🎓 Lesson 4
D3
Vapor-Liquid Equilibrium and Raoult’s Law Limitations
Vapor-liquid equilibrium is when a liquid and its vapor coexist at the same temperature and pressure without changing over time — like boiling water in a sealed pot where steam and water stay balanced.
🎯 Learning Objectives
- ✓ Calculate vapor-phase composition using Raoult’s Law for binary ideal mixtures
- ✓ Analyze deviations from Raoult’s Law using activity coefficients and relative volatility
- ✓ Explain limitations of Raoult’s Law with respect to molecular polarity, size disparity, and hydrogen bonding
- ✓ Apply modified Raoult’s Law (e.g., with Wilson or NRTL models) to estimate bubble-point temperatures for non-ideal systems
- ✓ Evaluate when Raoult’s Law is sufficiently accurate for preliminary distillation design versus when advanced thermodynamic models are required
📖 Why This Matters
In mineral processing, distillation is used to recover solvents (e.g., kerosene in solvent extraction), purify reagents (e.g., sulfuric acid regeneration), or separate volatile metal complexes (e.g., nickel carbonyl). Misapplying Raoult’s Law—especially in systems involving polar solvents, ionic species, or high-boiling organics—leads to grossly inaccurate column sizing, energy overdesign, or failure to meet product purity specs. Understanding *when* and *why* Raoult’s Law fails is essential for safe, efficient, and economically viable separation design.
📘 Core Principles
VLE fundamentals begin with phase rule (F = C − P + 2) and equilibrium criteria: fugacity equality (f_i^L = f_i^V). For ideal solutions and vapors, Raoult’s Law emerges directly from this condition. But real systems violate assumptions of identical intermolecular forces (A–A ≈ A–B ≈ B–B) and zero volume/enthalpy change on mixing. Deviations manifest as azeotropes, immiscibility, or temperature-dependent non-ideality. Activity coefficients (γ_i) quantify these deviations; γ_i > 1 indicates positive deviation (e.g., ethanol–water), while γ_i < 1 signals negative deviation (e.g., chloroform–acetone). Hydrogen bonding, polarity mismatch (e.g., hydrocarbons vs. glycols), and large molecular size differences (>20% in van der Waals volume) are primary drivers of non-ideality in metallurgical solvent systems.
📐 Raoult’s Law and Modified Raoult’s Law
Raoult’s Law applies only to ideal liquid solutions and low-pressure vapor phases. When non-ideality is significant, the modified form incorporates activity coefficients to correct liquid-phase fugacity. This is critical for designing solvent recovery units handling acidic leach liquors or organic extractants.
Modified Raoult’s Law
y_i P = x_i \gamma_i P_i^{\text{sat}}Extends Raoult’s Law to non-ideal liquid solutions using activity coefficient γ_i.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| γ_i | Activity coefficient of component i | dimensionless | Thermodynamic correction for intermolecular non-ideality |
Typical Ranges:
Hydrocarbon–alcohol mixtures: 1.5 – 12.0
Acid–water systems (e.g., H2SO4–H2O): 0.05 – 0.85
💡 Worked Example
Problem: A binary mixture of 60 mol% n-hexane (A) and 40 mol% methanol (B) is at 50°C. Saturation pressures are P_A^sat = 46.0 kPa and P_B^sat = 12.2 kPa. Experimental activity coefficients at this composition are γ_A = 3.25 and γ_B = 1.88. Calculate total pressure and vapor composition.
1.
Step 1: Apply modified Raoult’s Law for each component: y_A·P = x_A·γ_A·P_A^sat and y_B·P = x_B·γ_B·P_B^sat
2.
Step 2: Sum equations: P = x_A·γ_A·P_A^sat + x_B·γ_B·P_B^sat = (0.60)(3.25)(46.0) + (0.40)(1.88)(12.2)
3.
Step 3: Compute: P = 89.7 + 9.19 = 98.9 kPa → y_A = (x_A·γ_A·P_A^sat)/P = 89.7 / 98.9 = 0.907; y_B = 1 − y_A = 0.093
Answer:
The total pressure is 98.9 kPa, and vapor composition is 90.7 mol% n-hexane and 9.3 mol% methanol — illustrating extreme positive deviation due to polarity mismatch.
🏗️ Real-World Application
At the Ravensthorpe Nickel Operation (Western Australia), a solvent extraction–electrowinning (SX-EW) plant recovered kerosene-based diluent from loaded organic phases via vacuum distillation. Initial design assumed Raoult’s Law for kerosene–oxime (LIX 84) mixtures, predicting a 140°C bubble point at 10 kPa. Actual pilot data showed 185°C — a 45°C error — due to strong hydrogen-bonding interactions between oxime groups and trace water. Redesign incorporated UNIFAC group-contribution estimates for γ_i, reducing predicted error to <3°C and avoiding under-specified reboiler duty and thermal degradation of extractant.
📋 Case Connection
📋 Bioethanol Dehydration Using Pervaporation Membranes
Azeotropic limitation of conventional distillation causing 30% energy penalty
📋 Wastewater Reclamation for Semiconductor Fab Using RO-NF Hybrid
High silica, boron, and trace metals (Cu, Ni) exceeding ultrapure water (UPW) specs (<0.1 ppb metals)