Heat capacity

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

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

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

At constant volume

δ Q = d U + p d V

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

C_{V} : = {\frac{δ Q}{\partial T} |}_{V} = {\frac{\partial U}{\partial T} |}_{V}

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

C_{p} : = {\frac{δ 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}