Maxwell relations

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The fundamental thermodynamic relation for a reversible process in a single-component system, where the only work term considered is pdV, is obtained from eq. dE \le TdS - \delta W from Thermodynamic property relations, i.e.,

dE = TdS - pdV\qquad \qquad(1)

which can also be rewritten in terms of enthalpy (H = E + pV), Helmholtz free energy (F = ETS), and Gibbs free energy (G = HTS) as

dH = TdS + Vdp\qquad \qquad(2)

dF =  - SdT - pdV\qquad \qquad(3)

dG =  - SdT + Vdp\qquad \qquad(4)

which all have the form of

dz = Mdx + Ndy\qquad \qquad(5)


M = {\left( {\frac{{\partial z}}{{\partial x}}} \right)_y}\qquad \qquad(6)

N = {\left( {\frac{{\partial z}}{{\partial y}}} \right)_x}\qquad \qquad(7)

and dz is an exact differential, as thermodynamic properties like E,H,F, and G are path-independent functions.

Since eq. (5) is the total differential of function z,M and N are related by

{\left( {\frac{{\partial M}}{{\partial y}}} \right)_x} = {\left( {\frac{{\partial N}}{{\partial x}}} \right)_y} = \frac{{{\partial ^2}z}}{{\partial x\partial y}}\qquad \qquad(8)

Applying eq. (8) to eqs. (1) – (4), the following relationships are obtained:

{\left( {\frac{{\partial T}}{{\partial V}}} \right)_S} =  - {\left( {\frac{{\partial p}}{{\partial S}}} \right)_V} \qquad \qquad (9)

{\left( {\frac{{\partial T}}{{\partial p}}} \right)_S} = {\left( {\frac{{\partial V}}{{\partial S}}} \right)_p}\qquad \qquad(10)

{\left( {\frac{{\partial S}}{{\partial V}}} \right)_T} = {\left( {\frac{{\partial p}}{{\partial T}}} \right)_V}\qquad \qquad(11)

{\left( {\frac{{\partial S}}{{\partial p}}} \right)_T} =  - {\left( {\frac{{\partial V}}{{\partial T}}} \right)_p}\qquad \qquad(12)

which are referred to as Maxwell relations. The goal of Maxwell relations is to find equivalent partial derivatives containing p,T, and V that can be physically measured and therefore provide a means of determining the change of entropy, which cannot be measured directly.


Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

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