humans will never have to experience such extremes.
The average normal body temperature is 98.6
   (37.0
  ), but people have been known to survive with body temperatures ranging from 75
   to 111
   (24
   to 44
  ).

### Converting Between Celsius, Kelvin, and Fahrenheit Scales

While the Fahrenheit scale is still the most commonly used scale in the United States, the majority of the world uses Celsius, and scientists prefer Kelvin. It’s often necessary to convert between these scales. For instance, if the TV meteorologist gave the local weather report in kelvins, there would likely be some confused viewers!  gives the equations for conversion between the three temperature scales.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Temperature is the quantity measured by a thermometer.
2. Temperature is related to the average kinetic energy of atoms and molecules in a system.
3. Absolute zero is the temperature at which there is no molecular motion.
4. There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
5. Temperatures on one scale can be converted into temperatures on another scale.

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# Thermal Energy, Heat, and Work

## Heat, Specific Heat, and Heat Transfer

### Section Key Terms

### Heat Transfer, Specific Heat, and Heat Capacity

We learned in the previous section that temperature is proportional to the average kinetic energy of atoms and molecules in a substance, and that the average internal kinetic energy of a substance is higher when the substance’s temperature is higher.
If two objects at different temperatures are brought in contact with each other, energy is transferred from the hotter object (that is, the object with the greater temperature) to the colder (lower temperature) object, until both objects are at the same temperature. There is no net heat transfer once the temperatures are equal because the amount of heat transferred from one object to the other is the same as the amount of heat returned. One of the major effects of heat transfer is temperature change: Heating increases the temperature while cooling decreases it. Experiments show that the heat transferred to or from a substance depends on three factors—the change in the substance’s temperature, the mass of the substance, and certain physical properties related to the phase of the substance.
The equation for heat transfer Q is

where m is the mass of the substance and ΔT is the change in its temperature, in units of Celsius or Kelvin. 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 1.00 ºC. The specific heat c is a property of the substance; its SI unit is J/(kg
  K) or J/(kg

  ). The temperature change (
  ) is the same in units of kelvins and degrees Celsius (but not degrees Fahrenheit). Specific heat is closely related to the concept of heat capacity. Heat capacity is the amount of heat necessary to change the temperature of a substance by 1.00
  . In equation form, heat capacity C is
, where m is mass and c is specific heat. Note that heat capacity is the same as specific heat, but without any dependence on mass. Consequently, two objects made up of the same material but with different masses will have different heat capacities. This is because the heat capacity is a property of an object, but specific heat is a property of any object made of the same material.
Values of specific heat must be looked up in tables, because there is no simple way to calculate them.  gives the values of specific heat for a few substances as a handy reference. We see from this table that the specific heat of water is five times that of glass, which means that it takes five times as much heat to raise the temperature of 1 kg of water than to raise the temperature of 1 kg of glass by the same number of degrees.

### Conduction, Convection, and Radiation

Whenever there is a temperature difference, heat transfer occurs. Heat transfer may happen rapidly, such as through a cooking pan, or slowly, such as through the walls of an insulated cooler.
There are three different heat transfer methods: conduction, convection, and radiation. At times, all three may happen simultaneously. See .
Conduction is heat transfer through direct physical contact. Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction. Sometimes, we try to control the conduction of heat to make ourselves more comfortable. Since the rate of heat transfer is different for different materials, we choose fabrics, such as a thick wool sweater, that slow down the transfer of heat away from our bodies in winter.
As you walk barefoot across the living room carpet, your feet feel relatively comfortable…until you step onto the kitchen’s tile floor. Since the carpet and tile floor are both at the same temperature, why does one feel colder than the other? This is explained by different rates of heat transfer: The tile material removes heat from your skin at a greater rate than the carpeting, which makes it feel colder.
Some materials simply conduct thermal energy faster than others. In general, metals (like copper, aluminum, gold, and silver) are good heat conductors, whereas materials like wood, plastic, and rubber are poor heat conductors.
 shows particles (either atoms or molecules) in two bodies at different temperatures. The (average) kinetic energy of a particle in the hot body is higher than in the colder body. If two particles collide, energy transfers from the particle with greater kinetic energy to the particle with less kinetic energy. When two bodies are in contact, many particle collisions occur, resulting in a net flux of heat from the higher-temperature body to the lower-temperature body. The heat flux depends on the temperature difference
  . Therefore, you will get a more severe burn from boiling water than from hot tap water.
Convection is heat transfer by the movement of a fluid. This type of heat transfer happens, for example, in a pot boiling on the stove, or in thunderstorms, where hot air rises up to the base of the clouds.
As the temperature of fluids increase, they expand and become less dense. For example,  could represent the wall of a balloon with different temperature gases inside the balloon than outside in the environment.  The hotter and thus faster moving gas particles inside the balloon strike the surface with more force than the cooler air outside, causing the balloon to expand. This decrease in density relative to its environment creates buoyancy (the tendency to rise). Convection is driven by buoyancy—hot air rises because it is less dense than the surrounding air.
Sometimes, we control the temperature of our homes or ourselves by controlling air movement. Sealing leaks around doors with weather stripping keeps out the cold wind in winter. The house in  and the pot of water on the stove in  are both examples of convection and buoyancy by human design. Ocean currents and large-scale atmospheric circulation transfer energy from one part of the globe to another, and are examples of natural convection.
Radiation is a form of heat transfer that occurs when electromagnetic radiation is emitted or absorbed. Electromagnetic radiation includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays, all of which have different wavelengths and amounts of energy (shorter wavelengths have higher frequency and more energy).
You can feel the heat transfer from a fire and from the sun. Similarly, you can sometimes tell that the oven is hot without touching its door or looking inside—it may just warm you as you walk by. Another example is thermal radiation from the human body; people are constantly emitting infrared radiation, which is not visible to the human eye, but is felt as heat.
Radiation is the only method of heat transfer where no medium is required, meaning that the heat doesn’t need to come into direct contact with or be transported by any matter. The space between Earth and the sun is largely empty, without any possibility of heat transfer by convection or conduction. Instead, heat is transferred by radiation, and Earth is warmed as it absorbs electromagnetic radiation emitted by the sun.
All objects absorb and emit electromagnetic radiation (see ). The rate of heat transfer by radiation depends mainly on the color of the object. Black is the most effective absorber and radiator, and white is the least effective. People living in hot climates generally avoid wearing black clothing, for instance. Similarly, black asphalt in a parking lot will be hotter than adjacent patches of grass on a summer day, because black absorbs better than green. The reverse is also true—black radiates better than green. On a clear summer night, the black asphalt will be colder than the green patch of grass, because black radiates energy faster than green. In contrast, white is a poor absorber and also a poor radiator. A white object reflects nearly all radiation, like a mirror.

