Article,
Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line
American Journal of Physics, 82 (4): 301-305 (2014)
DOI: 10.1119/1.4858403
Abstract
We obtain an expression for the second derivative of the line in a PT diagram denoting a first-order phase transition for a pure hydrostatic system. Our result goes beyond the classical Clausius–Clapeyron equation, which provides only the first derivative of the pressure with respect to the temperature along the transition line. We present two pedagogical derivations suitable for an undergraduate thermodynamics class; the first one uses derivatives of the entropy while the second one uses derivatives of the enthalpy. The final expression for the second derivative involves only standard thermodynamic quantities such as the specific heats, the isothermal compressibilities, and the coefficients of thermal expansion of the two phases at the transition line. As an illustration, we compute the second derivatives of the freezing and vaporization lines of water at atmospheric pressure, and show that at this pressure the freezing line is concave down (negative second derivative) while the vaporization line is concave up (positive second derivative).
