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 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)

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


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

From \(dU=TdS-PdV\)

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


\(-P=(\frac{\partial U}{\partial V})_{S}\) \(\frac{\partial }{\partial y}(\frac{\partial z}{\partial x})_{y}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})_{x}=\frac{\partial^2 z}{\partial y\partial x}=\frac{\partial^2 z}{\partial x\partial y}\) (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 \((\frac{\partial V}{\partial T})_{P}=-(\frac{\partial S}{\partial P})_{T}\)


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:

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

Solved Examples

Example 1:

Prove that \((\frac{\partial V}{\partial T})_{p}=T\frac{\alpha }{\kappa _{T}}-p\).


Combining first and second laws:

dU = TdS – pdV

Diving both the sides by dV 

\(\frac{\mathrm{d} U}{\mathrm{d} V}|_{T}=\frac{T\mathrm{d} S}{\mathrm{d} V}|_{T}-p\frac{\mathrm{d} V}{\mathrm{d} V}|_{T}\)

\(\frac{\mathrm{d} U}{\mathrm{d} V}|_{T}=(\frac{\partial U}{\partial V})_{T}\)

\(\frac{T\mathrm{d} S}{\mathrm{d} V}|_{T}=(\frac{\partial S}{\partial V})_{T}\)

\(\frac{\mathrm{d} V}{\mathrm{d} V}|_{T}=1\)

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

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

\((\frac{\partial U}{\partial V})_{T}=T(\frac{\partial p}{\partial T})_{V}-p\)

\((\frac{\partial p}{\partial T})_{V}=\frac{\alpha }{\kappa _{T}}\)

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