# First order phase transition and latent heat

Usually in physics, first order phase transitions are phase transitions where the free energy potential's first derivative is discontinuous when differentiated by some thermodynamic variable such as temperature or pressure. Usually, also it is assumed that first order phase transitions lead to latent heat that is either absorbed or released

However I was wondering if there are first order phase transitions where there is no release of latent heat, but the free energy potential has still a discontinuous first derivative when differentiated by some thermodynamic variable.

Another way to formulate it : given that latent heat depends on the variation of the free energy with respect to the temperature, since $$S=-\frac{\partial F}{\partial T}$$, is it possible that a discontinuity with respect to the pressure for instance $$V=\frac{dG}{dP}$$, doesn't lead to a discontinuity with respect to the temperature ?
The answer in that case lies in Clausius-Clapeyron relation : $$\frac{dP}{dT}=\frac{\Delta s}{\Delta v}= \frac{L}{T\Delta v}$$ where $$dP/dT$$ is the slope of the coexistence curve. Except for the case where the slope is infinite or $$0$$, latent heat and change in the volume imply each other
• Okay. I kept thinking about this. The Clausius-Clapeyron equation is derived from the Gibbs potential from my understanding. However there are phase transitions where pressure is not held constant, but volume (Helmholtz), In that case, $\Delta v=0$. So I was wondering if a phase transition under a Helmholtz potential could exist without latent heat release. – mathdummy Dec 18 '18 at 20:59