12.8

Heat and Phase Change: Latent Heat

Surprisingly, there are situations in which the addition or removal of heat does not cause a temperature change. Consider a well-stirred glass of iced tea that has come to thermal equilibrium. Even though heat enters the glass from the warmer room, the temperature of the tea does not rise above 0 °C as long as ice cubes are present. Apparently the heat is being used for some purpose other than raising the temperature. In fact, the heat is being used to melt the ice, and only when all of it is melted will the temperature of the liquid begin to rise.
An important point illustrated by the iced tea example is that there is more than one type or phase of matter. For instance, some of the water in the glass is in the solid phase (ice) and some in the liquid phase. The gas or vapor phase is the third familiar phase of matter. In the gas phase, water is referred to as water vapor or steam. All three phases of water are present in the scene depicted in Figure 12-27.

Figure 12-27 The three phases of water: solid ice is floating in liquid water, and water vapor (invisible) is present in the air. (Klein/Peter Arnold, Inc.)

Matter can change from one phase to another, and heat plays a role in the change. Figure 12-28 summarizes the various possibilities. A solid can melt or fuse into a liquid if heat is added, while the liquid can freeze into a solid if heat is removed. Similarly, a liquid can evaporate into a gas if heat is supplied, while the gas can condense into a liquid if heat is taken away. Rapid evaporation, with the formation of vapor bubbles within the liquid, is called boiling. Finally, a solid can sometimes change directly into a gas if heat is provided. We say that the solid sublimes into a gas. Examples of sublimation are (1) solid carbon dioxide, CO2 (dry ice), turning into gaseous CO2 and (2) solid naphthalene (moth balls) turning into naphthalene fumes. Conversely, if heat is removed under the right conditions, the gas will condense directly into a solid.

Figure 12-28 Three familiar phases of matter—solid, liquid, and gas—and the phase changes that can occur between any two of them.

Figure 12-29 displays a graph that indicates what typically happens when heat is added to a material that changes phase. The graph records temperature versus heat added and refers to water at the normal atmospheric pressure of . The water starts off as ice at the subfreezing temperature of −30 °C. As heat is added, the temperature of the ice increases, in accord with the specific heat capacity of ice []. Not until the temperature reaches the normal melting/freezing point of 0 °C does the water begin to change phase. Then, when heat is added, the solid changes into the liquid, the temperature staying at 0 °C until all the ice has melted. Once all the material is in the liquid phase, additional heat causes the temperature to increase again, now in accord with the specific heat capacity of liquid water []. When the temperature reaches the normal boiling/condensing point of 100 °C, the water begins to change from the liquid to the gas phase and continues to do so as long as heat is added. The temperature remains at 100 °C until all liquid is gone. When all of the material is in the gas phase, additional heat once again causes the temperature to rise, this time according to the specific heat capacity of water vapor at constant atmospheric pressure []. Conceptual Example 13 applies the information in Figure 12-29 to a familiar situation.

Figure 12-29 The graph shows the way the temperature of water changes as heat is added, starting with ice at −30 °C. The pressure is atmospheric pressure.

| Conceptual Example 13 | | | Saving Energy

Suppose you are cooking spaghetti for dinner, and the instructions say “boil the pasta in water for ten minutes.” To cook spaghetti in an open pot with the least amount of energy, should you turn up the burner to its fullest so the water vigorously boils, or should you turn down the burner so the water barely boils?
Reasoning and Solution The spaghetti needs to cook at a temperature of 100 °C for ten minutes. It doesn’t matter whether the water is vigorously boiling or barely boiling, because its temperature is still 100 °C. Remember, as long as there is boiling water in the pot, and the pot is open to one atmosphere of pressure, no amount of additional heat will cause the water temperature to rise above 100 °C. Additional heat only vaporizes the water and produces steam, which is of no use in cooking the spaghetti. So, save your money; turn down the heat, because the least amount of energy is expended when the water barely boils.

