Vapor-Liquid Equilibrium Theory

Introduction

Vapor-Liquid Equilibrium (VLE) describes the distribution of components between vapor and liquid phases at thermodynamic equilibrium. Understanding VLE is essential for designing distillation columns, flash drums, and other separation equipment in chemical engineering.

Phase Equilibrium Criterion

At equilibrium, the fugacity of each component must be equal in both phases:

f̂_i^V = f̂_i^L

This leads to the fundamental VLE equation:

y_iφ̂_i^VP = x_iγ_iP_i^sat

Raoult's Law

For ideal mixtures (γ = 1, φ = 1), VLE simplifies to Raoult's Law:

y_iP = x_iP_i^sat

The K-value (equilibrium ratio) for ideal systems:

K_i = y_i/x_i = P_i^sat/P

Raoult's Law works well for mixtures of similar molecules (e.g., benzene-toluene).

Modified Raoult's Law

For non-ideal liquid mixtures, activity coefficients (γ) account for molecular interactions:

y_iP = x_iγ_iP_i^sat

γ > 1:Positive deviation - molecules repel (more volatile)

γ < 1:Negative deviation - molecules attract (less volatile)

γ = 1:Ideal behavior

NRTL Model

The Non-Random Two-Liquid (NRTL) model predicts activity coefficients:

ln γ₁ = x₂²[τ₂₁(G₂₁/(x₁+x₂G₂₁))² + τ₁₂G₁₂/(x₂+x₁G₁₂)²]

G_ij = exp(-α_ijτ_ij), τ_ij = A_ij/RT

Parameters A₁₂, A₂₁, and α are fitted to experimental data.

Antoine Equation

Vapor pressure is calculated using the Antoine equation:

log₁₀(P^sat) = A - B/(C + T)

Where P is in mmHg and T is in °C. Constants A, B, C are substance-specific and available in literature (e.g., NIST, Perry's).

VLE Calculations

Bubble Point

First vapor bubble forms when heating liquid. Given x_i:

P^bubble = Σ x_iγ_iP_i^sat

y_i = x_iγ_iP_i^sat / P

Dew Point

First liquid drop forms when cooling vapor. Given y_i:

P^dew = 1 / Σ(y_i / γ_iP_i^sat)

x_i = y_iP / γ_iP_i^sat

Flash Calculation

Two-phase region. Solve Rachford-Rice equation for vapor fraction V/F:

Σ z_i(K_i-1) / (1 + (V/F)(K_i-1)) = 0

Azeotropes

An azeotrope occurs when x = y (no separation possible by simple distillation):

Minimum boiling: γ > 1, e.g., ethanol-water (95.6% EtOH at 78.1°C)
Maximum boiling: γ < 1, e.g., acetone-chloroform

At the azeotrope: α₁₂ = 1, meaning γ₁P₁^sat = γ₂P₂^sat

Relative Volatility

The relative volatility indicates ease of separation:

α₁₂ = K₁/K₂ = (γ₁P₁^sat) / (γ₂P₂^sat)

α > 1: Component 1 more volatile, concentrates in vapor
α = 1: No separation possible (azeotrope)
Higher α = easier separation

References

Smith, Van Ness, Abbott "Introduction to Chemical Engineering Thermodynamics"
Prausnitz, Lichtenthaler, Azevedo "Molecular Thermodynamics of Fluid-Phase Equilibria"
Seader, Henley "Separation Process Principles"
NIST Chemistry WebBook - Antoine Equation Parameters

Process Tools

VLE Calculator