---
title: "Thermodynamic potential"
firstLetter: "T"
publishDate: 2021-07-07
categories:
- Physics
- Thermodynamics

date: 2021-07-03T14:40:22+02:00
draft: false
markup: pandoc
---

# Thermodynamic potential

**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.

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.