### Solving Heat Transfer Problems

### Practice Problems

### Check Your Understanding

### Section Summary

1. Heat is thermal (internal) energy transferred due to a temperature difference.
2. The transfer of heat Q that leads to a change
   in the temperature of a body with mass m is
 , where c is the specific heat of the material.
3. Heat is transferred by three different methods: conduction, convection, and radiation.
4. Heat conduction is the transfer of heat between two objects in direct contact with each other.
5. Convection is heat transfer by the movement of mass.
6. Radiation is heat transfer by electromagnetic waves.

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# Thermal Energy, Heat, and Work

## Phase Change and Latent Heat

### Section Key Terms

### Phase Changes

So far, we have learned that adding thermal energy by heat increases the temperature of a substance. But surprisingly, there are situations where adding energy does not change the temperature of a substance at all! Instead, the additional thermal energy acts to loosen bonds between molecules or atoms and causes a phase change. Because this energy enters or leaves a system during a phase change without causing a temperature change in the system, it is known as latent heat (latent means hidden).
The three phases of matter that you frequently encounter are solid, liquid and gas (see ). Solid has the least energetic state; atoms in solids are in close contact, with forces between them that allow the particles to vibrate but not change position with neighboring particles. (These forces can be thought of as springs that can be stretched or compressed, but not easily broken.)
Liquid has a more energetic state, in which particles can slide smoothly past one another and change neighbors, although they are still held together by their mutual attraction.
Gas has a more energetic state than liquid, in which particles are broken free of their bonds. Particles in gases are separated by distances that are large compared with the size of the particles.
The most energetic state of all is plasma. Although you may not have heard much about plasma, it is actually the most common state of matter in the universe—stars are made up of plasma, as is lightning. The plasma state is reached by heating a gas to the point where particles are pulled apart, separating the electrons from the rest of the particle. This produces an ionized gas that is a combination of the negatively charged free electrons and positively charged ions, known as plasma.
During a phase change, matter changes from one phase to another, either through the addition of energy by heat and the transition to a more energetic state, or from the removal of energy by heat and the transition to a less energetic state.
Phase changes to a more energetic state include the following:
1. Melting—Solid to liquid
2. Vaporization—Liquid to gas (included boiling and evaporation)
3. Sublimation—Solid to gas
4. IonizationGas to plasma
Phase changes to a less energetic state are as follows:
1. Condensation—Gas to liquid
2. Freezing—Liquid to solid
3. Recombination—Plasma to gas
4. DepositionGas to solid
Energy is required to melt a solid because the bonds between the particles in the solid must be broken. Since the energy involved in a phase changes is used to break bonds, there is no increase in the kinetic energies of the particles, and therefore no rise in temperature. Similarly, energy is needed to vaporize a liquid to overcome the attractive forces between particles in the liquid. There is no temperature change until a phase change is completed. The temperature of a cup of soda and ice that is initially at 0
   stays at 0
   until all of the ice has melted. In the reverse of these processes—freezing and condensation—energy is released from the latent heat (see ).
The heat, Q, required to change the phase of a sample of mass m is

