4.9.3 Other Heat Capacities

Constant volume and constant pressure heat capacities are not the only types of heat capacities around (though they are the most useful.) The heat capacity during a process depends on the path taken by that process. Constant volume and constant pressure are two paths.

What about an adiabatic path? In that case no heat enters or leaves the system, though the temperature can change. So in that case q is always zero and the adiabatic heat capacity is always zero!

$$C_{adiabatic} = 0$$

And what about an isothermal path?^4.15 In that case there is never a change of temperature. A glance at the general definition of heat capacity,

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

shows that dT will always be zero. The result is that the heat capacity of an isothermal process is infinite! In fact, depending on whether you take heat out of or put heat into an isothermal system, the heat capacity can be either plus or minus infinity.

It isn't hard to invent paths in which the heat capacity is negative. That seems impossible, but it's not. The details of the path can cause heat gain or loss, independent of the change in temperature.