A modified electrolyte-Uniquac model for computing the activity coefficient and phase diagrams of electrolytes systems

Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

Contents lists available at ScienceDirect

Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad

A modified electrolyte-Uniquac model for computing the activity coefficient and phase diagrams of electrolytes systems B. Messnaoui a,∗ , S. Ouiazzane b , A. Bouhaouss c , T. Bounahmidi b a Université Mohammed V – Agdal – Rabat, Maroc b Laboratoire d’Analyse et Synthèse des Procédés Industriels, Ecole Mohammadia d’Ingénieurs, Université Mohammed V – Agdal – Rabat, Maroc c Laboratoire de Chimie Physique Générale I des matériaux, Nanomatériaux et Environnement, Université Mohammed V, Faculté des Sciences, Av. Ibn Battouta, Rabat, Maroc

article

info

Article history: Received 10 December 2007 Received in revised form 9 April 2008 Accepted 13 April 2008 Available online 8 May 2008 Keywords: Activity coefficient Osmotic coefficient Solubility Modelling Electrolyte solutions

a b s t r a c t A modified electrolyte-Uniquac model describing the behaviour of single and multicomponent electrolyte systems was developed. The thermodynamic properties of electrolyte solutions are considered to be the sum of two contributions: the long-range represented by Debye–Hückel contributions and the localcomposition expression (modified Uniquac type) that have been applied to account for short-range contributions. The local mole fractions are calculated based on the assumptions proposed by Chen and coworkers. The model has been tested on 86 aqueous electrolyte solutions at 298.15 K and results have been compared with those obtained from the Pitzer and electrolyte-NRTL models. The modified electrolyte-Uniquac model led to the accurate calculation of individual activity coefficient of ion. This new model can also be used to predict the excess properties and the salt solubility in aqueous multielectrolyte solutions with the interaction parameters obtained from binary data. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The computer-aided simulation and design of many industrial processes involving electrolytes require suitable models to represent and predict the thermodynamic and transport properties of electrolyte solutions. The correlation and prediction of phase equilibrium data are of major importance in the design of equipment for separation processes in the chemical industry. Much work has been done in the development of thermodynamic models for this purpose. For nonelectrolyte mixtures at ordinary temperatures and pressures reliable methods are available today. For mixtures with electrolytes, useful methods for practical phase equilibrium calculations have appeared in the last few decades, and progress has also been made in the development of statistical mechanical theories for electrolyte solutions, and Pitzer [1], Maurer [2] and Renon [3] give excellent reviews of the subject. Good models for the description of the thermodynamic properties of electrolyte systems are therefore highly needed. In recent years, different models have been proposed in the literature. All those models try to give a good representation of the behaviour of electrolytic solutions. The first of them is similar to the Bromley model [4] and is presented as an empirical expression while the others are based upon a theoretical background. The Debye–Hückel model [5] was the first model

∗ Corresponding author. Tel.: +212 70 91 36 89; fax: +212 37 77 88 53. E-mail address: [email protected] (B. Messnaoui). 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.04.003

employed to predict the phase behavior of aqueous electrolytes in solution, however, it can be used for very dilute solutions [2]. Guggenheim [6,7], based on earlier works of Bronstedt [8] suggested an empirical extension introducing a coefficient specific for the interactions between cation and anion. Bromley [4] suggested a one-parameter equation found empirically by fitting to experimental data. Using the pressure equation of statistical thermodynamics, Pitzer [9,10] pointed out that the interaction coefficient is a function of the ionic strength. Recently, different equations were proposed based on a local-composition model such as Uniquac, NRTL and Wilson models. Cruz and Renon [11] combined the Debye–Hückel expression with the NRTL model and a Born model contribution, taking into consideration the change of the dielectric constant with salt concentration. Four adjustable parameters are necessary to describe a system containing one solvent and one salt. A similar approach was suggested by Ball et al. [12]. In contrast to the model of Cruz and Renon, the calculation of the Born contribution was carried out in a less empirical way, which reduced the number of adjustable parameters to two for one binary system. Chen et al. [13,14] combined the electrostatic function of the Pitzer model with a local composition term, which is an extension of the NRTL equation to electrolyte solutions. The local mole fractions are calculated based on two assumptions: (1) repulsion of ions carrying charges of the same sign; (2) local electroneutrality around solvent molecules. Two interaction parameters are necessary to describe a binary system. Haghtalab and Vera [15] adopted the general ideas of Chen using the original Debye–Hückel expression

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

Nomenclature A aw B g hji hii Hji I Ksp m Mw MRSDa MRSDφ n q r R T uji x y z

β θ γ υc υa φ

Debye–Hückel constant water activity constant (b = 1.2) excess Gibbs energy enthalpies of interaction between j–i enthalpies  of interaction between i–i  u exp − Tji ionic strength in molality scale solubility product Molality molecular weight of the solvent mean standard deviation related to estimation of water activity data mean standard deviation related to estimation of osmotic coefficient data hydration number surface area parameter volume parameter gas constant Temperature interaction parameter mole fraction ionic strength fraction number of charge volume fraction surface area fraction activity coefficient stoichiometric numbers of cation stoichiometric numbers of anion osmotic coefficient

567

the excess Gibbs energy of an electrolyte solution can be written as a sum of the contribution of a long-range electrostatic term and a short-range interaction term. The long-range contribution is described by Debye–Hückel equation, while the short-range is described the contributions derived from Uniquac model. gex RT

=

gex,dh

+

RT

gex,C

+

RT

gex,R

(1)

RT

Hence ln γi = ln γidh + ln γiC + ln γiR .

