10.2 Vapor-Liquid Equilibrium and Fugacity

Vapor-Liquid Equilibrium Criterion

Vapor-liquid equilibrium (VLE) describes the condition where liquid and vapor phases coexist with no net mass transfer between them. Each component's tendency to escape one phase is exactly balanced by its tendency to escape the other. Understanding VLE is foundational for designing distillation columns, flash drums, and other separation equipment.

Fugacity quantifies a component's "escape tendency" from a phase. It plays the same role in phase equilibrium that temperature plays in thermal equilibrium: two phases reach equilibrium when fugacity is equalized, just as heat transfer stops when temperatures match.

Equilibrium Condition

For a system at vapor-liquid equilibrium, the fugacity of each component $i$ must be equal across both phases:

$f_i^L = f_i^V$

This holds for every component in the mixture simultaneously. If $f_i^L > f_i^V$ for some component, molecules of that species will preferentially transfer from liquid to vapor until the fugacities equalize.

For a pure component at moderate pressures, the fugacity at saturation is approximately equal to the vapor pressure:

$f_i^{pure} \approx P_i^{sat}$

This approximation breaks down at high pressures, where the Poynting correction factor and vapor-phase non-ideality become significant.

Fugacity in Mixtures

In a mixture, the fugacity of component $i$ depends on composition and on how much the system deviates from ideal behavior. That deviation is captured by the fugacity coefficient $\hat{\phi}_i$, which you can calculate from an equation of state (EOS).

Common EOS choices include:

• Virial equation (truncated at the second coefficient; works well at low to moderate pressures)
• Cubic equations of state: van der Waals, Redlich-Kwong (RK), Soave-Redlich-Kwong (SRK), and Peng-Robinson (PR)

The fugacity coefficient approaches unity ($\hat{\phi}_i \to 1$) as the mixture approaches ideal-gas behavior (low pressure, high temperature).

Fugacity Calculation

Pure Components

For a pure ideal gas, fugacity equals the pressure: $f = P$. For a pure real gas, the fugacity is corrected by the fugacity coefficient:

$f = \phi P$

At saturation (on the vapor-pressure curve), the liquid and vapor fugacities of a pure species are equal, giving the useful low-pressure approximation:

$f_i^{pure} \approx P_i^{sat}$

For a component in an ideal-gas mixture, the fugacity equals its partial pressure:

$\hat{f}_i = y_i P$

where $y_i$ is the vapor-phase mole fraction and $P$ is total pressure.

Mixtures

For a real mixture, fugacity expressions include a fugacity coefficient to account for intermolecular interactions:

• Vapor phase: $\hat{f}_i^V = y_i \hat{\phi}_i^V P$
• Liquid phase: $\hat{f}_i^L = x_i \hat{\phi}_i^L P$

Here $x_i$ and $y_i$ are liquid and vapor mole fractions, and $\hat{\phi}_i$ is the fugacity coefficient of component $i$ in the mixture (note the hat, distinguishing it from the pure-component coefficient).

There are two main modeling approaches:

1. $\phi$-$\phi$ approach: Use an EOS for both phases. You calculate $\hat{\phi}_i$ from the same equation of state in both liquid and vapor. This is common for high-pressure systems and hydrocarbon mixtures.
2. $\gamma$-$\phi$ approach: Use an activity coefficient model for the liquid phase and an EOS for the vapor phase. The equilibrium relation becomes $y_i \hat{\phi}_i^V P = x_i \gamma_i P_i^{sat}$ (with a Poynting correction at high pressures). This is the standard approach for low-to-moderate pressure systems with polar or non-ideal liquids.

Activity coefficient models for the liquid phase include Wilson, NRTL, and UNIQUAC. The choice depends on the system chemistry and available experimental data.

Ideal vs. Non-Ideal Solutions

Ideal Solutions (Raoult's Law)

An ideal solution is one where intermolecular forces between unlike molecules are the same as those between like molecules. In this case, the liquid-phase fugacity simplifies to Raoult's law:

$\hat{f}_i^L = x_i P_i^{sat}$

Setting this equal to the ideal-gas vapor fugacity ($y_i P$) gives the familiar form:

$y_i P = x_i P_i^{sat}$

Raoult's law works well for mixtures of chemically similar species, such as neighboring members of a homologous series (e.g., benzene-toluene). It serves as the reference state against which non-ideal behavior is measured.

