One-Component Phase Diagrams & Gibbs Phase Rule

Phase diagrams are maps of thermodynamic stability: for every combination of temperature, pressure (and composition for mixtures), they identify which phase or phases coexist at equilibrium. Mastering phase diagrams requires Gibbs's phase rule, chemical potential equality across phase boundaries, and the geometry of the P-V-T surface.

Gibbs Phase Rule

For a system with $C$ components and $P$ phases at thermodynamic equilibrium:

$$F = C - P + 2$$

where $F$ is the number of degrees of freedom (intensive variables that can be changed independently without altering the number of phases). For a single-component system ($C=1$):

• Single phase ($P=1$): $F=2$ — both $T$ and $p$ are free.
• Two-phase coexistence ($P=2$): $F=1$ — fixing $T$ fixes $p$ (the coexistence curve).
• Triple point ($P=3$): $F=0$ — unique $(T_3, p_3)$ point, no freedom.

Chemical Potential and Phase Equilibrium

Two phases $\alpha$ and $\beta$ coexist when their molar Gibbs energies (chemical potentials) are equal:

$$\mu^\alpha(T, p) = \mu^\beta(T, p)$$

This equality defines the coexistence curve. The slope of any coexistence curve follows from differentiating the equality of $G$:

$$\frac{dp}{dT}\bigg|_{\text{coex}} = \frac{\Delta S_m}{\Delta V_m} = \frac{\Delta H_m}{T\,\Delta V_m}$$

This is the Clausius-Clapeyron equation in exact form (see Lesson 2).

Key Features of a One-Component Phase Diagram

Feature	Water	CO₂	Significance
Triple point	273.16 K, 611.7 Pa	216.6 K, 5.18 atm	Defines the thermodynamic temperature scale
Critical point	647.1 K, 220.6 bar	304.2 K, 73.8 bar	Liquid–gas distinction vanishes; supercritical fluid
Melting slope	Negative (−13.5 MPa/K)	Positive	Water ice is less dense than liquid — anomalous
Sublimation	Below 611.7 Pa	Below 5.18 atm at 1 bar	CO₂ sublimes at atmospheric pressure (dry ice)

The P-V-T Surface

The three-dimensional surface $p = p(V_m, T)$ encodes all phase information. Projecting onto the $p$-$T$ plane yields the familiar phase diagram; projecting onto $p$-$V$ reveals the two-phase dome. Inside the dome, the Maxwell equal-area construction (see Lesson 4) replaces the van der Waals loop with the physical horizontal tie line at constant $p$, where liquid and vapour coexist.