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+---
+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.