Engineering & Science
NRTL (Non-Random Two-Liquid) VLE & LLE Activity Coefficients
Calculate Henri Renon and John Prausnitz’s 3-parameter NRTL activity coefficients gamma_1 and gamma_2 using energy parameters tau_12, tau_21 and non-randomness factor alpha_12 for partially miscible liquid-liquid and azeotropic VLE mixtures.
Why NRTL handles liquid-liquid phase splitting where Wilson’s equation fails!
Wilson’s equation cannot predict immiscible two-liquid phase splitting (LLE, such as butanol-water or ethyl-acetate-water decanters) because its second derivative of Gibbs mixing energy never turns negative! Henri Renon and John M. Prausnitz (AIChE Journal 1968) fused Scott’s two-liquid cell theory with a non-randomness parameter alpha_12 (typically 0.20–0.47) to create the Non-Random Two-Liquid (NRTL) equation—the default thermodynamic workhorse for polar/aqueous extraction and heterogeneous azeotropic distillation!
Defining Boltzmann factors G_12 = EXP(-alpha_12 * tau_12) and G_21 = EXP(-alpha_12 * tau_21) with x_2 = 1 - x_1 gives exact activity coefficients: ln(gamma_1) = x_2^2 * (tau_21 * (G_21 / (x_1 + x_2 * G_21))^2 + (tau_12 * G_12) / (x_2 + x_1 * G_12)^2) and symmetrically for gamma_2! For an aqueous-organic binary (tau_12 = 1.35, tau_21 = 0.65, alpha_12 = 0.30) at x_1 = 0.25, gamma_1 = 2.5393 and gamma_2 = 1.0932, climbing to gamma_1^inf = 4.7134 at infinite dilution x_1 = 0!
| Liquid Mole Fraction x1 (tau12 = 1.35, tau21 = 0.65, alpha = 0.30) | NRTL Activity gamma1 | NRTL Activity gamma2 | Phase Equilibrium Behavior |
|---|---|---|---|
| Infinite Dilution of 1 (x1 = 0.00) | 4.7134 | 1.0000 | High Organic Stripping Factor |
| Operating Tray / Decanter Feed (x1 = 0.25) | 2.5393 | 1.0932 | Strong Positive Deviation (VLE/LLE) |
| Invalid Composition (x1 < 0 or x1 > 1) | Mole-Fraction Guard | "Invalid NRTL Inputs" | Requires 0 <= x1 <= 1, alpha >= 0 |
NRTL Component 1 Activity Coefficient gamma_1
Evaluates Renon-Prausnitz gamma_1 directly from x_1 (B2), tau_12 (B3), tau_21 (B4), and alpha_12 (B5).
=IF(OR(B2 < 0, B2 > 1, B5 < 0), "Invalid NRTL Inputs", ROUND(EXP(((1 - B2) ^ 2) * (B4 * ((EXP(-B5 * B4) / (B2 + (1 - B2) * EXP(-B5 * B4))) ^ 2) + (B3 * EXP(-B5 * B3)) / (((1 - B2) + B2 * EXP(-B5 * B3)) ^ 2))), 4))NRTL Component 2 Activity Coefficient gamma_2
Evaluates symmetric Renon-Prausnitz gamma_2 obeying exact Gibbs-Duhem consistency.
=IF(OR(B2 < 0, B2 > 1, B5 < 0), "Invalid NRTL Inputs", ROUND(EXP((B2 ^ 2) * (B3 * ((EXP(-B5 * B3) / ((1 - B2) + B2 * EXP(-B5 * B3))) ^ 2) + (B4 * EXP(-B5 * B4)) / ((B2 + (1 - B2) * EXP(-B5 * B4)) ^ 2))), 4))Try the example
Use the original values or change them to test your understanding.