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,

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

where $S$ is the entropy.

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 $p$),

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

where $H$ is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by

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