# Temperature, Kinetic Theory, and the Gas Laws

## Thermal Expansion of Solids and Liquids

### Learning Objectives

By the end of this section, you will be able to:
1. Define and describe thermal expansion.
2. Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
3. Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
4. Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.
The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.
What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.
What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature, an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.
 lists representative values of the coefficient of linear expansion, which may have units of  or 1/K. Because the size of a kelvin and a degree Celsius are the same, both  and  can be expressed in units of kelvins or degrees Celsius. The equation  is accurate for small changes in temperature and can be used for large changes in temperature if an average value of  is used.

### Thermal Expansion in Two and Three Dimensions

Objects expand in all dimensions, as illustrated in . That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger).
In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases) when it is at temperatures greater than . However, it expands with decreasing temperature when it is between  and  to . Water is densest at . (See .) Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to  it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of . If the temperature in the surface layer drops below , the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in  water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.

### Thermal Stress

Thermal stress is created by thermal expansion or contraction (see Elasticity: Stress and Strain for a discussion of stress and strain). Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful: for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes.
Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints. (See .) Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. (Pyrex® is less susceptible because of its small coefficient of thermal expansion.) Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water.
Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.
Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion closer to those of teeth.

### Section Summary

1. Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
2. Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
3. Linear thermal expansion is

where  is the change in length ,  is the change in temperature, and  is the coefficient of linear expansion, which varies slightly with temperature.

4. The change in area due to thermal expansion is

where  is the change in area.

5. The change in volume due to thermal expansion is

where  is the coefficient of volume expansion and . Thermal stress is created when thermal expansion is constrained.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## The Ideal Gas Law

### Learning Objectives

By the end of this section, you will be able to:
1. State the ideal gas law in terms of molecules and in terms of moles.
2. Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.
3. Use Avogadro’s number to convert between number of molecules and number of moles.
In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, , and oxygen, , are composed of two or more atoms. We will primarily use the term “molecule” in discussing a gas because the term can also be applied to monatomic gases, such as helium.)
Gases are easily compressed. We can see evidence of this in , where you will note that gases have the largest coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the same rate, or have the same . This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.
The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in . Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.
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 an initially 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 we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See .)
At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.
The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product  is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of . The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that  is the total number of atoms and molecules, independent of the type of gas.)
Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure  is essentially equal to atmospheric pressure, and the volume  increases in direct proportion to the number of atoms and molecules  put into the tire. Once the volume of the tire is constant, the equation  predicts that the pressure should increase in proportion to the number N of atoms and molecules.

### Moles and Avogadro’s Number

It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A mole (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called Avogadro’s number, in recognition of Italian scientist Amedeo Avogadro (1776–1856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro’s number is

### The Ideal Gas Law Restated Using Moles

A very common expression of the ideal gas law uses the number of moles, , rather than the number of atoms and molecules, . We start from the ideal gas law,
and multiply and divide the equation by Avogadro’s number . This gives
Note that  is the number of moles. We define the universal gas constant , and obtain the ideal gas law in terms of moles.
The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.

### The Ideal Gas Law and Energy

Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.
The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in  is . This term is roughly the amount of translational kinetic energy of  atoms or molecules at an absolute temperature , as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is , which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods—similar to what occurs in gasoline or steam engines and turbines.

### Test Prep for AP Courses

### Section Summary

1. The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
2. The ideal gas law can be written in terms of the number of molecules of gas:

where  is pressure,  is volume,  is temperature,  is number of molecules, and  is the Boltzmann constant

3. A mole is the number of atoms in a 12-g sample of carbon-12.
4. The number of molecules in a mole is called Avogadro’s number ,

5. A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
6. The ideal gas law can also be written and solved in terms of the number of moles of gas:

where  is number of moles and  is the universal gas constant,

7. The ideal gas law is generally valid at temperatures well above the boiling temperature.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature

### Learning Objectives

By the end of this section, you will be able to:
1. Express the ideal gas law in terms of molecular mass and velocity.
2. Define thermal energy.
3. Calculate the kinetic energy of a gas molecule, given its temperature.
4. Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
5. Describe the distribution of speeds of molecules in a gas.
We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
 shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Things Great and Small feature below. The following relationship is found:
where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.
What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the previous expression of the ideal gas law:
Equating the right-hand side of this equation with the right-hand side of  gives
We can get the average kinetic energy of a molecule, , from the right-hand side of the equation by canceling  and multiplying by 3/2. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature.
The average translational kinetic energy of a molecule, , is called thermal energy. The equation  is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.
It is sometimes useful to rearrange , and solve for the average speed of molecules in a gas in terms of temperature,
where  stands for root-mean-square (rms) speed.

### Distribution of Molecular Speeds

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This distribution is called the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and has since been confirmed experimentally. (See .) The distribution has a long tail, because a few molecules may go several times the rms speed. The most probable speed  is less than the rms speed .  shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.
The distribution of thermal speeds depends strongly on temperature. As temperature increases, the speeds are shifted to higher values and the distribution is broadened.
What is the implication of the change in distribution with temperature shown in  for humans? All other things being equal, if a person has a fever, they are likely to lose more water molecules, particularly from linings along moist cavities such as the lungs and mouth, creating a dry sensation in the mouth.

### Test Prep for AP Courses

### Section Summary

1. Kinetic theory is the atomistic description of gases as well as liquids and solids.
2. Kinetic theory models the properties of matter in terms of continuous random motion of atoms and molecules.
3. The ideal gas law can also be expressed as

where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.

4. Thermal energy is defined to be the average translational kinetic energy  of an atom or molecule.
5. The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.

    or

6. The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Phase Changes

### Learning Objectives

By the end of this section, you will be able to:
1. Interpret a phase diagram.
2. State Dalton’s law.
3. Identify and describe the triple point of a gas from its phase diagram.
4. Describe the state of equilibrium between a liquid and a gas, a liquid and a solid, and a gas and a solid.
Up to now, we have considered the behavior of ideal gases. Real gases are like ideal gases at high temperatures. At lower temperatures, however, the interactions between the molecules and their volumes cannot be ignored. The molecules are very close (condensation occurs) and there is a dramatic decrease in volume, as seen in . The substance changes from a gas to a liquid. When a liquid is cooled to even lower temperatures, it becomes a solid. The volume never reaches zero because of the finite volume of the molecules.
High pressure may also cause a gas to change phase to a liquid. Carbon dioxide, for example, is a gas at room temperature and atmospheric pressure, but becomes a liquid under sufficiently high pressure. If the pressure is reduced, the temperature drops and the liquid carbon dioxide solidifies into a snow-like substance at the temperature . Solid  is called “dry ice.” Another example of a gas that can be in a liquid phase is liquid nitrogen .  is made by liquefaction of atmospheric air (through compression and cooling). It boils at 77 K  at atmospheric pressure.  is useful as a refrigerant and allows for the preservation of blood, sperm, and other biological materials. It is also used to reduce noise in electronic sensors and equipment, and to help cool down their current-carrying wires. In dermatology,  is used to freeze and painlessly remove warts and other growths from the skin.

