Phase diagrams
A phase diagram is an equilibrium map. Reading it well means separating regions, boundaries and the path followed when temperature or pressure changes.
Reading a pressure–temperature diagram
Stability regions
Inside a single-phase region, one phase is stable. On a boundary, two phases coexist in equilibrium. The solid–liquid line is therefore not a fuzzy transition zone: at each point on that line, solid and liquid have the same thermodynamic stability. The same idea applies to the liquid–vapour and solid–vapour boundaries.
Following a path
Do not start by naming every curve. Start from a point and ask what happens when you move.
- Locate the initial temperature and pressure. Identify the region.
- Change one variable at a time. Heating at constant pressure is a horizontal path on the usual pressure–temperature plot; compression at constant temperature is a vertical path.
- Watch for boundary crossings. Every crossing corresponds to a phase transition.
- Read the direction of the crossing. Solid → liquid is melting; liquid → gas is vaporisation; solid → gas is sublimation. Reverse the direction and the names reverse too.
At 1 bar, heating ordinary water from below 0 °C to above 100 °C crosses the solid–liquid boundary and later the liquid–vapour boundary. The diagram therefore predicts two equilibrium transitions along that path: melting, then boiling.
Triple point and critical point
Triple point
The three coexistence curves meet at the triple point. At that one pressure and temperature, solid, liquid and vapour can coexist in equilibrium. Move away from the point and at least one of those phases loses equilibrium stability.
The triple point is not the ordinary melting point. The melting point quoted in everyday tables normally refers to a specified pressure, often close to 1 bar. The triple point is a unique pair of pressure and temperature fixed by the substance.
Critical point
Follow the liquid–vapour coexistence curve toward higher temperature and pressure and it ends at the critical point. As that point is approached, the properties of the liquid and vapour become increasingly similar. Beyond it there is no sharp liquid–gas phase boundary to cross: the material is a supercritical fluid.
This is why “compress the gas until it becomes a liquid” stops being a valid description above the critical temperature. There is no separate liquid phase available there; density can change continuously without crossing a liquid–vapour coexistence line.
Water and carbon dioxide
Water
The melting curve has a negative slope near ordinary conditions because ice is less dense than liquid water.
Physical reading: increasing pressure favours the denser liquid.
CO₂
The triple-point pressure is about 5.18 bar. At atmospheric pressure, the liquid region is inaccessible.
Physical reading: dry ice sublimes.
For most substances, the solid is denser than the liquid, so the solid–liquid boundary slopes toward higher temperature as pressure rises. Water near its normal melting point behaves the other way around: ordinary ice is less dense than liquid water. Increasing pressure therefore favours the denser liquid phase, and the melting temperature decreases slightly.
The left-leaning melting curve on a water phase diagram is not a decorative anomaly. It encodes a molecular fact: the hydrogen-bonded structure of ice occupies more volume than the liquid. Pressure penalises that larger-volume solid structure.
Carbon dioxide has a triple-point pressure of about 5.18 bar, well above atmospheric pressure. At roughly 1 bar there is no pressure–temperature path through a stable liquid-CO₂ region. Warm solid CO₂ therefore crosses the solid–vapour boundary and sublimes instead of melting. That is why it is called dry ice.
The same geometry helps explain freeze-drying, but with water: if ice is kept below water’s triple-point pressure and heat is supplied carefully, the stable route out of the solid region can be sublimation rather than melting. Removing the vapour continuously drives dehydration without a bulk liquid-water stage.
Boundary slopes and the phase rule
Clapeyron relation
A coexistence line is not arbitrary. Its local slope is related to the entropy and volume changes of the phase transition. A compact form is the Clapeyron equation:
dP/dT = ΔHtrans / (T ΔVtrans)
For melting, ΔH is positive. The sign of the slope is therefore controlled by the volume change. If the liquid occupies more volume than the solid, ΔV is positive and the melting line normally slopes upward. For water near 0 °C, melting reduces the volume, so ΔV is negative and the slope is negative. The diagram is therefore a thermodynamic consequence, not just an empirical sketch.
Gibbs phase rule
For an equilibrium system, Gibbs’ phase rule is F = C − P + 2, where F is the number of independent intensive variables, C the number of components and P the number of phases present. For a pure substance, C = 1, so F = 3 − P.
- Inside a one-phase region, P = 1 and F = 2: temperature and pressure can both vary independently.
- On a two-phase boundary, P = 2 and F = 1: choosing the temperature fixes the equilibrium pressure, or vice versa.
- At the triple point, P = 3 and F = 0: neither temperature nor pressure can be changed while keeping all three phases in equilibrium.
This is the reason regions have area, coexistence conditions form lines, and the three-phase equilibrium is a point.
Limits of the model
A phase diagram describes equilibrium stability. It does not by itself tell you how quickly equilibrium will be reached. Water can be supercooled below its equilibrium freezing temperature; a liquid can remain metastable because crystal nucleation is slow; hysteresis can make the observed transition depend on the path taken. Those effects belong to kinetics and metastability, not to the equilibrium map alone.
Real substances may also have several solid structures. Water, for example, has multiple high-pressure ice phases that are omitted from the simplified classroom diagram. A diagram is only as complete as the phases and variables included in it.
The same idea extends beyond a pure substance. In mixtures, composition becomes an essential variable. Binary phase diagrams often plot temperature against composition at fixed pressure and reveal liquid–solid coexistence, eutectic points, solubility limits and phase fractions. Ternary diagrams add a third component. The common pressure–temperature diagram is therefore the simplest member of a much larger family of equilibrium maps.
Example and exercise
Compression at constant temperature
Suppose a sealed sample is initially a gas at a temperature below the critical temperature. You compress it at constant temperature. On a pressure–temperature diagram this is a vertical path. If the path crosses the liquid–vapour coexistence curve, liquid and vapour coexist at the crossing pressure; further compression moves the equilibrium state into the liquid region. If the chosen temperature is above the critical temperature, the same compression never crosses that curve because the curve has already ended: the fluid simply becomes denser continuously.
Reading a path
A pure substance has a triple-point pressure of 4 bar. Its solid is heated at 1 bar. Can the liquid become stable?
Solution
No. Below the triple-point pressure the liquid region is inaccessible, so heating crosses the solid–vapour boundary and the solid sublimes.