Thanks to physics, you know that when you increase the temperature of a solid or liquid, its volume will expand. If a solid or liquid undergoes a small temperature change of just a few degrees, you can say that its volume will change in a way proportionate to the temperature change.

As long as the temperature differences involved are small, the fraction by which a solid expands,

is proportional to the change in temperature,

With volume expansion, the constant involved is called the coefficient of volume expansion.

You’ve created the analog (or equivalent) of the equation

for linear expansion.

If the lengths and temperature changes are small, you find that

for most solids. This makes sense, because you go from one dimension to three. For example, for steel,

Here’s an example: Say you’re at the gasoline refinery when you notice that workers are filling all the 5,000-gallon tanker trucks to the very brim before driving off on a hot summer day. “Uh oh,” you think as you get your calculator out. For gasoline,

and you figure that it’s 10.0 degrees Celsius warmer in the sunshine than in the building, so here’s how much the volume of gasoline will increase:

Not good news for the refinery — those 5,000-gallon tankers of gasoline that are filled to the brim have to carry 5,047.5 gallons of gasoline after they go out in the sunshine.

of gasoline. Should you tell the refinery workers? Or should you ask for a bigger fee first?

First you negotiate your whopper fee, and then you go explain the problem to the foreman. “Holy smokes!” he cries. “We’d have gasoline pouring out of the caps on the top of our trucks.” He stops the trucks and gets some gasoline taken out of each before they’re sent on their way.