### PV Diagrams

We can examine aspects of the behavior of a substance by plotting a graph of pressure versus volume, called a . When the substance behaves like an ideal gas, the ideal gas law describes the relationship between its pressure and volume. That is,
Now, assuming the number of molecules and the temperature are fixed,
For example, the volume of the gas will decrease as the pressure increases. If you plot the relationship  on a  diagram, you find a hyperbola.  shows a graph of pressure versus volume. The hyperbolas represent ideal-gas behavior at various fixed temperatures, and are called isotherms. At lower temperatures, the curves begin to look less like hyperbolas—the gas is not behaving ideally and may even contain liquid. There is a critical point—that is, a critical temperature—above which liquid cannot exist. At sufficiently high pressure above the critical point, the gas will have the density of a liquid but will not condense. Carbon dioxide, for example, cannot be liquefied at a temperature above . Critical pressure is the minimum pressure needed for liquid to exist at the critical temperature.  lists representative critical temperatures and pressures.

### Phase Diagrams

The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so  graphs are called phase diagrams.  shows the phase diagram for water. Using the graph, if you know the pressure and temperature you can determine the phase of water. The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist (that is, they exist together in ratios, depending on pressure and temperature). For example, the boiling point of water is  at 1.00 atm. As the pressure increases, the boiling temperature rises steadily to  at a pressure of 218 atm. A pressure cooker (or even a covered pot) will cook food faster because the water can exist as a liquid at temperatures greater than  without all boiling away. The curve ends at a point called the critical point, because at higher temperatures the liquid phase does not exist at any pressure. The critical point occurs at the critical temperature, as you can see for water from . The critical temperature for oxygen is , so oxygen cannot be liquefied above this temperature.
Similarly, the curve between the solid and liquid regions in  gives the melting temperature at various pressures. For example, the melting point is  at 1.00 atm, as expected. Note that, at a fixed temperature, you can change the phase from solid (ice) to liquid (water) by increasing the pressure. Ice melts from pressure in the hands of a snowball maker. From the phase diagram, we can also say that the melting temperature of ice falls with increased pressure. When a car is driven over snow, the increased pressure from the tires melts the snowflakes; afterwards the water refreezes and forms an ice layer.
At sufficiently low pressures there is no liquid phase, but the substance can exist as either gas or solid. For water, there is no liquid phase at pressures below 0.00600 atm. The phase change from solid to gas is called sublimation. It accounts for large losses of snow pack that never make it into a river, the routine automatic defrosting of a freezer, and the freeze-drying process applied to many foods. Carbon dioxide, on the other hand, sublimates at standard atmospheric pressure of 1 atm. (The solid form of  is known as dry ice because it does not melt. Instead, it moves directly from the solid to the gas state.)
All three curves on the phase diagram meet at a single point, the triple point, where all three phases exist in equilibrium. For water, the triple point occurs at 273.16 K , and is a more accurate calibration temperature than the melting point of water at 1.00 atm, or 273.15 K . See  for the triple point values of other substances.

### Equilibrium

Liquid and gas phases are in equilibrium at the boiling temperature. (See .) If a substance is in a closed container at the boiling point, then the liquid is boiling and the gas is condensing at the same rate without net change in their relative amount. Molecules in the liquid escape as a gas at the same rate at which gas molecules stick to the liquid, or form droplets and become part of the liquid phase. The combination of temperature and pressure has to be “just right”; if the temperature and pressure are increased, equilibrium is maintained by the same increase of boiling and condensation rates.
One example of equilibrium between liquid and gas is that of water and steam at  and 1.00 atm. This temperature is the boiling point at that pressure, so they should exist in equilibrium. Why does an open pot of water at  boil completely away? The gas surrounding an open pot is not pure water: it is mixed with air. If pure water and steam are in a closed container at  and 1.00 atm, they would coexist—but with air over the pot, there are fewer water molecules to condense, and water boils. What about water at  and 1.00 atm? This temperature and pressure correspond to the liquid region, yet an open glass of water at this temperature will completely evaporate. Again, the gas around it is air and not pure water vapor, so that the reduced evaporation rate is greater than the condensation rate of water from dry air. If the glass is sealed, then the liquid phase remains. We call the gas phase a vapor when it exists, as it does for water at , at a temperature below the boiling temperature.

### Vapor Pressure, Partial Pressure, and Dalton’s Law

Vapor pressure is defined as the pressure at which a gas coexists with its solid or liquid phase. Vapor pressure is created by faster molecules that break away from the liquid or solid and enter the gas phase. The vapor pressure of a substance depends on both the substance and its temperature—an increase in temperature increases the vapor pressure.
Partial pressure is defined as the pressure a gas would create if it occupied the total volume available. In a mixture of gases, the total pressure is the sum of partial pressures of the component gases, assuming ideal gas behavior and no chemical reactions between the components. This law is known as Dalton’s law of partial pressures, after the English scientist John Dalton (1766–1844), who proposed it. Dalton’s law is based on kinetic theory, where each gas creates its pressure by molecular collisions, independent of other gases present. It is consistent with the fact that pressures add according to Pascal’s Principle. Thus water evaporates and ice sublimates when their vapor pressures exceed the partial pressure of water vapor in the surrounding mixture of gases. If their vapor pressures are less than the partial pressure of water vapor in the surrounding gas, liquid droplets or ice crystals (frost) form.

### Section Summary

1. Most substances have three distinct phases: gas, liquid, and solid.
2. Phase changes among the various phases of matter depend on temperature and pressure.
3. The existence of the three phases with respect to pressure and temperature can be described in a phase diagram.
4. Two phases coexist (i.e., they are in thermal equilibrium) at a set of pressures and temperatures. These are described as a line on a phase diagram.
5. The three phases coexist at a single pressure and temperature. This is known as the triple point and is described by a single point on a phase diagram.
6. A gas at a temperature below its boiling point is called a vapor.
7. Vapor pressure is the pressure at which a gas coexists with its solid or liquid phase.
8. Partial pressure is the pressure a gas would create if it existed alone.
9. Dalton’s law states that the total pressure is the sum of the partial pressures of all of the gases present.

