Our discussion of the liquid thermometer made use of one of the best-known changes in a substance: As its temperature increases, its volume almost always increases. (As we shall see shortly, in some substances the volume decreases when the temperature increases.) This phenomenon, known as thermal expansion, has an important role in numerous engineering applications. For example, thermal expansion joints, such as those shown in Figure 19.6, must be included in buildings, concrete highways, railroad tracks, brick walls, and bridges to compensate for dimensional changes that occur as the temperature changes.
Figure 19.6 (a) Thermal-expansion joints are used to separate sections of roadways on bridges. Without these joints, the surfaces would buckle due to thermal expansion on very hot days or crack due to contraction on very cold days. (b) The long, vertical joint is filled with a soft material that allows the wall to expand and contract as the temperature of the bricks changes.

Thermal expansion is a consequence of the change in the average separation between the constituent atoms in an object. To understand this, imagine that the atoms are connected by stiff springs, as shown in Figure 19.7. At ordinary temperatures, the atoms in a solid oscillate about their equilibrium positions with an amplitude of approximately 10^{-11} m and a frequency of approximately 10^{13} Hz. The average spacing between the atoms is about 10^{-10} m. As the temperature of the
solid increases, the atoms oscillate with greater amplitudes; as a result, the average separation between them increases. Consequently, the object expands.
Figure 19.7 A mechanical model of the atomic configuration in a substance. The atoms (spheres) are imagined to be attached to each other by springs that reflect the elastic nature of the interatomic forces.

If thermal expansion is sufficiently small relative to an object’s initial dimensions, the change in any dimension is, to a good approximation, proportional to the first power of the temperature change. Suppose that an object has an initial length L_{i} along some direction at some temperature and that the length increases by an amount ΔL for a change in temperature ΔT. Because it is convenient to consider the fractional change in length per degree of temperature change, we define the average coefficient of linear expansion as
Experiments show that α is constant for small changes in temperature. For purposes of calculation, this equation is usually rewritten as
where L_{f} is the final length, T_{i} and T_{f} are the initial and final temperatures, and the proportionality constant α is the average coefficient of linear expansion for a given material and has units of °C^{-1}.
It may be helpful to think of thermal expansion as an effective magnification or as a photographic enlargement of an object. For example, as a metal washer is heated (Fig. 19.8), all dimensions, including the radius of the hole, increase according to Equation 19.4.
Figure 19.8 Thermal expansion of a homogeneous metal washer. As the washer is heated, all dimensions increase. (The expansion is exaggerated in this figure.)

Table 19.2 lists the average coefficient of linear expansion for various materials. Note that for these materials α is positive, indicating an increase in length with increasing temperature. This is not always the case. Some substances—calcite (CaCO_{3}) is one example—expand along one dimension (positive α) and contract along another (negative α) as their temperatures are increased.
Because the linear dimensions of an object change with temperature, it follows that surface area and volume change as well. The change in volume at constant pressure is proportional to the initial volume V_{i} and to the change in temperature according to the relationship
where β is the average coefficient of volume expansion. For a solid, the average coefficient of volume expansion is approximately three times the average linear expansion coefficient: β = 3α. (This assumes that the average coefficient of linear expansion of the solid is the same in all directions.)
To see that β = 3α for a solid, consider a box of dimensions l, w, and h. Its volume at some temperature T_{i} is V_{i} = lwh. If the temperature changes to T_{i} + ΔT, its volume changes to V_{i} + ΔV where each dimension changes according to Equation 19.4. Therefore,
If we now divide both sides by V_{i} and then isolate the term ΔV/V_{i} , we obtain the fractional change in volume:
Because α ΔT << 1 for typical values of ΔT (< ~100°C), we can neglect the terms 3(α ΔT)^{2} and (α ΔT)^{3}. Upon making this approximation, we see that
Equation 19.6 shows that the right side of this expression is equal to β, and so we have 3α = β, the relationship we set out to prove. In a similar way, you can show that the change in area of a rectangular plate is given by ΔA = 2αA_{i}ΔT.
As Table 19.2 indicates, each substance has its own characteristic average coefficient of expansion. For example, when the temperatures of a brass rod and a steel rod of equal length are raised by the same amount from some common initial value, the brass rod expands more than the steel rod does because brass has a greater average coefficient of expansion than steel does. A simple mechanism
called a bimetallic strip utilizes this principle and is found in practical devices such as thermostats. It consists of two thin strips of dissimilar metals bonded together. As the temperature of the strip increases, the two metals expand by different amounts and the strip bends, as shown in Figure 19.9.
Figure 19.9 (a) A bimetallic strip bends as the temperature changes because the two metals have different expansion coefficients. (b) A bimetallic strip used in a thermostat to break or make electrical contact. (c) The interior of a thermostat, showing the coiled bimetallic strip. Why do you suppose the strip is coiled?

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