Critical-scaling theory for ternary systems with ‘‘polar’’ tielines

As used in the previous literature, critical-scaling theory for ternary (or pseudoternary) systems requires that the tielines be parallel, i.e., that one of the concentrations appears to be a field variable, rather than a density variable as it is known to be. However, for many chemical systems, the tielines in the conventional weight-fraction phase diagram are not parallel; instead, extensions of the tielines radiate from a single point (or are assumed to do so). For such cases, we show that there are two coordinate transforms (one to polar coordinates, the other to a second set of triangular coordinates) that convert one of the compositions from a density variable to a ‘‘hidden variable.’’ The latter transformation makes the tielines parallel. An association model that produces such parallel tielines is discussed, and the ability of critical-scaling equations to describe conjugate phase compositions in the system of transformed triangular coordinates is examined for a real ternary system, ethanol/n-hexane/water.