Engineering & Science
Margules Non-Ideal Liquid Activity Coefficients (gamma1 & gamma2 VLE)
Calculate asymmetric binary vapor-liquid equilibrium (VLE) activity coefficients gamma_1 = EXP((1 - x_1)^2 * (A_12 + 2*(A_21 - A_12)*x_1)) and gamma_2 = EXP(x_1^2 * (A_21 + 2*(A_12 - A_21)*(1 - x_1))) for azeotropic distillation columns.
Capture positive & negative Raoult’s Law deviations in non-ideal binary distillation!
Ideal Raoult’s Law (y_i * P = x_i * P_i^sat) assumes identical intermolecular forces between molecules A and B—failing completely for polar/nonpolar mixtures like acetone-chloroform, ethanol-water, or ethyl-acetate/water that form minimum- or maximum-boiling azeotropes! Max Margules (1895) expanded excess molar Gibbs energy G^E / (R * T) = x_1 * x_2 * (A_21 * x_1 + A_12 * x_2) so infinite-dilution limits ln(gamma_1^inf) = A_12 (at x_1 -> 0) and ln(gamma_2^inf) = A_21 (at x_1 -> 1) match experimental VLE data!
Applying the Gibbs-Duhem derivative yields exact thermodynamic consistency across all mole fractions 0 <= x_1 <= 1 (with x_2 = 1 - x_1): gamma_1 = EXP((1 - x_1)^2 * (A_12 + 2 * (A_21 - A_12) * x_1)) and gamma_2 = EXP(x_1^2 * (A_21 + 2 * (A_12 - A_21) * (1 - x_1)))! For a positively deviating polar-organic binary (A_12 = 1.45, A_21 = 0.95) at liquid mole fraction x_1 = 0.30, gamma_1 = 1.7568 (boosting component 1 volatility 75.7% above Raoult’s Law!) while gamma_2 = 1.1601; and at infinite dilution x_1 = 0, gamma_1 peaks at EXP(1.45) = 4.2631!
| Liquid Mole Fraction x1 (A12 = 1.45, A21 = 0.95) | Activity Coefficient gamma1 | Activity Coefficient gamma2 | Thermodynamic VLE Limit |
|---|---|---|---|
| Infinite Dilution of 1 (x1 = 0.00) | 4.2631 (= EXP(A12)) | 1.0000 (Pure Component 2) | Henry’s Law Infinite Dilution |
| Operating Tray Composition (x1 = 0.30) | 1.7568 (> 1 Positive Dev.) | 1.1601 | Enhanced Relative Volatility |
| Pure Component 1 (x1 = 1.00) | 1.0000 (Raoult’s Limit) | 2.5857 (= EXP(A21)) | Gibbs-Duhem Boundary Verified |
Asymmetric Margules Activity Coefficient gamma_1 for Component 1
Evaluates EXP((1 - x_1)^2 * (A_12 + 2 * (A_21 - A_12) * x_1)) across valid mole fractions 0 <= x_1 <= 1.
=IF(OR(B2 < 0, B2 > 1), "Invalid Mole Fraction (0-1)", ROUND(EXP(((1 - B2) ^ 2) * (B3 + 2 * (B4 - B3) * B2)), 4))Asymmetric Margules Activity Coefficient gamma_2 for Component 2
Evaluates EXP(x_1^2 * (A_21 + 2 * (A_12 - A_21) * (1 - x_1))) obeying exact Gibbs-Duhem symmetry.
=IF(OR(B2 < 0, B2 > 1), "Invalid Mole Fraction (0-1)", ROUND(EXP((B2 ^ 2) * (B4 + 2 * (B3 - B4) * (1 - B2))), 4))Try the example
Use the original values or change them to test your understanding.