# Second derivative of vapor pressure from a cubic equation of state

It is quite easy to compute the first derivative of vapor pressure with respect to temperature from a cubic equation of state at least at the critical point since there is a continuity with the critical isochore. But, is there any way to establish the second derivative or at least its value at the critical point? For simplicity, Van der Waals or Redlich-Kwong, Soave-Redlich-Kwong or Peng-Robinson equations of state are in my mind.

## 1 Answer

I believe the answer to your question can be found in this article : "On the continuity of isochore slopes, and the divergence of the curvature of the vaporization curve at the critical point of a simple fluid" by John Stephenson.

From what I understand, it can be shown that the second derivative of saturation pressure is proportional to the isochoric heat capacity of vaporization. Because of the divergence of this quantity at the critical temperature, the second derivative of saturation pressure will, theoretically, also go to infinity.

Anyway, cubic equations of state do not reproduce the divergence of isochoric heat capacity so you might obtain a finite value if you apply the equations from Stephenson's article and combine it with the isochoric heat capacity of vaporization from the cubic equation (which is also related to the second derivative of the alpha function in these equations).

• Thanks for your answer ! I know this paper from John. In fact, my idea is that the so-called $\alpha(T_R)$ function means probably more and that its relation to vapor pressure is hidding things. My goal was to try to extrapolate it for $T_r >1$ using, e.g., Padé approximants or something like that. – Claude Leibovici May 31 '18 at 10:28
• What do you mean by extrapolating? The $\alpha$ functions from the literature can be extrapolated without any problem so I believe you would like a theoretical trend for the $\alpha$ function in the supercritical temperature domain and thus demonstrate that the function must be decreasing and reaching a finite value (maybe 0) at the high temperature limit. Am I right? – yolegu May 31 '18 at 11:37
• You are totally correct. Up to now, the $\alpha$ functions are adjusted to better fit the vapor pressure. Few people care really about its extrapolation (think about Soave's original function which goes through a minimum and then increases to $\infty$. For a physical point of view, as Berthelot "almost" wrote it in 1875, the limit should be $0$. After having wasted more than 50 years with EoS, I am totally convinced that these functions hide something which is much deeper. – Claude Leibovici May 31 '18 at 15:25