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 (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 = \frac{\delta Q}{\partial T}}

At constant volume

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_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

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 = \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}