Phase Equilibrium and Coexistence Conditions

Core Idea

At equilibrium, different phases (solid, liquid, gas) coexist when their chemical potentials are equal: μ_solid = μ_liquid = μ_gas. The Clausius-Clapeyron equation dP/dT = L/(T·ΔV) relates the slope of phase boundaries to the latent heat L and volume change ΔV. The phase diagram (P-T plot) summarizes all equilibrium conditions and is essential for understanding when substances exist in different phases and the conditions for transitions.

How It's Best Learned

Use Clausius-Clapeyron to predict phase boundary slopes. Compare theory with experimental phase diagrams. Identify triple and critical points.

Common Misconceptions

• Confusing phase equilibrium with thermodynamic equilibrium (phase equilibrium is a special case).
• Assuming latent heat is constant over all temperatures (it weakly depends on T).
• Thinking the Clausius-Clapeyron equation is valid far from phase boundaries.

Explainer

From your study of chemical potential, you know that μ = (∂G/∂N)_{T,P} — it is the free energy cost of adding one particle to the system. At equilibrium, μ is uniform throughout the system: any inhomogeneity in μ drives a particle current to equalize it, just as a temperature gradient drives heat flow and a pressure gradient drives mechanical flow. When two phases coexist (ice and water in the same cup), they must satisfy all three equilibrium conditions simultaneously: T_solid = T_liquid, P_solid = P_liquid, and μ_solid(T,P) = μ_liquid(T,P). The last condition is the one that determines where in the (T,P) plane coexistence is possible.

The Clausius-Clapeyron equation dP/dT = L/(TΔV) tells you the slope of the phase boundary in the P-T diagram. Its derivation follows directly from the coexistence condition: if you shift T slightly along the phase boundary, P must shift to maintain μ_α = μ_β. Using dμ = −sdT + vdP (where s and v are entropy and volume per particle), the condition dμ_α = dμ_β gives −s_α dT + v_α dP = −s_β dT + v_β dP, rearranging to dP/dT = (s_β − s_α)/(v_β − v_α) = ΔS/ΔV = L/(TΔV), where L = TΔS is the latent heat. Notice what this tells you qualitatively: the slope of the phase boundary is large when ΔV is small (as for solid-liquid water, which is nearly incompressible), and it is positive unless ΔV is negative. Water is anomalous: ice is less dense than liquid water (ΔV < 0 on melting), so its solid-liquid boundary has a negative slope — increased pressure lowers the melting point. This is why ice skating is possible.

The phase diagram (P-T plot) organizes this information. The liquid-gas boundary ends at the critical point (Tc, Pc) where the distinction between liquid and gas vanishes — the latent heat goes to zero and ΔV → 0. Above the critical point, you can continuously convert liquid to gas without crossing a phase boundary. The solid-liquid boundary typically extends without a critical point (it is very hard to continuously convert solid to liquid). The solid, liquid, and gas regions meet at the triple point, the unique (T,P) combination where all three phases coexist simultaneously. For water, the triple point is at 273.16 K and 611.7 Pa — it defines the kelvin on the International Temperature Scale.

A practical skill from this topic: given a phase diagram, you can immediately determine the direction of phase change in response to any (T,P) perturbation. Decreasing pressure below the vapor pressure at fixed T → liquid boils (or solid sublimes). Increasing pressure at fixed T along the solid-liquid boundary of water → ice melts. These are not memorization tasks — they follow from asking "which phase has lower μ at the new (T,P)?" The phase with lower chemical potential is always thermodynamically favored, and the phase diagram is a map of which phase wins where.