Paul Peter Urone; Roger Hinrichs

Learning Objectives

By the end of this section, you will be able to:

Define and describe thermal expansion.
Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.

Figure 13.11 Thermal expansion joints like these in the Auckland Harbour Bridge in New Zealand allow bridges to change length without buckling. (credit: Ingolfson, Wikimedia Commons)

The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because air expands when its temperature increases, which causes the hot air’s density to be smaller than the density of surrounding air. As a result, the buoyant (upward) force on a given mass of air increases when the air is heated, resulting in a net upward force on that air mass. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.

What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.

What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature, an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.

Linear Thermal Expansion—Thermal Expansion in One Dimension

The change in length $Δ L$ is proportional to length $L$ . The dependence of thermal expansion on temperature, substance, and length is summarized in the equation

Δ L = αL Δ T,

13.7

where $Δ L$ is the change in length $L$ , $Δ T$ is the change in temperature, and $α$ is the coefficient of linear expansion, which varies slightly with temperature.

Table 13.2 lists representative values of the coefficient of linear expansion, which may have units of $1 / º C$ or 1/K. Because the size of a kelvin and a degree Celsius are the same, both $α$ and $Δ T$ can be expressed in units of kelvins or degrees Celsius. The equation $Δ L = αL Δ T$ is accurate for small changes in temperature and can be used for large changes in temperature if an average value of $α$ is used.

Material	Coefficient of linear expansion $α (1 / º C)$	Coefficient of volume expansion $β (1 / º C)$
Solids
Aluminum	$25 \times 10^{- 6}$	$75 \times 10^{- 6}$
Brass	$19 \times 10^{- 6}$	$56 \times 10^{- 6}$
Copper	$17 \times 10^{- 6}$	$51 \times 10^{- 6}$
Gold	$14 \times 10^{- 6}$	$42 \times 10^{- 6}$
Iron or Steel	$12 \times 10^{- 6}$	$35 \times 10^{- 6}$
Invar (Nickel-iron alloy)	$0 . 9 \times 10^{- 6}$	$2 . 7 \times 10^{- 6}$
Lead	$29 \times 10^{- 6}$	$87 \times 10^{- 6}$
Silver	$18 \times 10^{- 6}$	$54 \times 10^{- 6}$
Glass (ordinary)	$9 \times 10^{- 6}$	$27 \times 10^{- 6}$
Glass (Pyrex®)	$3 \times 10^{- 6}$	$9 \times 10^{- 6}$
Quartz	$0.4 \times 10^{- 6}$	$1 \times 10^{- 6}$
Concrete, Brick (approximate)	$12 \times 10^{- 6}$	$36 \times 10^{- 6}$
Marble (average)	$7 \times 10^{- 6}$	$2 . 1 \times 10^{- 5}$
Liquids
Ether		$1650 \times 10^{- 6}$
Ethyl alcohol		$1100 \times 10^{- 6}$
Petrol		$950 \times 10^{- 6}$
Glycerin		$500 \times 10^{- 6}$
Mercury		$180 \times 10^{- 6}$
Water		$210 \times 10^{- 6}$
Gases
Air and most other gases at atmospheric pressure		$3400 \times 10^{- 6}$

Table 13.2 Thermal Expansion Coefficients at

20 º C

¹

Example 13.3

Calculating Linear Thermal Expansion: The Golden Gate Bridge

The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from $- 15 º C$ to $40 º C$ . What is its change in length between these temperatures? Assume that the bridge is made entirely of steel.

Strategy

Use the equation for linear thermal expansion $Δ L = αL Δ T$ to calculate the change in length , $Δ L$ . Use the coefficient of linear expansion, $α$ , for steel from Table 13.2, and note that the change in temperature, $Δ T$ , is $55 º C$ .

Solution

Plug all of the known values into the equation to solve for $Δ L$ .

Δ L = αL Δ T = (\frac{12 \times 10^{- 6}}{º C}) (1275 m) (55 º C) = 0 . 84 m.

13.8

Discussion

Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.

Thermal Expansion in Two and Three Dimensions

Objects expand in all dimensions, as illustrated in Figure 13.12. That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger).

Thermal Expansion in Two Dimensions

For small temperature changes, the change in area $Δ A$ is given by

Δ A = 2 αA Δ T,

13.9

where $Δ A$ is the change in area $A$ , $Δ T$ is the change in temperature, and $α$ is the coefficient of linear expansion, which varies slightly with temperature.

Part a shows the outline of a flat metal plate before and after expansion. After expansion, it has the same shape and ratio of dimensions as before, but it takes up a greater area. Part b shows the outline of a flat metal plate with a hole in it, before and after expansion. The hole expands. Part c shows the outline of a rectangular box before and after expansion. After expansion, the box has the same proportions as before expansion, but it has a greater volume.

Figure 13.12 In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three dimensions increase.

Thermal Expansion in Three Dimensions

The change in volume $Δ V$ is very nearly $Δ V = 3 α V Δ T$ . This equation is usually written as

Δ V = βV Δ T,

13.10

where $β$ is the coefficient of volume expansion and $β \approx 3α$ . Note that the values of $β$ in Table 13.2 are almost exactly equal to $3α$ .

In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases) when it is at temperatures greater than $4 º C (40 º F)$ . However, it expands with decreasing temperature when it is between $+ 4 º C$ and $0 º C$ $(40 º F$ to $32 º F)$ . Water is densest at $+ 4 º C$ . (See Figure 13.13.) Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to $4 º C$ it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of $4 º C$ . If the temperature in the surface layer drops below $4 º C$ , the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in $4 º C$ water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.

A graph of density of freshwater in grams per cubic centimeter versus temperature in degrees Celsius. The line is convex up. At zero degrees C, the density is just under zero point nine nine nine five grams per cubic centimeter. The density then increases at a decreasing rate until it hits a peak of about zero point nine nine nine nine seven grams per cubic centimeter at about four degrees C. Above four degrees C, the density decreases with increasing temperature.

Figure 13.13 The density of water as a function of temperature. Note that the thermal expansion is actually very small. The maximum density at

+ 4 º C

is only 0.0075% greater than the density at

2 º C

, and 0.012% greater than that at

0 º C

.

Making Connections: Real-World Connections—Filling the Tank

Differences in the thermal expansion of materials can lead to interesting effects at the gas station. One example is the dripping of gasoline from a freshly filled tank on a hot day. Gasoline starts out at the temperature of the ground under the gas station, which is cooler than the air temperature above. The gasoline cools the steel tank when it is filled. Both gasoline and steel tank expand as they warm to air temperature, but gasoline expands much more than steel, and so it may overflow.

This difference in expansion can also cause problems when interpreting the gasoline gauge. The actual amount (mass) of gasoline left in the tank when the gauge hits “empty” is a lot less in the summer than in the winter. The gasoline has the same volume as it does in the winter when the “add fuel” light goes on, but because the gasoline has expanded, there is less mass. If you are used to getting another 40 miles on “empty” in the winter, beware—you may only get 38 miles in the summer.

Figure 13.14 Because the gas expands more than the gas tank with increasing temperature, you can’t drive as many miles on “empty” in the summer as you can in the winter. (credit: Hector Alejandro, Flickr)

Example 13.4

Calculating Thermal Expansion: Gas vs. Gas Tank

Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of $15.0 º C$ . How much gasoline has spilled by the time they warm to $35.0 º C$ ?