% miko:type: atomic
% miko:title: Clausius-Clapeyron Equation
% miko:summary: The slope of a coexistence curve in a phase diagram.
% miko:tags: thermo
% miko:difficulty: undergrad
% miko:related: phase-transition, gibbs-free-energy, thermodynamic-potentials
% miko:keywords: phase coexistence, latent heat, vapor pressure

\documentclass{article}
\usepackage{amsmath}
\title{Clausius-Clapeyron Equation}
\begin{document}
\maketitle

\section{Coexistence}

Along a phase boundary the molar Gibbs free energies of two phases are equal.
Differentiating this equality gives

$$
\frac{dp}{dT}=\frac{\Delta s}{\Delta v}
=\frac{L}{T\Delta v},
$$

where $L$ is the latent heat per mole and $\Delta v$ is the molar volume
change.

\section{Interpretation}

The equation tells how the transition pressure changes with temperature. It is
especially useful for liquid-vapor coexistence and vapor pressure curves.

\end{document}
