4.8 Heat Capacity

The heat capacity is directly related to q. It is defined as the amount of heat needed to raise the temperature of something by one degree Kelvin. In symbols, with C as the heat capacity

$$C = \frac{q}{T}$$

or, better, for small changes

$$C = \frac{dq}{dT}$$

But there are many ways to heat something. For example one can heat an object at constant pressure. We do that all the time when we heat something in the open air. Or we can heat something at constant volume, ^4.9 or in any one of an infinite other ways.

Usually we heat something in useful ways. Constant pressure is a common experimental condition. About the only things that is usually heated at constant volume are gases. And then one does it carefully.

What's the difference between these two heat capacities? The two quantities are not the same! Why? In the constant pressure case the system (usually) expands. That means that it does work against the surrounding atmosphere. At constant volume no such work is done. So more energy is needed in the constant pressure case; part to heat the system and part to expand it.

Of the two, heating at constant volume is the simpler so we shall discuss it first.