Thermodynamic derivation of scaling at the liquid-vapor critical point

TL;DR

This paper derives the scaling behavior at the liquid-vapor critical point using thermodynamics, confirming Widom's hypothesis and predicting the behavior of critical exponents and thermodynamic properties.

Contribution

It provides a thermodynamic derivation of the scaling form at the critical point, linking entropy, energy, and particle densities without relying on specific models.

Findings

Confirms the Widom scaling hypothesis from thermodynamics.

Predicts the critical isotherm has the same functional form as the coexistence curve.

Shows specific heat diverges while isothermal compressibility also diverges at criticality.

Abstract

With the use of thermodynamics and general equilibrium conditions only, we study the entropy of a fluid in the vicinity of the critical point of the liquid-vapor phase transition. By assuming a general form for the coexistence curve in the vicinity of the critical point, we show that the functional dependence of the entropy as a function of energy and particle densities necessarily obeys the scaling form hypothesized by Widom. Our analysis allows for a discussion on the properties of the corresponding scaling function, with the interesting prediction that the critical isotherm has the same functional dependence, between the energy and particles densities, as the coexistence curve. In addition to the derivation of the expected equalities of the critical exponents, the conditions that lead to scaling also imply that while the specific heat at constant volume can diverge at the critical point, the isothermal compressibility must do so.