Categories: Physics, Thermodynamics.

Maxwell relations

The Maxwell relations are a useful set of relations in thermodynamics. They arise from the fact that the order of differentiation is irrelevant for well-behaved functions (sometimes known as the Schwarz theorem), applied to the thermodynamic potentials.

We start by proving the general “recipe”. Given that the differential element of some \(z\) is defined in terms of two constant quantities \(A\) and \(B\) and two independent variables \(x\) and \(y\):

\[\begin{aligned} \dd{z} \equiv A \dd{x} + B \dd{y} \end{aligned}\]

Then the quantities \(A\) and \(B\) can be extracted by dividing by \(\dd{x}\) and \(\dd{y}\) respectively:

\[\begin{aligned} A = \Big( \pdv{z}{x} \Big)_y \qquad B = \Big( \pdv{z}{y} \Big)_x \end{aligned}\]

By differentiating \(A\) and \(B\), and using that the order of differentiation is irrelevant, we find:

\[\begin{aligned} \pdv{z}{y}{x} = \boxed{ \Big( \pdv{A}{y} \Big)_x = \Big( \pdv{B}{x} \Big)_y } = \pdv{z}{x}{y} \end{aligned}\]

Using this, all Maxwell relations are derived. Each relation also has a reciprocal form:

\[\begin{aligned} \Big( \pdv{A}{y} \Big)_x^{-1} = \boxed{ \Big( \pdv{y}{A} \Big)_x = \Big( \pdv{x}{B} \Big)_y } = \Big( \pdv{B}{x} \Big)_y^{-1} \end{aligned}\]

The following quantities are useful to rewrite some of the Maxwell relations: the iso-\(P\) thermal expansion coefficient \(\alpha\), the iso-\(T\) combressibility \(\kappa_T\), the iso-\(S\) combressibility \(\kappa_S\), the iso-\(V\) heat capacity \(C_V\), and the iso-\(P\) heat capacity \(C_P\):

\[\begin{gathered} \alpha \equiv \frac{1}{V} \Big( \pdv{V}{T} \Big)_{P,N} \\ \kappa_T \equiv - \frac{1}{V} \Big( \pdv{V}{P} \Big)_{T,N} \qquad \quad \kappa_S \equiv - \frac{1}{V} \Big( \pdv{V}{P} \Big)_{S,N} \\ C_V \equiv T \Big( \pdv{S}{T} \Big)_{V,N} \qquad \qquad C_P \equiv T \Big( \pdv{S}{T} \Big)_{P,N} \end{gathered}\]

Internal energy

The following Maxwell relations can be derived from the internal energy \(U(S, V, N)\):

\[\begin{gathered} \pdv{U}{V}{S} = \boxed{ \Big( \pdv{T}{V} \Big)_S = - \Big( \pdv{P}{S} \Big)_V } = \pdv{U}{S}{V} \\ \pdv{U}{V}{N} = \boxed{ \Big( \pdv{\mu}{V} \Big)_N = - \Big( \pdv{P}{N} \Big)_V } = \pdv{U}{N}{V} \\ \pdv{U}{S}{N} = \boxed{ \Big( \pdv{\mu}{S} \Big)_N = \Big( \pdv{T}{N} \Big)_S } = \pdv{U}{N}{S} \end{gathered}\]

And the corresponding reciprocal relations are then given by:

\[\begin{gathered} \boxed{ \Big( \pdv{V}{T} \Big)_S = - \Big( \pdv{S}{P} \Big)_V } \\ \boxed{ \Big( \pdv{V}{\mu} \Big)_N = - \Big( \pdv{N}{P} \Big)_V } \\ \boxed{ \Big( \pdv{S}{\mu} \Big)_N = \Big( \pdv{N}{T} \Big)_S } \end{gathered}\]

Enthalpy

The following Maxwell relations can be derived from the enthalpy \(H(S, P, N)\):

\[\begin{gathered} \pdv{H}{P}{S} = \boxed{ \Big( \pdv{T}{P} \Big)_S = \Big( \pdv{V}{S} \Big)_P } = \pdv{H}{S}{P} \\ \pdv{H}{P}{N} = \boxed{ \Big( \pdv{\mu}{P} \Big)_N = \Big( \pdv{V}{N} \Big)_P } = \pdv{H}{N}{P} \\ \pdv{H}{N}{S} = \boxed{ \Big( \pdv{T}{N} \Big)_S = \Big( \pdv{\mu}{S} \Big)_N } = \pdv{H}{S}{N} \end{gathered}\]

