Van Laar Activity Coefficients (gamma1 & gamma2) for Azeotropic Distillation

Fit strongly skewed alcohol-water azeotropes better than polynomial Margules curves!

When two liquid components differ substantially in molecular size and hydrogen-bonding polarity—such as Ethanol (1) + Water (2) or Isopropanol (1) + Water (2)—excess Gibbs energy is sharply skewed toward the dilute organic end, where polynomial Margules expansions can create spurious oscillations! Johannes van Laar derived a rational excess-Gibbs function from Van der Waals mixing rules whose infinite-dilution intercepts A_12 = ln(gamma_1^inf) and A_21 = ln(gamma_2^inf) remain smooth and monotonic across the entire composition range!

For interior liquid compositions 0 < x_1 < 1 (with boundary limits gamma_1(0) = EXP(A_12), gamma_1(1) = 1), Van Laar’s equations read: ln(gamma_1) = A_12 / (1 + (A_12 * x_1) / (A_21 * (1 - x_1)))^2 and ln(gamma_2) = A_21 / (1 + (A_21 * (1 - x_1)) / (A_12 * x_1))^2 (which algebraically clears 0/0 boundary division as A_12 * (A_21 * x_2)^2 / (A_12 * x_1 + A_21 * x_2)^2 so even x_1 = 0 and x_1 = 1 evaluate with zero division errors!)! For Ethanol-Water near atmospheric boiling (A_12 = 1.62, A_21 = 1.18) at x_1 = 0.25, gamma_1 = 2.1436 and gamma_2 = 1.1233!

Singularity-Free Van Laar Activity Coefficient gamma_1 (Valid Including x_1 = 0 and 1)

Uses the cleared denominator form EXP((A_12 * (A_21 * (1 - x_1))^2) / (A_12 * x_1 + A_21 * (1 - x_1))^2) so x_1 = 0 and x_1 = 1 never divide by zero!

=IF(OR(B2 < 0, B2 > 1, B3 = 0, B4 = 0), "Invalid Van Laar Inputs", ROUND(EXP((B3 * ((B4 * (1 - B2)) ^ 2)) / ((B3 * B2 + B4 * (1 - B2)) ^ 2)), 4))

Singularity-Free Van Laar Activity Coefficient gamma_2

Evaluates EXP((A_21 * (A_12 * x_1)^2) / (A_12 * x_1 + A_21 * (1 - x_1))^2) symmetrically.

=IF(OR(B2 < 0, B2 > 1, B3 = 0, B4 = 0), "Invalid Van Laar Inputs", ROUND(EXP((B4 * ((B3 * B2) ^ 2)) / ((B3 * B2 + B4 * (1 - B2)) ^ 2)), 4))

Further reading

• Microsoft Support — EXP function