[[Thermodynamics MOC]]
# Thermodynamic potential

A thermodynamic potential is an [[Extensive and intensive parameters|extensive]] quantity which is naturally[^n1] a function of the natural[^n2] independent variables and whose minimization or maximization follows from the [[Entropy#^P4]] when these independent variables are held constant.

  [^n1]: i.e. by the [[First law of thermodynamics]].
  [^n2]: a natural choice of variables for the situation at hand, i.e. the [[Extensive and intensive parameters|intensive or extensive parameters]] that can be controlled directly.

## Equations of state
To show that a quantity $M$ is naturally a function of parameters $(x_{i})_{i=1}^n$, one takes the [[exterior derivative]] $dM$ and shows $dM = \sum_{i=1}^n f_{i}dx_{i}$, where $f_{i}$ are 0-forms.
One then arrives at the **equations of state**
$$
\begin{align*}
\frac{ \partial M }{ \partial x_{i} } = f_{i}
\end{align*}
$$

## Examples

- [[Entropy#As a thermodynamic potential|Entropy]]
- [[Helmholtz free energy]]
- [[Gibbs free energy]]
- [[Enthalpy]]
- [[Landau free energy]]



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#state/tidy  | #lang/en | #SemBr