And the corresponding reciprocal relations are then given by:

\[\begin{gathered} \boxed{ \Big( \pdv{P}{T} \Big)_S = \Big( \pdv{S}{V} \Big)_P } \\ \boxed{ \Big( \pdv{P}{\mu} \Big)_N = \Big( \pdv{N}{V} \Big)_P } \\ \boxed{ \Big( \pdv{N}{T} \Big)_S = \Big( \pdv{S}{\mu} \Big)_N } \end{gathered}\]

Helmholtz free energy

The following Maxwell relations can be derived from the Helmholtz free energy \(F(T, V, N)\):

\[\begin{gathered} - \pdv{F}{V}{T} = \boxed{ \Big( \pdv{S}{V} \Big)_T = \Big( \pdv{P}{T} \Big)_V } = - \pdv{F}{T}{V} \\ \pdv{F}{V}{N} = \boxed{ \Big( \pdv{\mu}{V} \Big)_N = - \Big( \pdv{P}{N} \Big)_V } = \pdv{F}{N}{V} \\ \pdv{F}{T}{N} = \boxed{ \Big( \pdv{\mu}{T} \Big)_N = - \Big( \pdv{S}{N} \Big)_T } = \pdv{F}{N}{T} \end{gathered}\]

And the corresponding reciprocal relations are then given by:

\[\begin{gathered} \boxed{ \Big( \pdv{V}{S} \Big)_T = \Big( \pdv{T}{P} \Big)_V } \\ \boxed{ \Big( \pdv{V}{\mu} \Big)_N = - \Big( \pdv{N}{P} \Big)_V } \\ \boxed{ \Big( \pdv{T}{\mu} \Big)_N = - \Big( \pdv{N}{S} \Big)_T } \end{gathered}\]

Gibbs free energy

The following Maxwell relations can be derived from the Gibbs free energy \(G(T, P, N)\):

\[\begin{gathered} \pdv{G}{T}{P} = \boxed{ \Big( \pdv{V}{T} \Big)_P = - \Big( \pdv{S}{P} \Big)_T } = \pdv{G}{P}{T} \\ \pdv{G}{N}{P} = \boxed{ \Big( \pdv{V}{N} \Big)_P = \Big( \pdv{\mu}{P} \Big)_N } = \pdv{G}{P}{N} \\ \pdv{G}{T}{N} = \boxed{ \Big( \pdv{\mu}{T} \Big)_N = - \Big( \pdv{S}{N} \Big)_T } = \pdv{G}{N}{T} \end{gathered}\]

And the corresponding reciprocal relations are then given by:

\[\begin{gathered} \boxed{ \Big( \pdv{T}{V} \Big)_P = - \Big( \pdv{P}{S} \Big)_T } \\ \boxed{ \Big( \pdv{N}{V} \Big)_P = \Big( \pdv{P}{\mu} \Big)_N } \\ \boxed{ \Big( \pdv{T}{\mu} \Big)_N = - \Big( \pdv{N}{S} \Big)_T } \end{gathered}\]

Landau potential

The following Maxwell relations can be derived from the Gibbs free energy \(\Omega(T, V, \mu)\):

\[\begin{gathered} - \pdv{\Omega}{V}{T} = \boxed{ \Big( \pdv{S}{V} \Big)_T = \Big( \pdv{P}{T} \Big)_V } = - \pdv{\Omega}{T}{V} \\ - \pdv{\Omega}{\mu}{V} = \boxed{ \Big( \pdv{P}{\mu} \Big)_V = \Big( \pdv{N}{V} \Big)_\mu } = - \pdv{\Omega}{V}{\mu} \\ - \pdv{\Omega}{T}{\mu} = \boxed{ \Big( \pdv{N}{T} \Big)_\mu = \Big( \pdv{S}{\mu} \Big)_T } = - \pdv{\Omega}{\mu}{T} \end{gathered}\]

And the corresponding reciprocal relations are then given by:

\[\begin{gathered} \boxed{ \Big( \pdv{V}{S} \Big)_T = \Big( \pdv{T}{P} \Big)_V } \\ \boxed{ \Big( \pdv{\mu}{P} \Big)_V = \Big( \pdv{V}{N} \Big)_\mu } \\ \boxed{ \Big( \pdv{T}{N} \Big)_\mu = \Big( \pdv{\mu}{S} \Big)_T } \end{gathered}\]

References

  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.

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