Engineering & Science
Wilson Local-Composition Binary VLE Activity Coefficients (gamma1 & gamma2)
Calculate George M. Wilson’s local-composition activity coefficients gamma_1 and gamma_2 for strongly non-ideal polar/alcohol-hydrocarbon distillation systems using binary interaction parameters Lambda_12 and Lambda_21.
Model extreme polar-nonpolar VLE skew accurately with local molecular compositions!
George M. Wilson’s landmark 1964 JACS paper launched modern local-composition thermodynamics (inspiring NRTL and UNIQUAC) by recognizing that around a central hydrogen-bonding ethanol molecule in hexane, local ethanol concentration far exceeds bulk mole fraction x_1! Defining positive binary Boltzmann-volume ratios Lambda_12 > 0 and Lambda_21 > 0 gives excess Gibbs energy G^E / (R * T) = -x_1 * LN(x_1 + Lambda_12 * x_2) - x_2 * LN(x_2 + Lambda_21 * x_1)!
Differentiating yields exact binary activity coefficients across 0 <= x_1 <= 1 (with x_2 = 1 - x_1): ln(gamma_1) = -LN(x_1 + Lambda_12 * x_2) + x_2 * ((Lambda_12 / (x_1 + Lambda_12 * x_2)) - (Lambda_21 / (Lambda_21 * x_1 + x_2))) and symmetrically for gamma_2! For an alcohol-hydrocarbon mixture (Lambda_12 = 0.32, Lambda_21 = 0.78) at x_1 = 0.35, gamma_1 = 1.5021 and gamma_2 = 1.1915, rising sharply at infinite dilution x_1 = 0 to gamma_1^inf = 3.8940!
| Liquid Mole Fraction x1 (Lambda12 = 0.32, Lambda21 = 0.78) | Wilson Activity gamma1 | Wilson Activity gamma2 | Distillation VLE Regime |
|---|---|---|---|
| Infinite Dilution of 1 (x1 = 0.00) | 3.8940 | 1.0000 | Steep Dilute Stripping Volatility |
| Feed Tray Composition (x1 = 0.35) | 1.5021 | 1.1915 | Homogeneous Non-Ideal Binary |
| Invalid Parameters (Lambda <= 0 or x1 outside 0-1) | Thermodynamic Guard | "Invalid Wilson Inputs" | Requires 0 <= x1 <= 1, Lambda > 0 |
Wilson Local-Composition Activity Coefficient gamma_1
Evaluates EXP(-LN(x_1 + L_12*(1-x_1)) + (1-x_1)*(L_12/(x_1 + L_12*(1-x_1)) - L_21/(L_21*x_1 + (1-x_1)))).
=IF(OR(B2 < 0, B2 > 1, B3 <= 0, B4 <= 0), "Invalid Wilson Inputs", ROUND(EXP(-LN(B2 + B3 * (1 - B2)) + (1 - B2) * ((B3 / (B2 + B3 * (1 - B2))) - (B4 / (B4 * B2 + (1 - B2))))), 4))Wilson Local-Composition Activity Coefficient gamma_2
Evaluates EXP(-LN((1-x_1) + L_21*x_1) - x_1*(L_12/(x_1 + L_12*(1-x_1)) - L_21/(L_21*x_1 + (1-x_1)))).
=IF(OR(B2 < 0, B2 > 1, B3 <= 0, B4 <= 0), "Invalid Wilson Inputs", ROUND(EXP(-LN((1 - B2) + B4 * B2) - B2 * ((B3 / (B2 + B3 * (1 - B2))) - (B4 / (B4 * B2 + (1 - B2))))), 4))Try the example
Use the original values or change them to test your understanding.