🎓 Lesson 8
D5
Fugacity and Fugacity Coefficients: The Bridge to Non-Ideality
Fugacity is a corrected 'effective pressure' that tells us how a real gas behaves in chemical reactions and phase changes—like how hard it 'tries' to escape a liquid or enter a vapor, accounting for molecular stickiness and space.
🎯 Learning Objectives
- ✓ Calculate fugacity coefficients for pure components using generalized compressibility charts or cubic equations of state
- ✓ Explain how fugacity governs vapor–liquid equilibrium (VLE) in multicomponent mixtures relevant to solvent recovery in mine dewatering or acid leaching processes
- ✓ Apply fugacity-based equilibrium criteria to analyze phase behavior of CO₂–H₂O–salt systems in carbon mineralization or heap leach modeling
- ✓ Analyze the impact of temperature, pressure, and composition on fugacity deviations in high-pressure vent gas streams from underground mining operations
📖 Why This Matters
In mining and metallurgical operations—such as pressure oxidation of refractory gold ores, CO₂ sequestration in abandoned mines, or solvent extraction with supercritical fluids—gases and vapors rarely behave ideally. Assuming ideal gas law (PV = nRT) leads to dangerous errors in predicting phase split, solubility, or reaction extent. Fugacity bridges this gap: it’s the 'real-world pressure' engineers use to safely design scrubbers, separators, and leach circuits where non-ideality dominates. Without it, a VLE calculation for H₂S–water in tailings gas treatment could underestimate vapor-phase concentration by >40%, risking HSE noncompliance.
📘 Core Principles
Fugacity emerges from the need to retain the simple functional form of chemical potential (μ = μ° + RT ln f) while accommodating intermolecular forces and finite molecular volume. We begin with the fundamental relation dG = V dP − S dT, then integrate along an isotherm to define f via the residual Gibbs energy. For pure substances, the fugacity coefficient φ is derived from the compressibility factor Z = PV/RT; deviations from Z = 1 directly quantify non-ideality. In mixtures, we extend to partial molar properties and use mixing rules (e.g., van der Waals one-fluid) within cubic EOS like Peng–Robinson—critical for modeling CO₂–CH₄–N₂ vent gas in deep mining environments. Finally, phase equilibrium requires equal fugacities of each component across phases (f_i^L = f_i^V), not equal pressures or concentrations.
📐 Fugacity Coefficient from Compressibility Chart
For quick engineering estimates—especially when EOS parameters are unavailable—the generalized compressibility chart (Nelson–Obert) provides φ from reduced temperature (T_r) and reduced pressure (P_r). This method is widely used in preliminary design of gas-handling systems for mine ventilation and emissions control.
💡 Worked Example
Problem: Estimate the fugacity coefficient φ of methane (CH₄) at 310 K and 8.5 MPa. Critical properties: T_c = 190.6 K, P_c = 4.60 MPa.
1.
Step 1: Compute reduced properties: T_r = 310 / 190.6 = 1.627; P_r = 8.5 / 4.60 = 1.848.
2.
Step 2: Use Nelson–Obert chart (or Lee–Kesler tables): At T_r ≈ 1.63 and P_r ≈ 1.85, φ ≈ 0.72.
3.
Step 3: Calculate fugacity: f = φ × P = 0.72 × 8.5 MPa = 6.12 MPa — meaning CH₄ behaves as if under only ~72% of its actual pressure due to attractive forces.
Answer:
The fugacity coefficient is 0.72, and fugacity is 6.12 MPa, falling within the typical range of 0.5–0.9 for subcritical hydrocarbons at P_r < 2.
🏗️ Real-World Application
At the Cadia East copper-gold mine (NSW, Australia), pressure oxidation (POX) autoclaves operate at 200°C and 2.2 MPa to oxidize sulfide concentrates. The off-gas contains O₂, CO₂, SO₂, and water vapor. Accurate prediction of SO₂ solubility in acidic condensate requires fugacity-based Henry’s law (C = k_H · f_SO₂), not partial pressure. Using Peng–Robinson EOS with binary interaction parameters, engineers calculated φ_SO₂ = 0.81 at process conditions—leading to a 19% lower predicted dissolved SO₂ than ideal assumptions. This corrected design of the quench tower and downstream scrubber, preventing under-sizing and ensuring SO₂ emissions remained below NSW EPA limit of 150 mg/m³.