Phase rule and criteria for phase equilibrium

Why it matters

Before computing anything about a separator, crystalliser or reactor at equilibrium, you need to know how many variables you are allowed to fix. Specify too few and the problem has no unique answer; specify too many and it is impossible. The phase rule counts them, and the equilibrium criteria (equal T, P and chemical potentials, minimum Gibbs energy) give the equations that every VLE, LLE and reaction-equilibrium calculation solves.

Key ideas

Criteria for equilibrium. For a closed system at constant T and P, any spontaneous change lowers the total Gibbs energy: (dG^t)_{T,P} ≤ 0. Equilibrium is the state of minimum G^t. For several phases α, β, … this requires:

• thermal equilibrium: equal temperature in all phases;
• mechanical equilibrium: equal pressure (flat interfaces);
• chemical (phase) equilibrium: μᵢ^α = μᵢ^β = … for every species i, equivalently f̂ᵢ^α = f̂ᵢ^β = … at the same T. Mass moves from a phase where μᵢ is higher to one where it is lower, just as heat flows down a temperature gradient.

Intensive variables and degrees of freedom. The phase rule counts only intensive variables (T, P and the phase compositions), not amounts of phases. The number of degrees of freedom F is the number of these you must fix to fix all the others.

Gibbs phase rule (non-reacting). Each phase has T, P and N − 1 independent mole fractions, giving 2 + π(N − 1) variables. The equality of μᵢ between phases supplies (π − 1)N equations. Subtracting: F = 2 − π + N.

Reacting systems. Each independent chemical reaction adds one equilibrium equation; special constraints (for example a fixed ratio of products because they all come from one reactant, or electroneutrality) remove further freedom: F = 2 − π + N − r − s. Duhem's theorem complements this: for a closed system formed from given initial amounts, the equilibrium state is completely fixed by any two independent variables (intensive or extensive).

Examples to remember.

• Pure water, one phase: F = 2 (choose T and P). Liquid-vapour: F = 1 (fix T and P_sat follows). Triple point: F = 0.
• Binary VLE: F = 2. Fix T and P and both phase compositions are fixed (the basis of T-x-y and P-x-y diagrams).
• Binary azeotrope: the extra condition x₁ = y₁ makes it F = 1 at a given P, so the azeotrope occurs at one T and composition.
• Binary VLLE (two liquids plus vapour): F = 1.

Stability and phase splitting. A single liquid phase is stable only if its Gibbs energy of mixing curve is convex: d²(ΔG_mix/RT)/dx₁² > 0 at constant T and P. Where the curvature becomes negative, the liquid lowers its G by splitting into two phases, which is how liquid-liquid equilibrium arises. For the one-parameter model G^E/RT = A·x₁x₂, splitting first occurs when A exceeds 2.

Formulas

(dG^t)_{T,P} ≤ 0 (spontaneous change; equality at equilibrium) T^α = T^β = …, P^α = P^β = … μᵢ^α = μᵢ^β = …, equivalently f̂ᵢ^α = f̂ᵢ^β = … (i = 1 … N) F = 2 − π + N (non-reacting) F = 2 − π + N − r − s (r independent reactions, s special constraints) d²(ΔG_mix/RT)/dx₁² > 0 (stability of a binary liquid, constant T and P) ΔG_mix/RT = x₁ ln x₁ + x₂ ln x₂ + G^E/RT

Symbols: G^t total Gibbs energy (J); μᵢ chemical potential (J/mol); f̂ᵢ fugacity of i in a phase (Pa); F degrees of freedom; π number of phases; N number of chemical species (components); r number of independent reactions; s number of special constraints; x₁, x₂ mole fractions; R = 8.314 J/(mol·K); T (K).

Worked examples

Example 1 (standard): counting degrees of freedom. Find F for (a) water at its triple point, (b) a vapour in equilibrium with liquid for a ternary mixture, (c) a binary system with two liquid phases and a vapour.

1. (a) N = 1, π = 3: F = 2 − 3 + 1 = 0. T and P are fixed by nature (273.16 K and 611.7 Pa for water).
2. (b) N = 3, π = 2: F = 2 − 2 + 3 = 3. For example fix P and two liquid mole fractions, and T and the vapour composition follow (a bubble-point calculation).
3. (c) N = 2, π = 3: F = 2 − 3 + 2 = 1. Fixing P fixes T and all three phase compositions (a heterogeneous azeotrope at that pressure). Answer: (a) 0, (b) 3, (c) 1

Example 2 (GATE level): reacting systems and stability. (i) CaCO₃(s) is heated in a closed vessel and partly decomposes: CaCO₃(s) ⇌ CaO(s) + CO₂(g). Find F.

1. Species: CaCO₃, CaO, CO₂, so N = 3. Phases: two separate solids and one gas, so π = 3. Reactions: r = 1.
2. Special constraints: none that link compositions, because each solid is a pure phase and the gas is pure CO₂ (moles of CaO = moles of CO₂ is a mass relation, not an intensive one). s = 0.
3. F = 2 − 3 + 3 − 1 − 0 = 1. Fixing T fixes the decomposition pressure of CO₂.

(ii) NH₄Cl(s) decomposes alone into NH₃(g) + HCl(g). Find F.

1. N = 3, π = 2 (solid and gas), r = 1.
2. The gas forms only from NH₄Cl, so y_NH₃ = y_HCl: s = 1.
3. F = 2 − 2 + 3 − 1 − 1 = 1.

(iii) A binary liquid follows G^E/RT = 2.5·x₁x₂. Is an equimolar liquid stable?

1. ΔG_mix/RT = x₁ ln x₁ + x₂ ln x₂ + 2.5·x₁x₂.
2. Second derivative: d²(ΔG_mix/RT)/dx₁² = 1/x₁ + 1/x₂ − 2 × 2.5 = 1/x₁ + 1/x₂ − 5.
3. At x₁ = 0.5: 2 + 2 − 5 = −1 < 0, so the single phase is unstable and splits into two liquids. Answer: (i) F = 1, (ii) F = 1, (iii) unstable (two liquid phases form)

Common mistakes

• Counting amounts of phases or total moles as degrees of freedom; the phase rule counts intensive variables only.
• Treating two immiscible solids as one phase; each pure solid is a separate phase.
• Forgetting special constraints (such as products formed in fixed ratio from one reactant) in reacting systems.
• Using F = N − π + 1 (the "condensed phase rule") without stating that pressure has been fixed.
• Saying equilibrium means equal concentrations in all phases; it means equal chemical potentials (equal fugacities).

For GATE CH

• MCQs and NAT questions counting F for pure substances, VLE, VLLE, azeotropes and reacting systems, often with solids.
• Conceptual questions on the criteria of equilibrium (equal μᵢ or f̂ᵢ, minimum G at constant T and P).
• Stability: deciding from a given G^E/RT whether a liquid splits, and the critical value of a model parameter.
• Practise listing species, phases, reactions and special constraints explicitly before using the formula.

Quick check

1. How many degrees of freedom does a pure substance have at its triple point?
2. What is F for a binary in vapour-liquid equilibrium?
3. State the chemical-potential condition for phase equilibrium.
4. For G^E/RT = A·x₁x₂, above what value of A does phase splitting occur?
5. Which thermodynamic function is minimised at equilibrium at constant T and P?

Answers: 1. 0. 2. 2. 3. μᵢ is the same in all phases for every species i. 4. A > 2. 5. The total Gibbs energy.