Activity coefficient models: Margules, van Laar, Wilson, NRTL

Margules, van Laar, Wilson and NRTL models for G^E/RT and activity coefficients, their limits, and fitting parameters from azeotrope or infinite-dilution data.

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Why it matters

Distillation, extraction and azeotrope design need activity coefficients at every tray composition, but they are measured at only a few points. Activity-coefficient models are compact equations for G^E/RT, fitted to a little data, that give γ at any composition and temperature. Choosing the right model (and knowing what it cannot do) decides whether a simulated column matches the real one.

Key ideas

Start from G^E/RT. Every model is an expression for G^E/RT as a function of composition (and temperature through its parameters). Activity coefficients follow as partial molar properties: ln γᵢ = [∂(nG^E/RT)/∂nᵢ]_{T,P,nⱼ}. Any valid model therefore satisfies summability (G^E/RT = Σxᵢ ln γᵢ), Gibbs-Duhem, and the limits G^E = 0 at x₁ = 0 and x₁ = 1, with γᵢ → 1 as xᵢ → 1.

Margules. A polynomial in mole fraction. The two-suffix (one-parameter) form G^E/RT = A·x₁x₂ is symmetric: ln γ₁ = A·x₂², ln γ₂ = A·x₁², and ln γ₁^∞ = ln γ₂^∞ = A. It suits mixtures of similar-size molecules. The three-suffix (two-parameter) form allows asymmetry; its parameters are the logarithms of the infinite-dilution activity coefficients, ln γ₁^∞ = A₁₂ and ln γ₂^∞ = A₂₁.

van Laar. G^E/RT = A′₁₂A′₂₁x₁x₂/(A′₁₂x₁ + A′₂₁x₂). Derived originally from the van der Waals equation, it now serves as a flexible empirical two-parameter fit; again ln γ₁^∞ = A′₁₂ and ln γ₂^∞ = A′₂₁. It cannot represent a maximum and a minimum in ln γ, and it fails when the parameters have opposite signs.

Wilson. A local-composition model: around a molecule of type 1, the local mole fractions differ from the bulk because of different interaction energies. It uses two parameters Λ₁₂ and Λ₂₁ (dimensionless, built from liquid molar volumes and interaction energies), extends to multicomponent systems with only binary parameters, and represents strongly non-ideal miscible systems such as alcohol-hydrocarbon mixtures well. Its key limitation: it cannot predict liquid-liquid phase splitting.

NRTL (non-random two-liquid). Also local-composition, with interaction parameters τ₁₂, τ₂₁ and a non-randomness parameter α (commonly 0.2 to 0.47). It handles both VLE and LLE and extends to multicomponent systems using binary parameters, which is why it is a default choice in process simulators for polar and partially miscible systems. UNIQUAC and the group-contribution method UNIFAC (for prediction without data) extend the same ideas.

Getting parameters. From infinite-dilution activity coefficients (direct for Margules and van Laar), from a single azeotrope (where γᵢ = P/Pᵢ^sat at low pressure), or by regressing a whole P-x-y data set. Always check that the fitted model reproduces the data points it came from.

Sign of deviation. γ > 1 (positive G^E) gives positive deviation from Raoult's law and, if large, maximum-pressure azeotropes or phase splitting. γ < 1 gives negative deviation and possibly a minimum-pressure azeotrope.

