Thermodynamic relations

These thermodynamic relations are also known as coefficient relations.

$T=\left({\frac {\partial U}{\partial S}}\right)_{V}$

$p=-\left({\frac {\partial U}{\partial V}}\right)_{S}$

$T=\left({\frac {\partial H}{\partial S}}\right)_{p}$

$V=\left({\frac {\partial H}{\partial p}}\right)_{S}$

$S=-\left({\frac {\partial A}{\partial T}}\right)_{V}$

$p=-\left({\frac {\partial A}{\partial V}}\right)_{T}$

$S=-\left({\frac {\partial G}{\partial T}}\right)_{p}$

$V=\left({\frac {\partial G}{\partial p}}\right)_{T}$

where

$H$ is the enthalpy
$T$ is the temperature
$G$ is the Gibbs energy function
$U$ is the internal energy,
$V$ is the volume
$A$ is the Helmholtz energy function
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See also