Generalized Korteweg equations for phase separation near the critical point in binary gas–liquid mixtures

Abstract

This paper deals with the definition of the order parameter in a binary gas–liquid system in the neighborhood of the critical point. The isothermal volume-composition (or V,x) diagram of a binary gas–liquid system may show one (or more) critical point(s). If a critical point is present, we determine the orientation of the tieline in the neighborhood of this point from the equation of state, using algebraic means. This orientation is needed to determine the proposed order parameter: the length of the tieline. The method is similar to the method based on the work of Korteweg, however, his calculation was restricted to the special case that the critical point lies close to the critical point of the pure component. Consequently, the character of such a critical point is mainly vapor–liquid-like in his work. For this reason Korteweg’s approach deals with a special situation and cannot be applied to the general case in which the phase separation is neither pure liquid–vapor, nor exclusively liquid–liquid, i.e., without vapor. The present method is based on a coordinate rotation in the V,x plane. It is shown that the end points of the tieline for a given pressure can be expressed in terms of the Taylor series coefficients of the equation of state evaluated around the critical point.