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Thermal expansion in three dimensions

The relationship between volume and temperature d V d T is given by d V d T = β V Δ T , where β is the coefficient of volume expansion    . As you can show in [link] , β = 3 α . This equation is usually written as

Δ V = β V Δ T .

Note that the values of β in [link] are equal to 3 α except for rounding.

Volume expansion is defined for liquids, but linear and area expansion are not, as a liquid’s changes in linear dimensions and area depend on the shape of its container. Thus, [link] shows liquids’ values of β but not α .

In general, objects expand with increasing temperature. Water is the most important exception to this rule. Water does expand with increasing temperature (its density decreases ) at temperatures greater than 4 ° C ( 40 ° F ) . However, it is densest at + 4 ° C and expands with decreasing temperature between + 4 ° C and 0 ° C ( 40 ° F to 32 ° F ), as shown in [link] . A striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4 ° C, it is denser than the remaining water and thus sinks to the bottom. This “turnover” leaves a layer of warmer water near the surface, which is then cooled. However, if the temperature in the surface layer drops below 4 ° C , that water is less dense than the water below, and thus stays near the top. As a result, the pond surface can freeze over. The layer of ice insulates the liquid water below it from low air temperatures. Fish and other aquatic life can survive in 4 ° C water beneath ice, due to this unusual characteristic of water.

Figure shows a graph of density of fresh water in grams per cubic centimeter versus temperature in degree Celsius. The graph starts at 0.99985 at 0 degrees and rises to a maximum y value of just under 1 at 4 degrees Celsius. It then curves down to 0.99950 at 12 degrees Celsius.
This curve shows the density of water as a function of temperature. Note that the thermal expansion at low temperatures is very small. The maximum density at 4 ° C is only 0.0075 % greater than the density at 2 ° C , and 0.012 % greater than that at 0 ° C . The decrease of density below 4 ° C occurs because the liquid water approachs the solid crystal form of ice, which contains more empty space than the liquid.

Calculating thermal expansion

Suppose your 60.0-L ( 15.9 -gal -gal) steel gasoline tank is full of gas that is cool because it has just been pumped from an underground reservoir. Now, both the tank and the gasoline have a temperature of 15.0 ° C . How much gasoline has spilled by the time they warm to 35.0 ° C ?


The tank and gasoline increase in volume, but the gasoline increases more, so the amount spilled is the difference in their volume changes. We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank. (The gasoline tank can be treated as solid steel.)


  1. Use the equation for volume expansion to calculate the increase in volume of the steel tank:
    Δ V s = β s V s Δ T .
  2. The increase in volume of the gasoline is given by this equation:
    Δ V gas = β gas V gas Δ T .
  3. Find the difference in volume to determine the amount spilled as
    V spill = Δ V gas Δ V s .

Alternatively, we can combine these three equations into a single equation. (Note that the original volumes are equal.)

V spill = ( β ga s β s ) V Δ T = [ ( 950 35 ) × 10 −6 / ° C ] ( 60.0 L ) ( 20.0 ° C ) = 1.10 L .


This amount is significant, particularly for a 60.0-L tank. The effect is so striking because the gasoline and steel expand quickly. The rate of change in thermal properties is discussed later in this chapter.

If you try to cap the tank tightly to prevent overflow, you will find that it leaks anyway, either around the cap or by bursting the tank. Tightly constricting the expanding gas is equivalent to compressing it, and both liquids and solids resist compression with extremely large forces. To avoid rupturing rigid containers, these containers have air gaps, which allow them to expand and contract without stressing them.

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Questions & Answers

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Practice Key Terms 4

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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