Modeling electrolyte

solutions with the extended

universal quasi-chemical

(UNIQUAC) model

UNIQUAC (short for UNIversal QUAsi-Chemical )

is an activity coefficient model used in description of

phase equilibria . The model is a so-called lattice

model.

The extended universal quasi-chemical

(UNIQUAC) model is a thermodynamic model for

solutions containing electrolytes and

nonelectrolytes. The model is a Gibbs excess

function consisting of a Debye–Huckel term and a

standard UNIQUAC term.

EXTENDED UNIQUAC MODEL :

In this work, the extended UNIQUAC model is applied for

modeling solutions containing electrolytes.

In 1986 , Sander et al. first introduced an extended

UNIQUAC model for electrolytes.

This model was later (1993) modified by Nicolaisen et al.

by replacing the modified UNIQUAC term used by

Sander et al. with a standard UNIQUAC term.

The current version of the extended UNIQUAC model

that is presented in this work was first presented by

Thomsen et al. (1997).

The extended UNIQUAC model

consists of three terms :

a combinatorial or entropic term.

a residual or enthalpic term.

and an electrostatic term.

G ex = G ex Combinatorial + G ex

Residual + G ex Extended Debye-Huckel

↓ ↓ ↓

G E,C G E,R G E,D-H

Model equations :

The Debye–Hückel contribution to the excess Gibbs energy of

the extended UNIQUAC model is:

G E,D-H / (RT) = - xw Mw 4A [ ln (1+ b I1/2) - b I1/2 + 0.5 b2 I ] / b3

G E = the molar excess Gibbs energy

xw = the mole fraction of water

Mw = the molar mass of water ( kg mol –1 )

b = a constant = 1.5 (kg mol –1) 1/2

I = ionic strength

A = the temperature- and pressure-dependent Debye–Hückelparameter

At the saturation pressure of water, the following equation

gives the temperature dependence of A at temperatures up to

500 K (T 0 is equal to 273.15 K) :

A= [ 1.131+1.335 x 10 -3 (T - T 0 ) +1.164 x 10 -5 (T - T 0)2 ]

( kg mol -1 ) ½

I is the ionic strength calculated as a function of

concentrations and the ionic charges z i :

I = 0.5 ∑i xizi2 /( x w M w ) ( kg mol -1 )

By proper differentiation of G E,D-H , the electrostatic

contributions to the activity coefficients are obtained.

For ions, this contribution is:

ln γiD-H = zi

2AI1/2/(1+bI1/2)

γiD-H : an unsymmetric mole fraction activity coefficient

The corresponding term for water is:

γwD-H = Mw2A[1+bI1/2-(1+bI1/2)-1- 2ln(1+bI1/2)] / b3

The UNIQUAC contribution to the excess Gibbs energy consists

of a combinatorial part and a residual part.

The combinatorial part is marked by superscript C and given by:

G E,C / (RT) = ∑iln (фi / xi) – 5.0 ∑i qi xi ln (фi / θi)

The combinatorial, entropic term is independent of temperature and

only depends on the relative sizes of the species.

xi is the mole fraction, φi is the volume fraction, and θi is the surface area fraction of component i:

фi = xiri / ∑jxjrj θi = xiqi/∑jxjqj

The volume parameter ri and the surface area parameter qi are

treated as adjustable parameters in this work.

The combinatorial contribution to the activity

coefficient of component i is:

lnγiC = ln(фi/xi) + 1- фi/xi - 5.0 qi [ ln (фi/θi) +1 - фi/θi

γiC = symmetric activity coefficient

The residual part of the excess Gibbs function is

marked by superscript R and given by:

G E,R / (RT) = - ∑i xiqi ln (∑j θjψji )

The residual, enthalpic term is dependent on temperature

through the parameter ψji .

ψji is defined by the equation:

ψji = exp [ - ( uji – uii )/ T ]

The interaction energy parameters uji and uii are

independent of composition, but are temperature-

dependent:

uji = uji0 + uji

t ( T – 298.15 )

The two parameters uji0 and uji

t are adjustable parameters. The value of these parameters can be determined from experimental data.

By differentiation of G E,R the residual contribution to the activity coefficient is obtained:

lnγiR = qi[ 1- ln(∑k θk ψki) - ∑j ( θj ψji / ∑k θk ψkj )

γiR = symmetric activity coefficient

Model parameters :The parameters needed in order to perform calculations with the extended UNIQUAC model are the Debye–Hückel A parameter, which is given in eq. 3th as a function of temperature. The Debye–Hückel b parameter is given the constant value 1.5 (kg mol –1) 1/2 .