### Conceptual Questions


# Temperature, Kinetic Theory, and the Gas Laws

## Humidity, Evaporation, and Boiling

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the relationship between vapor pressure of water and the capacity of air to hold water vapor.
2. Explain the relationship between relative humidity and partial pressure of water vapor in the air.
3. Calculate vapor density using vapor pressure.
4. Calculate humidity and dew point.
The expression “it’s not the heat, it’s the humidity” makes a valid point. We keep cool in hot weather by evaporating sweat from our skin and water from our breathing passages. Because evaporation is inhibited by high humidity, we feel hotter at a given temperature when the humidity is high. Low humidity, on the other hand, can cause discomfort from excessive drying of mucous membranes and can lead to an increased risk of respiratory infections.
When we say humidity, we really mean relative humidity. Relative humidity tells us how much water vapor is in the air compared with the maximum possible. At its maximum, denoted as saturation, the relative humidity is 100%, and evaporation is inhibited. The amount of water vapor in the air depends on temperature. For example, relative humidity rises in the evening, as air temperature declines, sometimes reaching the dew point. At the dew point temperature, relative humidity is 100%, and fog may result from the condensation of water droplets if they are small enough to stay in suspension. Conversely, if you wish to dry something (perhaps your hair), it is more effective to blow hot air over it rather than cold air, because, among other things, the increase in temperature increases the energy of the molecules, so the rate of evaporation increases.
The amount of water vapor in the air depends on the vapor pressure of water. The liquid and solid phases are continuously giving off vapor because some of the molecules have high enough speeds to enter the gas phase; see (a). If a lid is placed over the container, as in (b), evaporation continues, increasing the pressure, until sufficient vapor has built up for condensation to balance evaporation. Then equilibrium has been achieved, and the vapor pressure is equal to the partial pressure of water in the container. Vapor pressure increases with temperature because molecular speeds are higher as temperature increases.  gives representative values of water vapor pressure over a range of temperatures.
Relative humidity is related to the partial pressure of water vapor in the air. At 100% humidity, the partial pressure is equal to the vapor pressure, and no more water can enter the vapor phase. If the partial pressure is less than the vapor pressure, then evaporation will take place, as humidity is less than 100%. If the partial pressure is greater than the vapor pressure, condensation takes place. In everyday language, people sometimes refer to the capacity of air to “hold” water vapor, but this is not actually what happens. The water vapor is not held by the air. The amount of water in air is determined by the vapor pressure of water and has nothing to do with the properties of air.
We can use this and the data in  to do a variety of interesting calculations, keeping in mind that relative humidity is based on the comparison of the partial pressure of water vapor in air and ice.
Why does water boil at ? You will note from  that the vapor pressure of water at  is , or 1.00 atm. Thus, it can evaporate without limit at this temperature and pressure. But why does it form bubbles when it boils? This is because water ordinarily contains significant amounts of dissolved air and other impurities, which are observed as small bubbles of air in a glass of water. If a bubble starts out at the bottom of the container at , it contains water vapor (about 2.30%). The pressure inside the bubble is fixed at 1.00 atm (we ignore the slight pressure exerted by the water around it). As the temperature rises, the amount of air in the bubble stays the same, but the water vapor increases; the bubble expands to keep the pressure at 1.00 atm. At , water vapor enters the bubble continuously since the partial pressure of water is equal to 1.00 atm in equilibrium. It cannot reach this pressure, however, since the bubble also contains air and total pressure is 1.00 atm. The bubble grows in size and thereby increases the buoyant force. The bubble breaks away and rises rapidly to the surface—we call this boiling! (See .)

### Section Summary

1. Relative humidity is the fraction of water vapor in a gas compared to the saturation value.
2. The saturation vapor density can be determined from the vapor pressure for a given temperature.
3. Percent relative humidity is defined to be

4. The dew point is the temperature at which air reaches 100% relative humidity.

### Conceptual Questions

### Problems & Exercises


# Heat and Heat Transfer Methods

## Connection for AP® Courses

Heat is one of the most intriguing of the many ways in which energy goes from one place to another. Heat is often hidden, as it only exists when energy is in transit, and the methods of transfer are distinctly different. Energy transfer by heat touches every aspect of our lives, and helps us to understand how the universe functions. It explains the chill you feel on a clear breezy night, and why Earth’s core has yet to cool.
In this chapter, the ideas of temperature and thermal energy are used to examine and define heat, how heat is affected by the thermal properties of materials, and how the various mechanisms of heat transfer function. These topics are fundamental and practical, and will be returned to in future chapters. Big Idea 4 of the AP® Physics Curriculum Framework is supported by a discussion of how systems interact through energy transfer by heat and how this leads to changes in the energy of each system. Big Idea 5 is supported by exploration of the law of energy conservation that governs any changes in the energy of a system. Heat involves the transfer of thermal energy, or internal energy, from one system to another or to its surroundings, and so leads to a change in the internal energy of the system. This is analogous to the way work transfers mechanical energy to a mass to change its kinetic or potential energy. However, heat occurs as a spontaneous process, in which thermal energy is transferred from a higher temperature system to a lower temperature system.
Big Idea 1 is supported by examination of the internal structure of systems, which determines the nature of those energy changes and the mechanism of heat transfer. Macroscopic properties, such as heat capacity, latent heat, and thermal conductivity, depend on the arrangements and interactions of the atoms or molecules in a substance. The arrangement of these particles also determines whether thermal energy will be transferred through direct physical contact between systems (conduction), through the motion of fluids with different temperatures (convection), or through emission or absorption of radiation.
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material.
Essential Knowledge 1.E.3 Matter has a property called thermal conductivity.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.3 Energy is transferred spontaneously from a higher temperature system to a lower temperature system. The process through which energy is transferred between systems at different temperatures is called heat.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.6 Energy can be transferred by thermal processes involving differences in temperature; the amount of energy transferred in this process of transfer is called heat.


# Heat and Heat Transfer Methods

## Heat

### Learning Objectives

By the end of this section, you will be able to:
1. Define heat as transfer of energy.
In Work, Energy, and Energy Resources, we defined work as force times distance and learned that work done on an object changes its kinetic energy. We also saw in  Temperature, Kinetic Theory, and the Gas Laws that temperature is proportional to the (average) kinetic energy of atoms and molecules. We say that a thermal system has a certain internal energy: its internal energy is higher if the temperature is higher. If two objects at different temperatures are brought in contact with each other, energy is transferred from the hotter to the colder object until equilibrium is reached and the bodies reach thermal equilibrium (i.e., they are at the same temperature). No work is done by either object, because no force acts through a distance. The transfer of energy is caused by the temperature difference, and ceases once the temperatures are equal. These observations lead to the following definition of heat: Heat is the spontaneous transfer of energy due to a temperature difference.
As noted in Temperature, Kinetic Theory, and the Gas Laws, heat is often confused with temperature. For example, we may say the heat was unbearable, when we actually mean that the temperature was high. Heat is a form of energy, whereas temperature is not. The misconception arises because we are sensitive to the flow of heat, rather than the temperature.
Owing to the fact that heat is a form of energy, it has the SI unit of joule (J). The calorie (cal) is a common unit of energy, defined as the energy needed to change the temperature of 1.00 g of water by  —specifically, between  and , since there is a slight temperature dependence. Perhaps the most common unit of heat is the kilocalorie (kcal), which is the energy needed to change the temperature of 1.00 kg of water by . Since mass is most often specified in kilograms, kilocalorie is commonly used. Food calories (given the notation Cal, and sometimes called “big calorie”) are actually kilocalories (), a fact not easily determined from package labeling.

### Mechanical Equivalent of Heat

It is also possible to change the temperature of a substance by doing work. Work can transfer energy into or out of a system. This realization helped establish the fact that heat is a form of energy. James Prescott Joule (1818–1889) performed many experiments to establish the mechanical equivalent of heat—the work needed to produce the same effects as heat transfer. In terms of the units used for these two terms, the best modern value for this equivalence is
We consider this equation as the conversion between two different units of energy.
The figure above shows one of Joule’s most famous experimental setups for demonstrating the mechanical equivalent of heat. It demonstrated that work and heat can produce the same effects, and helped establish the principle of conservation of energy. Gravitational potential energy (PE) (work done by the gravitational force) is converted into kinetic energy (KE), and then randomized by viscosity and turbulence into increased average kinetic energy of atoms and molecules in the system, producing a temperature increase. His contributions to the field of thermodynamics were so significant that the SI unit of energy was named after him.
Heat added or removed from a system changes its internal energy and thus its temperature. Such a temperature increase is observed while cooking. However, adding heat does not necessarily increase the temperature. An example is melting of ice; that is, when a substance changes from one phase to another. Work done on the system or by the system can also change the internal energy of the system. Joule demonstrated that the temperature of a system can be increased by stirring. If an ice cube is rubbed against a rough surface, work is done by the frictional force. A system has a well-defined internal energy, but we cannot say that it has a certain “heat content” or “work content”. We use the phrase “heat transfer” to emphasize its nature.

