Phase diagrams: reading pressure-temperature maps

A phase diagram is a map. Two axes (pressure and temperature) partition the plane into regions where solid, liquid, or gas is thermodynamically stable. The boundaries between regions are curves along which two phases coexist, and the slopes of those curves carry the enthalpies of transition. This guide reads a P-T diagram end to end: the three lines, the triple point, the critical point, and the anomalies that make water’s diagram look different from CO₂’s.

Prerequisites: Gibbs free energy in three intuitions. Every phase boundary in this guide is an isolevel line of ΔG = 0 between two phases.

The three regions and three lines

A single-component phase diagram in P-T space has three regions and three lines:

• Solid region (low T, high P) — molecules locked in a lattice, translational freedom near zero.
• Liquid region (moderate T, moderate P) — molecules close-packed but mobile.
• Gas region (high T, low P) — molecules dilute, kinetic energy dominates.
• Melting line (solid ↔ liquid): typically steep and vertical, small slope.
• Vaporisation line (liquid ↔ gas): curved, running up from the triple point to the critical point.
• Sublimation line (solid ↔ gas): curved, running from the triple point down toward P=0, T=0.

The three lines meet at exactly one point — the triple point — where all three phases coexist in equilibrium. This point is a defining property of the substance. For water: T_tp = 273.16 K, P_tp = 611.657 Pa (IAPWS R7-97, 2012). For CO₂: T_tp = 216.55 K, P_tp = 517 kPa.

The vaporisation line terminates at the critical point (T_c, P_c). Above the critical point, no interface distinguishes liquid from gas — the substance is supercritical. Water’s critical point is 647.10 K / 22.06 MPa; CO₂’s is 304.13 K / 7.38 MPa (CRC 97e §6).

Reading the diagram — three questions

At any (P, T) point on the diagram, three questions have answers:

1. Which phase is stable? Look at the region enclosing the point.

2. What happens as I change P at fixed T? Move vertically. Cross a line and the phase changes.

3. What happens as I change T at fixed P? Move horizontally. Cross a line and the phase changes.

For water at atmospheric pressure (101 kPa): starting at 250 K (solid, ice), heat isobarically. At 273.15 K, cross the melting line — ice melts. At 373.15 K, cross the vaporisation line — water boils. Same horizontal traverse gives you the entire freeze-melt-boil sequence.

Because 101 kPa is well above the triple-point pressure of 611 Pa, the horizontal traverse crosses two lines. If you were at 100 Pa instead (well below P_tp), the isobar would cross only the sublimation line: ice → gas directly. This is why freeze-dried food (lyophilisation) works: pull the pressure below the triple point, and water evaporates from the frozen state without melting.

Clausius-Clapeyron — where the slopes come from

The slope of any coexistence line is set by the Clapeyron equation:

$\frac{dP}{dT} = \frac{\Delta S_{trans}}{\Delta V_{trans}} = \frac{\Delta H_{trans}}{T \Delta V_{trans}}$

For a liquid-gas transition, ΔV is enormous (gas is ~1000× the molar volume of the liquid), so dP/dT is small and positive: the line is nearly horizontal near low pressure and curves upward toward the critical point. For a solid-liquid transition, ΔV is small (liquid is typically only a few percent denser than solid), so dP/dT is very large — the melting line is nearly vertical.

The sign of dP/dT for melting depends on the sign of ΔV_fusion. For most substances, the solid is denser than the liquid (ΔV_fusion > 0), so dP/dT > 0 — increase pressure and the melting temperature rises. For water, the solid is less dense than the liquid (ice floats; ΔV_fusion < 0), so dP/dT < 0 — increase pressure and the melting temperature falls. This tiny anomaly is why water’s phase diagram has a backward-sloping melting line, and why liquid water can exist at very low temperatures under high pressure at the bottom of glaciers and in geological subduction zones.

When ΔV is dominated by the gas phase and the gas is treated as ideal, Clapeyron collapses to Clausius-Clapeyron:

$\frac{d \ln P}{d T} = \frac{\Delta H_{vap}}{RT^2}$

or, integrated between two points:

$\ln \frac{P_2}{P_1} = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

This is the equation the Clausius-Clapeyron calculator implements. Two (P, T) points on the vaporisation line and you can extract ΔH_vap, or take one (P, T) plus ΔH_vap and predict boiling temperature at any other pressure.

