Thermal Expansion in Two and Three Dimensions

Objects expand in all dimensions, as illustrated in Figure 13.11. 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.

Figure 13.11 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.

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 $4\text{º}\text{C}(\text{40}\text{º}\text{F})$. However, it expands with decreasing temperature when it is between $+4\text{º}\text{C}$ and $0\text{º}\text{C}$$(\text{40}\text{º}\text{F}$ to $\text{32}\text{º}\text{F})$. Water is densest at $+4\text{º}\text{C}$ (see Figure 13.12). Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to $4\text{º}\text{C}$, 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 $4\text{º}\text{C.}$ If the temperature in the surface layer drops below $4\text{º}\text{C,}$ 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 $4\text{º}\text{C}$ 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.

Figure 13.12 The density of water as a function of temperature. Note that the thermal expansion is actually very small. The maximum density at $+4\text{º}\text{C}$ is only 0.0075 percent greater than the density at $2\text{º}\text{C,}$ and 0.012 percent greater than that at $0\text{º}\text{C.}$

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 the 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, or 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 will probably run out much more quickly in the summer.

Figure 13.13 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. (Hector Alejandro, Flickr)

Thermal Stress

Thermal stress is created by thermal expansion or contraction [see Elasticity: Stress and Strain for a discussion of stress and strain]. Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful, for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes.

Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints (see Figure 13.14). Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. Pyrex® is less susceptible because of its small coefficient of thermal expansion. Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water.

Figure 13.14 Thermal stress contributes to the formation of potholes. (credit: Editor5807, Wikimedia Commons)

Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.

Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings—gold, silver, etc.—are being replaced by composite fillings—porcelain—which have smaller coefficients of expansion and are closer to those of teeth.

Check Your Understanding

Two blocks, A and B, are made of the same material. Block A has dimensions $l\times w\times h=L\times 2L\times L$ and Block B has dimensions $2L\times 2L\times 2L$. If the temperature changes, what is (a) the change in the volume of the two blocks, (b) the change in the cross-sectional area $l\times w$, and (c) the change in the height $h$ of the two blocks?

Figure 13.15

Solution

(a) The change in volume is proportional to the original volume. Block A has a volume of $L\times 2L\times L={2L}^3\text{.}$. Block B has a volume of $2L\times 2L\times 2L={8L}^3,$ which is four times that of Block A. Thus the change in volume of Block B should be four times the change in volume of Block A.

(b) The change in area is proportional to the area. The cross-sectional area of Block A is $L\times 2L={2L}^2,$ while that of Block B is $2L\times 2L={4L}^2\text{.}$ Because cross-sectional area of Block B is twice that of Block A, the change in the cross-sectional area of Block B is twice that of Block A.

(c) The change in height is proportional to the original height. Because the original height of Block B is twice that of A, the change in the height of Block B is twice that of Block A.