Vapor-Liquid Equilibrium: Phase Rule

Vapor-Liquid Equilibrium

• Phase equilibrium signifies a stable state where multiple phases coexist.
• Vapor-liquid equilibrium (VLE) involves the equilibrium between a liquid and a vapor phase.

Equilibrium Criteria

• Temperature Equilibrium: All phases must have the same temperature, represented as $T^{\alpha} = T^{\beta} = T^{\gamma} =...$
• Pressure Equilibrium: All phases must have the same pressure, represented as $P^{\alpha} = P^{\beta} = P^{\gamma} =...$
• Chemical Potential Equilibrium: Each species must have the same chemical potential in all phases, represented as $\mu_{i}^{\alpha} = \mu_{i}^{\beta} = \mu_{i}^{\gamma} =...$
• $\alpha, \beta, \gamma$ denote different phases
• $i$ specifies a chemical species within the mixture

The Phase Rule

• The phase rule, developed by J. Willard Gibbs, relates degrees of freedom, components, and phases in equilibrium: $F = 2 - \pi + N$.
• $F$ represents degrees of freedom, the number of independent variables that can be changed without altering the number of phases in equilibrium.
• $\pi$ represents the number of phases in equilibrium.
• $N$ is the number of components in the system.
• Components refer to the independent chemical species required to define the composition of all phases present.

Understanding the Phase Rule

• Degrees of Freedom (F): Number of intensive variables (temperature, pressure, composition) that can be independently changed without altering the number of phases present.
• Phases ($\pi$): Number of physically distinct and homogeneous parts of the system.
• Components (N): Minimum number of independent chemical constituents needed to specify the composition of each phase.

Examples

• Single Component System (Water): At the triple point, ice, liquid water, and steam coexist
  • $N = 1$
  • $\pi = 3$
  • $F = 2 - 3 + 1 = 0$
  • No degrees of freedom at the triple point; temperature and pressure are fixed.
• Binary System with Vapor and Liquid Phases (Ethanol and Water):
  • $N = 2$
  • $\pi = 2$
  • $F = 2 - 2 + 2 = 2$
  • Two intensive properties can be independently varied.

Importance

• The phase rule is crucial for designing and operating separation processes like distillation, extraction, and crystallization.
• Aids in determining conditions to achieve desired separations and purifications.

Understanding vapor-liquid equilibrium (VLE) involving equilibrium between liquid and vapor phases. Key criteria include temperature, pressure, and chemical potential equilibrium across phases. The Phase Rule, relates degrees of freedom, components, and phases in equilibrium.