(2)

2.1. The Debye–Hückel contribution The long-range interaction term accounts for the electrostatic interactions between ions and the short-range interaction term considers the non-electrostatic interactions between all species (ion and solvent). In this work, the Debye–Hückel model [5] is used to represent the contribution of the long-range ion–ion interactions. The Debye–Hückel equation (DH) for excess Gibbs energy can be written as: !  √ √ b2 I 4A gex,dh = −xw Mw 3 ln 1. + b I − b I + (3) RT b 2 where xw is the mole fraction of water, Mw is the molar mass of water. A is the Debye–Hückel parameter, b is dependent on the size of the involved ions, but is usually considered constant. and I is the  P ionic strength I = 21 i zi2 mi . By partial molal differentiation of the Debye–Hückel excess Gibbs energy term (3), one obtains for water:   √ √  1 2A √ − 2 ln 1. + b I (4) ln γwdh = Mw 3 1. + b I − b 1. + b I and for the ions:

√ instead of the Pitzer–Debye–Hückel formula, and expressed the local composition through nonrandom factors. Liu et al. [16] introduced a modified Debye–Hückel term, which covers only the long-range interactions between the central ion and all ions outside the first coordination shell. The electrostatic interactions between ions within the first coordination shell are included in the short-range contribution, which is based on a previous derivation of the three-parameter Wilson equation [17]. Sander et al. [18] and Macedo et al. [19] combined the Debye–Hückel expression with a modified Uniquac equation. These authors introduce the concentration dependence in the interaction energy parameters between ion and solvent and the interaction ion-specific for the model parameters. The Debye–Hückel term has been modified by Macedo and co-workers to ensure a more correct representation of the long-range forces. The developed models are applied to calculate the vapor–liquid equilibria in mixed solvent/salts systems [18,19]. The object of this paper is to use a modified electrolyte-Uniquac model for the calculation of single and mixed salts systems using a large data base and to compare the results with those of other wellknown models. For single systems, the equations of Pitzer [9,10] and Electrolyte-NRTL [13,14] models were chosen, mainly because parameters from these models are given for most aqueous systems in the literature. 2. Thermodynamic model The thermodynamic model used in this work is based on the local composition model derived from the original Uniquac model [20] by adding a Debye–Hückel term. The present model thus consists of three terms: a combinatorial or entropic term, a residual or enthalpic term and an electrostatic term. Therefore,

ln γ

dh i

A I

√ .

2

= −zi

(5)

1. + b I

2.2. Combinatorial contribution The combinatorial term is identical to the term used in the traditional Uniquac equation. The combinatorial, entropic term is: gex,C RT

=

X

 Xi ln

βi Xi

i



−

z X

2

 qi Xi ln

i

βi θi



(6)

where Xi = Ci xi (Ci = Zi for ions Ci = unity for solvent), z = 10 is the coordination number. xi is the mole fraction, βi is the volume fraction, and θi is the surface area fraction of component i Xj rj

βj = P

Xj rj

Xj qj

and θj = P

j

Xj qj

(7)

j

rj and qj are volume and surface area parameters for component j.

By partial molal differentiation of Eq. (6), the combinatorial parts of symmetrical activity coefficients are obtained:            βi βi z βi βi ln γiC = ln +1− − qi ln +1− . (8) Xi Xi 2 θi θi The infinite dilution terms are obtained by setting Xw = 1 in Eq. (8):       ri ri ∞,C +1− ln γi = ln rw

rw      ri qw ri qw − qi ln +1− . 2 rw qi rw qi z

(9)

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

568

2.3. Residual contribution Following Chen et al. [13,14], one assumes the existence of three types of local cells. One type consists of a central solvent molecule with other solvent molecules, cations and anions in its immediate neighborhood and the other two types of cells consist of either a central cation or anion with solvent molecules and ions of opposite charge in its immediate neighborhood. The first type of cell follows the local electroneutrality assumption; which states that, around a central solvent molecule, the arrangement of cations and anions will be such that the net ionic charge is zero. And the cells with central ions follow the like-ion assumption, that is, due to large repulsive forces between ions of the same charge and no ions of like charge exist near each other. Thus, the effective local surface area θji and θii with central component i, are defined by:

θji θj = Hji θii θi

(10) 

where Hji = exp −

uji

θm0 m hm0 m +

hm =

and uji = hji − hii .

T

θji θj = Hji,ki θki θk

θmc hmc +

X

hc = Zc

(11)

uji,ki



and uji,ki = hji − hki .

