% miko:type: atomic
% miko:title: Thermodynamic Potentials
% miko:summary: Legendre transforms that match the variables controlled by an experiment.
% miko:tags: thermo
% miko:difficulty: undergrad
% miko:related: thermodynamic-first-law, gibbs-free-energy, chemical-potential, maxwell-relations
% miko:keywords: Helmholtz, Gibbs, enthalpy, Legendre transform

\documentclass{article}
\usepackage{amsmath}
\title{Thermodynamic Potentials}
\begin{document}
\maketitle

\section{Natural Variables}

The internal energy has natural variables $S,V,N$:

$$
dU=T\,dS-p\,dV+\mu\,dN.
$$

Legendre transforms exchange inconvenient variables for controlled variables.
The Helmholtz free energy

$$
F=U-TS
$$

has natural variables $T,V,N$, while the Gibbs free energy

$$
G=U-TS+pV
$$

has natural variables $T,p,N$.

\section{Equilibrium Criteria}

At fixed $T$ and $V$, equilibrium minimizes $F$. At fixed $T$ and $p$,
equilibrium minimizes $G$.

\end{document}
