Heat capacity

From the first law of thermodynamics one has

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

where $Q$ is the heat, $U$ is the internal energy, $p$ is the pressure and $V$ is the volume. The heat capacity is given by the differential of the heat with respect to the temperature,

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

where $S$ is the entropy.

At constant volume

At constant volume (denoted by the subscript $V$),

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

At constant pressure

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

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 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 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 H} is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume 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_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p}