### Test Prep for AP Courses

### Summary

1. Heat and work are the two distinct methods of energy transfer.
2. Heat is energy transferred solely due to a temperature difference.
3. Any energy unit can be used for heat transfer, and the most common are kilocalorie (kcal) and joule (J).
4. Kilocalorie is defined to be the energy needed to change the temperature of 1.00 kg of water between  and .
5. The mechanical equivalent of this heat transfer is

### Conceptual Questions


# Heat and Heat Transfer Methods

## Temperature Change and Heat Capacity

### Learning Objectives

By the end of this section, you will be able to:
1. Observe heat transfer and change in temperature and mass.
2. Calculate final temperature after heat transfer between two objects.
One of the major effects of heat transfer is temperature change: heating increases the temperature while cooling decreases it. We assume that there is no phase change and that no work is done on or by the system. Experiments show that the transferred heat depends on three factors—the change in temperature, the mass of the system, and the substance and phase of the substance.
The dependence on temperature change and mass are easily understood. Owing to the fact that the (average) kinetic energy of an atom or molecule is proportional to the absolute temperature, the internal energy of a system is proportional to the absolute temperature and the number of atoms or molecules. Owing to the fact that the transferred heat is equal to the change in the internal energy, the heat is proportional to the mass of the substance and the temperature change. The transferred heat also depends on the substance so that, for example, the heat necessary to raise the temperature is less for alcohol than for water. For the same substance, the transferred heat also depends on the phase (gas, liquid, or solid).
Values of specific heat must generally be looked up in tables, because there is no simple way to calculate them. In general, the specific heat also depends on the temperature.  lists representative values of specific heat for various substances. Except for gases, the temperature and volume dependence of the specific heat of most substances is weak. We see from this table that the specific heat of water is five times that of glass and ten times that of iron, which means that it takes five times as much heat to raise the temperature of water the same amount as for glass and ten times as much heat to raise the temperature of water as for iron. In fact, water has one of the largest specific heats of any material, which is important for sustaining life on Earth.
Note that  is an illustration of the mechanical equivalent of heat. Alternatively, the temperature increase could be produced by a blow torch instead of mechanically.

### Summary

1. The transfer of heat  that leads to a change  in the temperature of a body with mass  is , where  is the specific heat of the material. This relationship can also be considered as the definition of specific heat.

### Conceptual Questions

### Problems & Exercises


# Heat and Heat Transfer Methods

## Phase Change and Latent Heat

### Learning Objectives

By the end of this section, you will be able to:
1. Examine heat transfer.
2. Calculate final temperature from heat transfer.
So far we have discussed temperature change due to heat transfer. No temperature change occurs from heat transfer if ice melts and becomes liquid water (i.e., during a phase change). For example, consider water dripping from icicles melting on a roof warmed by the Sun. Conversely, water freezes in an ice tray cooled by lower-temperature surroundings.
Energy is required to melt a solid because the cohesive bonds between the molecules in the solid must be broken apart such that, in the liquid, the molecules can move around at comparable kinetic energies; thus, there is no rise in temperature. Similarly, energy is needed to vaporize a liquid, because molecules in a liquid interact with each other via attractive forces. There is no temperature change until a phase change is complete. The temperature of a cup of soda initially at  stays at  until all the ice has melted. Conversely, energy is released during freezing and condensation, usually in the form of thermal energy. Work is done by cohesive forces when molecules are brought together. The corresponding energy must be given off (dissipated) to allow them to stay together .
The energy involved in a phase change depends on two major factors: the number and strength of bonds or force pairs. The number of bonds is proportional to the number of molecules and thus to the mass of the sample. The strength of forces depends on the type of molecules. The heat  required to change the phase of a sample of mass  is given by
where the latent heat of fusion, , and latent heat of vaporization, , are material constants that are determined experimentally. See ().
Latent heat is measured in units of J/kg. Both  and  depend on the substance, particularly on the strength of its molecular forces as noted earlier.  and  are collectively called latent heat coefficients. They are latent, or hidden, because in phase changes, energy enters or leaves a system without causing a temperature change in the system; so, in effect, the energy is hidden.  lists representative values of  and , together with melting and boiling points.
The table shows that significant amounts of energy are involved in phase changes. Let us look, for example, at how much energy is needed to melt a kilogram of ice at  to produce a kilogram of water at . Using the equation for a change in temperature and the value for water from , we find that  is the energy to melt a kilogram of ice. This is a lot of energy as it represents the same amount of energy needed to raise the temperature of 1 kg of liquid water from  to . Even more energy is required to vaporize water; it would take 2256 kJ to change 1 kg of liquid water at the normal boiling point ( at atmospheric pressure) to steam (water vapor). This example shows that the energy for a phase change is enormous compared to energy associated with temperature changes without a phase change.
Phase changes can have a tremendous stabilizing effect even on temperatures that are not near the melting and boiling points, because evaporation and condensation (conversion of a gas into a liquid state) occur even at temperatures below the boiling point. Take, for example, the fact that air temperatures in humid climates rarely go above , which is because most heat transfer goes into evaporating water into the air. Similarly, temperatures in humid weather rarely fall below the dew point because enormous heat is released when water vapor condenses.
We examine the effects of phase change more precisely by considering adding heat into a sample of ice at  (). The temperature of the ice rises linearly, absorbing heat at a constant rate of  until it reaches . Once at this temperature, the ice begins to melt until all the ice has melted, absorbing 79.8 cal/g of heat. The temperature remains constant at  during this phase change. Once all the ice has melted, the temperature of the liquid water rises, absorbing heat at a new constant rate of . At , the water begins to boil and the temperature again remains constant while the water absorbs 539 cal/g of heat during this phase change. When all the liquid has become steam vapor, the temperature rises again, absorbing heat at a rate of .
Water can evaporate at temperatures below the boiling point. More energy is required than at the boiling point, because the kinetic energy of water molecules at temperatures below  is less than that at , hence less energy is available from random thermal motions. Take, for example, the fact that, at body temperature, perspiration from the skin requires a heat input of 2428 kJ/kg, which is about 10 percent higher than the latent heat of vaporization at . This heat comes from the skin, and thus provides an effective cooling mechanism in hot weather. High humidity inhibits evaporation, so that body temperature might rise, leaving unevaporated sweat on your brow.
We have seen that vaporization requires heat transfer to a liquid from the surroundings, so that energy is released by the surroundings. Condensation is the reverse process, increasing the temperature of the surroundings. This increase may seem surprising, since we associate condensation with cold objects—the glass in the figure, for example. However, energy must be removed from the condensing molecules to make a vapor condense. The energy is exactly the same as that required to make the phase change in the other direction, from liquid to vapor, and so it can be calculated from .
Sublimation is the transition from solid to vapor phase. You may have noticed that snow can disappear into thin air without a trace of liquid water, or the disappearance of ice cubes in a freezer. The reverse is also true: Frost can form on very cold windows without going through the liquid stage. A popular effect is the making of “smoke” from dry ice, which is solid carbon dioxide. Sublimation occurs because the equilibrium vapor pressure of solids is not zero. Certain air fresheners use the sublimation of a solid to inject a perfume into the room. Moth balls are a slightly toxic example of a phenol (an organic compound) that sublimates, while some solids, such as osmium tetroxide, are so toxic that they must be kept in sealed containers to prevent human exposure to their sublimation-produced vapors.
All phase transitions involve heat. In the case of direct solid-vapor transitions, the energy required is given by the equation , where  is the heat of sublimation, which is the energy required to change 1.00 kg of a substance from the solid phase to the vapor phase.  is analogous to  and , and its value depends on the substance. Sublimation requires energy input, so that dry ice is an effective coolant, whereas the reverse process (i.e., frosting) releases energy. The amount of energy required for sublimation is of the same order of magnitude as that for other phase transitions.
The material presented in this section and the preceding section allows us to calculate any number of effects related to temperature and phase change. In each case, it is necessary to identify which temperature and phase changes are taking place and then to apply the appropriate equation. Keep in mind that heat transfer and work can cause both temperature and phase changes.