   (for melting/freezing),

   (for vaporization/condensation),
where
   is the latent heat of fusion, and
   is the latent heat of vaporization. The latent heat of fusion is the amount of heat needed to cause a phase change between solid and liquid. The latent heat of vaporization is the amount of heat needed to cause a phase change between liquid and gas.
   and
   are coefficients that vary from substance to substance, depending on the strength of intermolecular forces, and both have standard units of J/kg. See  for values of
   and
   of different substances.
Let’s consider the example of adding heat to ice to examine its transitions through all three phases—solid to liquid to gas. A phase diagram indicating the temperature changes of water as energy is added is shown in . The ice starts out at −20
 , and its temperature rises linearly, absorbing heat at a constant rate until it reaches 0
   Once at this temperature, the ice gradually melts, absorbing 334 kJ/kg. The temperature remains constant at 0
   during this phase change. Once all the ice has melted, the temperature of the liquid water rises, absorbing heat at a new constant rate. At 100
 , the water begins to boil and the temperature again remains constant while the water absorbs 2256 kJ/kg during this phase change. When all the liquid has become steam, the temperature rises again at a constant rate.
We have seen that vaporization requires heat transfer to a substance from its surroundings. Condensation is the reverse process, where heat in transferred away from a substance to its surroundings. This release of latent heat increases the temperature of the surroundings. Energy must be removed from the condensing particles to make a vapor condense. This is why condensation occurs on cold surfaces: the heat transfers energy away from the warm vapor to the cold surface. The energy is exactly the same as that required to cause the phase change in the other direction, from liquid to vapor, and so it can be calculated from
  . Latent heat is also released into the environment when a liquid freezes, and can be calculated from
  .

### Solving Thermal Energy Problems with Phase Changes

### Practice Problems

### Check Your Understanding

### Section Summary

1. Most substances have four distinct phases: solid, liquid, gas, and plasma.
2. Gas is the most energetic state and solid is the least.
3. During a phase change, a substance undergoes transition to a higher energy state when heat is added, or to a lower energy state when heat is removed.
4. Heat is added to a substance during melting and vaporization.
5. Latent heat is released by a substance during condensation and freezing.
6. Phase changes occur at fixed temperatures called boiling and freezing (or melting) points for a given substance.

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# Thermodynamics

## Introduction

Energy can be transferred to or from a system, either through a temperature difference between it and another system (i.e., by heat) or by exerting a force through a distance (work). In these ways, energy can be converted into other forms of energy in other systems. For example, a car engine burns fuel for heat transfer into a gas. Work is done by the gas as it exerts a force through a distance by pushing a piston outward. This work converts the energy into a variety of other forms—into an increase in the car’s kinetic or gravitational potential energy; into electrical energy to run the spark plugs, radio, and lights; and back into stored energy in the car’s battery. But most of the thermal energy transferred by heat from the fuel burning in the engine does not do work on the gas. Instead, much of this energy is released into the surroundings at lower temperature (i.e., lost through heat), which is quite inefficient. Car engines are only about 25 to 30 percent efficient. This inefficiency leads to increased fuel costs, so there is great interest in improving fuel efficiency. However, it is common knowledge that modern gasoline engines cannot be made much more efficient. The same is true about the conversion to electrical energy in large power stations, whether they are coal, oil, natural gas, or nuclear powered. Why is this the case?
The answer lies in the nature of heat. Basic physical laws govern how heat transfer for doing work takes place and limit the maximum possible efficiency of the process. This chapter will explore these laws as well their applications to everyday machines. These topics are part of thermodynamics—the study of heat and its relationship to doing work.


# Thermodynamics

## Zeroth Law of Thermodynamics: Thermal Equilibrium

### Section Key Terms

We learned in the previous chapter that when two objects (or systems) are in contact with one another, heat will transfer thermal energy from the object at higher temperature to the one at lower temperature until they both reach the same temperature. The objects are then in thermal equilibrium, and no further temperature changes will occur if they are isolated from other systems. The systems interact and change because their temperatures are different, and the changes stop once their temperatures are the same. Thermal equilibrium is established when two bodies are in thermal contact with each other—meaning heat transfer (i.e., the transfer of energy by heat) can occur between them. If two systems cannot freely exchange energy, they will not reach thermal equilibrium. (It is fortunate that empty space stands between Earth and the sun, because a state of thermal equilibrium with the sun would be too toasty for life on this planet!)
If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C. This statement may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the zeroth law of thermodynamics.
You may be wondering at this point, why the wacky name? Shouldn’t this be called the first law of thermodynamics rather than the zeroth? The explanation is that this law was discovered after the first and second laws of thermodynamics but is so fundamental that scientists decided it should logically come first.
As an example of the zeroth law in action, consider newborn babies in neonatal intensive-care units in hospitals. Prematurely born or sick newborns are placed in special incubators. These babies have very little covering while in the incubators, so to an observer, they look as though they may not be warm enough. However, inside the incubator, the temperature of the air, the cot, and the baby are all the same—that is, they are in thermal equilibrium. The ambient temperature is just high enough to keep the baby safe and comfortable.

### Check Your Understanding

### Section Summary

1. Systems are in thermal equilibrium when they have the same temperature.
2. Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
3. The zeroth law of thermodynamics states that when two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.

