Heat capacity

From the first law of thermodynamics we have

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

the heat capacity is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C={\frac {\delta Q}{\partial T}}}

At constant volume

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}