### Problem-Solving Strategies for the Effects of Heat Transfer

1. Examine the situation to determine that there is a change in the temperature or phase. Is there heat transfer into or out of the system? When the presence or absence of a phase change is not obvious, you may wish to first solve the problem as if there were no phase changes, and examine the temperature change obtained. If it is sufficient to take you past a boiling or melting point, you should then go back and do the problem in steps—temperature change, phase change, subsequent temperature change, and so on.
2. Identify and list all objects that change temperature and phase.
3. Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful.
4. Make a list of what is given or what can be inferred from the problem as stated (identify the knowns).
5. Solve the appropriate equation for the quantity to be determined (the unknown). If there is a temperature change, the transferred heat depends on the specific heat (see ) whereas, for a phase change, the transferred heat depends on the latent heat. See .
6. Substitute the knowns along with their units into the appropriate equation and obtain numerical solutions complete with units. You will need to do this in steps if there is more than one stage to the process (such as a temperature change followed by a phase change).
7. Check the answer to see if it is reasonable: Does it make sense? As an example, be certain that the temperature change does not also cause a phase change that you have not taken into account.

### Summary

1. Most substances can exist either in solid, liquid, and gas forms, which are referred to as “phases.”
2. Phase changes occur at fixed temperatures for a given substance at a given pressure, and these temperatures are called boiling and freezing (or melting) points.
3. During phase changes, heat absorbed or released is given by:
    where

### Conceptual Questions

### Problems & Exercises


# Heat and Heat Transfer Methods

## Heat Transfer Methods

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the different methods of heat transfer.
Equally as interesting as the effects of heat transfer on a system are the methods by which this occurs. Whenever there is a temperature difference, heat transfer occurs. Heat transfer may occur rapidly, such as through a cooking pan, or slowly, such as through the walls of a picnic ice chest. We can control rates of heat transfer by choosing materials (such as thick wool clothing for the winter), controlling air movement (such as the use of weather stripping around doors), or by choice of color (such as a white roof to reflect summer sunlight). So many processes involve heat transfer, so that it is hard to imagine a situation where no heat transfer occurs. Yet every process involving heat transfer takes place by only three methods:
1. Conduction is heat transfer through stationary matter by physical contact. (The matter is stationary on a macroscopic scale—we know there is thermal motion of the atoms and molecules at any temperature above absolute zero.) Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction.
2. Convection is the heat transfer by the macroscopic movement of a fluid. This type of transfer takes place in a forced-air furnace and in weather systems, for example.
3. Heat transfer by radiation occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed. An obvious example is the warming of the Earth by the Sun. A less obvious example is thermal radiation from the human body.
We examine these methods in some detail in the three following modules. Each method has unique and interesting characteristics, but all three do have one thing in common: they transfer heat solely because of a temperature difference .

### Summary

1. Heat is transferred by three different methods: conduction, convection, and radiation.

### Conceptual Questions


# Heat and Heat Transfer Methods

## Conduction

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate thermal conductivity.
2. Observe conduction of heat in collisions.
3. Study thermal conductivities of common substances.
Your feet feel cold as you walk barefoot across the living room carpet in your cold house and then step onto the kitchen tile floor. This result is intriguing, since the carpet and tile floor are both at the same temperature. The different sensation you feel is explained by the different rates of heat transfer: the heat loss during the same time interval is greater for skin in contact with the tiles than with the carpet, so the temperature drop is greater on the tiles.
Some materials conduct thermal energy faster than others. In general, good conductors of electricity (metals like copper, aluminum, gold, and silver) are also good heat conductors, whereas insulators of electricity (wood, plastic, and rubber) are poor heat conductors.  shows molecules in two bodies at different temperatures. The (average) kinetic energy of a molecule in the hot body is higher than in the colder body. If two molecules collide, an energy transfer from the molecule with greater kinetic energy to the molecule with less kinetic energy occurs. The cumulative effect from all collisions results in a net flux of heat from the hot body to the colder body. The heat flux thus depends on the temperature difference . Therefore, you will get a more severe burn from boiling water than from hot tap water. Conversely, if the temperatures are the same, the net heat transfer rate falls to zero, and equilibrium is achieved. Owing to the fact that the number of collisions increases with increasing area, heat conduction depends on the cross-sectional area. If you touch a cold wall with your palm, your hand cools faster than if you just touch it with your fingertip.
A third factor in the mechanism of conduction is the thickness of the material through which heat transfers. The figure below shows a slab of material with different temperatures on either side. Suppose that  is greater than , so that heat is transferred from left to right. Heat transfer from the left side to the right side is accomplished by a series of molecular collisions. The thicker the material, the more time it takes to transfer the same amount of heat. This model explains why thick clothing is warmer than thin clothing in winters, and why Arctic mammals protect themselves with thick blubber.
Lastly, the heat transfer rate depends on the material properties described by the coefficient of thermal conductivity. All four factors are included in a simple equation that was deduced from and is confirmed by experiments. The rate of conductive heat transfer through a slab of material, such as the one in , is given by
where  is the rate of heat transfer in watts or kilocalories per second,  is the thermal conductivity of the material,  and  are its surface area and thickness, as shown in , and  is the temperature difference across the slab.  gives representative values of thermal conductivity.
A combination of material and thickness is often manipulated to develop good insulators—the smaller the conductivity  and the larger the thickness , the better. The ratio of  will thus be large for a good insulator. The ratio  is called the .  The rate of conductive heat transfer is inversely proportional to . The larger the value of , the better the insulation.  factors are most commonly quoted for household insulation, refrigerators, and the like—unfortunately, it is still in non-metric units of ft2·°F·h/Btu, although the unit usually goes unstated (1 British thermal unit [Btu] is the amount of energy needed to change the temperature of 1.0 lb of water by 1.0 °F). A couple of representative values are an  factor of 11 for 3.5-in-thick fiberglass batts (pieces) of insulation and an  factor of 19 for 6.5-in-thick fiberglass batts. Walls are usually insulated with 3.5-in batts, while ceilings are usually insulated with 6.5-in batts. In cold climates, thicker batts may be used in ceilings and walls.
Note that in , the best thermal conductors—silver, copper, gold, and aluminum—are also the best electrical conductors, again related to the density of free electrons in them. Cooking utensils are typically made from good conductors.

