2. 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.
3. 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).
4. 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
5. 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–Hückel
parameter
6. 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 xizi
2 /( x w M w ) ( kg mol -1 )
7. By proper differentiation of G E,D-H , the electrostatic
contributions to the activity coefficients are obtained.
For ions, this contribution is:
ln γi
D-H = zi
2AI1/2/(1+bI1/2)
γi
D-H : an unsymmetric mole fraction activity coefficient
The corresponding term for water is:
γw
D-H = Mw2A[1+bI1/2-(1+bI1/2)-1- 2ln(1+bI1/2)] / b3
8. 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.
9. The combinatorial contribution to the activity
coefficient of component i is:
lnγi
C = ln(фi/xi) + 1- фi/xi - 5.0 qi [ ln (фi/θi) +1 - фi/θi
γi
C = 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 θj ψji )
The residual, enthalpic term is dependent on temperature
through the parameter ψji .
10. 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 = uji
0 + uji
t ( T – 298.15 )
The two parameters uji
0 and uji
t are adjustable
parameters. The value of these parameters can be
determined from experimental data.
11. By differentiation of G E,R the residual contribution
to the activity coefficient is obtained:
lnγi
R = qi[ 1- ln(∑k θk ψki) - ∑j ( θj ψji / ∑k θk ψkj )
γi
R = symmetric activity coefficient
12. 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 uji
0 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.
13. 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) µ0
Na2 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.
14. 2- µ w = µ0
w + 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 ]=
(µ0
Na2 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.
15. 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)
16. 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:
µig
SO2 + 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.
17. 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 :
18. µ*Na+ + µ*Cl- + RT ln(xI
Na+ γ*,I
Na+ xI
Cl- γ*,I
Cl-) =
µ*Na+ + µ*Cl- + RT ln(xII
Na+ γ*,II
Na+ xII
Cl- γ*,II
Cl-)
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:
xI
Na+ γ*,I
Na+ xI
Cl- γ*,I
Cl- = xII
Na+ γ*,II
Na+ xII
Cl- γ*,II
Cl-
A similar equilibrium equation is written for each of the
other two components.
19. 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.
