12.5

Volume Thermal Expansion

The volume of a normal material increases as the temperature increases. Most solids and liquids behave in this fashion. By analogy with linear thermal expansion, the change in volume is proportional to the change in temperature and to the initial volume V0, provided the change in temperature is not too large. These two proportionalities can be converted into Equation 12.3 with the aid of a proportionality constant , known as the coefficient of volume expansion. The algebraic form of this equation is similar to that for linear expansion, .

| Volume Thermal Expansion
| The volume V0 of an object changes by an amount when its temperature changes by an amount :

| (12.3)

where is the coefficient of volume expansion.
Common Unit for the Coefficient of Volume Expansion: (C°)−1

The unit for , like that for , is (C°)−1. Values for depend on the nature of the material, and Table 12-1 lists some examples measured near 20 °C. The values of for liquids are substantially larger than those for solids, because liquids typically expand more than solids, given the same initial volumes and temperature changes. Table 12-1 also shows that, for most solids, the coefficient of volume expansion is three times as much as the coefficient of linear expansion: .
If a cavity exists within a solid object, the volume of the cavity increases when the object expands, just as if the cavity were filled with the surrounding material. The expansion of the cavity is analogous to the expansion of a hole in a sheet of material. Accordingly, the change in volume of a cavity can be found using the relation , where is the coefficient of volume expansion of the material that surrounds the cavity. Example 8 illustrates this point.

| Example 8 | | | An Automobile Radiator

A small plastic container, called the coolant reservoir, catches the radiator fluid that overflows when an automobile engine becomes hot (see Figure 12-19). The radiator is made of copper, and the coolant has a coefficient of volume expansion of . If the radiator is filled to its 15-quart capacity when the engine is cold (6.0 °C), how much overflow from the radiator will spill into the reservoir when the coolant reaches its operating temperature of 92 °C?

Figure 12-19 An automobile radiator and a coolant reservoir for catching the overflow from the radiator.

Reasoning When the temperature increases, both the coolant and the radiator expand. If they were to expand by the same amount, there would be no overflow. However, the liquid coolant expands more than the radiator, and the overflow volume is the amount of coolant expansion minus the amount of the radiator cavity expansion.
Solution When the temperature increases by 86 C°, the coolant expands by an amount

| (12.3)

The radiator cavity expands as if it were filled with copper [; see Table 12-1]. The expansion of the radiator cavity is

The overflow volume is .

The physics of the overflow of an automobile radiator.

Problem solving insight The way in which the level of a liquid in a container changes with temperature depends on the change in volume of both the liquid and the container.

Although most substances expand when heated, a few do not. For instance, if water at 0 °C is heated, its volume decreases until the temperature reaches 4 °C. Above 4 °C water behaves normally, and its volume increases as the temperature increases. Because a given mass of water has a minimum volume at 4°C, the density (mass per unit volume) of water is greatest at 4 °C, as Figure 12-20 shows.

Figure 12-20 The density of water in the temperature range from 0 to 10 °C. At 4 °C water has a maximum density of 999.973 kg/m3. (This value is equivalent to the often-quoted density of 1.000 00 grams per milliliter.)

The fact that water has its greatest density at 4 °C, rather than at 0 °C, has important consequences for the way in which a lake freezes. When the air temperature drops, the surface layer of water is chilled. As the temperature of the surface layer drops toward 4 °C, this layer becomes more dense than the warmer water below. The denser water sinks and pushes up the deeper and warmer water, which in turn is chilled at the surface. This process continues until the temperature of the entire lake reaches 4 °C. Further cooling of the surface water below 4 °C makes it less dense than the deeper layers; consequently, the surface layer does not sink but stays on top. Continued cooling of the top layer to 0 °C leads to the formation of ice that floats on the water, because ice has a smaller density than water at any temperature. Below the ice, however, the water temperature remains above 0 °C. The sheet of ice acts as an insulator that reduces the loss of heat from the lake, especially if the ice is covered with a blanket of snow, which is also an insulator. As a result, lakes usually do not freeze solid, even during prolonged cold spells, so fish and other aquatic life can survive.

The physics of ice formation and the survival of aquatic life.

Need more practice?

Interactive LearningWare12.1
A can is filled with liquid to 97.0% of its capacity. The temperature of the can and liquid is 0.0 °C. The material from which the can is made has a coefficient of volume expansion of . At a temperature of 100.0 °C, the can is filled to the brim. Determine the coefficient of volume expansion of the liquid.
Related Homework: Problems 30, 32

The physics of bursting water pipes. The fact that the density of ice is smaller than the density of water has an important consequence for homeowners, who have to contend with the possibility of bursting water pipes during severe winters. Water often freezes in a section of pipe exposed to unusually cold temperatures. The ice can form an immovable plug that prevents the subsequent flow of water, as Figure 12-21 illustrates. When water (larger density) turns to ice (smaller density), its volume expands by 8.3%. Therefore, when more water freezes at the left side of the plug, the expanding ice pushes liquid back into the pipe leading to the street connection, and no damage is done. However, when ice forms on the right side of the plug, the expanding ice pushes liquid to the right. But it has nowhere to go if the faucet is closed. As ice continues to form and expand, the water pressure between the plug and faucet rises. Even a small increase in the amount of ice produces a large increase in the pressure. This situation is analogous to the thermal stress discussed in Multi-Concept Example 4, where a small change in the length of the steel beam produces a large stress on the concrete supports. The entire section of pipe to the right of the blockage experiences the same elevated pressure, according to Pascal’s principle (Section 11.5). Therefore, the pipe can burst at any point where it is structurally weak, even within the heated space of the building. If you should lose heat during the winter, there is a simple way to prevent pipes from bursting. Simply open the faucet, so it drips a little. The excessive pressure will be relieved.

Figure 12-21 As water freezes and expands, enormous pressure is applied to the liquid water between the ice and the faucet.

Test your understanding of the material in Sections 12.1–12.5:

· | Temperature Scales and Thermometers
· | Linear Thermal Expansion
· | Volume Thermal Expansion

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