UNIFAC (Original) Activity Coefficient Model: DETAILS

The following expressions define the terms leading to the equation for G^EX.

φ₁ = x₁ ×r₁/(x₁ ×r₁ + x₂ ×r₂); and φ₂ = x₂ ×r₂/(x₁ ×r₁ + x₂ ×r₂)
θ₁ = x₁ ×q₁/(x₁ ×r₁ + x₂ ×q₂); and θ₂ = x₂ ×q₂/(x₁ ×r₁ + x₂ ×q₂)
ln(γ_1c) = ln(φ₁/x₁) + 5×q₁ ×ln(θ₁/φ₁) + l₁ - φ₁/x₁ ×(x₁ ×l₁ + x₂ ×l₂)
ln(γ_2c) = ln(φ₂/x₂) + 5×q₂ ×ln(θ₂/φ₂) + l₂ - φ₂/x₂ ×(x₁ ×l₁ + x₂ ×l₂)
x_1,i = n_1,i / Σn_1,i, where the summation is from i = 1 to N.
θ_1,i = x_1,i ×Q_type(i)/Σx_1,i ×Q_type(i), where the summation is from i = 1 to N.

x_2,i and θ_2,i are calculated analogously, as are x_M,i and θ_M,i for the mixture. The mixture is indicated by the subscript M.

Ψ_ij = exp[-UNIFAC Parameter{MainGroup(i),MainGroup(j)}/T]
ln(γ_1,k) = Q_type(k) ×[1 - Σθ_1,i ×Ψ_ki /Σθ_1,j ×Ψ_ji - lnΣθ_1,j ×Ψ_ki], where the summations are from i = 1 to N, j = 1 to N, and i = 1 to N, respectively.

Similarly, γ_2,k and γ_M,k vectors are calculated for component 2 and the mixture. γ_M,k is based on the mole fractions of structural groups in the mixture:

x_M,i = (x₁ ×n_1,i + x₂ ×n_2,i) / Σ(x₁ ×n_1,i + x₂ ×n_2,i), where the summation is from i = 1 to N, N is the dimension of the vector of distinct structural group types in the molecules, and n₁ and n₂ are vectors representing the number of structural groups of each type.

ln(γ₁) = ln(γ_1,c) + Σn_1,i ×{lnγ_i) - ln(γ_1,i)}
ln(γ₂) = ln(γ_2,c) + Σn_2,i ×{ln(γ_i) - ln(γ_2,i)}
G^EX / (R ×T) = x₁ ×ln(γ₁) + x₂ ×ln(γ₂)