path: root/source/know/concept/thermodynamic-potential/index.md

---
title: "Thermodynamic potential"
sort_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.