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+---
+title: "Thermodynamic potential"
+date: 2021-07-07
+categories:
+- Physics
+- Thermodynamics
+layout: "concept"
+---
+
+**Thermodynamic potentials** are state functions
+whose minima or maxima represent equilibrium states of a system.
+Such functions are either energies (hence *potential*) or entropies.
+
+Which potential (of many) decides the equilibrium states for a given system?
+That depends which variables are assumed to already be in automatic equilibrium.
+Such variables are known as the **natural variables** of that potential.
+For example, if a system can freely exchange heat with its surroundings,
+and is consequently assumed to be at the same temperature $T = T_{\mathrm{sur}}$,
+then $T$ must be a natural variable.
+
+The link from natural variables to potentials
+is established by [thermodynamic ensembles](/know/category/thermodynamic-ensembles/).
+
+Once enough natural variables have been found,
+the appropriate potential can be selected from the list below.
+All non-natural variables can then be calculated
+by taking partial derivatives of the potential
+with respect to the natural variables.
+
+Mathematically, the potentials are related to each other
+by [Legendre transformation](/know/concept/legendre-transform/).
+
+
+## Internal energy
+
+The **internal energy** $U$ represents
+the capacity to do both mechanical and non-mechanical work,
+and to release heat.
+It is simply the integral
+of the [fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/):
+
+$$\begin{aligned}
+    \boxed{
+        U(S, V, N) \equiv T S - P V + \mu N
+    }
+\end{aligned}$$
+
+It is a function of the entropy $S$, volume $V$, and particle count $N$:
+these are its natural variables.
+An infinitesimal change $\dd{U}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N}
+    }
+\end{aligned}$$
+
+The non-natural variables are
+temperature $T$, pressure $P$, and chemical potential $\mu$.
+They can be recovered by differentiating $U$
+with respect to the natural variables $S$, $V$, and $N$:
+
+$$\begin{aligned}
+    \boxed{
+        T = \Big( \pdv{U}{S} \Big)_{V,N}
+    \qquad
+        P = - \Big( \pdv{U}{V} \Big)_{S,N}
+    \qquad
+        \mu = \Big( \pdv{U}{N} \Big)_{S,V}
+    }
+\end{aligned}$$
+
+It is convention to write those subscripts,
+to help keep track of which function depends on which variables.
+They are meaningless; these are normal partial derivatives.
+
+
+## Enthalpy
+
+The **enthalpy** $H$ of a system, in units of energy,
+represents its capacity to do non-mechanical work,
+plus its capacity to release heat.
+It is given by:
+
+$$\begin{aligned}
+    \boxed{
+        H(S, P, N) \equiv U + P V
+    }
+\end{aligned}$$
+
+It is a function of the entropy $S$, pressure $P$, and particle count $N$:
+these are its natural variables.
+An infinitesimal change $\dd{H}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{H} = T \dd{S} + V \dd{P} + \mu \dd{N}
+    }
+\end{aligned}$$
+
+The non-natural variables are
+temperature $T$, volume $V$, and chemical potential $\mu$.
+They can be recovered by differentiating $H$
+with respect to the natural variables $S$, $P$, and $N$:
+
+$$\begin{aligned}
+    \boxed{
+        T = \Big( \pdv{H}{S} \Big)_{P,N}
+    \qquad
+        V = \Big( \pdv{H}{P} \Big)_{S,N}
+    \qquad
+        \mu = \Big( \pdv{H}{N} \Big)_{S,P}
+    }
+\end{aligned}$$
+
+
+## Helmholtz free energy
+
+The **Helmholtz free energy** $F$ represents
+the capacity of a system to
+do both mechanical and non-mechanical work,
+and is given by:
+
+$$\begin{aligned}
+    \boxed{
+        F(T, V, N) \equiv U - T S
+    }
+\end{aligned}$$
+
+It depends on the temperature $T$, volume $V$, and particle count $N$:
+these are natural variables.
+An infinitesimal change $\dd{H}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{F} = - P \dd{V} - S \dd{T} + \mu \dd{N}
+    }
+\end{aligned}$$
+
+The non-natural variables are
+entropy $S$, pressure $P$, and chemical potential $\mu$.
