Heat capacity

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

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

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

At constant volume

From the first law of thermodynamics one has

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

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

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 }

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

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