UNIQUAC

Equations

In the UNIQUAC model the activity coefficients of the i^th component of a two component mixture are described by a combinatorial and a residual contribution.

ln ⁡ γ i = ln ⁡ γ i C + ln ⁡ γ i R

The first is an entropic term quantifying the deviation from ideal solubility as a result of differences in molecule shape. The latter is an enthalpic correction caused by the change in interacting forces between different molecules upon mixing.

Combinatorial contribution

The combinatorial contribution accounts for shape differences between molecules and affects the entropy of the mixture and is based on the lattice theory. The excess entropy γ^C is calculated exclusively from the pure chemical parameters, using the relative Van der Waals volumes r_i and surface areas q_i of the pure chemicals.

ln ⁡ γ i C = ( 1 − V i + ln ⁡ V i ) − z 2 q i ( 1 − V i F i + ln ⁡ V i F i )

With the volume fraction per mixture mole fraction, V_i, for the i^th component given by:
V i = r i ∑ j r j x j
And the surface area fraction per mixture molar fraction, F_i, for the i^th component given by:
F i = q i ∑ j q j x j
The first 3 terms on the right hand side of the combinatorial term form the Flory-Huggins contribution, while the left terms, the Guggenhem-Staverman correction, reduce this because a connecting segments cannot be placed in all direction in space. This spatial correction shifts the result of the Flory-Huggins term about 5% towards an ideal solution. The coordination number, z, i.e. the number of close interacting molecules around a central molecule, is frequently set to 10. It can be regarded as an average value that lies between cubic (z=6) and hexagonal packing (z=12) of molecules that are simplified by spheres.

In the limit of infinite dilution and a binary mixture the equations for the combinatorial contribution reduces to:

{ ln ⁡ γ 1 C , ∞ = 1 − r 1 r 2 + ln ⁡ r 1 r 2 − z 2 q 1 ( 1 − r 1 q 2 r 2 q 1 + ln ⁡ r 1 q 2 r 2 q 1 ) ln ⁡ γ 2 C , ∞ = 1 − r 2 r 1 + ln ⁡ r 2 r 1 − z 2 q 2 ( 1 − r 2 q 1 r 1 q 2 + ln ⁡ r 2 q 1 r 1 q 2 )

This pair of equation shows that molecules of same shape, i.e. same r and q parameters, have γ 1 C , ∞ = γ 2 C , ∞ = 1

Residual contribution

The residual, enthalpic term contains an empirical parameter, τ i j , which is determined from the binary interaction energy parameters. The expression for the residual activity coefficient for molecule i is:

ln ⁡ γ i R = q i ( 1 − ln ⁡ ∑ j q j x j τ j i ∑ j q j x j − ∑ j q j x j τ i j ∑ k q k x k τ k j )

with

τ i j = e − Δ u i j / R T

Δu_ij [J/mol] is the binary interaction energy parameter. Theory defines Δu_ij = u_ij – u_ii, and Δu_ji = u_ji – u_jj, where u_ij is the interaction energy between molecules i and j. The interaction energy parameters are usually determined from activity coefficients, vapor-liquid, liquid-liquid, or liquid-solid equilibrium data.

Usually Δu_ij ≠ Δu_ji, because the energies of evaporation (i.e. u_ii), are in many cases different, while the energy of interaction between molecule i and j is symmetric, and therefore u_ij=u_ji. If the interactions between the j molecules and i molecules is the same as between molecules i and j, than mixing has no excess energy effect upon mixing, Δu_ij=Δu_ji=0. And thus γ i R = 1

Alternatively, in some process simulation software τ i j can be expressed as follows :

ln ⁡ τ i j = A i j + B i j / T + C i j ln ⁡ ( T ) + D i j T + E i j / T 2 .

The "C", "D", and "E" coefficients are primarily used in fitting liquid–liquid equilibria (with "D" and "E" rarely used at that). The "C" coefficient is useful in vapor-liquid equilibria as well. The use of such an expression ignores the fact that on a molecular level the energy, Δu_ij, is temperature independent. It is a correction to repair the simplifications, which were applied in the derivation of the model.

Applications

Activity coefficients can be used to predict simple phase equilibria (vapour–liquid, liquid–liquid, solid–liquid), or to estimate other physical properties (e.g. viscosity of mixtures). Models such as UNIQUAC allow chemical engineers to predict the phase behavior of multicomponent chemical mixtures. They are commonly used in process simulation programs to calculate the mass balance in and around separation units.

Parameters

UNIQUAC requires two basic underlying parameters:

1. Relative surface and volume fractions are chemical constants, which must be known for all chemicals.
2. An empirical parameter between components that describes the intermolecular behaviour. This parameter must be known for all binary pairs in the mixture. In a quaternary mixture there are six such parameters (1–2,1–3,1–4,2–3,2–4,3–4) and the number rapidly increases with additional chemical components. The empirical parameters are derived from experimental activity coefficients, or from phase diagrams, from which the activity coefficients themselves can be calculated. An alternative is to obtain activity coefficients with a method such as UNIFAC, and the UNIFAC parameters can then be simplified by fitting to obtain the UNIQUAC parameters. This method allows for the more rapid calculation of activity coefficients, rather than direct usage of the more complex method.

Newer developments

UNIQUAC has been extended by several research groups. Some selected derivatives are: