Heat capacity

From the first law of thermodynamics we have

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

the heat capacity is given by

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

At constant volume

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

where U is the internal energy, T is the temperature, and V is the volume.

At constant pressure

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

where p is the pressure.

We have

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