Worked example — boiling point on a mountain

The critical point — where the line ends

Move up the vaporisation curve toward the critical point. The liquid gets less dense (thermal expansion) and the gas gets denser (rising pressure). At (T_c, P_c) the two densities equal each other and the interface disappears. Above T_c, no isobaric compression will liquefy the gas — you can only compress it into a supercritical fluid.

Supercritical CO₂ (above 304 K, 7.4 MPa) is the industrial workhorse of the family: it has liquid-like density (dissolves things), gas-like viscosity (diffuses like a gas), and is chemically inert. It is used for decaffeinating coffee (extracting caffeine from beans), dry-cleaning, and green-chemistry reactions. Supercritical water (above 647 K, 22 MPa) becomes an oxidising, non-polar solvent — the reverse of ordinary water, dissolving hydrocarbons but not salts.

Approaching a critical point, macroscopic properties fluctuate wildly: density fluctuations reach visible-wavelength scales and the fluid turns opalescent, the “critical opalescence” phenomenon.

Water’s oddities

Water’s phase diagram is famously not the generic textbook one. Two features:

1. Negative-slope melting line. As noted, ice is less dense than liquid water because the tetrahedral hydrogen-bond lattice is open. Compress ice and it turns to water; release pressure and water refreezes. This effect is small (about 1 K per 130 atm), so the “pressure melting” of ice under ice-skate blades is largely a myth (the actual mechanism is frictional heating), but it is real thermodynamically.

2. A dozen crystalline ice polymorphs. Ice-Ih is the ambient form. At higher pressures, water reorganises into ice-II, ice-III, ice-V, ice-VI, ice-VII, and so on up to ice-XX (Salzmann 2019). Each polymorph has its own region on the extended P-T diagram, meeting at additional triple points. This is why “the” phase diagram for water is really a mosaic of ice-polymorph domains.

Most substances have simpler diagrams — CO₂, for example, has only one solid form and the standard three-region diagram. Water is the exception, not the rule.

The Gibbs phase rule

For a single component (C=1), the phase rule states:

$F = C - P + 2 = 3 - P$

where F is the number of degrees of freedom, P is the number of phases in equilibrium. Regions (P=1): 2 degrees of freedom (T and P can vary independently). Lines (P=2): 1 degree of freedom (fix T and P is determined, or fix P and T is determined). The triple point (P=3): 0 degrees of freedom — a single point in P-T space.

This is why the triple-point temperature of water was, until 2019, one of the defining fixed points of the SI temperature scale: it is reproducible to microkelvin because there is nowhere else on the diagram where all three phases coexist. The 2019 SI redefinition anchored temperature to a fixed value of the Boltzmann constant, but the triple point retains its practical role in thermometry.

Common pitfalls

Confusing “vapor” with “gas” above the critical point. Below T_c, condensing a vapor gives a liquid interface — the transition is discontinuous. Above T_c, the substance is a supercritical fluid; density changes continuously with pressure and there is no phase transition to cross.

Using Clausius-Clapeyron across a wide temperature range with fixed ΔH_vap. ΔH_vap varies with temperature — it goes to zero at the critical point. For temperature ranges wider than about 30 K, a temperature-dependent ΔH_vap or the full Antoine equation is more accurate.

Misreading water’s melting line. The negative slope is real, but it is very small: about −7 MPa/K. On a normal pressure axis (0-1 atm), the melting line looks vertical to the eye.

Assuming the diagram of one substance predicts another. Every substance has its own P_c, T_c, and P_tp, T_tp. Boiling points at atmospheric pressure vary from 4 K (helium) to 6000 K (tungsten). Only the topology of the three-region diagram is universal.

Practice

Where this leads

The one-component P-T diagram is a base camp. Extensions:

• Multi-component diagrams — binary mixtures on T-x plots (azeotropes, eutectics), covered in unit operations for distillation.
• Phase transitions of higher order — the liquid-liquid critical point of supercooled water, the λ transition of superfluid helium, ferromagnetic ordering. These have discontinuities in second derivatives of G, not the first, and the Clapeyron equation does not describe them.
• Metastable states — supercooled liquids, superheated liquids, and the kinetic barriers that hold them there.

Every one of them starts by locating (P, T) on the equilibrium diagram and asking which phase has the lowest G.