T

(13)

Due to the like-ion repulsion assumption, the local area fraction θc0 c = θa0 a = 0. The local composition is thus: Around a central solvent molecule X X X θam + θcm + θm0 m = 1 c

Around a central anion

X

θma +

X

m

Around a central cation

X

θca = 1

(14)

θmc +

X

, href c = Zc

(Xa0 qa0 ha0 c )

X

href a = Za

(Xc0 qc0 hc0 a )

X

(15)

θa . P θa0 Ha0 c,ac + θm Hmc,ac

m

+

X

qc Xc

X a0

qm Xm (hm − hm

=

)+

m

+

qc Xc (hc − hc

)

c

X

qa Xa (ha − href a )

R

a ref hi

where is the reference enthalpies of the cells with a central species i. hi is the residual molar enthalpies of cells with a central

c00

qc00 Xc00

j

qa0 Xa0 X θjc,a0 c ujc,a0 c . P qa00 Xa00 j

(19)

a00

T

0

m

−

X

j

Xc qc

X

X a0

Xa qa

X c0

X qa0 Xa0 ln θj Hjc,a0 c P qa00 Xa00 j

!

a00

! X qc0 Xc0 ln θj Hja,c0 a . P qc00 Xc00 j

(21)

c00

Since for the contribution of long-range interactions to the excess Gibbs energy (the Pitzer–Debye–Huckel model) the unsymmetrical normalization is used, it is necessary to normalize the contribution due to short-range interactions on the same basis. The normalization equation is:

=

gex,R RT

−

X c

Xc ln γc∞ −

X a

Xa ln γa∞ −

X

∞ Xm ln γm

(22)

m

where γc∞ and γa∞ are the infinite dilution activity coefficients of the cation and anion, respectively. By applying a local electroneutrality assumption to a central solvent molecule cell: X X θcm . (23) θam = a

(16)

c0

This yields the symmetric expression: ! X X gex,R =− Xm qm ln θj Hjm

RT

The residual molar excess enthalpy of a system containing one completely dissociated electrolyte in a single solvent, hex,R , can be calculated by assuming the excess enthalpies of all the cells as: X X ex,R ref ref h

a

j

The excess molar Gibbs energy, gex,res , can then be obtained by combining Eq. (19) and the following exact thermodynamic relation:   Z gex,res 1 1/T ex,res 1 = h d . (20)

gex∗,R

m

Xc00 qc00 .

c00

a

m

θac = P a0

θc P θc0 Hc0 a,ca + θm Hma,ca

(18)

Considering the above assumptions, the following equation is obtained for hex,lc of electrolyte solutions: X X X X qc0 Xc0 X θja,c0 a uja,c0 a hex,R = qm Xm θjm ujm + qa Xa P

−

m0

X

c0

a

c

Xa00 qa00

a00

c

c0

X ,

Using Eqs. (10), (12) and (14), the following expressions for the local area fractions may be derived:

θca = P

θca hca .

c

href m = hmm

θac = 1.

θi Him,mm θim = P P P θa Ham,mm + θc Hcm,mm + θm0 Hcm0 ,mm

(17)

Following the Chen et al. [13,14] approximation, the reference states were considered as pure solvent and completely ionized pure electrolyte, wich may be hypothetical. The reference state enthalpies of the cells with a central species i are thus expressed as:

RT

c

m

a

X

m

RT

m0

!

!

θma hma +

c

where Hji,ki = exp −

θac hac

a

X

ha = Za

θam ham

a

X

m

(12) 

θcm hcm +

c

m0

a0



T the absolute temperature, R the universal gas constant, hji and hii are the enthalpies of interaction between j–i and i–i. It should be noted that the interaction enthalpies are symmetric, i.e., hji = hij . The effective local surface area θji and θki , respectively, in the neighborhood of species i are defined by:

a

species i, which can be expressed in terms of the local area fractions as: X X X

c

It can be seen that, for a multicomponent solution of solvent m and salts with common anion (c0 a, c00 a, . . .): X Xa qa Ham = Xc qc Hcm . (24) c

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

Fig. 1. Activity coefficient of sodium and chloride ions in NaCl solutions.

569

Fig. 4. Activity coefficient of lithium and chloride ions in LiCl solutions.

Fig. 5. Activity coefficient of potassium and Bromide ions in KBr solutions. Fig. 2. Activity coefficient of sodium and Bromide ions in NaBr solutions.

Therefore, by combining Eqs. (23) and (10), it may be written for a single electrolyte solution: qa Ham = qc Hcm .

(25)

The interaction enthalpies are symetric, it may be inferred from this results that:  qc   Hcm = Hcam Ham = qc Hcm = qa Ham qa ⇔ (26) q  qc Hmc,ac = qa Hma,ca Hma,ca = c Hmc,ac = Hmca . qa

So, by combining Eqs. (24) and (26): , X X Ham =

Xc00 .

Xc qc Hcam

c

(27)

c00

Similar results are obtained by considering multicomponent solutions of salts with common cation: , X qa X Hcm = Xa Hcam Xa00 . (28) Fig. 3. Activity coefficient of potassium and chloride ions in KCl solutions.

Since, like-ion repulsion is assumed; the local compositions around each cation should be the same as in a single electrolyte solution.

a

qc

a00

The molecule–ion interaction parameters may also be expressed in terms of the salts molecule binary parameters: Hma,ca = Ham Hmca /Hcam

and Hmc,ac = Hcm Hmca /Hcam .

(29)

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

570

Proper partial differentiation of the excess Gibbs energy expression allows us to obtain expressions for the activity coefficients of the different species in the solution. The expression for the solvent is:  !   X X X Hmm0  R ln γm = qm 1 − ln θj Hjm − θm0 P m

X

−

qc

X

c

a0

m0

j

j

θj Hjm

qa0 Xa0 θc Hmc,a0 c P P qa00 Xa00 θj Hjc,a0 c a00

j



−

X

qa

X

a

c0

qc0 Xc0 θc Hma,c0 a  P P  qc00 Xc00 θj Hja,c0 a c00

j

and for the anion:    X θc0 1 1 ln γaR = qa  − P Za

Za

−

c0

XX c

a0

(30)

c00

θc00

ln

X

θ

!

j Hja,c0 a

−

j

X θm Ham P θj Hjm m j

Fig. 6. Activity coefficient of lithium and Bromide ions in LiBr solutions.

