Engineering & Science
UNIQUAC Combinatorial Activity Coefficient (Staverman-Guggenheim VLE)
Calculate the entropic size-and-shape UNIQUAC combinatorial activity coefficient γ_1^C from van der Waals volume (r_i) and surface-area (q_i) parameters in asymmetric solvent-heavy hydrocarbon or oligomer mixtures.
Why mixing small acetone molecules into long alkane chains breaks Raoult’s Law even at ΔH_mix = 0!
When molecules of starkly different sizes mix—such as small acetone (van der Waals volume r_1 = 2.57, surface area q_1 = 2.34) dissolved in long-chain n-hexadecane (r_2 = 11.24, q_2 = 9.26)—configurational entropy of mixing deviates strongly from ideal mole fractions x_i even before energetic interactions are counted! Denis Abrams and John M. Prausnitz (AIChE Journal, 1975) built the universal UNIQUAC and UNIFAC models atop Staverman-Guggenheim lattice statistics using volume fraction Φ_1 = x_1 r_1 / Σ(x_j r_j), surface fraction θ_1 = x_1 q_1 / Σ(x_j q_j), and bulkiness factor l_i = (z/2)(r_i - q_i) - (r_i - 1) (with coordination z = 10)!
In ln(γ_1^C) = ln(Φ_1 / x_1) + 5 * q_1 * ln(θ_1 / Φ_1) + l_1 - (r_1 / Σ x_j r_j) * (x_1 l_1 + x_2 l_2), size asymmetry alone drives γ_1^C < 1! At x_1 = 0.20 acetone in 80% hexadecane, acetone’s volume fraction is just Φ_1 = 0.0541, yielding ln(γ_1^C) = -0.5279 and combinatorial activity coefficient γ_1^C = 0.5898 (whereas equal-sized molecules r_1 = r_2, q_1 = q_2 collapse identically to ln γ_1^C = 0, γ_1^C = 1.0000)!
| Binary Liquid System at x1 = 0.20 (z = 10) | Volume Fraction Φ1 | Combinatorial ln(γ1^C) | Activity Coeff γ1^C |
|---|---|---|---|
| Acetone (r1=2.57, q1=2.34) + Hexadecane (r2=11.24, q2=9.26) | 0.0541 | -0.5279 | 0.5898 (< 1 Negative Dev.) |
| Symmetric Size Limit (r2 = 2.57, q2 = 2.34) | 0.2000 (= x1) | 0.0000 | 1.0000 (Ideal Entropy) |
| Boundary Mole Fraction (x1 <= 0 or x1 >= 1) | Pure Component Guard | "Invalid Mole Fraction" | Requires 0 < x1 < 1 |
UNIQUAC Volume Fraction Φ_1 of Component 1
Computes x_1 * r_1 / (x_1 * r_1 + (1 - x_1) * r_2) from van der Waals segment volumes.
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B5 <= 0), "Invalid Mole Fraction", ROUND((B2 * B3) / (B2 * B3 + (1 - B2) * B5), 4))Staverman-Guggenheim Combinatorial Activity Coefficient γ_1^C (z = 10)
Evaluates EXP(ln(Φ_1/x_1) + 5*q_1*ln(θ_1/Φ_1) + l_1 - (r_1/r_bar)*(x_1*l_1 + x_2*l_2)) directly from x_1, r_1, q_1, r_2, q_2.
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Mole Fraction", ROUND(EXP(LN(B3 / (B2 * B3 + (1 - B2) * B5)) + 5 * B4 * LN((B4 / (B2 * B4 + (1 - B2) * B6)) / (B3 / (B2 * B3 + (1 - B2) * B5))) + (5 * (B3 - B4) - (B3 - 1)) - (B3 / (B2 * B3 + (1 - B2) * B5)) * (B2 * (5 * (B3 - B4) - (B3 - 1)) + (1 - B2) * (5 * (B5 - B6) - (B5 - 1)))), 4))Try the example
Use the original values or change them to test your understanding.