+They can be recovered by differentiating $F$
+with respect to the natural variables $T$, $V$, and $N$:
+
+$$\begin{aligned}
+    \boxed{
+        S = - \Big( \pdv{F}{T} \Big)_{V,N}
+    \qquad
+        P = - \Big( \pdv{F}{V} \Big)_{T,N}
+    \qquad
+        \mu = \Big( \pdv{F}{N} \Big)_{T,V}
+    }
+\end{aligned}$$
+
+
+## Gibbs free energy
+
+The **Gibbs free energy** $G$ represents
+the capacity of a system to do non-mechanical work:
+
+$$\begin{aligned}
+    \boxed{
+        G(T, P, N)
+        \equiv U + P V - T S
+    }
+\end{aligned}$$
+
+It depends on the temperature $T$, pressure $P$, and particle count $N$:
+they are natural variables.
+An infinitesimal change $\dd{G}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{G} = V \dd{P} - S \dd{T} + \mu \dd{N}
+    }
+\end{aligned}$$
+
+The non-natural variables are
+entropy $S$, volume $V$, and chemical potential $\mu$.
+These can be recovered by differentiating $G$
+with respect to the natural variables $T$, $P$, and $N$:
+
+$$\begin{aligned}
+    \boxed{
+        S = - \Big( \pdv{G}{T} \Big)_{P,N}
+    \qquad
+        V = \Big( \pdv{G}{P} \Big)_{T,N}
+    \qquad
+        \mu = \Big( \pdv{G}{N} \Big)_{T,P}
+    }
+\end{aligned}$$
+
+
+## Landau potential
+
+The **Landau potential** or **grand potential** $\Omega$, in units of energy,
+represents the capacity of a system to do mechanical work,
+and is given by:
+
+$$\begin{aligned}
+    \boxed{
+        \Omega(T, V, \mu) \equiv U - T S - \mu N
+    }
+\end{aligned}$$
+
+It depends on temperature $T$, volume $V$, and chemical potential $\mu$:
+these are natural variables.
+An infinitesimal change $\dd{\Omega}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\Omega} = - P \dd{V} - S \dd{T} - N \dd{\mu}
+    }
+\end{aligned}$$
+
+The non-natural variables are
+entropy $S$, pressure $P$, and particle count $N$.
+These can be recovered by differentiating $\Omega$
+with respect to the natural variables $T$, $V$, and $\mu$:
+
+$$\begin{aligned}
+    \boxed{
+        S = - \Big( \pdv{\Omega}{T} \Big)_{V,\mu}
+    \qquad
+        P = - \Big( \pdv{\Omega}{V} \Big)_{T,\mu}
+    \qquad
+        N = - \Big( \pdv{\Omega}{\mu} \Big)_{T,V}
+    }
+\end{aligned}$$
+
+
+## Entropy
+
+The **entropy** $S$, in units of energy over temperature,
+is an odd duck, but nevertheless used as a thermodynamic potential.
+It is given by:
+
+$$\begin{aligned}
+    \boxed{
+        S(U, V, N) \equiv \frac{1}{T} U + \frac{P}{T} V - \frac{\mu}{T} N
+    }
+\end{aligned}$$
+
+It depends on the internal energy $U$, volume $V$, and particle count $N$:
+they are natural variables.
+An infinitesimal change $\dd{S}$ is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{S} = \frac{1}{T} \dd{U} + \frac{P}{T} \dd{V} - \frac{\mu}{T} \dd{N}
+    }
+\end{aligned}$$
+
+The non-natural variables are $1/T$, $P/T$, and $\mu/T$.
+These can be recovered by differentiating $S$
+with respect to the natural variables $U$, $V$, and $N$:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{1}{T} = \Big( \pdv{S}{U} \Big)_{V,N}
+    \qquad
+        \frac{P}{T} = \Big( \pdv{S}{V} \Big)_{U,N}
+    \qquad
+        \frac{\mu}{T} = - \Big( \pdv{S}{N} \Big)_{U,V}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Gould, J. Tobochnik,
+    *Statistical and thermal physics*, 2nd edition,
+    Princeton.