Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young’s and Laplace’s equations

Abstract

The fundamental laws governing the mechanical equilibrium of solid-fluid systems were formulated in 1805 and 1806. They are Laplace’s law, which describes the pressure drop across an interface, and Young’s equation for the contact angle. At that time, these laws were justified on purely mechanical grounds. In 1880 Gibbs used thermodynamics to show that these laws were necessary conditions for the equilibrium of heterogeneous systems. We revisit Gibbs’ derivation and simplify it for possible use at the undergraduate level. In addition, we present derivations of Young’s and Laplace’s equations, which involve energy balance on a volume element located at the surface. In particular, it is shown that the derivations are simpler, allow the analysis of nonequilibrium situations, and give a natural identification of the surface energy with the surface tension of the liquid-vapor interface.