When a substance changes from one phase to another, the amount of heat that must be added or removed depends on the type of material and the nature of the phase change. The heat per kilogram associated with a phase change is referred to as latent heat:

| Heat Supplied or Removed in Changing the Phase of a Substance
| The heat Q that must be supplied or removed to change the phase of a mass m of a substance is

| (12.5)

where L is the latent heat of the substance.
SI Unit of Latent Heat: J/kg

The latent heat of fusion Lf refers to the change between solid and liquid phases, the latent heat of vaporization Lv applies to the change between liquid and gas phases, and the latent heat of sublimation Ls refers to the change between solid and gas phases.
Table 12-3 gives some typical values of latent heats of fusion and vaporization. For instance, the latent heat of fusion for water is . Thus, of heat must be supplied to melt one kilogram of ice at 0 °C into liquid water at 0 °C; conversely, this amount of heat must be removed from one kilogram of liquid water at 0 °C to freeze the liquid into ice at 0 °C. In comparison, the latent heat of vaporization for water has the much larger value of . When water boils at 100 °C, of heat must be supplied for each kilogram of liquid turned into steam. And when steam condenses at 100 °C, this amount of heat is released from each kilogram of steam that changes back into liquid. Liquid water at 100 °C is hot enough by itself to cause a bad burn, and the additional effect of the large latent heat can cause severe tissue damage if condensation occurs on the skin.

Table 12-3 Latent Heats* of Fusion and Vaporization

Substance | Melting Point (°C) | Latent Heat of Fusion, Lf (J/kg) | Boiling Point (°C) | Latent Heat of Vaporization, Lv (J/kg)
Ammonia | −77.8 |  | −33.4 |
Benzene | 5.5 |  | 80.1 |
Copper | 1083 |  | 2566 |
Ethyl alcohol | −114.4 |  | 78.3 |
Gold | 1063 |  | 2808 |
Lead | 327.3 |  | 1750 |
Mercury | −38.9 |  | 356.6 |
Nitrogen | −210.0 |  | −195.8 |
Oxygen | −218.8 |  | −183.0 |
Water | 0.0 |  | 100.0 |

a) |  | The values pertain to 1 atm pressure.

The physics of steam burns.

By taking advantage of the latent heat of fusion, designers can now engineer clothing that can absorb or release heat to help maintain a comfortable and approximately constant temperature close to your body. As the photograph in Figure 12-30 shows, the fabric in this type of clothing is coated with microscopic balls of heat-resistant plastic that contain a substance known as a “phase-change material” (PCM). When you are enjoying your favorite winter sport, for example, it is easy to become overheated. The PCM prevents this by melting, absorbing excess body heat in the process. When you are taking a break and cooling down, however, the PCM freezes and releases heat to keep you warm. The temperature range over which the PCM can maintain a comfort zone is related to its melting/freezing temperature, which is determined by its chemical composition.

Figure 12-30 This highly magnified image shows a fabric that has been coated with microscopic balls of heat-resistant plastic. The balls contain a substance known as a “phase-change material,” the melting and freezing of which absorbs and releases heat. Such fabrics automatically adjust in reaction to your body heat to help maintain a constant temperature next to your skin. (Courtesy Outlast Technologies, Boulder, CO)

The physics of high-tech clothing.

Examples 14 and 15 and Examples 14 and 15 illustrate how to take into account the effect of latent heat when using the conservation-of-energy principle.

