Training Phase Diagrams — Thermodynamic Stability & Phase Transitions Binary Phase Diagrams — Eutectic Systems & Lever Rule
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Binary Phase Diagrams — Eutectic Systems & Lever Rule

35 min Phase Diagrams — Thermodynamic Stability & Phase Transitions

Binary phase diagrams describe the equilibrium behaviour of two-component mixtures. They are essential in materials science, metallurgy, pharmaceutical formulation, and chemical separations.

Gibbs Phase Rule for Binary Systems

With $C=2$ components: $F = 2 - P + 2 = 4 - P$. At fixed pressure (isobaric diagram), subtract 1: $F = 3 - P$.

  • Single phase: $F=2$ — both $T$ and $x$ (mole fraction) are free.
  • Two phases (tie line): $F=1$ — fixing $T$ fixes the compositions of both phases.
  • Three phases (eutectic point): $F=0$ — unique eutectic temperature and compositions.

The Lever Rule

At a given temperature $T$ inside a two-phase region, a mixture of overall composition $z$ splits into phase $\alpha$ (composition $x^\alpha$) and phase $\beta$ (composition $x^\beta$). The mole fraction of phase $\beta$ is:

$$f^\beta = \frac{z - x^\alpha}{x^\beta - x^\alpha}$$

Geometrically: the amount of each phase is proportional to the length of the tie line on the opposite side of $z$ — hence “lever rule.”

The Eutectic System

A simple eutectic phase diagram (e.g., Pb-Sn, Bi-Cd) has:

  • Two liquidus curves descending from the pure-component melting points to the eutectic point.
  • A eutectic point where three phases coexist: liquid + solid A + solid B.
  • A horizontal solidus line (eutectic isotherm) — the invariant reaction: L → α + β

Raoult's Law and Ideal Solutions

For an ideal binary mixture the partial vapour pressures obey Raoult's law:

$$p_A = x_A p_A^*, \qquad p_B = x_B p_B^*$$

The total pressure is linear in $x_A$: $p = p_B^* + (p_A^* - p_B^*)x_A$. The vapour is enriched in the more volatile component, giving the vapour composition $y_A = x_A p_A^*/p > x_A$ when $p_A^* > p_B^*$. This is the basis of fractional distillation.

Real Solutions: Azeotropes

Non-ideal solutions can form minimum-boiling azeotropes (negative deviations from Raoult, e.g. ethanol–water) or maximum-boiling azeotropes (positive deviations, e.g. HCl–water). At the azeotropic composition the liquid and vapour compositions are identical, so distillation cannot separate them beyond that point.

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