Key Concepts of Phase Diagrams

Why This Matters

Phase diagrams are the roadmaps of physical chemistry—they tell you exactly what state a substance will be in under any combination of pressure and temperature, and they reveal the conditions where dramatic transformations occur. You're being tested on your ability to read these diagrams, predict phase behavior, and apply fundamental principles like the Gibbs phase rule to explain why systems behave the way they do. Whether it's understanding why ice floats, why dry ice sublimes, or how to design an alloy with specific melting properties, phase diagrams connect thermodynamic theory to real-world phenomena.

The concepts here span single-component equilibria, binary mixtures, ternary systems, and the mathematical framework governing degrees of freedom. Exam questions will ask you to interpret diagrams, calculate variables using the phase rule, and compare the behavior of different substances. Don't just memorize what each diagram looks like—know what physical principles each feature represents and how to use tie lines, phase boundaries, and critical points to extract quantitative information.

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Single-Component Phase Diagrams

These diagrams show how a pure substance responds to changes in pressure and temperature. The key insight is that phase boundaries represent conditions of two-phase equilibrium, where the chemical potentials of both phases are equal.

Pure Substance Phase Diagram (P-T Diagram)

• Phase boundaries indicate two-phase coexistence—along these lines, the chemical potential of adjacent phases is equal, satisfying equilibrium conditions
• The slope of each boundary follows the Clapeyron equation $\frac{dP}{dT} = \frac{\Delta S}{\Delta V} = \frac{\Delta H}{T\Delta V}$, connecting thermodynamic quantities to observable phase behavior
• Regions between boundaries represent single-phase stability—within these areas, $F = 2$ (you can vary both P and T independently)

Water Phase Diagram

• Negative solid-liquid boundary slope reflects that ice is less dense than liquid water ($\Delta V < 0$ for melting), a consequence of hydrogen bonding geometry
• Triple point at 0.01°C and 611.657 Pa—this invariant point ($F = 0$) is used to define the Kelvin temperature scale
• Critical point at 374°C and 22.06 MPa marks where liquid and gas become indistinguishable; beyond this, only supercritical fluid exists

CO₂ Phase Diagram

• No liquid phase at 1 atm because the triple point pressure (5.11 atm) exceeds atmospheric pressure—this is why dry ice sublimes directly
• Triple point at -56.6°C and 5.11 atm explains why pressurized CO₂ containers can contain liquid while atmospheric exposure produces only solid or gas
• Critical point at 31.1°C and 73.8 atm is accessible under mild conditions, making supercritical CO₂ practical for extraction and green chemistry applications

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Binary Phase Diagrams

These diagrams add composition as a variable, showing how two-component mixtures behave. The lever rule and tie lines become essential tools for determining the amounts and compositions of coexisting phases.

Binary Liquid-Liquid Phase Diagram

• Two-phase region indicates immiscibility—within this dome, the system separates into two liquid phases with compositions given by the tie line endpoints
• Tie lines connect coexisting phase compositions at constant temperature; the lever rule determines the relative amounts of each phase
• Upper critical solution temperature (UCST) marks where the miscibility gap closes and the system becomes single-phase upon heating

Binary Solid-Liquid Phase Diagram

• Liquidus line shows where solidification begins upon cooling; compositions above this line are entirely liquid
• Eutectic point represents the lowest melting temperature—at this invariant point, liquid transforms directly into two solid phases simultaneously
• Solidus line indicates complete solidification—below this, only solid phases exist; between liquidus and solidus, solid and liquid coexist

Temperature-Composition (T-x) Diagram

• Horizontal tie lines connect equilibrium compositions—at any temperature in a two-phase region, the endpoints give the compositions of coexisting phases
• Lever rule quantifies phase fractions: $\frac{n_\alpha}{n_\beta} = \frac{x_\beta - x_0}{x_0 - x_\alpha}$ where $x_0$ is the overall composition
• Liquidus and solidus curves define the "mushy zone"—critical for predicting solidification behavior in alloys and purification by zone refining

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Multi-Component and Alternative Representations

Beyond simple binary systems, phase diagrams can represent three components or use different variables to highlight specific thermodynamic relationships.

Ternary Phase Diagram

• Triangular format with pure components at vertices—any point inside represents a mixture, with composition read by perpendicular distances to opposite sides
• Two-phase and three-phase regions appear as areas and triangular tie-triangles, respectively; tie lines still connect coexisting compositions
• Critical for formulation science—understanding emulsion stability, extraction efficiency, and product design in pharmaceuticals and materials

Pressure-Volume (P-V) Diagram

• Isotherms reveal phase transition behavior—horizontal segments indicate two-phase coexistence where pressure remains constant during vaporization
• The critical isotherm shows an inflection point at the critical point, where $\left(\frac{\partial P}{\partial V}\right)_T = 0$ and $\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0$
• Area under curves represents work: $w = -\int P \, dV$, essential for analyzing thermodynamic cycles and engine efficiency

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Fundamental Principles

These concepts provide the mathematical and theoretical foundation for interpreting all phase diagrams.

Gibbs Phase Rule

• $F = C - P + 2$ determines degrees of freedom—the number of intensive variables (T, P, composition) you can change independently while maintaining the same number of phases
• At a triple point, $F = 1 - 3 + 2 = 0$—no variables can change; this is why triple points are invariant and useful for calibration standards
• In two-phase binary systems, $F = 2 - 2 + 2 = 2$—you can independently vary T and overall composition, but phase compositions are then fixed by the diagram

Triple Point and Critical Point

• Triple point: unique invariant condition where solid, liquid, and gas coexist; used to define thermodynamic temperature scales (water's triple point formerly defined the Kelvin)
• Critical point: end of liquid-gas distinction—above $T_c$ and $P_c$, the substance exists as a supercritical fluid with properties intermediate between liquid and gas
• Critical behavior follows universal scaling laws—near the critical point, properties like density difference between phases vanish according to power-law relationships

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Quick Reference Table

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Self-Check Questions

1. Using the Gibbs phase rule, calculate the degrees of freedom at the eutectic point of a binary solid-liquid system. What does this tell you about the eutectic temperature?

2. Which two phase diagrams both feature "domes" representing two-phase regions, and what physical phenomenon does each dome represent?

3. Compare and contrast the solid-liquid phase boundary in water versus CO₂. How does the Clapeyron equation explain the difference, and what molecular property of water is responsible?

4. On a T-x diagram for a binary alloy, you have an overall composition $x_0 = 0.4$ at a temperature where the liquidus composition is $x_L = 0.6$ and solidus composition is $x_S = 0.2$. Use the lever rule to determine the fraction of liquid present.

5. An FRQ asks you to explain why CO₂ fire extinguishers release a white solid rather than a liquid when discharged. Using the CO₂ phase diagram, construct your response including specific reference to the triple point.