18.4 Thermal Expansion

Objects expand when heated. This thermal expansion is why the liquid rises in a thermometer and why pipes, highways, and bridges have expansion joints. FIGURE 18.4 shows an object of length $L$ that changes by $Δ L$ when the temperature is changed from $T$ to $T + Δ T$ . For most solids, the fractional change in length, $Δ L / L$ , is proportional to the temperature change $Δ T$ with a proportionality coefficient that depends on the material. That is,

\frac{Δ L}{L} = α Δ T

(18.8)

where $α$ (Greek alpha) is the material’s coefficient of linear expansion. Equation 18.8 characterizes both expansion ( $Δ L > 0$ if the temperature increases) and contraction ( $Δ L < 0$ if the temperature falls).

The figure consists of two parts that illustrate the Before and After lengths of a horizontally oriented rod.

FIGURE 18.4 Full Alternative Text

TABLE 18.4 gives the coefficients of linear expansion for a few common materials and also for a metal alloy called invar that is specifically designed to have extremely low thermal expansion. Because the fractional change in length is dimensionless, $α$ has units of $° C^{- 1}$ , read as “per degree Celsius.” The units may also be written $K^{- 1}$ , because a temperature change $Δ T$ is the same in $° C$ and K, but practical measurements are made in $° C$ , not K.

TABLE 18.4 Coefficients of linear and volume expansion

Material	$α (° C^{- 1})$
Aluminum	$2.3 \times 10^{- 5}$
Brass	$1.9 \times 10^{- 5}$
Concrete	$1.2 \times 10^{- 5}$
Steel	$1.1 \times 10^{- 5}$
Invar	$0.09 \times 10^{- 5}$
Material	$β (° C^{- 1})$
Gasoline	$9.6 \times 10^{- 4}$
Mercury	$1.8 \times 10^{- 4}$
Ethyl alcohol	$1.1 \times 10^{- 3}$

EXAMPLE 18.3 An expanding pipe

A 55-m-long steel pipe runs from one side of a refinery to the other. By how much does the pipe expand on a $5 ° C$ winter day when $155 ° C$ oil is pumped through it?

SOLVE The expansion is given by Equation 18.8, with the coefficient of linear expansion for steel taken from Table 18.4:

\begin{matrix} Δ L & = & α L Δ T = (1.1 \times 10^{- 5} ° C^{- 1}) (55 m) (150 ° C) \\ = & 0.091 m = 9.1 cm \end{matrix}

REVIEW 9.1 cm is a very small fraction of 55 m, so the pipe as a whole has expanded very little. Nevertheless, $9.1 cm \approx 3.5 in$ is a huge expansion in terms of the engineering of the pipe. Pipes like this have to be designed with flexible expansion joints that allow for thermal expansion and contraction.

Volume expansion is treated the same way. If an object’s volume changes by $Δ V$ during a temperature change $Δ T$ , the fractional change in volume is

\frac{Δ V}{V} = β Δ T

(18.9)

where $β$ (Greek beta) is the material’s coefficient of volume expansion.

Liquids are constrained by the shape of their container, so liquids are characterized by a volume-expansion coefficient but not by a linear-expansion coefficient. A few values are given in Table 18.4, where you can see that $β$ , like $α$ , has units of $° C^{- 1}$ .

Solids expand linearly in all three directions, as given by Equation 18.8, and in the process a solid changes its volume. Imagine a cube of edge length $L$ and volume $V = L^{3}$ . If the edge length changes by a very small amount $d L$ , the volume changes by

d V = 3 L^{2} d L

(18.10)

If we divide both sides by $V = L^{3}$ , we see that

\frac{d V}{V} = 3 \frac{d L}{L}

(18.11)

That is, the fractional change in volume is three times the fractional change in the length of the sides. Consequently, a solid’s coefficient of volume expansion is

β_{solid} = 3 α

(18.12)

You may have noticed that Table 18.4 does not include water. Water is a very common substance, but—due to its molecular structure—water has some unusual properties that set it apart from other liquids. If you lower the temperature of water, its volume contracts as expected. But only until the temperature reaches $4 ° C.$ If you continue cooling water, from $4 ° C$ down to the freezing point at $0 ° C$ , the volume expands! Water has maximum density at $4 ° C$ , with slightly less density—due to the expanded volume—at both higher and lower temperatures. Consequently, the thermal expansion of water cannot be characterized by a single coefficient of thermal expansion.

A photograph of a system of tubes that are connected by rubber molds. — Any structure subject to temperature changes has to be able to respond to thermal expansion. Highways, bridges, and railroads have *thermal expansion joints*. These joints are also needed in pipes and ducts. The flexible bellows seen in this photo can handle large temperature changes. Molded rubber segments can deal with less extreme temperature swings.