Gibbs phase rule

Remarks

Each phase is described by the temperature, the pressure and $C - 1$ independent mole fractions, so there are $P(C - 1) + 2$ intensive variables. Equilibrium requires that the chemical potential of each component is the same in all phases, which gives $C(P - 1)$ independent equations. Their difference is $P(C - 1) + 2 - C(P - 1) = C - P + 2$. For a pure substance, $C = 1$. One phase has $F = 2$ and fills a region of the phase diagram, and two phases have $F = 1$ and lie on a curve. Three phases have $F = 0$ and meet at an isolated triple point. In particular a pure substance cannot have four phases in equilibrium, since $F$ would be negative. If the components take part in $R$ independent chemical reactions, $C$ is the number of species minus $R$.

Going deeper

For a student

The rule is pure counting of unknowns and equations, so it applies to any system that meets the hypotheses, however complicated the chemistry. It is the first thing to check when reading a phase diagram: regions, curves and points must match $F = 2, 1, 0$ for one component. For a binary alloy at fixed pressure, the rule becomes $F = C - P + 1 = 3 - P$, which explains why two-phase regions have fixed compositions at each temperature. It counts only intensive variables; the amounts of each phase are not fixed by it but by mass balance, as in the lever rule.