Non-Ideal Solutions

Real mixtures deviate from Raoult's law because molecules of different species interact differently than molecules of the same species. The activity coefficient $\gamma_i$ quantifies this deviation:

$\hat{f}_i^L = \gamma_i \, x_i \, f_i^{pure}$

• When $\gamma_i = 1$, the solution is ideal.
• When $\gamma_i > 1$, you have a positive deviation from Raoult's law.
• When $\gamma_i < 1$, you have a negative deviation.

Positive deviations ($\gamma_i > 1$) occur when unlike molecules repel each other more than like molecules do. The components "want to escape" the mixture more readily, so vapor pressures are higher than Raoult's law predicts. A classic example is ethanol-water: hydrogen bonding between ethanol molecules and between water molecules is stronger than the cross-interactions, pushing both species into the vapor phase more easily.

Negative deviations ($\gamma_i < 1$) occur when unlike molecules attract each other more strongly than like molecules do. Vapor pressures end up lower than Raoult's law predicts. Acetone-chloroform is the textbook example: a hydrogen bond forms between chloroform's C-H and acetone's carbonyl oxygen, an interaction that doesn't exist in either pure component.

Analyzing Vapor-Liquid Equilibrium Data

Equations of State and Activity Coefficient Models

Choosing the right thermodynamic model is a practical skill you'll use repeatedly in process design:

• Cubic EOS (SRK, Peng-Robinson) with mixing rules work well for hydrocarbon systems and high-pressure applications. They can model both phases with a single equation ($\phi$-$\phi$ approach).
• Activity coefficient models (Wilson, NRTL, UNIQUAC) are preferred for polar and strongly non-ideal liquid mixtures at low to moderate pressures. They require binary interaction parameters, which are typically regressed from experimental VLE data.
• NRTL handles partially miscible systems (liquid-liquid equilibrium) better than Wilson, which cannot predict phase splitting.
• UNIQUAC has a built-in combinatorial term that accounts for molecular size and shape differences, making it more physically grounded for asymmetric systems.

The quality of any model depends on having reliable experimental data for parameter fitting. Without good data, even a sophisticated model will give poor predictions.

Graphical Representation and Interpretation

VLE data are commonly displayed on $xy$ diagrams, which plot vapor mole fraction $y_i$ against liquid mole fraction $x_i$ at constant temperature or pressure. A 45° diagonal line represents $y = x$; the farther the equilibrium curve sits from this diagonal, the easier the separation.

Azeotropes appear on an $xy$ diagram where the equilibrium curve crosses the diagonal. At an azeotropic composition, $x_i = y_i$, so the mixture boils at a constant composition and cannot be separated further by ordinary distillation. Ethanol-water forms a minimum-boiling azeotrope at about 95.6 wt% ethanol (1 atm), which is why you can't distill pure ethanol from an aqueous mixture without special techniques (e.g., pressure-swing distillation, entrainers).

Relative volatility provides a single number to gauge separation difficulty:

$\alpha_{ij} = \frac{y_i / x_i}{y_j / x_j}$

• $\alpha_{ij} \gg 1$: easy separation; fewer trays or stages needed.
• $\alpha_{ij} \approx 1$: difficult separation; many stages required, and the system may be near-azeotropic.
• $\alpha_{ij} = 1$: azeotropic point; conventional distillation cannot achieve further separation.

Thermodynamic consistency tests verify that experimental VLE data obey the Gibbs-Duhem equation. The two most common are:

• Redlich-Kister area test: integrates $\ln(\gamma_1 / \gamma_2)$ over the full composition range. For consistent data, the net area should be close to zero.
• Herington test: compares the ratio of positive and negative areas to a correction term based on the boiling point range of the system.

If data fail these tests, the measurements likely contain systematic errors and should not be used for model parameter regression without careful scrutiny.