θa0 θc Hac,a0 c P P θa00 θj Hjc,a0 c a00

j

 X 0 0 θ ln H 1 ( ) c ma,c a   − − − Ham  P 

Za

c0

c00

and for the cation:    X θa0 1 1 ln γcR = qc  − P Zc

Zc

−

a0

XX a

c0

a00

!

θa00

(31)

θc00

ln

X

θj Hjc,a0 c −

j

X θm Hcm P θj Hjm m j

θc0 θa Hca,c0 a P P θc00 θj Hja,c0 a c00

j

 X 0 0 θ ln H 1 ( ) a mc,a c   − Hcm  . − − P 

Zc

a0

a00

θa00

(32)

Eqs. (31), (8) and (4) were combined to give the complete local composition model for activity coefficient for solvent. Eq. (31) for anion (or (32) for cation), (9), (8) and (5) may be combined to give the complete local composition model for activity coefficient for ion. The molality-based osmotic coefficient φ can be calculated using the following relation:

φ=−

1000. Ms υm

ln(aw ).

(33)

Also the mean ionic activity coefficient of an electrolyte with cation c and anion a, γ±ca can be calculated using the following relation: x ln γ± ca =

υc ln γc + υa ln γa υ

(34)

where υc and υa are the stoichiometric numbers of the cation and the anion of the electrolyte, respectively, υ = υc + υa , Ms is the molar mass of water, and m is the molality of the electrolyte solution. The mean ionic activity coefficient experimental data available in the literature are normalized in molality scale and the activity coefficients calculated from the model are normalized in mole fraction scale. Therefore, the following relation was used to m obtain the molal mean ionic activity coefficient γ± from the mole x fraction mean ionic activity coefficient γ± .   (Ms mυ) m ln γ± = ln γ±x − ln 1. + . 1000

(35)

Fig. 7. Activity coefficient of Barium and chloride ions in BaCl2 solutions.

From the above description of the electrolyte-Uniquac model, it can be seen that the parameters in the model are the volume and surface area parameters r and q for species i and the binary interaction energy parameter ucam and umca . The values of the r and q volume and surface area parameters for non-electrolyte species are usually calculated on the basis of the geometry of the species [18]. For electrolyte species, the dimension of an ion in a crystal lattice probably only has little to do with its effective dimension in an aqueous solution, due to hydration. The r and q parameters have therefore in this work been treated as adjustable parameters. The model presents two binary interaction energy parameters ucam and umca , which become the correlation variables in treating the thermodynamic properties of a system containing one electrolyte in a single solvent. As can be seen from Eq. (11), the electrolyte–molecule parameter, ucam , is the difference of the interaction enthalpies between the ion–molecule pair and the molecule–molecule pair and the molecule electrolyte parameter, umca , is the difference of the interaction enthalpies between the molecule–ion pair and the cation–anion pair. The all parameters were fitted to experimental data.

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

571

Fig. 8. Activity coefficient of calcium and chloride ions in CaCl2 solutions.

Fig. 10. Solubility of salts in mixed electrolyte at 298.15 K. NaCl–NaOH–H2 O system.

Fig. 9. Activity coefficient of Magnesium and Bromide ions in MgBr2 solutions.

3. Results and discussion 3.1. Correlation of mean ionic activity and osmotic coefficients for singles electrolytes systems The experimental data of mean ionic activity and osmotic coefficients for singles electrolytes systems were collected from the literature [21–25]. The model developed in this work was applied to correlate these thermodynamic properties. In the parameter determination, a nonlinear mean square method using the algorith minimization of Maquardt [26] is employed by fiting the model to the experimental mean ionic activity and osmotic coefficients (γ± and φ). The minimization is carried out on sums of squares of deviation calculated and experimental quantities following the objective function F :
F ucl ak m , umcl ak , . . . , rl , ql , . . .

=

NX data X 2  i=1 j=1

exp ζical ,j ucl ak m , umcl ak , . . . , rl , ql , . . . − ζi,j




2 

(36)

Fig. 11. Solubility of salts in mixed electrolyte at 298.15 K. NaCl–HCl–H2 O System.

where Ndata is the number of data points, ζi,1 = ln γ±,i , ζi,2 = φi . The corresponding standard deviation σζ is: 

Ndata

X σζj =   i=1



exp ζical ,j ucl ak m , umcl ak , . . . , rl , ql , . . . − ζi,j




Ndata − NP − 1

2 1/2  

.

