Phase rule — Gibbs phase rule

WHAT is the phase rule?

Key words decoded:

• Phase: a physically distinct, mechanically separable, homogeneous region (ice, liquid water, vapour are 3 phases of 1 substance).
• Component: the minimum number of independent chemical species needed to specify the composition of every phase.
• Degree of freedom: an intensive variable (not amount) you may change freely.

WHY is it true? — Derivation from scratch

We just count variables, then subtract equations. That's the whole game.

Step 1 — Count the intensive variables. Each phase's intensive state is fixed by $T$, $P$, and its composition. With $C$ components, the mole fractions in one phase are $x_1, x_2, \dots, x_C$, but they sum to 1: $\sum_{i=1}^{C} x_i = 1$ Why this step? Mole fractions are not all independent — one is determined by the rest. So each phase has $C-1$ independent composition variables.

Across $P$ phases: composition variables $= P(C-1)$. Add the two shared variables $T$ and $P$: $\text{Total variables} = P(C-1) + 2$

Step 2 — Count the equilibrium constraints. At equilibrium, the chemical potential of each component must be equal in every phase (else matter would flow until it is): $\mu_i^{(\alpha)} = \mu_i^{(\beta)} = \dots = \mu_i^{(P)} \quad \text{for each component } i$ Why this step? Mechanical equilibrium already forces $P$ equal everywhere, thermal forces $T$ equal everywhere — but chemical equilibrium adds the new constraints we must count.

For ONE component, $P$ phases give a chain of equalities — that's $P-1$ independent equations. For all $C$ components: $\text{Total constraints} = C(P-1)$

Step 3 — Subtract. $F = \underbrace{[P(C-1)+2]}_{\text{variables}} - \underbrace{C(P-1)}_{\text{constraints}}$

Expand: $F = PC - P + 2 - CP + C = C - P + 2$

HOW to use it — counting components & the $+2$

Counting $C$: $C = (\text{number of chemical species}) - (\text{independent chemical reactions}) - (\text{additional constraints})$.

Worked examples

Steel-manned mistakes

Active-recall flashcards

Connections

• Chemical potential — the equality $\mu_i^{(\alpha)}=\mu_i^{(\beta)}$ that supplies every constraint.
• Phase diagrams of pure substances — triple point, sublimation/fusion/vaporisation curves.
• Clausius–Clapeyron equation — the slope of the $F=1$ coexistence lines.
• Components and independent reactions — how to count $C$ correctly.
• Condensed phase rule — fixed-pressure version $F=C-P+1$.
• Gibbs free energy and equilibrium — why $\mu$ equality = minimum $G$.

Concept Map

Hinglish (regional understanding)