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

$\delta Q=dU+pdV$

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

$C_V:={\frac{\delta Q}{\partial T}|}_V={\frac{\partial U}{\partial T}|}_V$

At constant pressure

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

$C_p:={\frac{\delta Q}{\partial T}|}_p={\frac{\partial H}{\partial T}|}_p={\frac{\partial U}{\partial T}|}_p+p{\frac{\partial V}{\partial T}|}_p$

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

$C_p-C_V=(p+{\frac{\partial U}{\partial V}|}_T){\frac{\partial V}{\partial T}|}_p$

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

$c_v=\frac{12{\pi }^4}{5}n{k}_B{(\frac{T}{{\mathrm{\Theta }}_D})}^3$

where $k_B$ is the Boltzmann constant, $T$ is the temperature and ${\mathrm{\Theta }}_D$ is an empirical parameter known as the Debye temperature.

See also

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