Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line

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).

Thermodynamics and phase transitions in galaxy clustering

“…The absence of any cusps except for at this point suggests the existence of a system containing a mixed phase, consistent with the conditions for the application of the Clausius–Clapeyron equation. the finding of a mixed phase and hysteresis in the system previously reported in Iqbal et al () was reproduced in this work via an independent free energy analysis. The free energy analysis in this work somewhat differs from a standard Ginzburg Landau free energy analysis, as discussed in Section 4. the results of a recent work deriving an equation for not only the slope, but the curvature of a phase coexistence curve in pressure–temperature space (see Krafcik & Velasco ()) were applied to the Cosmological Many Body Problem. an equation for the curvature of the phase coexistence curve in the context of the Cosmological Many Body Problem was derived in this work for a more general parameter space, namely p − b space.…”

“…Reinserting Equation (64) into Equation (65), and exploiting the Maxwell thermodynamics relations yields, as in Krafcik & Velasco (), …”

“…There has been a recent work that derived an equation not only for the slope but the curvature of the equation of state along the phase coexistence curve in pressure–temperature space (Krafcik & Velasco ), going beyond the Clausius–Clapeyron equation. Recall the Clausius–Clapeyron equation is given by where ℓ = T Δ s is the specific latent heat.…”

“…The Clausius–Clapeyron equation yields the slope on the phase transition line in pressure–temperature space for systems exhibiting a first‐order phase transition for this mixed phase: where ℓ is the specific latent heat, T is the temperature, and is the specific volume (Greiner et al ). The authors of (Krafcik & Velasco ) went beyond the Clausius–Clapeyron equation to derive an equation to describe the curvature of the phase transition line of this mixed phase in pressure–temperature space. where is the specific heat at constant pressure, is the specific volume, is the coefficient of thermal expansion, and is the isothermal compressibility.…”

“…where is the specific latent heat, T is the temperature, and ≡ is the specific volume (Greiner et al 1994). The authors of (Krafcik & Velasco 2014) went beyond the Clausius-Clapeyron equation to derive an equation to describe the curvature of the phase transition line of this mixed phase in pressure-temperature space.…”