(37)

NP is the number of adjusted parameters. To determine both the interactions parameters ucam , umca and the structural parameters r and q, the data of activity and osmotic coefficients for all salts with the same ion are required. Hence, all parameters are simultaneously determined using all of experimental data to minimize the absolute deviation between experimental and calculated values. However, the use of such a

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

572

Table 1 Modified e-Uniquac r and q parameters Na+

K+

Li+

Cs+

H+

NH+ 4

Ag+

r 3.240 3.471 5.619 3.040 5.763 3.240 3.151 q 3.514 2.385 5.216 2.050 6.041 3.285 3.684

Br−

Cl−

F−

I−

OH−

NO− 3

ClO− 4

r 3.440 3.006 3.832 4.122 4.838 2.517 3.044 q 2.615 2.132 2.491 3.452 3.954 1.594 2.683

Ba2+

Be2+

Ca2+

Cd2+

Co2+

Cu2+

Fe2+

3.606 3.240 3.092 3.895

5.140 4.124

5.020 4.693 5.105 5.073

5.608 5.083

4.051 4.507

4.256 5.965 4.585 5.767

ClO− 3

H2 AsO− 4

SO24−

HAsO24−

CO23−

SO23−

3.437 3.277

3.435 3.649 2.562 2.715

3.858 2.838

3.316 2.451

2.802 2.972 1.700 1.741

Rb+

H2 PO− 4

3.652 3.879 2.215 4.177

Fig. 12. Solubility of salts in mixed electrolyte at 298.15 K. KCl–HCl–H2 O System.

HPO24−

Mg2+

Mn2+ Ni2+ 4.390 2.999

Pb2+ Sr2+

Zn2+

6.155 5.953 3.243 4.652 5.930 4.990 3.389 4.341

S2 O23−

Fig. 14. Solubility of salts in mixed electrolyte at 298.15 K. NaCl–CaCl2 –H2 O system.

Fig. 13. Solubility of salts in mixed electrolyte at 298.15 K. NaCl–MgCl2 –H2 O System.

Fig. 15. Solubility of salts in mixed electrolyte at 298.15 K. CsBr–KBr–H2 O system.

large number of parameters resulted in unrealistic parameters. In this work, therefore the parameter optimization process was estimated by using the following procedure. In the first step, both the interaction parameters between a pair ion and water

and the structural parameters r and q for ions were determined for the electrolyte with common ion for a series of 3 or 4 salts, for example (NaCl, KCl, CsCl, RbCl) or (NaCl, Na2 SO4 , Na2 Br, NaI). On the basis of these reliable parameters, the other parameters

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

573

Table 2 Regressed interaction parameters for the modified e-Uniquac model, and a comparison with results obtained with the Electrolyte–NRTL and Pitzer models to the experimental data of mean ionic activity and osmotic coefficients Electrolyte

HCl HBr HI LiCl LiBr LiI LiOH LiClO4 LiNO3 NaF NaCl NaBr NaI NaOH NaClO3 NaClO4 NaBrO3 NaNO3 NaH2 PO4 NaH2 AsO4 NaCNS KF KCl KBr KI KOH KClO3 KBrO3 KNO3 KH2 PO4 KH2 AsO4 KCNS RbF RbCl RbBr RbI RbNO3 CsF CsCl CsBr CsI CsOH CsNO3 AgNO3 NH4 Cl NH4 ClO4 NH4 NO3 MgCl2 MgBr2 MgI2 CaCl2 CaBr2 CaI2 SrCl2 SrBr2 SrI2 BaCl2 BaBr2 BaI2 ZnF2 ZnCl2 Zn(ClO4 )2 ZnBr2 Zn(NO3 )2 Cd(ClO4 )2 Cd(NO2 )2 Cd(NO3 )2 Li2 SO4 Na2 SO3 Na2 SO4 Na2 HPO4 Na2 CO3 Na2 HAsO4

mmax

6. 6. 6. 6. 6. 3. 5. 4.5 6. 1. 6. 6. 6. 6. 3. 6. 2.61 6. 6. 1.3 6. 6. 5. 5.5 4.5 6.0 0.7 0.5 3.5 1.8 1.3 5.0 3.5 7.8 5.0 5.0 4.5 3.5 6.0 5.0 3.0 1.2 1.5 6.0 7.4 2.1 6.0 4.0 3.0 3.5 6.0 3.0 6.0 3.5 2.12 1.97 1.7 2.0 1.998 0.142 6.0 4.0 6.0 6.0 1.25 6.0 2.63 3.0 2.058 4.445 2.121 2.767 1.029

nexp

29 29 29 29 29 23 26 26 29 16 29 29 29 29 23 29 23 29 29 19 29 29 28 28 26 29 13 11 24 20 18 27 24 32 27 27 26 24 29 27 24 17 19 29 31 22 29 40 36 38 38 36 48 38 40 38 35 38 38 20 48 40 48 48 29 48 35 36 33 43 33 36 29

umca

−56.64 −28.65 88.45

−103.78 −124.36 −138.35 −10.30 −129.08 −344.89 19.40

−9.04 43.48 154.36 255.38 −214.24 303.46 −404.52 −453.56 −136.53 451.22 139.57 −243.50 −244.09 −199.77 −144.54 −86.89 −207.21 −214.42 −216.97 −193.67 296.58 −172.72 −258.25 −191.72 −140.74 −58.15 −11.47 −262.76 −177.46 −121.26 −21.050 −164.89 −152.16 −319.46 −154.70 85.21 −112.34 −111.56 −54.79 50.32 −50.13 4.77 208.79 −100.70 −70.99 −9.04 288.84 5.40 99.57 55.04 −222.20 −172.04 −260.39 −275.69 −91.06 −352.86 −298.16 −190.76 −52.11 236.42 141.93 145.74 122.92

ucam

−1115.13 −1460.12 −974.43 −1124.16 −1749.19 −182.08 −24.81 −522.82 −239.92 −550.36 −1225.72 −1469.38 −988.74 −595.95 74.39

