Not only does length change with a change in temperature, but area and volume change also. Thus, Δ A = γ A0Δ T, where Δ A is the change in the original area A0. The Greek letter gamma (γ) is the average coefficient of area expansion, which equals 2α. For change in volume, Δ V = β V0, Δ T, where Δ V is the change in the original volume V0. The Greek letter beta (β) is the average coefficient of volume expansion, which is equal to 3α.
Example 1: As an example of the application of these equations, consider heating a steel washer. What will be the area of the washer hole with original cross-sectional area of 10 mm2 if the steel has α = 1.1 × 10−5 per °C and is heated from 20 degrees C to 70 degrees C?
Solution: The hole will expand the same as a piece of the material having the same dimensions. The equation for increase in area leads to the following:
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Water is an exception to the usual increase in volume with increasing temperature. Note in Figure 2 that the maximum density of water occurs at 4 degrees Celsius.
This characteristic of water explains why a lake freezes at the surface. To see this, imagine that the air cools from 10 degrees Celsius to 5 degrees Celsius. The surface water in equilibrium with the air at these temperatures is denser than the slightly warmer water below it; therefore, the colder water sinks and warmer water from below comes to the surface. This occurs until the air temperature decreases to below 4 degrees when the surface water is less dense than the deeper water of about 4 degrees; then, the mixing ceases. As the temperature of the air continues to fall, the surface water freezes. The less dense ice remains on top of the water. Under these conditions, life near the bottom of the lake can continue to survive because only the water at or near the surface is frozen. Life on earth might have evolved quite differently if a pool of water froze from the bottom up.
