% miko:type: atomic
% miko:title: Maxwell Relations
% miko:summary: Identities between thermodynamic derivatives following from state functions.
% miko:tags: thermo
% miko:difficulty: undergrad
% miko:related: thermodynamic-potentials, entropy-second-law, gibbs-free-energy
% miko:keywords: derivative, state function, thermodynamic identity

\documentclass{article}
\usepackage{amsmath}
\title{Maxwell Relations}
\begin{document}
\maketitle

\section{Origin}

Thermodynamic potentials are state functions. Their mixed second derivatives
therefore commute. For the Helmholtz free energy,

$$
dF=-S\,dT-p\,dV,
$$

so

$$
\left(\frac{\partial S}{\partial V}\right)_T
=
\left(\frac{\partial p}{\partial T}\right)_V.
$$

\section{Use}

Maxwell relations convert difficult entropy derivatives into measurable
pressure, volume, or temperature derivatives.

\end{document}