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# Thermodynamics

## First law of Thermodynamics: Thermal Energy and Work

### Section Key Terms

### Pressure, Volume, Temperature, and the Ideal Gas Law

Before covering the first law of thermodynamics, it is first important to understand the relationship between pressure, volume, and temperature. Pressure, P, is defined as
where F is a force applied to an area, A, that is perpendicular to the force.
Depending on the area over which it is exerted, a given force can have a significantly different effect, as shown in .
The SI unit for pressure is the pascal, where

Pressure is defined for all states of matter but is particularly important when discussing fluids (such as air). You have probably heard the word pressure being used in relation to blood (high or low blood pressure) and in relation to the weather (high- and low-pressure weather systems). These are only two of many examples of pressures in fluids.
The relationship between the pressure, volume, and temperature for an ideal gas is given by the ideal gas law. A gas is considered ideal at low pressure and fairly high temperature, and forces between its component particles can be ignored. The ideal gas law states that
where P is the pressure of a gas, V is the volume it occupies, N is the number of particles (atoms or molecules) in the gas, and T is its absolute temperature. The constant k is called the Boltzmann constant and has the value
   For the purposes of this chapter, we will not go into calculations using the ideal gas law. Instead, it is important for us to notice from the equation that the following are true for a given mass of gas:
1. When volume is constant, pressure is directly proportional to temperature.
2. When temperature is constant, pressure is inversely proportional to volume.
3. When pressure is constant, volume is directly proportional to temperature.
This last point describes thermal expansion—the change in size or volume of a given mass with temperature. What is the underlying cause of thermal expansion? An increase in temperature means that there’s an increase in the kinetic energy of the individual atoms. Gases are especially affected by thermal expansion, although liquids expand to a lesser extent with similar increases in temperature, and even solids have minor expansions at higher temperatures. This is why railroad tracks and bridges have expansion joints that allow them to freely expand and contract with temperature changes.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into a deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If you continue to pump air into tire (which now has a nearly constant volume), the pressure increases with increasing temperature (see ).

### Pressure–Volume Work

Pressure–volume work is the work that is done by the compression or expansion of a fluid. Whenever there is a change in volume and external pressure remains constant, pressure–volume work is taking place. During a compression, a decrease in volume increases the internal pressure of a system as work is done on the system. During an expansion (), an increase in volume decreases the internal pressure of a system as the system does work.
Recall that the formula for work is
   We can rearrange the definition of pressure,
   to get an expression for force in terms of pressure.
Substituting this expression for force into the definition of work, we get
Because area multiplied by displacement is the change in volume,

, the mathematical expression for pressure–volume work is
Just as we say that work is force acting over a distance, for fluids, we can say that work is the pressure acting through the change in volume. For pressure–volume work, pressure is analogous to force, and volume is analogous to distance in the traditional definition of work.

### The First Law of Thermodynamics

Heat (Q) and work (W) are the two ways to add or remove energy from a system. The processes are very different. Heat is driven by temperature differences, while work involves a force exerted through a distance. Nevertheless, heat and work can produce identical results. For example, both can cause a temperature increase. Heat transfers energy into a system, such as when the sun warms the air in a bicycle tire and increases the air’s temperature. Similarly, work can be done on the system, as when the bicyclist pumps air into the tire. Once the temperature increase has occurred, it is impossible to tell whether it was caused by heat or work. Heat and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy, U, of a system.
Internal energy is the sum of the kinetic and potential energies of a system’s atoms and molecules. It can be divided into many subcategories, such as thermal and chemical energy, and depends only on the state of a system (that is, P, V, and T), not on how the energy enters or leaves the system.
In order to understand the relationship between heat, work, and internal energy, we use the first law of thermodynamics. The first law of thermodynamics applies the conservation of energy principle to systems where heat and work are the methods of transferring energy into and out of the systems. It can also be used to describe how energy transferred by heat is converted and transferred again by work.
The first law of thermodynamics states that the change in internal energy of a closed system equals the net heat transfer into the system minus the net work done by the system. In equation form, the first law of thermodynamics is
Here,

 is the change in internal energy, U, of the system. As shown in , Q is the net heat transferred into the system—that is, Q is the sum of all heat transfers into and out of the system. W is the net work done by the system—that is, W is the sum of all work done on or by the system. By convention, if Q is positive, then there is a net heat transfer into the system; if W is positive, then there is net work done by the system. So positive Q adds energy to the system by heat, and positive W takes energy from the system by work. Note that if heat transfers more energy into the system than that which is done by work, the difference is stored as internal energy.
It follows also that negative Q indicates that energy is transferred away from the system by heat and so decreases the system’s internal energy, whereas negative W is work done on the system, which increases the internal energy.

### Solving Problems Involving the First Law of Thermodynamics

### Practice Problems

### Check Your Understanding

### Section Summary

1. Pressure is the force per unit area over which the force is applied perpendicular to the area.
2. Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
3. The ideal gas law relates the pressure and volume of a gas to the number of gas particles (atoms or molecules) and the absolute temperature of the gas.
4. Heat and work are the two distinct methods of energy transfer.
5. Heat is energy transferred solely due to a temperature difference.
6. The first law of thermodynamics is given as

, where
   is the change in internal energy of a system, Q is the net energy transfer into the system by heat (the sum of all transfers by heat into and out of the system), and W is the net work done by the system (the sum of all energy transfers by work out of or into the system).
7. Both Q and W represent energy in transit; only
   represents an independent quantity of energy capable of being stored.
8. The internal energy U of a system depends only on the state of the system, and not how it reached that state.

