A fourth transform that gives pressure as a function of temperature and the three chemical potentials. Temperature and

are fixed at values giving the equimolar critical-point composition.

the base form is

U = f(S,N1,N2,N3,V)

the fourth transform is

A´´´ = U – TS - µ1N1 - µ2N2 - µ3N3 = -PV

scaling with respect to volume gives

As a fourth transform in a ternary system this function has the form of a swallowtail catastrophe. Phase-equilibrium states are given by points along the self-intersection curve where the variables T,P,µ1,µ2,µ3 are common to both the vapor and liquid phases. Again, the red unstable region has been omitted because it would cover the intersection curve and the metastable states.

Iowa State University