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} \mpdv{z}{y}{x} = \boxed{ \Big( \pdv{A}{y} \Big)_x = \Big( \pdv{B}{x} \Big)_y } = \mpdv{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 $U(S, V, N)$:

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

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 $H(S, P, N)$:

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

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 $F(T, V, N)$:

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

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 $G(T, P, N)$:

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

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 $\Omega(T, V, \mu)$:

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

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.