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 $dV=0$,

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

At constant pressure

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

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

where $H$ is the enthalpy. The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by

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

Liquids

• Claudio A. Cerdeiriña, Diego González-Salgado, Luis Romani, María del Carmen Delgado, Luis A. Torres and Miguel Costas "Towards an understanding of the heat capacity of liquids. A simple two-state model for molecular association", Journal of Chemical Physics 120 pp. 6648-6659 (2004)

Solids

Debye

A low temperatures on has

$c_{v}={\frac {12\pi ^{4}}{5}}nk_{B}\left({\frac {T}{\Theta _{D}}}\right)^{3}$

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature and $\Theta _{D}$ is an empirical parameter known as the Debye temperature.

See also

• Heat capacity of an ideal gas
• Yang-Yang anomaly