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# Thermodynamics

## Second Law of Thermodynamics: Entropy

### Section Key Terms

### Entropy

Recall from the chapter introduction that it is not even theoretically possible for engines to be 100 percent efficient. This phenomenon is explained by the second law of thermodynamics, which relies on a concept known as entropy. Entropy is a measure of the disorder of a system. Entropy also describes how much energy is not available to do work. The more disordered a system and higher the entropy, the less of a system's energy is available to do work.
Although all forms of energy can be used to do work, it is not possible to use the entire available energy for work. Consequently, not all energy transferred by heat can be converted into work, and some of it is lost in the form of waste heat—that is, heat that does not go toward doing work. The unavailability of energy is important in thermodynamics; in fact, the field originated from efforts to convert heat to work, as is done by engines.
The equation for the change in entropy,
 , is
where Q is the heat that transfers energy during a process, and T is the absolute temperature at which the process takes place.
Q is positive for energy transferred into the system by heat and negative for energy transferred out of the system by heat. In SI, entropy is expressed in units of joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take T to be the average temperature in order to avoid trickier math (calculus).

### Second Law of Thermodynamics

Have you ever played the card game 52 pickup? If so, you have been on the receiving end of a practical joke and, in the process, learned a valuable lesson about the nature of the universe as described by the second law of thermodynamics. In the game of 52 pickup, the prankster tosses an entire deck of playing cards onto the floor, and you get to pick them up. In the process of picking up the cards, you may have noticed that the amount of work required to restore the cards to an orderly state in the deck is much greater than the amount of work required to toss the cards and create the disorder.
The second law of thermodynamics states that the total entropy of a system either increases or remains constant in any spontaneous process; it never decreases. An important implication of this law is that heat transfers energy spontaneously from higher- to lower-temperature objects, but never spontaneously in the reverse direction. This is because entropy increases for heat transfer of energy from hot to cold (). Because the change in entropy is Q/T, there is a larger change in
   at lower temperatures (smaller T). The decrease in entropy of the hot (larger T) object is therefore less than the increase in entropy of the cold (smaller T) object, producing an overall increase in entropy for the system.
Another way of thinking about this is that it is impossible for any process to have, as its sole result, heat transferring energy from a cooler to a hotter object. Heat cannot transfer energy spontaneously from colder to hotter, because the entropy of the overall system would decrease.
Suppose we mix equal masses of water that are originally at two different temperatures, say

 and
  . The result will be water at an intermediate temperature of
  . Three outcomes have resulted: entropy has increased, some energy has become unavailable to do work, and the system has become less orderly. Let us think about each of these results.
First, why has entropy increased? Mixing the two bodies of water has the same effect as the heat transfer of energy from the higher-temperature substance to the lower-temperature substance. The mixing decreases the entropy of the hotter water but increases the entropy of the colder water by a greater amount, producing an overall increase in entropy.
Second, once the two masses of water are mixed, there is no more temperature difference left to drive energy transfer by heat and therefore to do work. The energy is still in the water, but it is now unavailable to do work.
Third, the mixture is less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a broad distribution of molecular speeds, the average of which yields an intermediate temperature.
These three results—entropy, unavailability of energy, and disorder—not only are related but are, in fact, essentially equivalent. Heat transfer of energy from hot to cold is related to the tendency in nature for systems to become disordered and for less energy to be available for use as work.
Based on this law, what cannot happen? A cold object in contact with a hot one never spontaneously transfers energy by heat to the hot object, getting colder while the hot object gets hotter. Nor does a hot, stationary automobile ever spontaneously cool off and start moving.
Another example is the expansion of a puff of gas introduced into one corner of a vacuum chamber. The gas expands to fill the chamber, but it never regroups on its own in the corner. The random motion of the gas molecules could take them all back to the corner, but this is never observed to happen ().
We've explained that heat never transfers energy spontaneously from a colder to a hotter object. The key word here is spontaneously. If we do work on a system, it is possible to transfer energy by heat from a colder to hotter object. We'll learn more about this in the next section, covering refrigerators as one of the applications of the laws of thermodynamics.
Sometimes people misunderstand the second law of thermodynamics, thinking that based on this law, it is impossible for entropy to decrease at any particular location. But, it actually is possible for the entropy of one part of the universe to decrease, as long as the total change in entropy of the universe increases. In equation form, we can write this as

Based on this equation, we see that
   can be negative as long as
   is positive and greater in magnitude.
How is it possible for the entropy of a system to decrease? Energy transfer is necessary. If you pick up marbles that are scattered about the room and put them into a cup, your work has decreased the entropy of that system. If you gather iron ore from the ground and convert it into steel and build a bridge, your work has decreased the entropy of that system. Energy coming from the sun can decrease the entropy of local systems on Earth—that is,
   is negative. But the overall entropy of the rest of the universe increases by a greater amount—that is,
   is positive and greater in magnitude. In the case of the iron ore, although you made the system of the bridge and steel more structured, you did so at the expense of the universe. Altogether, the entropy of the universe is increased by the disorder created by digging up the ore and converting it to steel. Therefore,
and the second law of thermodynamics is not violated.
Every time a plant stores some solar energy in the form of chemical potential energy, or an updraft of warm air lifts a soaring bird, Earth experiences local decreases in entropy as it uses part of the energy transfer from the sun into deep space to do work. There is a large total increase in entropy resulting from this massive energy transfer. A small part of this energy transfer by heat is stored in structured systems on Earth, resulting in much smaller, local decreases in entropy.