### Test Prep for AP Courses

### Summary

1. Heat conduction is the transfer of heat between two objects in direct contact with each other.
2. The rate of heat transfer  (energy per unit time) is proportional to the temperature difference  and the contact area  and inversely proportional to the distance  between the objects:

### Conceptual Questions

### Problems & Exercises


# Heat and Heat Transfer Methods

## Convection

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss the method of heat transfer by convection.
Convection is driven by large-scale flow of matter. In the case of Earth, the atmospheric circulation is caused by the flow of hot air from the tropics to the poles, and the flow of cold air from the poles toward the tropics. (Note that Earth’s rotation causes the observed easterly flow of air in the northern hemisphere). Car engines are kept cool by the flow of water in the cooling system, with the water pump maintaining a flow of cool water to the pistons. The circulatory system is used the body: when the body overheats, the blood vessels in the skin expand (dilate), which increases the blood flow to the skin where it can be cooled by sweating. These vessels become smaller when it is cold outside and larger when it is hot (so more fluid flows, and more energy is transferred).
The body also loses a significant fraction of its heat through the breathing process.
While convection is usually more complicated than conduction, we can describe convection and do some straightforward, realistic calculations of its effects. Natural convection is driven by buoyant forces: hot air rises because density decreases as temperature increases. The house in  is kept warm in this manner, as is the pot of water on the stove in . Ocean currents and large-scale atmospheric circulation transfer energy from one part of the globe to another. Both are examples of natural convection.
A cold wind is much more chilling than still cold air, because convection combines with conduction in the body to increase the rate at which energy is transferred away from the body. The table below gives approximate wind-chill factors, which are the temperatures of still air that produce the same rate of cooling as air of a given temperature and speed. Wind-chill factors are a dramatic reminder of convection’s ability to transfer heat faster than conduction. For example, a 15.0 m/s wind at  has the chilling equivalent of still air at about .
Although air can transfer heat rapidly by convection, it is a poor conductor and thus a good insulator. The amount of available space for airflow determines whether air acts as an insulator or conductor. The space between the inside and outside walls of a house, for example, is about 9 cm (3.5 in) —large enough for convection to work effectively. The addition of wall insulation prevents airflow, so heat loss (or gain) is decreased. Similarly, the gap between the two panes of a double-paned window is about 1 cm, which prevents convection and takes advantage of air’s low conductivity to prevent greater loss. Fur, fiber, and fiberglass also take advantage of the low conductivity of air by trapping it in spaces too small to support convection, as shown in the figure. Fur and feathers are lightweight and thus ideal for the protection of animals.
Some interesting phenomena happen when convection is accompanied by a phase change. It allows us to cool off by sweating, even if the temperature of the surrounding air exceeds body temperature. Heat from the skin is required for sweat to evaporate from the skin, but without air flow, the air becomes saturated and evaporation stops. Air flow caused by convection replaces the saturated air by dry air and evaporation continues.
Another important example of the combination of phase change and convection occurs when water evaporates from the oceans. Heat is removed from the ocean when water evaporates. If the water vapor condenses in liquid droplets as clouds form, heat is released in the atmosphere. Thus, there is an overall transfer of heat from the ocean to the atmosphere. This process is the driving power behind thunderheads, those great cumulus clouds that rise as much as 20.0 km into the stratosphere. Water vapor carried in by convection condenses, releasing tremendous amounts of energy. This energy causes the air to expand and rise, where it is colder. More condensation occurs in these colder regions, which in turn drives the cloud even higher. Such a mechanism is called positive feedback, since the process reinforces and accelerates itself. These systems sometimes produce violent storms, with lightning and hail, and constitute the mechanism driving hurricanes.
The movement of icebergs is another example of convection accompanied by a phase change. Suppose an iceberg drifts from Greenland into warmer Atlantic waters. Heat is removed from the warm ocean water when the ice melts and heat is released to the land mass when the iceberg forms on Greenland.

### Test Prep for AP Courses

### Summary

1. Convection is heat transfer by the macroscopic movement of mass. Convection can be natural or forced and generally transfers thermal energy faster than conduction.  gives wind-chill factors, indicating that moving air has the same chilling effect of much colder stationary air. Convection that occurs along with a phase change can transfer energy from cold regions to warm ones.

### Conceptual Questions

### Problems & Exercises


# Heat and Heat Transfer Methods

## Radiation

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss heat transfer by radiation.
2. Explain the power of different materials.
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. The space between the Earth and the Sun is largely empty, without any possibility of heat transfer by convection or conduction. In these examples, heat is transferred by radiation. That is, the hot body emits electromagnetic waves that are absorbed by our skin: no medium is required for electromagnetic waves to propagate. Different names are used for electromagnetic waves of different wavelengths: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.
The energy of electromagnetic radiation depends on the wavelength (color) and varies over a wide range: a smaller wavelength (or higher frequency) corresponds to a higher energy. Because more heat is radiated at higher temperatures, a temperature change is accompanied by a color change. Take, for example, an electrical element on a stove, which glows from red to orange, while the higher-temperature steel in a blast furnace glows from yellow to white. The radiation you feel is mostly infrared, which corresponds to a lower temperature than that of the electrical element and the steel. The radiated energy depends on its intensity, which is represented in the figure below by the height of the distribution.
Electromagnetic Waves explains more about the electromagnetic spectrum and Introduction to Quantum Physics discusses how the decrease in wavelength corresponds to an increase in energy.
All objects absorb and emit electromagnetic radiation. The rate of heat transfer by radiation is largely determined by the color of the object. Black is the most effective, 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 gray sidewalk on a summer day, because black absorbs better than gray. The reverse is also true—black radiates better than gray. Thus, on a clear summer night, the asphalt will be colder than the gray sidewalk, because black radiates the energy more rapidly than gray. An ideal radiator is the same color as an ideal absorber, and captures all the radiation that falls on it. In contrast, white is a poor absorber and is also a poor radiator. A white object reflects all radiation, like a mirror. (A perfect, polished white surface is mirror-like in appearance, and a crushed mirror looks white.)
Gray objects have a uniform ability to absorb all parts of the electromagnetic spectrum. Colored objects behave in similar but more complex ways, which gives them a particular color in the visible range and may make them special in other ranges of the nonvisible spectrum. Take, for example, the strong absorption of infrared radiation by the skin, which allows us to be very sensitive to it.
The rate of heat transfer by emitted radiation is determined by the Stefan-Boltzmann law of radiation:
where  is the Stefan-Boltzmann constant,  is the surface area of the object, and  is its absolute temperature in kelvin. The symbol  stands for the emissivity of the object, which is a measure of how well it radiates. An ideal jet-black (or black body) radiator has , whereas a perfect reflector has
. Real objects fall between these two values. Take, for example, tungsten light bulb filaments which have an
 of about
, and carbon black (a material used in printer toner), which has the (greatest known) emissivity of about
.
The radiation rate is directly proportional to the fourth power of the absolute temperature—a remarkably strong temperature dependence. Furthermore, the radiated heat is proportional to the surface area of the object. If you knock apart the coals of a fire, there is a noticeable increase in radiation due to an increase in radiating surface area.
Skin is a remarkably good absorber and emitter of infrared radiation, having an emissivity of 0.97 in the infrared spectrum. Thus, we are all nearly (jet) black in the infrared, in spite of the obvious variations in skin color. This high infrared emissivity is why we can so easily feel radiation on our skin. It is also the basis for the use of night scopes used by law enforcement and the military to detect human beings. Even small temperature variations can be detected because of the  dependence. Images, called thermographs, can be used medically to detect regions of abnormally high temperature in the body, perhaps indicative of disease. Similar techniques can be used to detect heat leaks in homes , optimize performance of blast furnaces, improve comfort levels in work environments, and even remotely map the Earth’s temperature profile.
All objects emit and absorb radiation. The net rate of heat transfer by radiation (absorption minus emission) is related to both the temperature of the object and the temperature of its surroundings. Assuming that an object with a temperature  is surrounded by an environment with uniform temperature , the net rate of heat transfer by radiation is
where  is the emissivity of the object alone. In other words, it does not matter whether the surroundings are white, gray, or black; the balance of radiation into and out of the object depends on how well it emits and absorbs radiation. When , the quantity  is positive; that is, the net heat transfer is from hot to cold.
The Earth receives almost all its energy from radiation of the Sun and reflects some of it back into outer space. Because the Sun is hotter than the Earth, the net energy flux is from the Sun to the Earth. However, the rate of energy transfer is less than the equation for the radiative heat transfer would predict because the Sun does not fill the sky. The average emissivity () of the Earth is about 0.65, but the calculation of this value is complicated by the fact that the highly reflective cloud coverage varies greatly from day to day. There is a negative feedback (one in which a change produces an effect that opposes that change) between clouds and heat transfer; greater temperatures evaporate more water to form more clouds, which reflect more radiation back into space, reducing the temperature. The often mentioned greenhouse effect is directly related to the variation of the Earth’s emissivity with radiation type (see the figure given below). The greenhouse effect is a natural phenomenon responsible for providing temperatures suitable for life on Earth. The Earth’s relatively constant temperature is a result of the energy balance between the incoming solar radiation and the energy radiated from the Earth. Most of the infrared radiation emitted from the Earth is absorbed by carbon dioxide () and water () in the atmosphere and then re-radiated back to the Earth or into outer space. Re-radiation back to the Earth maintains its surface temperature about  higher than it would be if there was no atmosphere, similar to the way glass increases temperatures in a greenhouse.
The greenhouse effect and its causes were first predicted by Eunice Newton Foote after she designed and conducted experiments on heating of different gases. After filling flasks with carbon dioxide, hydrogen, and regular air, then also modifying moisture, she placed them in the sun and carefully measured their heating and, especially, their heat retention. She discovered that the CO2 flask gained the most temperature and held it the longest. After subsequent research, her paper "Circumstances affecting the Heat of the Sun’s Rays" included conclusions that an atmosphere consisting of more carbon dioxide would be hotter resulting from the gas trapping radiation.
The greenhouse effect is also central to the discussion of global warming due to emission of carbon dioxide and methane (and other so-called greenhouse gases) into the Earth’s atmosphere from industrial production and farming. Changes in global climate could lead to more intense storms, precipitation changes (affecting agriculture), reduction in rain forest biodiversity, and rising sea levels.
Heating and cooling are often significant contributors to energy use in individual homes. Mária Telkes, a Hungarian-born American scientist, was among the foremost developers of solar energy applications in industrial and community use. After inventing a widely deployed solar seawater distiller used on World War II life rafts, she partnered with architect Eleanor Raymond to design the first modern home to be completely heated by solar power. Air warmed on rooftop collectors transferred heat to salts, which stored the heat for later use. Telkes later worked with the Department of Energy to develop the first solar-electrically powered home. Current research efforts into developing environmentally friendly homes quite often focus on reducing conventional heating and cooling through better building materials, strategically positioning windows to optimize radiation gain from the Sun, and opening spaces to allow convection. It is possible to build a zero-energy house that allows for comfortable living in most parts of the United States with hot and humid summers and cold winters.
Conversely, dark space is very cold, about , so that the Earth radiates energy into the dark sky. Owing to the fact that clouds have lower emissivity than either oceans or land masses, they reflect some of the radiation back to the surface, greatly reducing heat transfer into dark space, just as they greatly reduce heat transfer into the atmosphere during the day. The rate of heat transfer from soil and grasses can be so rapid that frost may occur on clear summer evenings, even in warm latitudes.