Formulas

G^E/(R·T) = Σ xᵢ·ln γᵢ G^E/(R·T) = A·x₁·x₂; ln γ₁ = A·x₂², ln γ₂ = A·x₁² (two-suffix Margules) G^E/(R·T) = x₁·x₂·(A₂₁·x₁ + A₁₂·x₂) (three-suffix Margules) ln γ₁ = x₂²·[A₁₂ + 2(A₂₁ − A₁₂)·x₁], ln γ₂ = x₁²·[A₂₁ + 2(A₁₂ − A₂₁)·x₂] G^E/(R·T) = A′₁₂·A′₂₁·x₁·x₂ / (A′₁₂·x₁ + A′₂₁·x₂) (van Laar) ln γ₁ = A′₁₂·(1 + A′₁₂·x₁/(A′₂₁·x₂))^(−2), ln γ₂ = A′₂₁·(1 + A′₂₁·x₂/(A′₁₂·x₁))^(−2) A′₁₂ = ln γ₁·(1 + x₂·ln γ₂/(x₁·ln γ₁))², A′₂₁ = ln γ₂·(1 + x₁·ln γ₁/(x₂·ln γ₂))² (from one data point) G^E/(R·T) = −x₁·ln(x₁ + x₂·Λ₁₂) − x₂·ln(x₂ + x₁·Λ₂₁) (Wilson) ln γ₁ = −ln(x₁ + x₂·Λ₁₂) + x₂·[Λ₁₂/(x₁ + x₂·Λ₁₂) − Λ₂₁/(x₂ + x₁·Λ₂₁)] ln γ₁ = x₂²·[τ₂₁·(G₂₁/(x₁ + x₂·G₂₁))² + τ₁₂·G₁₂/(x₂ + x₁·G₁₂)²], Gᵢⱼ = exp(−α·τᵢⱼ) (NRTL) ln γ₁^∞ = A₁₂ (Margules) = A′₁₂ (van Laar)

Symbols: xᵢ liquid mole fractions; γᵢ activity coefficients; A, A₁₂, A₂₁, A′₁₂, A′₂₁ dimensionless model parameters (for G^E/RT); Λ₁₂, Λ₂₁ Wilson parameters (dimensionless, positive); τᵢⱼ NRTL interaction parameters and α non-randomness parameter (dimensionless); γᵢ^∞ infinite-dilution activity coefficient. ln γ₂ for Wilson and NRTL follows by swapping indices 1↔2. Parameters depend on temperature; take them from data books or regress them from VLE data.

Worked examples

Example 1 (standard): three-suffix Margules. Given: A₁₂ = 0.8, A₂₁ = 1.2. Find γ₁, γ₂ at x₁ = 0.3 and the infinite-dilution values.

  1. ln γ₁ = x₂²·[A₁₂ + 2(A₂₁ − A₁₂)·x₁] = 0.49 × [0.8 + 2 × 0.4 × 0.3] = 0.49 × 1.04 = 0.5096, so γ₁ = 1.665.
  2. ln γ₂ = x₁²·[A₂₁ + 2(A₁₂ − A₂₁)·x₂] = 0.09 × [1.2 − 0.8 × 0.7] = 0.09 × 0.64 = 0.0576, so γ₂ = 1.059.
  3. Check: G^E/RT = x₁x₂(A₂₁x₁ + A₁₂x₂) = 0.21 × (0.36 + 0.56) = 0.1932, and x₁ ln γ₁ + x₂ ln γ₂ = 0.1529 + 0.0403 = 0.1932. Consistent.
  4. γ₁^∞ = e^0.8 = 2.23; γ₂^∞ = e^1.2 = 3.32. Answer: γ₁ = 1.665, γ₂ = 1.059, γ₁^∞ = 2.23, γ₂^∞ = 3.32

Example 2 (GATE level): van Laar parameters from an azeotrope. Given: a binary forms a maximum-pressure azeotrope at x₁ = 0.6 and P = 101.3 kPa, where P₁^sat = 80 kPa and P₂^sat = 60 kPa. Assume modified Raoult's law. Find the van Laar constants and γ₁, γ₂ at x₁ = 0.2.