### Solving Problems Involving the Second Law of Thermodynamics

Entropy is related not only to the unavailability of energy to do work; it is also a measure of disorder. For example, in the case of a melting block of ice, a highly structured and orderly system of water molecules changes into a disorderly liquid, in which molecules have no fixed positions (). There is a large increase in entropy for this process, as we'll see in the following worked example.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Entropy is a measure of a system's disorder: the greater the disorder, the larger the entropy.
2. Entropy is also the reduced availability of energy to do work.
3. The second law of thermodynamics states that, for any spontaneous process, the total entropy of a system either increases or remains constant; it never decreases.
4. Heat transfers energy spontaneously from higher- to lower-temperature bodies, but never spontaneously in the reverse direction.

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# Thermodynamics

## Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators

### Section Key Terms

### Heat Engines, Heat Pumps, and Refrigerators

In this section, we’ll explore how heat engines, heat pumps, and refrigerators operate in terms of the laws of thermodynamics.
One of the most important things we can do with heat is to use it to do work for us. A heat engine does exactly this—it makes use of the properties of thermodynamics to transform heat into work. Gasoline and diesel engines, jet engines, and steam turbines that generate electricity are all examples of heat engines.
 illustrates one of the ways in which heat transfers energy to do work. Fuel combustion releases chemical energy that heat transfers throughout the gas in a cylinder. This increases the gas temperature, which in turn increases the pressure of the gas and, therefore, the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Thus, heat transfer of energy to the gas in the cylinder results in work being done.
To repeat this process, the piston needs to be returned to its starting point. Heat now transfers energy from the gas to the surroundings, so that the gas’s pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance.
A cyclical process brings a system, such as the gas in a cylinder, back to its original state at the end of every cycle. All heat engines use cyclical processes.
Heat engines do work by using part of the energy transferred by heat from some source. As shown in , heat transfers energy,
 , from the high-temperature object (or hot reservoir), whereas heat transfers unused energy,
 , into the low-temperature object (or cold reservoir), and the work done by the engine is W. In physics, a reservoir is defined as an infinitely large mass that can take in or put out an unlimited amount of heat, depending upon the needs of the system. The temperature of the hot reservoir is
   and the temperature of the cold reservoir is
  .
As noted, a cyclical process brings the system back to its original condition at the end of every cycle. Such a system’s internal energy, U, is the same at the beginning and end of every cycle—that is,

. The first law of thermodynamics states that
   where Q is the net heat transfer during the cycle, and W is the net work done by the system. The net heat transfer is the energy transferred in by heat from the hot reservoir minus the amount that is transferred out to the cold reservoir (
  ). Because there is no change in internal energy for a complete cycle (

), we have
so that
Therefore, the net work done by the system equals the net heat into the system, or
for a cyclical process.
Because the hot reservoir is heated externally, which is an energy-intensive process, it is important that the work be done as efficiently as possible. In fact, we want W to equal
 , and for there to be no heat to the environment (that is,
  ). Unfortunately, this is impossible. According to the second law of thermodynamics, heat engines cannot have perfect conversion of heat into work. Recall that entropy is a measure of the disorder of a system, which is also how much energy is unavailable to do work. The second law of thermodynamics requires that the total entropy of a system either increases or remains constant in any process. Therefore, there is a minimum amount of
   that cannot be used for work. The amount of heat rejected to the cold reservoir,
   depends upon the efficiency of the heat engine. The smaller the increase in entropy,