−327.20 589.17 542.08 497.21 48.93 −1237.06 −1083.02 −211.42 −49.37 130.60 −1164.87 −496.22 −83.24 −134.27 695.69 −188.95 5.00 17.32 −105.91 −13.05 56.31 −209.17 −839.71 −800.42 −300.45 −102.66 263.72 −81.49 390.72 −38.47 98.22 −65.50 −1011.11 −1146.69 −1167.07 −1293.07 −1047.51 −1191.95 −1253.26 −636.43 −448.65 −721.17 −262.53 −439.04 126.77 15.77 −949.19 134.29 −1137.24 −192.02 520.13 −43.26 25.67 −156.83 −97.32 17.62 −94.51 124.38

The present model

Chen model

σφ

σln γ±

σφ

σln γ±

σφ

Pitzer model

σln γ±

0.0101 0.0432 0.0361 0.0160 0.0213 0.0151 0.0201 0.0110 0.0041 0.0026 0.0032 0.0025 0.0065 0.0095 0.0008 0.0017 0.0013 0.0030 0.0029 0.0032 0.0045 0.0042 0.0025 0.0017 0.0014 0.0071 0.0019 0.0008 0.0066 0.0037 0.0028 0.0028 0.0036 0.0056 0.0050 0.0051 0.0083 0.0062 0.0097 0.0076 0.0090 0.0015 0.0122 0.0086 0.0050 0.0190 0.0106 0.0655 0.0496 0.0699 0.0444 0.0444 0.0907 0.0341 0.0283 0.0334 0.0137 0.0169 0.0351 0.0099 0.0687 0.1924 0.0897 0.0590 0.0220 0.0261 0.0203 0.0243 0.0062 0.0307 0.0106 0.0132 0.0154

0.0120 0.0452 0.0381 0.0170 0.0221 0.0181 0.0210 0.0140 0.0042 0.0069 0.0035 0.0030 0.0077 0.0100 0.0008 0.0017 0.0010 0.0031 0.0053 0.0037 0.0055 0.0570 0.0088 0.0027 0.0018 0.0087 0.0016 0.0013 0.0125 0.0051 0.0040 0.0042 0.0048 0.0095 0.0075 0.0074 0.0197 0.0078 0.0153 0.0129 0.0104 0.0021 0.0185 0.0141 0.0071 0.0358 0.0191 0.0555 0.0327 0.0680 0.0318 0.0317 0.0675 0.0428 0.0234 0.0277 0.0085 0.0115 0.0205 0.0192 0.0816 0.2280 0.0747 0.0490 0.0225 0.0758 0.0168 0.0233 0.0200 0.0560 0.0156 0.0199 0.0199

0.0281 0.1262 0.1320 0.0323 0.0381 0.0192 0.0181 0.0230 0.0091 0.0026 0.0161 0.0316 0.0481 0.0201 0.0035 0.0076 0.0868 0.0023 0.0012 0.0033 0.0469 0.0225 0.0045 0.0037 0.0029 0.0203 0.0010 0.0005 0.0058 0.0042 0.0012 0.0218 0.0022 0.0027 0.0026 0.0027 0.0076 0.0032 0.0057 0.0047 0.0069 0.0029 0.0019 0.0081 0.0015 0.0045 0.0045 0.0398 0.0452 0.0852 0.0783 0.0410 0.1376 0.0353 0.0220 0.0242 0.0045 0.0107 0.0206 0.0032 0.0585 0.1279 0.0661 0.0914 0.0167 0.0342 0.0322 0.0093 0.0051 0.0169 0.0066 0.0086 0.0143

0.0331 0.1532 0.1670 0.0363 0.0430 0.0221 0.0202 0.0261 0.0113 0.0077 0.0180 0.0384 0.0597 0.0228 0.0051 0.0096 0.1842 0.0038 0.0054 0.0154 0.0598 0.0272 0.0127 0.0052 0.0046 0.0213 0.0020 0.0011 0.0067 0.0045 0.0033 0.0030 0.0024 0.0035 0.0039 0.0031 0.0090 0.0156 0.0085 0.0056 0.0075 0.0150 0.0023 0.0096 0.0034 0.0215 0.0221 0.1133 0.1196 0.1999 0.1753 0.1120 0.2979 0.0845 0.0858 0.0964 0.0137 0.0414 0.0826 0.0104 0.1716 0.2892 0.2005 0.2425 0.0572 0.0186 0.1352 0.0198 0.0137 0.0308 0.0224 0.0235 0.0576

0.0033 0.0122 0.0130 0.0028 0.0042 0.0083 0.0313 0.0030 0.0025 0.0026 0.0010 0.0058 0.0050 0.0061 0.0016 0.0010 0.0010 0.0026 0.0045 0.0020 0.0238 0.0009 0.0008 0.0008 0.0037 0.0105 0.0008 0.0007 0.0037 0.0030 0.0032 0.0010 0.0022 0.0051 0.0009 0.0010 0.0034 0.0028 0.0010 0.0027 0.0027 0.0041 0.0033 0.0022 0.0103 0.0024 0.0022 0.0070 0.0031 0.0033 0.0122 0.0107 0.1051a 0.0069 0.0028 0.0146 0.0050 0.0025 0.0063 – 0.0062b 0.0031c 0.0052d 0.0024c – – 0.0033 0.0057 – 0.0064e 0.0030f 0.0321g 0.0024

