Fugacity and fugacity coefficient of pure species and mixtures

Why it matters

Phase equilibrium is the equality of chemical potentials, but chemical potential goes to minus infinity as pressure goes to zero and has no absolute value, which makes it awkward to compute. Fugacity is a "corrected pressure" that does the same job with ordinary numbers. High-pressure VLE in gas processing, flash calculations in simulators and the gas-phase side of every equilibrium-constant problem all run on fugacities and fugacity coefficients.

Key ideas

Definition for a pure species. For an ideal gas at constant T, dG = RT d ln P. For a real fluid we keep the same form and define fugacity fᵢ so that dGᵢ = RT d ln fᵢ at constant T, with the limit fᵢ/P → 1 as P → 0. Fugacity has units of pressure. The fugacity coefficient φᵢ = fᵢ/P is dimensionless; φ = 1 for an ideal gas, φ < 1 when attractive forces dominate, φ > 1 at very high pressure when repulsion dominates.

Link to residual properties. ln φᵢ = Gᵢ^R/RT, so ln φ follows from Z: ln φ = ∫(Z − 1) dP/P at constant T. Any equation of state or generalised correlation that gives Z gives φ. With the two-term virial equation, ln φ = BP/RT. With the Pitzer correlation, ln φ = (Pr/Tr)(B⁰ + ωB¹).

Fugacity of a pure liquid. Coexisting saturated phases have equal G, so f^l = f^v = φ^sat·P^sat at the saturation pressure. To go from P^sat to a higher pressure P, integrate dG = V^l dP for the nearly incompressible liquid. This gives the Poynting factor exp[V^l(P − P^sat)/RT], which is close to 1 unless the pressure difference is tens of bar or more. At low pressure f^l ≈ P^sat.

Effect of temperature and pressure. (∂ ln f/∂P)_T = V/RT, so fugacity always rises with pressure. (∂ ln f/∂T)_P = −H^R/RT².

Species in a mixture. The fugacity of species i in solution, f̂ᵢ, is defined by dμᵢ = RT d ln f̂ᵢ at constant T, with f̂ᵢ → yᵢP as P → 0. The fugacity coefficient of i in solution is φ̂ᵢ = f̂ᵢ/(yᵢP). In an ideal-gas mixture f̂ᵢ = yᵢP, the partial pressure. ln φ̂ᵢ is the partial molar property of ln φ of the mixture (with respect to G^R/RT), so the summability rule ln φ_mix = Σyᵢ ln φ̂ᵢ holds.

Virial mixtures. For a binary gas at moderate pressure, B_mix = y₁²B₁₁ + 2y₁y₂B₁₂ + y₂²B₂₂, and the species coefficients follow in closed form using δ₁₂ = 2B₁₂ − B₁₁ − B₂₂.

Criterion of phase equilibrium. Equal T, equal P and equal f̂ᵢ of each species in every phase. This is the starting point of the gamma-phi and phi-phi formulations of VLE.

Formulas

dGᵢ = R·T·d ln fᵢ (constant T) φᵢ = fᵢ / P, lim(P→0) φᵢ = 1 ln φᵢ = Gᵢ^R/(R·T) = ∫₀^P (Zᵢ − 1) dP/P (constant T) ln φᵢ = Bᵢᵢ·P/(R·T) (two-term virial) ln φ = (Pr/Tr)·(B⁰ + ω·B¹) (Pitzer) fᵢ^l = φᵢ^sat·Pᵢ^sat·exp[Vᵢ^l·(P − Pᵢ^sat)/(R·T)] (compressed liquid, Poynting) f̂ᵢ = φ̂ᵢ·yᵢ·P ln φ̂₁ = (P/RT)·(B₁₁ + y₂²·δ₁₂), ln φ̂₂ = (P/RT)·(B₂₂ + y₁²·δ₁₂), δ₁₂ = 2B₁₂ − B₁₁ − B₂₂ ln φ_mix = Σ yᵢ·ln φ̂ᵢ f̂ᵢ^α = f̂ᵢ^β = … (phase equilibrium)

Symbols: fᵢ pure-species fugacity (Pa or bar); f̂ᵢ fugacity of i in solution (same units); φᵢ, φ̂ᵢ fugacity coefficients (dimensionless); P pressure; Pᵢ^sat vapour pressure; Vᵢ^l liquid molar volume (m³/mol); B virial coefficients (m³/mol); yᵢ vapour mole fraction; Tr, Pr reduced temperature and pressure; ω acentric factor; R = 8.314 J/(mol·K); T (K).