, the smaller the value of
 , and the more heat energy is available to do work.
Heat pumps, air conditioners, and refrigerators utilize heat transfer of energy from low to high temperatures, which is the opposite of what heat engines do. Heat transfers energy
   from a cold reservoir and delivers energy
   into a hot one. This requires work input, W, which produces a transfer of energy by heat. Therefore, the total heat transfer to the hot reservoir is
The purpose of a heat pump is to transfer energy by heat to a warm environment, such as a home in the winter. The great advantage of using a heat pump to keep your home warm rather than just burning fuel in a fireplace or furnace is that a heat pump supplies
  . Heat
   comes from the outside air, even at a temperature below freezing, to the indoor space. You only pay for W, and you get an additional heat transfer of
   from the outside at no cost. In many cases, at least twice as much energy is transferred to the heated space as is used to run the heat pump. When you burn fuel to keep warm, you pay for all of it. The disadvantage to a heat pump is that the work input (required by the second law of thermodynamics) is sometimes more expensive than simply burning fuel, especially if the work is provided by electrical energy.
The basic components of a heat pump are shown in . A working fluid, such as a refrigerant, is used. In the outdoor coils (the evaporator), heat
   enters the working fluid from the cold outdoor air, turning it into a gas.
The electrically driven compressor (work input W) raises the temperature and pressure of the gas and forces it into the condenser coils that are inside the heated space. Because the temperature of the gas is higher than the temperature inside the room, heat transfers energy to the room, and the gas condenses into a liquid. The liquid then flows back through an expansion (pressure-reducing) valve. The liquid, having been cooled through expansion, returns to the outdoor evaporator coils to resume the cycle.
The quality of a heat pump is judged by how much energy is transferred by heat into the warm space (
  ) compared with how much input work (W) is required.
Air conditioners and refrigerators are designed to cool substances by transferring energy by heat
   out of a cool environment to a warmer one, where heat
   is given up. In the case of a refrigerator, heat is moved out of the inside of the fridge into the surrounding room. For an air conditioner, heat is transferred outdoors from inside a home. Heat pumps are also often used in a reverse setting to cool rooms in the summer.
As with heat pumps, work input is required for heat transfer of energy from cold to hot. The quality of air conditioners and refrigerators is judged by how much energy is removed by heat
   from a cold environment, compared with how much work, W, is required. So, what is considered the energy benefit in a heat pump, is considered waste heat in a refrigerator.

### Thermal Efficiency

In the conversion of energy into work, we are always faced with the problem of getting less out than we put in. The problem is that, in all processes, there is some heat
   that transfers energy to the environment—and usually a very significant amount at that. A way to quantify how efficiently a machine runs is through a quantity called thermal efficiency.
We define thermal efficiency, Eff, to be the ratio of useful energy output to the energy input (or, in other words, the ratio of what we get to what we spend). The efficiency of a heat engine is the output of net work, W, divided by heat-transferred energy,
 , into the engine; that is

An efficiency of 1, or 100 percent, would be possible only if there were no heat to the environment (

).

### Solving Thermal Efficiency Problems

### Practice Problems

### Check Your Understanding

### Section Summary

1. Heat engines use the heat transfer of energy to do work.
2. Cyclical processes are processes that return to their original	state at the end of every cycle.
3. The thermal efficiency of a heat engine is the ratio of work output divided by the amount of energy input.
4. The amount of work a heat engine can do is determined by the net heat transfer of energy during a cycle; more waste heat leads to less work output.
5. Heat pumps draw energy by heat from cold outside air and use it to heat an interior room.
6. A refrigerator is a type of heat pump; it takes energy from the warm air from the inside compartment and transfers it to warmer exterior air.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Performance Task

### Test Prep Extended Response


# Waves and Their Properties

## Introduction

Recall from the chapter on Motion in Two Dimensions that oscillations—the back-and-forth movement between two points—involve force and energy. Some oscillations create waves, such as the sound waves created by plucking a guitar string. Other examples of waves include earthquakes and visible light. Even subatomic particles, such as electrons, can behave like waves. You can make water waves in a swimming pool by slapping the water with your hand. Some of these waves, such as water waves, are visible; others, such as sound waves, are not. But every wave is a disturbance that moves from its source and carries energy. In this chapter, we will learn about the different types of waves, their properties, and how they interact with one another.


# Waves and Their Properties

## Types of Waves

### Section Key Terms

### Mechanical Waves

What do we mean when we say something is a wave? A wave is a disturbance that travels or propagates from the place where it was created. Waves transfer energy from one place to another, but they do not necessarily transfer any mass. Light, sound, and waves in the ocean are common examples of waves. Sound and water waves are mechanical waves; meaning, they require a medium to travel through. The medium may be a solid, a liquid, or a gas, and the speed of the wave depends on the material properties of the medium through which it is traveling. However, light is not a mechanical wave; it can travel through a vacuum such as the empty parts of outer space.
A familiar wave that you can easily imagine is the water wave. For water waves, the disturbance is in the surface of the water, an example of which is the disturbance created by a rock thrown into a pond or by a swimmer splashing the water surface repeatedly. For sound waves, the disturbance is caused by a change in air pressure, an example of which is when the oscillating cone inside a speaker creates a disturbance. For earthquakes, there are several types of disturbances, which include the disturbance of Earth’s surface itself and the pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves. Because water waves are common and visible, visualizing water waves may help you in studying other types of waves, especially those that are not visible.
Water waves have characteristics common to all waves, such as amplitude, period, frequency, and energy, which we will discuss in the next section.

### Pulse Waves and Periodic Waves

If you drop a pebble into the water, only a few waves may be generated before the disturbance dies down, whereas in a wave pool, the waves are continuous. A pulse wave is a sudden disturbance in which only one wave or a few waves are generated, such as in the example of the pebble. Thunder and explosions also create pulse waves. A periodic wave repeats the same oscillation for several cycles, such as in the case of the wave pool, and is associated with simple harmonic motion. Each particle in the medium experiences simple harmonic motion in periodic waves by moving back and forth periodically through the same positions.
Consider the simplified water wave in . This wave is an up-and-down disturbance of the water surface, characterized by a sine wave pattern. The uppermost position is called the crest and the lowest is the trough. It causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs pass under the bird.

