Theory of coexisting states: calculation of binodals

A set of differential equations is established that determine the positions of the tielines for phase separation at constant temperature, provided the equation of state for a binary mixture is given in analytical form. Despite the fact that a large amount of work has been done on this problem, there is little information on the general theory, except for some papers that were written at the turn of the century. In most cases tielines are calculated by a “brute force” method, i.e. after the problem is formulated a non-linear simultaneous equation-solving program is called upon, with little or no attention to the specifics of the problem on hand. It is shown that there exists a very useful relation between the orientation of the tieline and the tangents to the binodals at each end. This relation persists near the almost pure liquid limit despite the fact that the Helmholtz free energy is a singular function in this case. In addition, a differential equation is given for the binodal as well as a theorem on the termination of a set of tielines in the neighborhood of a three-phase (or triple) point. The equations given here are different from the Gibbs Konowalow expressions. The latter hold at constant concentration, while the equations presented here are at constant temperature.