 # Maxwell’s Relations

The Maxwell relations are derived from Euler’s reciprocity relation. The relations are expressed in partial differential form. The Maxwell relations consists of the characteristic functions: internal energy U, enthalpy H, Helmholtz free energy F, and Gibbs free energy G and thermodynamic parameters: entropy S, pressure P, volume V, and temperature T. Following is the table of Maxwell relations for secondary derivatives:

 $+(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}=\frac{\partial^2 U}{\partial S\partial V}$ $+(\frac{\partial T}{\partial P})_{S}=+(\frac{\partial V}{\partial S})_{P}=\frac{\partial^2 H}{\partial S\partial P}$ $+(\frac{\partial S}{\partial V})_{T}=+(\frac{\partial P}{\partial T})_{V}=\frac{\partial^2 F}{\partial T\partial V}$ $-(\frac{\partial S}{\partial P})_{T}=+(\frac{\partial V}{\partial T})_{V}=\frac{\partial^2 G}{\partial T\partial P}$

## What are Maxwell’s relations?

These are the set of thermodynamics equations which derived from a symmetry of secondary derivatives and from thermodynamic potentials. These relations are named after James Clerk Maxwell, who was a 19th-century physicist.

### Derivation of Maxwell’s relations

Maxwell’s relations can be derived as:

$dU=TdS-PdV$ (differential form of internal energy)

$dU=(\frac{\partial z}{\partial x})_{y}dx+(\frac{\partial z}{\partial y})_{x}dy$ (total differential form)

$dz=Mdx+Ndy$ (other way of showing the equation)

$M=(\frac{\partial z}{\partial x})_{y}$

And

$N=(\frac{\partial z}{\partial y})_{x}$

From $dU=TdS-PdV$

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

And

$-P=(\frac{\partial U}{\partial V})_{S}$ $\frac{\partial }{\partial y}(\frac{\partial z}{\partial x})_{y}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})_{x}=\frac{\partial^2 z}{\partial y\partial x}=\frac{\partial^2 z}{\partial x\partial y}$ (symmetry of second derivatives)

$\frac{\partial }{\partial V}(\frac{\partial U}{\partial S})_{V}=\frac{\partial }{\partial S}(\frac{\partial U}{\partial V})_{S}$ $(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}$

### Common forms of Maxwell’s relations

 Function Differential Natural variables Maxwell Relation U dU = TdS – PdV S, V $(\frac{\partial T}{\partial V})_{S}=-(\frac{\partial P}{\partial S})_{V}$ H dH = TdS + VdP S, P $(\frac{\partial T}{\partial P})_{S}=(\frac{\partial V}{\partial S})_{P}$ F dF = -PdV – SdT V, T $(\frac{\partial P}{\partial T})_{V}=(\frac{\partial S}{\partial V})_{T}$ G dG = VdP – SdT P, T $(\frac{\partial V}{\partial T})_{P}=-(\frac{\partial S}{\partial P})_{T}$

Where,

T is the temperature

S is the entropy

P is the pressure

V is the volume

U is the internal energy

H is the entropy

G is the Gibbs free energy

F is the Helmholtz free energy

With respect to pressure and particle number, enthalpy and Maxwell’s relation can be written as:

$(\frac{\partial \mu }{\partial P})_{S,N} = (\frac{\partial V}{\partial N})_{S,P} = (\frac{\partial^2 H}{\partial P\partial N})$

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