9.2 Phase equilibria and phase diagrams

Phase equilibria is a crucial concept in thermodynamics. It occurs when multiple phases of a substance coexist in balance, with no net mass transfer. This requires equal temperature, pressure, and chemical potential for each component across all phases.

Phase diagrams visually represent phase equilibria. They show regions of stability for different phases and boundaries where phases coexist. Understanding these diagrams is key to predicting phase behavior and solving real-world thermodynamic problems.

Phase Equilibria and Phase Diagrams

Concept of phase equilibria

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• Occurs when two or more phases of a substance coexist in a state of dynamic balance with no net transfer of mass between phases
• Phases have equal chemical potentials ($\mu_i$) for each component $i$
• Requires thermal equilibrium with equal temperature ($T$) among phases, mechanical equilibrium with equal pressure ($P$) among phases, and chemical equilibrium with equal chemical potential ($\mu_i$) for each component $i$ in all phases
• Examples: Ice and water coexisting at 0°C and 1 atm, liquid water and water vapor coexisting at 100°C and 1 atm

Construction of phase diagrams

• Single-component phase diagrams
  • Pressure-temperature ($P$-$T$) diagrams show regions of stability for solid, liquid, and gas phases with lines representing phase boundaries where two phases coexist
    • Triple point represents coexistence of all three phases
    • Critical point is the terminus of the liquid-gas phase boundary
  • Pressure-volume ($P$-$V$) diagrams use isotherms to represent constant temperature, regions to represent single-phase states, and tie lines to connect coexisting phases
• Binary phase diagrams
  • Temperature-composition ($T$-$x$) diagrams at constant pressure show phase behavior of two-component mixtures
    • Liquidus line indicates temperature at which a solid phase begins to form
    • Solidus line indicates temperature at which the last liquid phase disappears
    • Eutectic point represents the lowest melting temperature of the mixture
  • Pressure-composition ($P$-$x$) diagrams at constant temperature show phase behavior of two-component mixtures under varying pressure with vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) curves

Types of phase transitions

• First-order phase transitions involve a discontinuous change in properties (density, enthalpy) with latent heat associated with the transition
  • Examples: Melting, vaporization, sublimation
• Second-order phase transitions involve a continuous change in properties with no latent heat associated with the transition
  • Examples: Superconductivity, ferromagnetism
• Solid-solid phase transitions involve transitions between different crystal structures
  • Examples: Allotropic transformations (graphite to diamond)

Application of Gibbs phase rule

• Gibbs phase rule: $F = C - P + 2$, where $F$ is degrees of freedom (variance), $C$ is number of components, and $P$ is number of phases
• Degrees of freedom represent the number of intensive variables (temperature, pressure, composition) that can be independently varied without changing the number of phases in equilibrium
• Applications
  • Single-component systems: $F = 3 - P$
    1. Triple point: $F = 0$ (no degrees of freedom)
    2. Phase boundaries: $F = 1$ (univariant)
  • Binary systems: $F = 4 - P$
    • Eutectic point: $F = 1$ (univariant)

Problem-solving with phase diagrams

• Reading phase diagrams involves identifying phases present at given conditions and determining phase compositions and relative amounts
• Lever rule is used to calculate the relative amounts of phases in a two-phase region, where lever arm lengths are inversely proportional to the mass fractions of the phases
• Tie lines connect coexisting phases in a two-phase region with endpoints representing the compositions of the individual phases
• Phase transformations involve
  1. Heating and cooling processes to determine the sequence of phase changes and calculate the amount of heat absorbed or released
  2. Pressure changes to predict phase changes along isotherms and determine the effect on phase equilibria

Thermodynamic Principles in Phase Equilibria

• Gibbs free energy ($G$) is the thermodynamic potential at constant temperature and pressure
  • Criterion for phase stability: Minimum Gibbs free energy
  • Phase equilibria: Equal Gibbs free energy for each component in all phases
• Chemical potential ($\mu_i$) is the partial molar Gibbs free energy of component $i$ and measures the tendency of a component to change phases
  • Phase equilibria: Equal chemical potential for each component in all phases
• Fugacity ($f_i$) and activity ($a_i$) are the effective partial pressure and effective concentration of component $i$, related to chemical potential by $\mu_i = \mu_i^0 + RT \ln (f_i/f_i^0) = \mu_i^0 + RT \ln a_i$
  • Phase equilibria: Equal fugacity or activity for each component in all phases
• Clapeyron equation relates the slope of a phase boundary ($dP/dT$) to the changes in molar volume ($\Delta V_m$) and molar entropy ($\Delta S_m$) during a phase transition: $dP/dT = \Delta S_m / \Delta V_m = \Delta H_m / (T \Delta V_m)$
  • Useful for solid-liquid and solid-solid phase transitions
• Clausius-Clapeyron equation is a specific form of the Clapeyron equation for vaporization, assuming ideal gas behavior and negligible volume of the condensed phase: $\ln (P_2/P_1) = -(\Delta H_\text{vap}/R)(1/T_2 - 1/T_1)$
  • Useful for estimating vapor pressures and boiling points