0.0033 0.0164 0.0255 0.0032 0.0078 0.0191 0.0499 0.0074 0.0036 0.0082 0.0036 0.0064 0.0079 0.0099 0.0081 0.0041 0.0050 0.0059 0.0141 0.0081 0.0324 0.0041 0.0115 0.0043 0.0048 0.0163 0.0021 0.0012 0.0124 0.0136 0.0150 0.0016 0.0043 0.0057 0.0038 0.0040 0.0104 0.0088 0.0088 0.0143 0.0101 0.0177 0.0124 0.0102 0.0052 0.0099 0.0087 0.0182 0.0063 0.0076 0.0360 0.0261 0.2270a 0.0276 0.0176 0.0387 0.0166 0.0037 0.0301 – 0.0181b 0.0093c 0.0189d 0.0120c – – 0.0038 0.0075 – 0.0109e 0.0077f 0.0641g 0.0126

(continued on next page)

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

574 Table 2 (continued) Electrolyte

mmax

K2 SO4 0.692 K2 HPO4 0.873 K2 HAsO4 0.886 Rb2 SO4 1.707 Cs2 SO4 1.50 BeSO4 4.0 MgSO4 2.0 MnSO4 3.0 NiSO4 2.5 CdSO4 3.5 ZnSO4 3.6 CuSO4 1.42 CoSO4 1.0 Mean standard deviation a m max b m max c m max d m max e m max f m max g m max

nexp

25 27 27 31 31 29 21 16 21 31 34 24 23

umca

−81.11 −89.82 −161.45 −18.78 −104.08 −155.83 8.73

−173.33 −116.77 31.57 113.50 −73.27 −71.85

ucam

−194.18 −482.69 −260.14 −149.70 −247.62 −110.21 −93.22 −12.13 −137.02 −282.95 −405.96 −450.35 −226.26

The present model

Chen model

σφ

σln γ±

σφ

σln γ±

σφ

Pitzer model

σln γ±

0.0169 0.0098 0.0299 0.0094 0.0118 0.1284 0.0232 0.0111 0.0272 0.0646 0.0341 0.0370 0.0381 0.0224

0.0376 0.0073 0.0265 0.0251 0.0277 0.0838 0.1303 0.1392 0.1439 0.1473 0.1850 0.1460 0.1645 0.0343

0.0067 0.0016 0.0150 0.0030 0.0049 0.0525 0.0261 0.0278 0.0261 0.0301 0.0481 0.0234 0.0277 0.0253

0.0191 0.0042 0.0570 0.0130 0.0201 0.0998 0.1709 0.1629 0.1573 0.1557 0.1735 0.2087 0.1806 0.0617

0.0081 0.0025 0.0044 0.0064 0.0088 0.0051 0.0086 0.0051 0.0088 0.0091 0.0109 0.0122 0.0221 0.0070

0.0278 0.0065 0.0176 0.0244 0.0253 0.0894 0.1466 0.1947 0.0917 0.0641 0.1544 0.0941 0.1011 0.0260

= 2. = 1.2. = 2. = 1.6. = 4. = 1. = 1.5.

Fig. 16. Solubility of salts in mixed electrolyte at 298.15 K. RbCl–NaCl–H2 O System.

for other pair ions were determined. The two-step determination successfully correlated the data of binary activity and osmotic coefficients. The Table 1 shows the obtained results of r and q. Table 2 presents the evaluated interaction parameters and the results of the representation of mean ionic activity and osmotic coefficients experimental data for 86 single electrolytes systems at 298.15, compared to those obtained from the Electrolyte-NRTL model developed by Chen et al. [13,14] and the Pitzer model [9,10]. The regression results are given as the standard deviations of the molal mean ionic activity coefficient and the osmotic coefficient. As can be seen from Table 2, the model developed here correlates the experimental data, related to the mean ionic activity and to the osmotic coefficients, with comparable than the other two models and in most cases with better accuracy than the Electrolyte-NRTL model. Indeed, for the logarithmic mean ionic activity coefficients, the obtained standard deviation for all studied binary systems is 0.0224 in the present work, 0.0253 from Electrolyte-NRTL model and 0.007 from Pitzer model. For the osmotic coefficients, the obtained standard deviation is 0.0343 in the present work, 0.026

Fig. 17. Solubility of salts in mixed electrolyte at 298.15 K. KCl–NaCl–H2 O System.

from Pitzer model and 0.0617 from Electrolye-NRTL model. We noted that the electrolyte-Uniquac model requires more adjustable parameters than electrolyte-NRTL model. Figs. 1–9 show that the model developed in this paper permit to lead to an accurate representation of the individual activity coefficient of ion in the binary system. For a single electrolyte system, the approximation used by Chen et al. [13,14] led to equality of activity coefficient of anion and cation in the case of using of Electrolyte-NRTL. In terms of the NaCl–H2 O, NaBr–H2 O, KCl–H2 O and LiCl–H2 O systems, Figs. 1–4 show that the calculated results related to the individual activity coefficient of ion were in good agreement with the experimental data and the calculated results of Lin and Lee [27]. It can be seen from Figs. 4 and 5 that the calculated result in the present work for individual activity coefficient of ion, present a less difference than that of Lin and Lee [27] for KBr–H2 O system and the results obtained by Taghikhani and coworkers [28] in the case of LiBr–H2 O system. For BaCl2 –H2 O system (Fig. 7), the calculated results were in good

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

575

Table 3 Standard deviation of the predicted water activity data and osmotic coefficient data for mixed electrolyte aqueous solutions at 298.15 K Binary mixture A–B

