In thermodynamics, Maxwell’s relations, Maxwell’s reciprocal relations, not to be confused with Maxwell’s equations (of the electromagnetic field) are sets of partial differential equations, derived by James Maxwell, that can be derived if one takes the partial derivative of the Pfaffian form of one of state equations for the thermodynamic potentials or other exact differential state equations. The four main Maxwell relations are: [1]

Relation

Derived from
Name

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V \qquad$
$dU = T dS - P dV \,$ Maxwell internal energy relation
$\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P \qquad$
$dH = T dS + V dP \,$ Maxwell enthalpy relation
$\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T$
$dF = -P dV - S dT \,$ Maxwell isothermal-isochoric free energy relation
$\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T$
$dG = dH - S dT \,$ Maxwell isothermal-isobaric free energy relation

Relation	Derived from	Name

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V \qquad$	$dU = T dS - P dV \,$	Maxwell internal energy relation
$\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P \qquad$	$dH = T dS + V dP \,$	Maxwell enthalpy relation
$\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T$	$dF = -P dV - S dT \,$	Maxwell isothermal-isochoric free energy relation
$\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T$	$dG = dH - S dT \,$	Maxwell isothermal-isobaric free energy relation

There are, however, more relations than this. [4]


Elementary graphical derivation of the Maxwell isothermal-isobaric free energy relation. [2]

Elementary graphical derivation of the Maxwell isothermal-isobaric free energy relation. [2]

Etymology
The name Maxwell’s relations, according to Polish-born American thermodynamicist Joseph Kestin, was assigned to Scottish physicit James Maxwell because he was the first who succeeded in writing down expressions that make the partial derivative relationships explicit. [2]

The publication would seem to have been Maxwell’s Theory of Heat, although this remains to be determined. Moreover, the name of the person who used the name “Maxwell’s relations” or Maxwell’s equations” needs to be tracked down. [3]

Value
The interest in Maxwell’s equations, according to French thermodynamics lexicographer Pierre Perrot, is that they lead to the partial derivative of entropy as a function of physical quantities, such as pressure, volume, and temperature, which can be directly measured by experiment in the laboratory [1]

See also
● Maxwell’s demon
● Maxwell’s thermodynamic surface
● Maxwell-Boltzmann distribution

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics (Maxwell’s equations, pgs. 195-97). Oxford University Press.
2. Kestin, Joseph. (1966). A Course in Thermodynamics (Maxwell’s relations, pgs. 506 (graph), 531, 526, 544). Blaisdell Publishing Co.
3. Maxwell, James. (1871). Theory of Heat. Publisher.
4. Oates, Gordon. (1997). Aerothermodynamics of Gas Turbine and Rocket Propulsion, Volume 1 (Maxwell’s relations, pg. 26). AIAA.

External links
● Maxwell relations – Wikipedia.