Mathematical double points according to the simplified–perturbed-hard-chain theory

Abstract

Critical curves of a binary fluid mixture are usually plotted as a function of pressure and temperature (p,T projection), temperature and composition (T,x projection) or as a function of the reduced densities of the two components (y1,y2 projection). In these three ways of representation, we will show the structure of the critical curves around a mathematical double point. Moreover, it will be shown that mathematical double points can conveniently be divided into two groups: the (meta)stable and unstable mathematical double points. To date, the (meta)stable mathematical double points have not been investigated with a pressure-explicit equation of state. In this paper, it will be shown how the two types of mathematical double points can be calculated. We will digress on the thermodynamic conditions to be obeyed for the calculation of the two types of mathematical double points.

Global phase behavior based on the simplified-perturbed hard-chain equation of state

“…To determine A 2x , A vx , A 2V , A 3x ,..., etc., we have derived the following expression for the molar Helmholtz energy, A(V,T,x) from the SPHCT equation: 25 A͑V,T,x ͒ϭ͑ 1Ϫx ͒A 1 *͑V 0 ,T ͒ϩxA 2 *͑V 0 ,T͒…”

“…A mathematical double point ͑MDP͒ is a point in the V,T,x-space of a binary mixture where the critical curve is self crossing. 25 In the global phase diagram the MDP curve contains binary mixtures that exhibit a MDP. This MDP curve in the global phase diagram is not a boundary between two types of phase behavior.…”

“…This MDP curve in the global phase diagram is not a boundary between two types of phase behavior. 25 It goes through the van Laar point 26 and is thermodynamically stable at this point only.…”

“…The closed critical curve is always partly unstable, i.e., G 4x Ͻ0, and partly ͑meta͒stable. 25 The stable part of the closed critical curve Fig. 8.…”

“…The boundary state between type V h and the new type VIII is formed by a phase diagram with a mathematical double point of the second kind. 24,25 van Pelt and de Loos 26 have shown that the ''ordinary'' types II, III, IV, and IV* can also be calculated using the SPHCT equation of state. Finally, Green and Jackson 27 extensively discussed the phase behavior of waterϩalkanol mixtures and the global phase behavior that is calculated from their own equation of state.…”