Heat capacity

From the first law of thermodynamics we have

$\delta Q=dU+pdV$

the heat capacity is given by

$C=\frac{\delta Q}{\partial T}$

At constant volume

$C_V={\frac{\delta Q}{\partial T}|}_V={\frac{\partial U}{\partial T}|}_V$

where U is the internal energy, T is the temperature, and V is the volume.

At constant pressure

$C_p={\frac{\delta Q}{\partial T}|}_p={\frac{\partial U}{\partial T}|}_p+p{\frac{\partial V}{\partial T}|}_p$

where p is the pressure.

We have

$C_p-C_V=(p+{\frac{\partial U}{\partial V}|}_T){\frac{\partial V}{\partial T}|}_p$