Heat capacity

The heat capacity is defined as the differential of heat with respect to the temperature $T$ ,

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

where $Q$ is heat and $S$ is the entropy.

At constant volume

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

thus at constant volume, denoted by the subscript $V$ , then $dV=0$ ,

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

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}

A low temperatures on has

c_{v}={\frac {12\pi ^{4}}{5}}nk_{B}\left({\frac {T}{\Theta _{D}}}\right)^{3}

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature and $\Theta _{D}$ is an empirical parameter known as the Debye temperature.