### Longitudinal Waves and Transverse Waves

Mechanical waves are categorized by their type of motion and fall into any of two categories: transverse or longitudinal. Note that both transverse and longitudinal waves can be periodic. A transverse wave propagates so that the disturbance is perpendicular to the direction of propagation. An example of a transverse wave is shown in , where a woman moves a toy spring up and down, generating waves that propagate away from herself in the horizontal direction while disturbing the toy spring in the vertical direction.
In contrast, in a longitudinal wave, the disturbance is parallel to the direction of propagation.  shows an example of a longitudinal wave, where the woman now creates a disturbance in the horizontal direction—which is the same direction as the wave propagation—by stretching and then compressing the toy spring.
Waves may be transverse, longitudinal, or a combination of the two. The waves on the strings of musical instruments are transverse (as shown in ), and so are electromagnetic waves, such as visible light. Sound waves in air and water are longitudinal. Their disturbances are periodic variations in pressure that are transmitted in fluids.
Sound in solids can be both longitudinal and transverse. Essentially, water waves are also a combination of transverse and longitudinal components, although the simplified water wave illustrated in  does not show the longitudinal motion of the bird.
Earthquake waves under Earth’s surface have both longitudinal and transverse components as well. The longitudinal waves in an earthquake are called pressure or P-waves, and the transverse waves are called shear or S-waves. These components have important individual characteristics; for example, they propagate at different speeds. Earthquakes also have surface waves that are similar to surface waves on water.

### Check Your Understanding

### Section Summary

1. A wave is a disturbance that moves from the point of creation and carries energy but not mass.
2. Mechanical waves must travel through a medium.
3. Sound waves, water waves, and earthquake waves are all examples of mechanical waves.
4. Light is not a mechanical wave since it can travel through a vacuum.
5. A periodic wave is a wave that repeats for several cycles, whereas a pulse wave has only one crest or a few crests and is associated with a sudden disturbance.
6. Periodic waves are associated with simple harmonic motion.
7. A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

### Concept Items

### Critical Thinking

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Waves and Their Properties

## Wave Properties: Speed, Amplitude, Frequency, and Period

### Section Key Terms

### Wave Variables

In the chapter on motion in two dimensions, we defined the following variables to describe harmonic motion:
1. Amplitude—maximum displacement from the equilibrium position of an object oscillating around such equilibrium position
2. Frequency—number of events per unit of time
3. Period—time it takes to complete one oscillation
For waves, these variables have the same basic meaning. However, it is helpful to word the definitions in a more specific way that applies directly to waves:
1. Amplitude—distance between the resting position and the maximum displacement of the wave
2. Frequency—number of waves passing by a specific point per second
3. Period—time it takes for one wave cycle to complete
In addition to amplitude, frequency, and period, their wavelength and wave velocity also characterize waves. The wavelength
   is the distance between adjacent identical parts of a wave, parallel to the direction of propagation. The wave velocity
   is the speed at which the disturbance moves.
Consider the periodic water wave in . Its wavelength is the distance from crest to crest or from trough to trough. The wavelength can also be thought of as the distance a wave has traveled after one complete cycle—or one period. The time for one complete up-and-down motion is the simple water wave’s period T. In the figure, the wave itself moves to the right with a wave velocity vw. Its amplitude X is the distance between the resting position and the maximum displacement—either the crest or the trough—of the wave. It is important to note that this movement of the wave is actually the disturbance moving to the right, not the water itself; otherwise, the bird would move to the right. Instead, the seagull bobs up and down in place as waves pass underneath, traveling a total distance of 2X in one cycle. However, as mentioned in the text feature on surfing, actual ocean waves are more complex than this simplified example.

### The Relationship between Wave Frequency, Period, Wavelength, and Velocity

Since wave frequency is the number of waves per second, and the period is essentially the number of seconds per wave, the relationship between frequency and period is
or
just as in the case of harmonic motion of an object. We can see from this relationship that a higher frequency means a shorter period. Recall that the unit for frequency is hertz (Hz), and that 1 Hz is one cycle—or one wave—per second.
The speed of propagation vw is the distance the wave travels in a given time, which is one wavelength in a time of one period. In equation form, it is written as
or
From this relationship, we see that in a medium where vw is constant, the higher the frequency, the smaller the wavelength. See .
These fundamental relationships hold true for all types of waves. As an example, for water waves, vw is the speed of a surface wave; for sound, vw is the speed of sound; and for visible light, vw is the speed of light. The amplitude X is completely independent of the speed of propagation vw and depends only on the amount of energy in the wave.

### Solving Wave Problems

### Practice Problems

### Check Your Understanding

### Section Summary

1. A wave is a disturbance that moves from the point of creation at a wave velocity vw.
2. A wave has a wavelength
 , which is the distance between adjacent identical parts of the wave.
3. The wave velocity and the wavelength are related to the wave’s frequency and period by
   or

4. The time for one complete wave cycle is the period T.
5. The number of waves per unit time is the frequency ƒ.
6. The wave frequency and the period are inversely related to one another.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response