  1. At an azeotrope xᵢ = yᵢ, so γᵢ = P/Pᵢ^sat: γ₁ = 101.3/80 = 1.2663, γ₂ = 101.3/60 = 1.6883.
  2. ln γ₁ = 0.23606, ln γ₂ = 0.52374.
  3. A′₁₂ = ln γ₁·(1 + x₂ ln γ₂/(x₁ ln γ₁))² = 0.23606 × (1 + 0.4 × 0.52374/(0.6 × 0.23606))² = 0.23606 × 2.4791² = 1.4508.
  4. A′₂₁ = ln γ₂·(1 + x₁ ln γ₁/(x₂ ln γ₂))² = 0.52374 × (1 + 0.14164/0.20950)² = 0.52374 × 1.6761² = 1.4713.
  5. At x₁ = 0.2: ln γ₁ = 1.4508 × (1 + 1.4508 × 0.2/(1.4713 × 0.8))^(−2) = 0.9337, γ₁ = 2.544; ln γ₂ = 1.4713 × (1 + 1.4713 × 0.8/(1.4508 × 0.2))^(−2) = 0.0575, γ₂ = 1.059.
  6. Check: substituting x₁ = 0.6 back gives ln γ₁ = 0.2361 and ln γ₂ = 0.5237, the data. Answer: A′₁₂ = 1.451, A′₂₁ = 1.471; at x₁ = 0.2, γ₁ = 2.54 and γ₂ = 1.06

Common mistakes

  • Swapping A₁₂ and A₂₁ in the three-suffix Margules expressions; check with the limit ln γ₁^∞ = A₁₂.
  • Writing γ = exp(G^E/RT) for a component; G^E/RT is a mixture property, ln γᵢ is its partial property.
  • Using Wilson for a partially miscible system; it can never predict two liquid phases.
  • Forgetting that model parameters are for G^E/RT (dimensionless) in some books and for G^E (J/mol) in others.
  • Fitting van Laar to data with a maximum or minimum in ln γ, or with opposite-sign deviations.

For GATE CH

  • NAT problems: γ from Margules or van Laar at a composition, γ at infinite dilution, or G^E/RT from given γ values.
  • Parameters from an azeotrope or from infinite-dilution data, then predicting VLE elsewhere.
  • Conceptual MCQs: which model cannot represent LLE, which models are local-composition, Gibbs-Duhem consistency of given ln γ expressions.
  • Practise deriving ln γ₁, ln γ₂ from a short G^E/RT expression.

Quick check

  1. For G^E/RT = 1.5x₁x₂, what is γ₁ at x₁ = 0.5?
  2. Which of Margules, van Laar, Wilson and NRTL cannot predict liquid-liquid immiscibility?
  3. In the three-suffix Margules model, what does A₂₁ equal?
  4. At a low-pressure azeotrope, how is γᵢ obtained from P and Pᵢ^sat?
  5. Name the two local-composition models in this topic.

Answers: 1. e^0.375 = 1.455. 2. Wilson. 3. ln γ₂^∞. 4. γᵢ = P/Pᵢ^sat. 5. Wilson and NRTL.

Interview questions

All Chemical Engineering Thermodynamics interview questions

Try answering each one aloud before you open it.

  1. 1.What is an activity coefficient and why is it important in chemical engineering thermodynamics?Concept

    An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture. It is important because it helps in predicting the behavior of real solutions, which often deviate from ideal solutions due to interactions between molecules. Understanding activity coefficients allows engineers to design processes more accurately, especially in separation processes like distillation.

  2. 2.Explain the Margules activity coefficient model.Concept

    Margules writes G^E/RT as a polynomial in mole fraction. The two-suffix form G^E/RT = A·x₁x₂ is symmetric, giving ln γ₁ = A·x₂² and ln γ₂ = A·x₁². The three-suffix form G^E/RT = x₁x₂(A₂₁x₁ + A₁₂x₂) allows asymmetry, with ln γ₁ = x₂²[A₁₂ + 2(A₂₁ − A₁₂)x₁]. Its parameters are directly the logarithms of the infinite-dilution activity coefficients, ln γ₁^∞ = A₁₂ and ln γ₂^∞ = A₂₁.

  3. 3.Describe the van Laar activity coefficient model and its applications.Concept

    Van Laar uses G^E/RT = A′₁₂A′₂₁x₁x₂/(A′₁₂x₁ + A′₂₁x₂), giving ln γ₁ = A′₁₂(1 + A′₁₂x₁/(A′₂₁x₂))^(−2) and the symmetric expression for γ₂. It was derived from the van der Waals equation but is used today as an empirical two-parameter fit, with ln γᵢ^∞ equal to its parameters. It is convenient for fitting from a single azeotrope or infinite-dilution data, but cannot represent extrema in ln γ or parameters of opposite sign.