### Test Prep for AP Courses

### Summary

1. Radiation is the rate of heat transfer through the emission or absorption of electromagnetic waves.
2. The rate of heat transfer depends on the surface area and the fourth power of the absolute temperature:
    where where

### Conceptual Questions

### Problems & Exercises


# Thermodynamics

## Connection for AP® Courses

Heat is energy in transit, and it can be used to do work. It can also be converted to any other form of energy. When a car engine burns fuel, for example, heat transfers into a gas. Work is done by the heated gas as it exerts a force through a distance (Essential Knowledge 5.B.5), converting its energy into a variety of other forms—into the car's kinetic or gravitational potential energy; into electrical energy to run the spark plugs, radio, and lights; and into stored energy in the car's battery. But most of the heat produced from burning fuel in the engine does not do work. Rather, the heat is released into the environment, implying that the engine is quite inefficient.
It is often said that modern gasoline engines cannot be made to be significantly more efficient. We hear the same about heat transfer to electrical energy in large power stations, whether they are coal, oil, natural gas, or nuclear powered. Why is that the case? Is the inefficiency caused by design problems that could be solved with better engineering and superior materials? Is it part of some money-making conspiracy by those who sell energy? Actually, the truth is more interesting, and reveals much about the nature of heat transfer. Basic physical laws govern how heat transfer for doing work takes place and place insurmountable limits onto its efficiency. This chapter will explore these laws as well as many applications and concepts associated with them. These topics are part of thermodynamics—the study of heat transfer and its relationship to doing work.
This chapter discusses thermodynamics in practical contexts including heat engines, heat pumps, and refrigerators, which support Big Idea 4, and that interactions between systems can result in changes in those systems. As systems either do work or have work done on them, the total energy of a system can change (Enduring Understanding 4.C). These ideas are based on the previous understanding of heat as the process of energy transfer from a higher temperature system to a lower temperature system (Essential Knowledge 4.C.3). You will learn about the first law of thermodynamics, which supports Big Idea 5, that changes that occur as a result of interactions are constrained by conservation laws. The first law of thermodynamics is a special case of energy conservation that explains the relationship between changes in the internal energy of a system (Essential Knowledge 5.B.4) and energy transfer in the form of heat or work (Essential Knowledge 5.B.7). Note that the energy of a system is conserved (Enduring Understanding 5.B). You will also learn about the second law of thermodynamics and entropy. These are applications of Big Idea 7, that the mathematics of probability can be used to describe the behavior of complex systems. For example, an isolated system will reach thermal equilibrium (Enduring Understanding 7.B), a state with higher disorder. This process has a probabilistic nature (Essential Knowledge 7.B.1) and is described by the second law of thermodynamics. The second law of thermodynamics describes the change of entropy for reversible and irreversible processes (Essential Knowledge 7.B.2). Entropy is considered qualitatively at this level.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.3 Energy is transferred spontaneously from a higher temperature system to a lower temperature system. The process through which energy is transferred between systems at different temperatures is called heat.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.4 The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the conﬁguration of the objects that make up the system.
Essential Knowledge 5.B.5 Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is deﬁned as the rate of energy transfer into, out of, or within a system. [A piston filled with gas getting compressed or expanded is treated in Physics 2 as a part of thermodynamics.]
Essential Knowledge 5.B.7 The ﬁrst law of thermodynamics is a speciﬁc case of the law of conservation of energy involving the internal energy of a system and the possible transfer of energy through work and/or heat. Examples should include P–V diagrams — isochoric process, isothermal process, isobaric process, adiabatic process. No calculations of heat or internal energy from temperature change; and in this course, examples of these relationships are qualitative and/or semi–quantitative.
Big Idea 7 The mathematics of probability can be used to describe the behavior of complex systems and to interpret the behavior of quantum mechanical systems.
Enduring Understanding 7.B The tendency of isolated systems to move toward states with higher disorder is described by probability.
Essential Knowledge 7.B.1 The approach to thermal equilibrium is a probability process.
Essential Knowledge 7.B.2 The second law of thermodynamics describes the change in entropy for reversible and irreversible processes. Only a qualitative treatment is considered in this course.


