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
• $p$ is the pressure
• $S$ is the entropy

See also

• Maxwell's relations