Q7(a). Maximum number of coexisting phases from the phase rule

Principle: Gibbs phase rule, F=C-P+2; the 2 counts temperature and pressure. When one of them is held fixed, the rule becomes F'=C-P+1; with both fixed, F”=C-P. The number of phases is largest when no freedom is left, F=0 (degrees of freedom cannot be negative).

1. (i) C=3, P and T fixed: F”=C-P=3-P. Setting F”=0: Pₘₐₓ=3 4pt] Example: in a ternary (triangular) diagram at fixed T and P, a three-phase triangle, such as three mutually saturated liquid or solid phases; the compositions of all three phases are then fixed.
2. (ii) C=2, T fixed: F'=C-P+1=3-P. Setting F'=0: Pₘₐₓ=3 4pt] At the chosen temperature three phases coexist only at one definite pressure, for example liquid A + liquid B + vapour for two partly miscible liquids, or ice + salt + saturated solution in a salt–water system at its eutectic temperature.
3. (iii) C=1, T and P fixed: F”=C-P=1-P. Setting F”=0: Pₘₐₓ=1 4pt] For a pure substance, choosing T and P independently fixes a point in one area of the phase diagram: only one phase is stable (water at 25°C and 1 atm is liquid only).
4. Contrast: two phases of a one-component system coexist only along a line (T and P linked, F=1), and three only at the triple point (F=0), which requires both T and P to be left unfixed so the system can choose them. Fixing them in advance leaves room for one phase.
5. Unconstrained maxima: without fixing anything the maximum is C+2: 3 for one component, 4 for two components, 5 for three. Each condition imposed removes one phase from the maximum.

Application. the phase rule fixes the number of phases at invariant points used in metallurgy and geology: a binary alloy at its eutectic (solid A + solid B + melt at fixed pressure) solidifies at constant temperature, and mineral assemblages in rocks rarely exceed the phase-rule limit.