| Example 14 | | | Ice-Cold Lemonade

Ice at 0 °C is placed in a Styrofoam cup containing 0.32 kg of lemonade at 27 °C. The specific heat capacity of lemonade is virtually the same as that of water; that is, . After the ice and lemonade reach an equilibrium temperature, some ice still remains. The latent heat of fusion for water is . Assume that the mass of the cup is so small that it absorbs a negligible amount of heat, and ignore any heat lost to the surroundings. Determine the mass of ice that has melted.
Reasoning According to the principle of energy conservation, the heat gained by the melting ice equals the heat lost by the cooling lemonade. According to Equation 12.5, the heat gained by the melting ice is , where m is the mass of the melted ice, and Lf is the latent heat of fusion for water. The heat lost by the lemonade is given by , where is the higher temperature of 27 °C minus the lower equilibrium temperature. The equilibrium temperature is 0 °C, because there is some ice remaining, and ice is in equilibrium with liquid water when the temperature is 0 °C.
Solution

The mass mice of ice that has melted is

| Example 15 | | | Getting Ready for a Party

A 7.00-kg glass bowl [] contains 16.0 kg of punch at 25.0 °C. Two-and-a-half kilograms of ice [] are added to the punch. The ice has an initial temperature of −20.0 °C, having been kept in a very cold freezer. The punch may be treated as if it were water [], and it may be assumed that there is no heat flow between the punch bowl and the external environment. The latent heat of fusion for water is . When thermal equilibrium is reached, all the ice has melted, and the final temperature of the mixture is above 0 °C. Determine this temperature.
Reasoning The final temperature can be found by using the conservation of energy principle: the heat gained is equal to the heat lost. Heat is gained (a) by the ice in warming up to the melting point, (b) by the ice in changing phase from a solid to a liquid, and (c) by the liquid that results from the ice warming up to the final temperature; heat is lost (d) by the punch and (e) by the bowl in cooling down. The heat gained or lost by each component in changing temperature can be determined from the relation , where is the higher temperature minus the lower temperature. The heat gained when water changes phase from a solid to a liquid at 0 °C is , where m is the mass of water and Lf is the latent heat of fusion.
Solution The heat gained or lost by each component is listed as follows:

a. |
b. |
c. |
d. |
e. |

Setting the heat gained equal to the heat lost gives:

This equation can be solved to show that .

An interesting application of the phase change between a solid and a gas is found in one kind of color printer used with computers. A dye-sublimation printer uses a thin plastic film coated with separate panels of cyan (blue), yellow, and magenta pigment or dye. A full spectrum of colors is produced by using combinations of tiny spots of these dyes. As Figure 12-31 shows, the coated film passes in front of a print head that extends across the width of the paper and contains 2400 heating elements. When a heating element is turned on, the dye in front of it absorbs heat and goes from a solid to a gas—it sublimes—with no liquid phase in between. A coating on the paper absorbs the gaseous dye on contact, producing a small spot of color. The intensity of the spot is controlled by the heating element, since each element can produce 256 different temperatures; the hotter the element, the greater the amount of dye transferred to the paper. The paper makes three separate passes across the print head, once for each of the dyes. The final result is an image of near-photographic quality. Some printers also employ a fourth pass, in which a clear plastic coating is deposited over the photograph. This coating makes the print waterproof and also helps to prevent premature fading.

Figure 12-31 A dye-sublimation printer. As the plastic film passes in front of the print head, the heat from a given heating element causes one of three pigments or dyes on the film to sublime from a solid to a gas. The gaseous dye is absorbed onto the coated paper as a dot of color. The size of the dots on the paper has been exaggerated for clarity.

The physics of a dye-sublimation color printer.

Check Your Understanding 4

When ice cubes are used to cool a drink, both their mass and temperature are important in how effective they are. The table below lists several possibilities for the mass and temperature of the ice cubes used to cool a particular drink. Rank the possibilities in descending order (best first), according to their cooling effectiveness. (The answer is given at the end of the book.)

| Mass of ice cubes | Temperature of ice cubes
(a) | m | −6.0 °C
(b) | ½m | −12 °C
(c) | 2m | −3.0 °C

Background: Both the latent heat of phase change and the specific heat capacity must be considered in answering this question.
For similar questions (including calculational counterparts), consult Self-Assessment Test 12.2. This test is described at the end of Section 12.10.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.