Key Concepts of Phase Equilibria Diagrams

Why This Matters

Phase equilibria diagrams are the roadmaps of physical chemistry—they tell you exactly what happens to matter when you change temperature, pressure, or composition. You're being tested on your ability to read these diagrams, predict phase behavior, and apply fundamental principles like the Gibbs phase rule to real systems. These concepts connect directly to thermodynamic stability, chemical potential, colligative properties, and intermolecular forces, all of which appear repeatedly on exams.

Don't just memorize what a triple point is or where to find an azeotrope on a diagram. Know why these features exist, how to use mathematical tools like the lever rule to extract quantitative information, and when different diagram types apply. The difference between a good answer and a great one is connecting diagram features to underlying thermodynamic principles.

---

Fundamental Diagram Types

Different phase diagrams serve different purposes depending on whether you're analyzing a pure substance or a mixture, and whether pressure or temperature is your variable of interest. The choice of axes determines what questions the diagram can answer.

Pressure-Temperature (P-T) Phase Diagrams

• Map single-component phase behavior—show which phase (solid, liquid, gas) is stable at any given P and T combination
• Phase boundaries represent conditions where two phases coexist in equilibrium, with slopes determined by the Clapeyron equation
• Most useful for pure substances—the diagram's simplicity breaks down when you add components to the system

Temperature-Composition (T-x) Phase Diagrams

• Essential for binary mixtures at constant pressure—typically 1 atm for liquid-vapor or solid-liquid systems
• Liquidus and solidus curves show the temperatures where phase transitions begin and end for each composition
• Reading tie lines horizontally gives you the composition of coexisting phases at any temperature within a two-phase region

Pressure-Composition (P-x) Phase Diagrams

• Analyze vapor-liquid equilibrium at constant temperature—complementary to T-x diagrams
• Raoult's law deviations become visible as departures from ideal straight-line behavior
• Critical for understanding distillation—the vertical distance between liquid and vapor curves indicates separation efficiency

---

Critical Features and Special Points

Certain locations on phase diagrams have profound thermodynamic significance. These points represent unique thermodynamic states where the system's degrees of freedom reach limiting values.

Triple Point

• Unique invariant point—the only P-T combination where solid, liquid, and gas coexist in equilibrium
• Zero degrees of freedom according to Gibbs phase rule ($F = 1 - 3 + 2 = 0$), meaning you cannot change any variable without losing a phase
• Defines absolute temperature standards—water's triple point (273.16 K, 611.7 Pa) was historically used to define the kelvin

Critical Point

• Terminus of the liquid-vapor boundary—beyond critical temperature $T_c$ and critical pressure $P_c$, liquid and gas become indistinguishable
• Supercritical fluids exhibit hybrid properties: gas-like diffusivity with liquid-like density, enabling applications like supercritical $CO_2$ extraction
• Divergent behavior—compressibility, heat capacity, and density fluctuations approach infinity as you approach the critical point

Vapor Pressure Curves

• Liquid-gas phase boundary on P-T diagrams—described quantitatively by the Clausius-Clapeyron equation: $\frac{d \ln P}{dT} = \frac{\Delta H_{vap}}{RT^2}$
• Slope reflects enthalpy of vaporization—steeper curves indicate stronger intermolecular forces requiring more energy to vaporize
• Terminates at critical point—the curve doesn't extend forever; it ends where the phase distinction disappears

---

Mixture Behavior and Special Systems

Real mixtures rarely behave ideally. Deviations from Raoult's law create distinctive diagram features that have practical consequences for separation processes.

Eutectic Systems

• Lowest melting point for any composition—the eutectic point represents where liquid transforms directly to two solid phases simultaneously
• Invariant point in binary systems—at the eutectic temperature, $F = 2 - 3 + 1 = 0$ (two components, three phases, constant pressure)
• Critical for alloy design—solder (Pb-Sn), dental amalgams, and casting alloys exploit eutectic compositions for controlled solidification

Azeotropes

• Constant-boiling mixtures—vapor and liquid have identical composition, making further separation by simple distillation impossible
• Positive azeotropes (minimum boiling) result from positive deviations from Raoult's law; negative azeotropes (maximum boiling) from negative deviations
• Practical barrier in separations—ethanol-water (95.6% ethanol) is a classic example requiring special techniques like molecular sieves to break

---

Quantitative Tools

Phase diagrams aren't just for qualitative analysis—they provide precise quantitative information when you apply the right mathematical tools.

Gibbs Phase Rule

• Fundamental constraint equation: $F = C - P + 2$, where $F$ = degrees of freedom, $C$ = components, $P$ = phases
• Predicts diagram dimensionality—a one-component system needs a 2D diagram; adding components requires simplification (fixing P or T) to visualize
• Explains invariant points—when $F = 0$, you're at a unique point (triple point, eutectic) that cannot shift without changing the system

Lever Rule

• Calculates phase fractions in two-phase regions—the fraction of phase $\alpha$ equals the length of the tie line segment on the opposite side divided by total tie line length
• Mathematical form: $\frac{n_\alpha}{n_\beta} = \frac{x_\beta - x_0}{x_0 - x_\alpha}$, where $x_0$ is overall composition
• Works on any two-phase region—applies equally to liquid-vapor, solid-liquid, and liquid-liquid equilibria

---

Quick Reference Table

---

Self-Check Questions

1. A binary system at constant pressure shows three phases in equilibrium. Using the Gibbs phase rule, calculate the degrees of freedom. What does this tell you about where you are on the phase diagram?

2. Compare and contrast the triple point and critical point: both are "special" locations on a P-T diagram, but what fundamentally distinguishes them in terms of phase behavior and thermodynamic significance?

3. You're examining a T-x diagram and find a horizontal tie line connecting two points at the same temperature but different compositions. How would you use the lever rule to determine the mass fraction of each phase if the overall composition is exactly midway between them?

4. Why can't you separate an ethanol-water mixture beyond 95.6% ethanol using simple distillation? What feature on a T-x diagram explains this limitation?

5. If you increase the pressure on a system at its triple point, predict what will happen to the phases present. Which phase diagram type would you consult, and how does the Gibbs phase rule inform your answer?