Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.33.
The thermodynamic identity for an infinitesimal process is
For constant volume processes , so we can derive the expression for the heat capacity by dividing both sides by :
For constant pressure processes, the heat capacity is defined in terms of the enthalpy. The enthalpy is defined as
It is the energy required to create the system, which is a combination of the internal energy of the system itself, plus the work required to clear the volume that the system occupies. If this work is done at constant pressure, the work is . In a system in which the only work done is from expansion, so the heat capacity at constant pressure is
The change in is
At constant pressure so dividing both sides by we get