Thermal Expansion of Solids and Liquids  518

20.2 Thermal Expansion of Solids and Liquids

As we saw in the preceding chapter, if the pressure is held constant, the volume of a given amount of gas will increase with the temperature. Such an increase in volume with temperature also occurs for solids and liquids; this phenomenon is called thermal expansion. However, the thermal expansion of solids and liquids is much less than that of gases. For example, if we raise the temperature of a piece of iron by 100°C, we will increase its volume by only 0.36%. During the thermal expansion the solid retains its shape, but all its dimensions increase in proportion. Figure 20.2a shows the thermal expansion of a piece of metal; for the sake of clarity, the expansion has been exaggerated. An expanding liquid does not, of course, retain its shape; the liquid will merely fill more of the container that holds it. Figure 20.2b shows the thermal expansion of a liquid.

The thermal expansion of a solid and a liquid arises from an increase of the random microscopic motions. The atoms or molecules of the solid are held together by interatomic or intermolecular forces which act rather like springs (see Figure 20.3).

The atoms oscillate about their equilibrium positions with an average amplitude that depends on the thermal energy. An increase of temperature leads to vibrational motions of larger amplitude. This, in itself, does not necessarily result in an expansion of the solid, since the size of the solid is determined by the average separation between the atoms, not by their amplitude of motion. However, the interatomic "springs" deviate somewhat from Hooke's Law -- they are somewhat easier to stretch than to compress. (This asymmetry of the interatomic force can be recognized from the plot of the interatomic potential energy given in Figure 8.5. The curve of potential energy is asymmetric; it is steeper on the near side of the equilibrium point than on the far side). Consequently, when an increase of temperature brings about an increase of the amplitude of motion, the maximum displacement attained during stretching of the spring exceeds the maximum displacement attained during compression, and therefore the average separation between the atoms increases, resulting in an expansion of the solid.

The thermal expansion of a solid can be described mathematically by the increase in the linear dimensions of the solid. The increment in the length is directly proportional to the increment of temperature and to the original length,

The constant of proportionality a is called the coefficient of linear expansion. Table 20.2 lists the values of this coefficient for a few materials.

The increment in the volume of the solid is directly proportional to the increment in the temperature and to the original volume,

Here the constant of proportionality b is called the coefficient of cubical expansion. This coefficient is three times the coefficient of linear expansion,

To see how this relationship comes about, consider a solid in the shape of a cube of edge L and volume V = L3. A small increment DL in the length can be treated as a differential and consequently DV = 3L2 DL, which gives

Comparing this with Eq. (3), we see that, indeed, b=3a .

The increment in the volume of a liquid can be described by the same equation [Eq. (3)] as the increment in the volume of a solid. Table 20.2 also lists values of coefficients of cubical expansion for some liquids.

Water has not been included in this table because its behavior is rather peculiar: from 0°C to 3.98°C, the volume decreases with temperature, but not uniformly; above 3.98°C, the volume increases with temperature. Figure 20.4 plots the volume of 1 kg of water as a function of the temperature. The strange behavior of the density of water at low temperatures can be traced to the crystal structure of ice. Water molecules have a rather angular shape that prevents a tight fit of these molecules; when they assemble in a solid, they adopt a very complicated crystal structure with large gaps. As a result, ice has a lower density than water -- the density of ice is 917 kg/m3, and the volume of 1 kg of ice is 1091 cm3. At a temperature slightly above the freezing point, water is liquid, but some of the water molecules already have assembled themselves into microscopic (and ephemeral) ice crystals; these microscopic crystals give the cold water an excess volume.

The maximum in the density of water at about 4°C has an important consequence for the ecology of lakes. In winter the layer of water on the surface of the lake cools, becomes denser than the lower layer, and sinks to the bottom. This process continues until the temperature of the entire body of the lake reaches 4°C. Beyond this point, the cooling of the surface layer will make it less dense than the lower layers; thus the surface layer stays in place, floating on top of the lake. Ultimately, this surface layer freezes, becoming a solid sheet of ice while the body of the lake remains at 4°C. The sheet of ice inhibits the heat loss from the lake, especially if covered with a blanket of snow. Besides, any further heat loss merely causes some thickening of the sheet of ice, without disturbing the deeper layers of water, which remain at a stable temperature of 4°C -- fish and other aquatic life can survive the winter in reasonable comfort.

EXAMPLE 2. A glass vessel of volume 200 cm3 is filled to the rim with mercury. How much of the mercury will overflow the vessel if we raise the temperature by 30°C?

SOLUTION: The volume of mercury will increase by

The volume of the glass vessel will increase just as though all of the vessel were filled with glass (solid glass):

The difference 1.08 cm3 - 0.20 cm3 = 0.88 cm3 is the volume of mercury that will overflow.

Thermal expansion must be taken into account in the design of long structures, such as bridges or railroad tracks. The decks of bridges usually have several expansion joints with gaps (see Figure 20.5) that permit changes of length and prevent the bridge from buckling. Likewise, gaps are left between the segments of rail in a railroad track; but if the temperature changes exceed the expectations of the designers, the results can be disastrous (see Figure 20.6).

Incidentally: Our ability to erect large buildings and other structures out of reinforced concrete hinges on the fortuitous coincidence of the coefficients of expansion of iron and concrete (see Table 20.2). Reinforced concrete consists of iron rods in a concrete matrix. If the coefficients of expansion for these two materials were appreciably different, then the daily and seasonal temperature changes would cause the iron rods to move relative to the concrete -- ultimately, the iron rods would work loose, and the reinforcement would come to an end.