# Thermodynamics

## The First Law of Thermodynamics

### Learning Objectives

By the end of this section, you will be able to:
1. Define the first law of thermodynamics.
2. Describe how conservation of energy relates to the first law of thermodynamics.
3. Identify instances of the first law of thermodynamics working in everyday situations, including biological metabolism.
4. Calculate changes in the internal energy of a system, after accounting for heat transfer and work done.
If we are interested in how heat transfer is converted into doing work, then the conservation of energy principle is important. The first law of thermodynamics applies the conservation of energy principle to systems where heat transfer and doing work are the methods of transferring energy into and out of the system. The first law of thermodynamics states that the change in internal energy of a 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  of the system.  is the net heat transferred into the system—that is,  is the sum of all heat transfer into and out of the system.  is the net work done by the system—that is,  is the sum of all work done on or by the system. We use the following sign conventions: if  is positive, then there is a net heat transfer into the system; if  is positive, then there is net work done by the system. So positive  adds energy to the system and positive  takes energy from the system. Thus . Note also that if more heat transfer into the system occurs than work done, the difference is stored as internal energy. Heat engines are a good example of this—heat transfer into them takes place so that they can do work. (See .) We will now examine , , and  further.

### Heat Q and Work W

Heat transfer () and doing work () are the two everyday means of bringing energy into or taking energy out of a system. The processes are quite different. Heat transfer, a less organized process, is driven by temperature differences. Work, a quite organized process, involves a macroscopic force exerted through a distance. Nevertheless, heat and work can produce identical results.For example, both can cause a temperature increase. Heat transfer into a system, such as when the Sun warms the air in a bicycle tire, can increase its temperature, and so can work 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 transfer or by doing work. This uncertainty is an important point. Heat transfer and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy  of a system. Internal energy is a form of energy completely different from either heat or work.

### Internal Energy U

We can think about the internal energy of a system in two different but consistent ways. The first is the atomic and molecular view, which examines the system on the atomic and molecular scale. The internal energy  of a system is the sum of the kinetic and potential energies of its atoms and molecules. Recall that kinetic plus potential energy is called mechanical energy. Thus internal energy is the sum of atomic and molecular mechanical energy. Because it is impossible to keep track of all individual atoms and molecules, we must deal with averages and distributions. A second way to view the internal energy of a system is in terms of its macroscopic characteristics, which are very similar to atomic and molecular average values.
Macroscopically, we define the change in internal energy  to be that given by the first law of thermodynamics:
Many detailed experiments have verified that , where  is the change in total kinetic and potential energy of all atoms and molecules in a system. It has also been determined experimentally that the internal energy  of a system depends only on the state of the system and not how it reached that state. More specifically,  is found to be a function of a few macroscopic quantities (pressure, volume, and temperature, for example), independent of past history such as whether there has been heat transfer or work done. This independence means that if we know the state of a system, we can calculate changes in its internal energy  from a few macroscopic variables.
To get a better idea of how to think about the internal energy of a system, let us examine a system going from State 1 to State 2. The system has internal energy  in State 1, and it has internal energy  in State 2, no matter how it got to either state. So the change in internal energy  is independent of what caused the change. In other words,  is independent of path. By path, we mean the method of getting from the starting point to the ending point. Why is this independence important? Note that . Both  and depend on path, but  does not. This path independence means that internal energy  is easier to consider than either heat transfer or work done.

### Human Metabolism and the First Law of Thermodynamics

Human metabolism is the conversion of food into heat transfer, work, and stored fat. Metabolism is an interesting example of the first law of thermodynamics in action. We now take another look at these topics via the first law of thermodynamics. Considering the body as the system of interest, we can use the first law to examine heat transfer, doing work, and internal energy in activities ranging from sleep to heavy exercise. What are some of the major characteristics of heat transfer, doing work, and energy in the body? For one, body temperature is normally kept constant by heat transfer to the surroundings. This means  is negative. Another fact is that the body usually does work on the outside world. This means  is positive. In such situations, then, the body loses internal energy, since  is negative.
Now consider the effects of eating. Eating increases the internal energy of the body by adding chemical potential energy (this is an unromantic view of a good steak). The body metabolizes all the food we consume. Basically, metabolism is an oxidation process in which the chemical potential energy of food is released. This implies that food input is in the form of work. Food energy is reported in a special unit, known as the Calorie. This energy is measured by burning food in a calorimeter, which is how the units are determined.
In chemistry and biochemistry, one calorie (spelled with a lowercase c) is defined as the energy (or heat transfer) required to raise the temperature of one gram of pure water by one degree Celsius.  Nutritionists and weight-watchers tend to use the dietary calorie, which is frequently called a Calorie (spelled with a capital C). One food Calorie is the energy needed to raise the temperature of one kilogram of water by one degree Celsius. This means that one dietary Calorie is equal to one kilocalorie for the chemist, and one must be careful to avoid confusion between the two.
Again, consider the internal energy the body has lost. There are three places this internal energy can go—to heat transfer, to doing work, and to stored fat (a tiny fraction also goes to cell repair and growth). Heat transfer and doing work take internal energy out of the body, and food puts it back. If you eat just the right amount of food, then your average internal energy remains constant. Whatever you lose to heat transfer and doing work is replaced by food, so that, in the long run, . If you overeat repeatedly, then  is always positive, and your body stores this extra internal energy as fat. The reverse is true if you eat too little. If  is negative for a few days, then the body metabolizes its own fat to maintain body temperature and do work that takes energy from the body. This process is how dieting produces weight loss.
Life is not always this simple, as any dieter knows. The body stores fat or metabolizes it only if energy intake changes for a period of several days. Once you have been on a major diet, the next one is less successful because your body alters the way it responds to low energy intake. Your basal metabolic rate (BMR) is the rate at which food is converted into heat transfer and work done while the body is at complete rest. The body adjusts its basal metabolic rate to partially compensate for over-eating or under-eating. The body will decrease the metabolic rate rather than eliminate its own fat to replace lost food intake. You will chill more easily and feel less energetic as a result of the lower metabolic rate, and you will not lose weight as fast as before. Exercise helps to lose weight, because it produces both heat transfer from your body and work, and raises your metabolic rate even when you are at rest. Weight loss is also aided by the quite low efficiency of the body in converting internal energy to work, so that the loss of internal energy resulting from doing work is much greater than the work done.It should be noted, however, that living systems are not in thermalequilibrium.
The body provides us with an excellent indication that many thermodynamic processes are irreversible. An irreversible process can go in one direction but not the reverse, under a given set of conditions. For example, although body fat can be converted to do work and produce heat transfer, work done on the body and heat transfer into it cannot be converted to body fat. Otherwise, we could skip lunch by sunning ourselves or by walking down stairs. Another example of an irreversible thermodynamic process is photosynthesis. This process is the intake of one form of energy—light—by plants and its conversion to chemical potential energy. Both applications of the first law of thermodynamics are illustrated in . One great advantage of conservation laws such as the first law of thermodynamics is that they accurately describe the beginning and ending points of complex processes, such as metabolism and photosynthesis, without regard to the complications in between.  presents a summary of terms relevant to the first law of thermodynamics.

### Test Prep for AP Courses

### Section Summary

1. The first law of thermodynamics is given as , where  is the change in internal energy of a system,  is the net heat transfer (the sum of all heat transfer into and out of the system), and  is the net work done (the sum of all work done on or by the system).
2. Both  and  are energy in transit; only  represents an independent quantity capable of being stored.
3. The internal energy  of a system depends only on the state of the system and not how it reached that state.
4. Metabolism of living organisms, and photosynthesis of plants, are specialized types of heat transfer, doing work, and internal energy of systems.

### Conceptual Questions

### Problems & Exercises