Heat capacity

From the first law of thermodynamics one has

$\left.\delta Q\right.=dU+pdV$

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 $V$),

$C_{V}:=\left.{\frac {\delta Q}{\partial T}}\right\vert _{V}=\left.{\frac {\partial U}{\partial T}}\right\vert _{V}$

At constant pressure

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

$C_{p}:=\left.{\frac {\delta Q}{\partial T}}\right\vert _{p}=\left.{\frac {\partial H}{\partial T}}\right\vert _{p}=\left.{\frac {\partial U}{\partial T}}\right\vert _{p}+p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

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

$C_{p}-C_{V}=\left(p+\left.{\frac {\partial U}{\partial V}}\right\vert _{T}\right)\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$