---
title: "Maxwell relations"
date: 2021-07-08
categories:
- Physics
- Thermodynamics
layout: "concept"
---

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](/know/concept/thermodynamic-potential/).

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.