Worked examples

Example 1 (standard): fugacity from the virial equation. Given: a pure gas at 400 K and 10 bar with B = −300 cm³/mol. Find φ and f.

1. ln φ = B·P/(R·T) = (−300 × 10⁻⁶ m³/mol)(10 × 10⁵ Pa)/(8.314 × 400) = −300/3325.6 = −0.0902.
2. φ = e^(−0.0902) = 0.9137.
3. f = φ·P = 0.9137 × 10 = 9.14 bar. Answer: φ = 0.914, f = 9.14 bar

Example 2 (GATE level): compressed liquid and a gas mixture. (a) Given: liquid water at 300 K compressed to 100 bar; P^sat = 0.0354 bar, V^l = 18.07 cm³/mol, φ^sat ≈ 1 (low pressure). Find f^l.

1. Poynting exponent = V^l(P − P^sat)/RT = 18.07 × 10⁻⁶ × (100 − 0.0354) × 10⁵ / (8.314 × 300) = 180.6/2494.2 = 0.0724.
2. Poynting factor = e^0.0724 = 1.0751.
3. f^l = 1 × 0.0354 × 1.0751 = 0.0381 bar. Answer: f^l = 0.0381 bar, only 7.5% above P^sat even at 100 bar.

(b) Given: a binary gas at 300 K and 20 bar with y₁ = 0.4; B₁₁ = −50, B₂₂ = −150, B₁₂ = −80 cm³/mol. Find φ̂₁, φ̂₂ and f̂₁, f̂₂.

1. δ₁₂ = 2(−80) − (−50) − (−150) = 40 cm³/mol.
2. P/RT = 20 × 10⁵ / (8.314 × 300) = 801.9 mol/m³.
3. ln φ̂₁ = (P/RT)(B₁₁ + y₂²δ₁₂) = 801.9 × (−50 + 0.36 × 40) × 10⁻⁶ = 801.9 × (−35.6 × 10⁻⁶) = −0.02855, so φ̂₁ = 0.9719.
4. ln φ̂₂ = (P/RT)(B₂₂ + y₁²δ₁₂) = 801.9 × (−150 + 0.16 × 40) × 10⁻⁶ = −0.11515, so φ̂₂ = 0.8912.
5. f̂₁ = 0.9719 × 0.4 × 20 = 7.78 bar; f̂₂ = 0.8912 × 0.6 × 20 = 10.69 bar.
6. Check by summability: B_mix = 0.16(−50) + 0.48(−80) + 0.36(−150) = −100.4 cm³/mol, ln φ_mix = 801.9 × (−100.4 × 10⁻⁶) = −0.0805 = 0.4(−0.02855) + 0.6(−0.11515). Consistent. Answer: φ̂₁ = 0.972, φ̂₂ = 0.891; f̂₁ = 7.78 bar, f̂₂ = 10.69 bar

Common mistakes

• Writing f̂ᵢ = φᵢ·yᵢ·P with the pure-species φᵢ instead of the mixture value φ̂ᵢ (that shortcut is the Lewis-Randall rule, an approximation).
• Unit slips in BP/RT: B in cm³/mol must be converted to m³/mol when P is in Pa.
• Forgetting the Poynting factor at high pressure, or applying it with the vapour volume instead of the liquid volume.
• Assuming φ < 1 always; at high reduced pressure φ can exceed 1.
• Using the two-term virial result for a liquid or near the critical point.

For GATE CH

• NAT problems on φ and f from the virial equation or the Pitzer correlation, and on liquid fugacity with a Poynting correction.
• Mixture fugacity coefficients from given B₁₁, B₂₂ and B₁₂.
• Conceptual MCQs on the definition of fugacity, its limit at zero pressure, the equilibrium criterion and the meaning of φ greater or less than 1.
• Practise deriving ln φ from a simple equation of state such as Z = 1 + BP/RT or V = RT/P + b.

Quick check

1. What is the limit of φ as P → 0?
2. For a gas obeying V = RT/P + b, what is ln φ?
3. Which correction converts the saturated-liquid fugacity to that of a compressed liquid?
4. In an ideal-gas mixture, what is f̂ᵢ?
5. What condition on fugacities defines phase equilibrium?

Answers: 1. 1. 2. bP/RT. 3. The Poynting factor exp[V^l(P − P^sat)/RT]. 4. The partial pressure yᵢP. 5. The fugacity of each species is equal in all phases at the same T and P.