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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.