  4. 4.What is the Wilson activity coefficient model and how does it differ from the Margules model?Concept

    Wilson is a local-composition model: it assumes the composition around a molecule differs from the bulk because of different interaction energies, and uses two parameters Λ₁₂ and Λ₂₁ that include liquid molar volumes. Unlike Margules, which is a purely empirical polynomial, it extends to multicomponent mixtures using only binary parameters and fits strongly non-ideal miscible systems such as alcohol-hydrocarbon mixtures well. Its main limitation is that it cannot predict liquid-liquid phase splitting.

  5. 5.Explain the NRTL (Non-Random Two-Liquid) model and its significance.Concept

    The NRTL model is a semi-empirical model used to describe the non-ideal behavior of liquid mixtures. It accounts for non-randomness in the distribution of molecules and uses parameters to describe the interaction between different components. The model is significant because it can accurately predict phase equilibria in systems with strong interactions, making it useful for designing separation processes.

  6. 6.Why is the NRTL model preferred over the Margules model in some cases?Application

    The NRTL model is preferred over the Margules model in cases where the mixture exhibits strong non-ideal behavior due to specific interactions between components. The NRTL model accounts for non-randomness in molecular distribution, which can provide more accurate predictions for phase equilibria in such systems. This makes it more suitable for complex mixtures where the Margules model may not be sufficient.

  7. 7.What happens if you use the van Laar model for a system with more than two components?Application

    Van Laar is essentially a binary model; its multicomponent extension is awkward and not consistent in the way local-composition models are, so predictions for ternary and higher systems built from binary van Laar constants are often poor. Wilson, NRTL and UNIQUAC are formulated so that multicomponent behaviour follows from binary parameters alone, which is why simulators use them for multicomponent VLE.

  8. 8.How would you determine the activity coefficients using the Wilson model for a binary mixture?Application

    Obtain Λ₁₂ and Λ₂₁ at the temperature of interest, from a data bank or by regressing VLE data. Then ln γ₁ = −ln(x₁ + x₂Λ₁₂) + x₂[Λ₁₂/(x₁ + x₂Λ₁₂) − Λ₂₁/(x₂ + x₁Λ₂₁)], and ln γ₂ follows by swapping indices with a sign change in the bracket. At infinite dilution ln γ₁^∞ = −ln Λ₁₂ + 1 − Λ₂₁, which is a quick check.

  9. 9.Calculate the activity coefficients for a binary mixture using the three-suffix Margules model with A12 = 1.5 and A21 = 0.5, at x1 = 0.4 and x2 = 0.6.Numerical

    Use ln γ₁ = x₂²[A₁₂ + 2(A₂₁ − A₁₂)x₁] and ln γ₂ = x₁²[A₂₁ + 2(A₁₂ − A₂₁)x₂]. ln γ₁ = 0.36 × [1.5 + 2(−1.0)(0.4)] = 0.36 × 0.7 = 0.252, so γ₁ = 1.287. ln γ₂ = 0.16 × [0.5 + 2(1.0)(0.6)] = 0.16 × 1.7 = 0.272, so γ₂ = 1.313. Check: x₁ ln γ₁ + x₂ ln γ₂ = 0.264 = x₁x₂(A₂₁x₁ + A₁₂x₂).

  10. 10.For a binary mixture, if the activity coefficient γ1 is found to be greater than 1 using the NRTL model, what does this indicate about the interactions in the mixture?Application

    If the activity coefficient γ1 is greater than 1, it indicates that the interactions between the molecules of component 1 and the other component are weaker than the interactions between the molecules of component 1 themselves. This suggests that component 1 is less stable in the mixture compared to its pure state, leading to positive deviations from Raoult's law.

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