Heat capacity

From the first law of thermodynamics one has

$\delta Q=dU+pdV$

where $Q$ is the heat, $U$ is the internal energy, $p$ is the pressure and $V$ is the volume. The heat capacity is given by the differential of the heat with respect to the temperature,

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

At constant volume

At constant volume (denoted by the subscript $V$),

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

At constant pressure

At constant pressure (denoted by the subscript $p$),

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

The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by

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