The only unknown parameters in the model are:

• UNIQUAC volume and surface area parameters ri and qi for each species, and

• UNIQUAC interaction energy parameters uji0 and uji

t for each pair of interacting species.ri and qi parameters assigned to water by the authors of the

UNIQUAC model were retained. All other parameters were determined on the basis of experimental data from the IVC-SEP databank for electrolyte solutions.

Solid–liquid equilibrium :

For equilibrium between crystalline glauber salt

[Na2 SO4 .10 H2O (c)] and an aqueous solution containing

sodium sulfate, it is required that the chemical potential of 2

mol sodium ions plus the chemical potential of 1 mol sulfate ions

and 10 mol water is identical to the chemical potential of 1 mol

crystalline glauber salt. The equilibrium condition for this solid–

liquid equilibrium can be expressed as:

1) µ0Na2 SO4 .10 H2O = 2 µNa++ µSO42- +10 µ H2O

Superscript 0 on the chemical potential of glauber salt indicates

that this is the chemical potential of a pure, crystalline phase.

2- µ w = µ0w + RT ln (xwγi)

3- µ i = µ*i + RT ln (xiγi*)

By using eqs. 2 & 3 , eq. 1 can be written as:

4) Ln [ (x Na+ γ*Na+ )2 x SO42- γ*SO42- (x w γ*w)10 ]=

(µ0Na2 SO4 .10 H2O - 2 µ*Na+ + µ*SO42- - 10 µ0

w) / RT

The right-hand side of eq. 4 can be calculated from the

tabulated values of the standard-state chemical

potentials. The concentrations on the left-hand side of

eq. 4 can then be adjusted by iteration

until the activity product yields the desired value.

Vapor–liquid equilibrium:

Equilibrium between volatile components in the gas

phase and in the liquid phase requires that the

chemical potentials of these volatile components are

identical in the two phases.

For equilibrium to exist between sulfur dioxide in the

gas phase and in an aqueous phase, it is required that

the chemical potential of sulfur dioxide is identical in

the two phases:

µSO2(g) = µSO2(aq)

The chemical potential of SO2 in the gas phase can be

expressed as an ideal gas chemical potential (superscript ig)

plus a term that varies with fugacity. Similarly, the chemical

potential of SO2 in the aqueous phase can be expressed in

terms of the standard-state chemical potential of solutes and

the activity coefficient:

µigSO2 + RTln (ySO2φ SO2 P) = µ*SO2 + RTln (xSO2 γ*SO2)

This is the so-called gamma-phi approach to vapor–liquid

equilibrium calculation.

φ SO2 is the fugacity coefficient of SO2 in the vapor phase,

ySO2 is the corresponding mole fraction.

Liquid–liquid equilibrium :

For liquid–liquid equilibrium to occur, the chemical potential of each independent component must be the same in both phases. In this connection, an independent component is a neutral species. For liquid– liquid equilibrium in a system consisting of NaCl, water, and iso-propanol, NaCl has to be considered an independent component. One equation can be written for the equilibrium of each of the three independent components between liquid phase I and liquid phase II. The

equation for NaCl can, byusing eq. µ i = µ*i + RT ln (xiγi*)

be expressed as :

µ*Na+ + µ*Cl- + RT ln(xINa+ γ*,I

Na+ xI

Cl-γ*,ICl-) =

µ*Na+ + µ*Cl- + RT ln(xIINa+ γ*,II

Na+ xII

Cl-γ*,IICl-)

Owing to the choice of standard states, the standard

chemical potentials cancel each other, and the

condition for equilibrium between the two phases for NaCl

is simplified to:

xINa+ γ*,I

Na+ xI

Cl-γ*,ICl- = xII

Na+ γ*,IINa+ x

IICl-γ*,II

Cl-

A similar equilibrium equation is written for each of the

other two components.

CONCLUSION:The extended UNIQUAC model is a very simple thermodynamic model for electrolytes. Yet it is able

to describe solid–liquid, liquid–liquid, and vapor–liquid equilibria using one set of parameters.

In addition,

thermal properties such as the heat of dilution and the heat capacities of electrolyte solutions are

calculated quite accurately by the model.

Thanks for your attention