Physical chemistry

Phase equilibria, phase diagrams

Reading phase boundaries, triple and critical points in p–T diagrams, and composition-dependent fields in binary diagrams.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: a map of which form of matter wins

Ice, liquid water and steam are three answers to the same question — how H₂O is best arranged — asked at different pressures and temperatures. A phase diagram is the map where boundaries mark the coin-flip conditions: along them two phases coexist in exact balance. Where three boundaries meet sits the triple point; where a boundary stops entirely sits the critical point.

Compare the two substances: water’s fusion line leans left, CO₂’s leans right; the grey line marks 1 atm.

Temperature versus composition for the classic solder system; the blue liquidus, red eutectic isotherm and the eutectic point are marked.

SchoolSchool: reading a phase diagram

Definition: Phase and coexistence line

A phase is a region of uniform properties — solid, liquid or gas for a pure substance. On a coexistence line the two neighbouring phases have equal chemical potentials and both are stable. The triple point is the unique (T, p) where all three coexist; the critical point ends the liquid–gas line because beyond it the two fluids merge into a single supercritical fluid.

The diagram explains everyday observations. Water boils at a temperature set by the point where its vapour-pressure curve meets the ambient pressure line: about 373 K at 1 atm, lower on a mountain, higher in a pressure cooker. CO₂ has its triple point at 216.6 K and 5.18 bar — above atmospheric pressure — so at 1 atm the liquid phase has no stable window and solid CO₂ sublimes directly (“dry ice”).

Example: Reading the water diagram

At 273.16 K and 611.7 Pa, what phases coexist for pure water? What happens to the state of water at 1 atm if it is cooled from 400 K to 250 K at constant pressure?

Solution

The given point is the triple point: solid, liquid and vapour coexist. On the 1 atm path (lg(p/bar) ≈ 0.006 for water’s triple point — but the path itself is the constant-pressure line through the diagram), cooling crosses the vaporisation boundary near 373 K (condensation) and the fusion boundary near 273 K (freezing): steam → liquid → ice.

UndergraduateUniversity: the thermodynamics of coexistence

Along a coexistence line the molar Gibbs energies of the two phases are equal. Differentiating this equality gives the Clapeyron equation, which fixes the slope of every boundary in terms of the transition enthalpy and volume change:

\frac{dP}{dT}=\frac{\Delta_{\mathrm{tr}}S}{\Delta_{\mathrm{tr}}V}=\frac{\Delta_{\mathrm{tr}}H}{T\,\Delta_{\mathrm{tr}}V}\qquad;\qquad \frac{d\ln P}{dT}\approx\frac{\Delta_{\mathrm{vap}}H}{RT^{2}}\;\text{(Clausius–Clapeyron)}

The sign of Δ_trV decides which way a boundary leans. For vaporisation, ΔV is large and positive, so the liquid–vapour curve rises steeply in a logarithmic pressure plot — which is why the y axis in the simulation is lg(p/bar). For the fusion of water, the liquid is denser than the solid (Δ_fusV ≈ −1.6 cm³ mol⁻¹), so dP/dT is negative: ice melts under added pressure, anomalous but crucial for glaciology and skating.

F=C-P+2\qquad(\text{Gibbs phase rule, non-reacting})

The Gibbs phase rule counts the degrees of freedom F of a system with C components and P phases in equilibrium. For a pure substance (C = 1): a single phase leaves F = 2 (T and p free — an area), a coexistence leaves F = 1 (a line), and three phases leave F = 0 (the invariant triple point). For the binary Pb–Sn diagram at fixed pressure the rule reduces to F = C − P + 1: the eutectic horizontal at 183 °C is a three-phase invariant, which is why the whole isotherm is flat.

Example: Applying the phase rule

For the Pb–Sn diagram at 1 atm, how many degrees of freedom remain (a) in the all-liquid region, (b) on the liquidus where liquid coexists with solid α, (c) at the eutectic point?

Solution

With C = 2 and pressure fixed, F = 3 − P. (a) P = 1 gives F = 2: temperature and composition are both free. (b) P = 2 gives F = 1: choosing T fixes the composition of each phase (the tie-line endpoints). (c) At the eutectic P = 3 gives F = 0: nothing can be varied — temperature, eutectic composition and both solid compositions are all fixed by nature.

AdvancedAdvanced: what diagrams leave out

Phase diagrams are equilibrium statements. Real cooling produces supercooling, glass formation, metastable phases and nucleation-limited paths that the equilibrium map does not show. At the critical point, densities and compositions fluctuate on all length scales (critical opalescence), the compressibility diverges, and the distinction between “liquid” and “gas” loses meaning. Near a triple point, tiny impurities measurably shift the fixed temperature, which is why triple-point cells of water were long used to define the kelvin (273.16 K exactly, before the 2019 redefinition).

References

On the Continuity of the Gaseous and Liquid States of Matter · T. Andrews, 1869
Phase Diagrams in Metallurgy · F. N. Rhines, 1956