A Stability Condition for Binaries

In this appendix we prove that a stable, one-phase, binary mixture must have values for component fugacities that are less than the corresponding pure-component values that is, we prove that a stable, one-phase, binary mixture must have 

Situation 1. If (F.0.2) is satisfied for all Xi between 0 and 1, then increases monoton-ically from 0 at = 0 to /pure i at = 1. The mixture remains a stable single phase at all compositions, and at every x (F.0.1) is obeyed. 

Situation 2. If the condition (F.0.2) is violated over some range of Xy then the mixture is not stable over some compositions and it may split into two phases a and p. The curve for /i(x ) either oscillates or it separates into distinct branches. The phase equilibrium conditions require 

Further, each of these phases is stable, so they each satisfy the stability requirement (F.0.2). 

Let P designate the phase that is rich in component 1. We presume pure 1 is a stable phase, so by continuity, mixtures from to = 1 are stable single phases, and because of (F.0.2) their fugacities must be less than /purei- Therefore they approach /pure 1 from below hence. 

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