10.2 Vapor-Liquid Equilibrium and Fugacity

Vapor-liquid equilibrium is all about balance. It's when the liquid and vapor phases of a mixture are in perfect harmony, with each component's fugacity equal in both phases. This concept is crucial for understanding phase behavior and separation processes.

Fugacity is like a component's "escape tendency" from a mixture. It's affected by temperature, pressure, and composition. In ideal solutions, fugacity follows simple rules, but real-world mixtures often deviate, making things more complex and interesting.

Vapor-Liquid Equilibrium Criterion

Equilibrium Condition

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• At vapor-liquid equilibrium, the fugacity of each component in the liquid phase is equal to the fugacity of that component in the vapor phase
• For a multicomponent system, the fugacity of each component in the liquid phase is equal to the fugacity of that component in the vapor phase, which can be expressed as: $f_i^L = f_i^V$
• The fugacity of a pure component is equal to its vapor pressure at the system temperature

Fugacity in Mixtures

• The fugacity of a component in a mixture is related to its mole fraction and the fugacity coefficient, which accounts for non-ideal behavior
• The fugacity coefficient can be calculated using equations of state, such as the Virial equation or cubic equations of state (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson)

Fugacity Calculation

Pure Components

• For a pure component, the fugacity is equal to the vapor pressure at the system temperature: $f_i^{pure} = P_i^{sat}$
• The fugacity of a component in an ideal gas mixture is equal to its partial pressure: $f_i = y_i P$, where $y_i$ is the mole fraction of component $i$ in the vapor phase and $P$ is the total pressure

Mixtures

• In a non-ideal mixture, the fugacity of a component is related to its mole fraction, total pressure, and fugacity coefficient: $f_i = x_i \phi_i P$ (liquid phase) or $f_i = y_i \phi_i P$ (vapor phase), where $x_i$ and $y_i$ are the mole fractions of component $i$ in the liquid and vapor phases, respectively, and $\phi_i$ is the fugacity coefficient
• The fugacity coefficient can be calculated using equations of state or empirical correlations, such as the Virial equation, cubic equations of state, or activity coefficient models (Wilson, NRTL, UNIQUAC)
• The fugacity coefficient for a component in a mixture depends on temperature, pressure, and composition, and it approaches unity as the system approaches ideal behavior

Ideal vs Non-ideal Solutions

Ideal Solutions

• An ideal solution is a hypothetical mixture in which the interactions between molecules of different components are identical to the interactions between molecules of the same component
• In an ideal solution, the fugacity of each component is directly proportional to its mole fraction, following Raoult's law: $f_i = x_i P_i^{sat}$, where $P_i^{sat}$ is the vapor pressure of the pure component at the system temperature

Non-ideal Solutions

• Non-ideal solutions exhibit deviations from Raoult's law due to differences in molecular interactions between components, such as size, shape, or polarity differences
• Positive deviations from Raoult's law occur when the interactions between unlike molecules are weaker than those between like molecules, leading to higher vapor pressures and increased volatility (ethanol-water)
• Negative deviations from Raoult's law occur when the interactions between unlike molecules are stronger than those between like molecules, resulting in lower vapor pressures and decreased volatility (acetone-chloroform)
• The extent of non-ideality in a solution can be quantified using activity coefficients, which relate the actual fugacity of a component to its ideal fugacity: $f_i = \gamma_i x_i f_i^{pure}$, where $\gamma_i$ is the activity coefficient of component $i$

Analyzing Vapor-Liquid Equilibrium Data

Equations of State and Activity Coefficient Models

• Equations of state, such as the Virial equation or cubic equations (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson), can be used to calculate fugacity coefficients and predict vapor-liquid equilibrium behavior
• Activity coefficient models, such as Wilson, NRTL, or UNIQUAC, can be used to calculate activity coefficients and account for non-ideal behavior in the liquid phase
• The choice of equation of state or activity coefficient model depends on the system's temperature, pressure, and composition, as well as the availability of experimental data for parameter estimation

Graphical Representation and Interpretation

• Vapor-liquid equilibrium data can be presented in the form of $xy$ diagrams, which plot the mole fractions of components in the vapor phase against those in the liquid phase at constant temperature and pressure
• Azeotropes, which are mixtures with a constant boiling point and composition, can be identified from $xy$ diagrams as points where the equilibrium line intersects the diagonal (ethanol-water, acetone-methanol)
• The relative volatility, $\alpha_{ij} = (y_i/x_i) / (y_j/x_j)$, can be used to compare the ease of separation of components in a mixture, with higher values indicating a more feasible separation
• The consistency of experimental vapor-liquid equilibrium data can be assessed using thermodynamic consistency tests, such as the Redlich-Kister or Herington tests, which verify that the data satisfy the Gibbs-Duhem equation