Molality range of electrolyte A

Molality range of electrolyte B

MRSDa

Reference

MRSDφ

y = 1/3

y = 1/2

y = 2/3

y = 1/3

y = 1/2

y = 2/3

0.0076 0.0040 0.0032 0.0004 0.0049 0.0004 0.0011 0.0029 0.0004 0.0006 0.0016 0.0013 0.0009

0.0624 0.0877 0.0350 0.0055 0.0049 0.0030 0.0080 0.0078 – 0.0121 0.0032 0.0052 –

0.0151 0.0075 0.0271 0.0060 0.0035 0.0035 0.0153 0.0040 – 0.0096 0.0156 0.0096 0.0210

0.0476 0.0229 0.0269 0.0039 0.0049 0.0029 0.0089 0.0060 – 0.0054 0.0145 0.0108 0.0181

CsCl–LiCl CsCl–NaCl LiCl–KCl NaCl–LiCl NH4 Cl–CsCl NH4 Cl–KCl NH4 Cl–LiCl NH4 Cl–NaCl NaCl–MgCl2 NaCl–BaCl2 MgSO4 –Na2 SO4 MgSO4 –MnSO4 MgSO4 –K2 SO4

0.10–5.33 0.10–5.33 0.10–4.00 0.10–4.00 0.10–6.67 0.10–3.80 0.10–4.00 0.10–4.00 0.15–3.60 0.15–3.60 0.10–2.40 0.10–2.80 0.10–1.65

0.20–5.33 0.20–5.33 0.10–4.00 0.10–4.00 0.10–6.00 0.10–4.00 0.10–4.00 0.10–4.00 0.10–1.20 0.10–1.40 0.10–2.13 0.10–2.80 0.10–0.80

0.0094 0.0137 0.0040 0.0005 0.0012 0.0004 0.0009 0.0007 0.0009 0.0012 0.0002 0.0004 –

0.0016 0.0007 0.0029 0.0005 0.0007 0.0005 0.0009 0.0063 0.0003 0.0010 0.0021 0.0009 0.0010

y = 1/5

y = 1/2

y = 4/5

y = 1/5

y = 1/2

y = 4/5

NH4 NO3 –KNO3 (NH4 )2 SO4 –Na2 SO4 (NH4 )2 SO4 –Li2 SO4 (NH4 )2 SO4 –K2 SO4

0.10–6.00 0.10–3.50 0.10–3.20 0.10–2.20

0.10–3.50 0.10–2.80 0.10–3.00 0.10–0.68

0.0003 0.0007 0.0010 0.0001

0.0020 0.0003 0.0031 0.0001

0.0020 0.0006 0.0021 0.0003

0.0045 0.0101 0.0065 0.0044

0.0103 0.0051 0.0193 0.0026

0.0093 0.0047 0.0164 0.0027

KCl–KH2 PO4 NaCl-CuCl2 CuCl2 –CuSO4 Na2 SO4 –CuSO4

0.18–1.2798 0. 00–4.2269 0.12–1.6846 0. 00–1.1968

0.19–1.6008 0. 00–1.9397 0. 00–1.3279 0.00–1.3511

Fig. 18. Solubility of salts in mixed electrolyte at 298.15 K. Cs2 SO4 –CsCl–H2 O system.

agreement with those determined by Rodil and Vera [29]. But, the calculated results for the systems CaCl2 –H2 O and MgBr2 –H2 O (Figs. 8 and 9) were between the results obtained by Rodil and Vera [30] and those estimated by Bates et al. [31] by using the Stokes and Robinson hydration model [32] for the calculation of activity coefficients of individual ions. 3.2. Prediction of the solubility and the activity coefficient for mixed electrolyte For many mixed aqueous solutions, experimental results neither for the osmotic and ionic mean activity coefficient nor for the solubility of salts in aqueous solution are available in the

– – – –

MRSDφ 0.0134 0.0041 0.0064 0.0070

[33] [33] [34] [42] [46] [47] [48] [49] [43] [40] [39] [38] [37] [50] [45] [45] [44] [35] [34] [34] [34]

Fig. 19. Solubility of salts in mixed electrolyte at 298.15 K. CsCl–KCl–H2 O system.

literature. These thermodynamic properties are therefore often used for testing new models. 3.2.1. Prediction of the activity coefficient for mixed electrolyte The activity coefficient for multiple components in aqueous solutions was predicted using the parameter determined above from the binary data. The experimental data related to mean activity coefficient of various solutes in mixed electrolytes systems are collected from literature [33–51]. Table 3 shows the standard deviation between predicted and experimental data. The accuracy of the prediction was around the standard deviations in the correlated results from binary systems. These results demonstrate that the interaction parameters estimated from binary data are useful for calculating the ionic mean activity coefficient for mixed electrolytes.

B. Messnaoui et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 566–576

576

Table 4 Value of solubility product used in the present work Salts

Molality of saturation (mol kg−1 )

ln Ksp

References

NaCl RbCl KCl CsCl KBr CsBr Cs2 SO4

6.198 7.848 4.861 11.327 5.815 5.813 5.105

3.650 2.956 2.080 3.488 2.637 1.881 1.9323

[52] [52] [52] [52] [52] [52] [52]

3.2.2. Prediction of the solubility for mixed electrolyte When a solid electrolyte Mvc Xva (H2 O)h precipitates from an aqueous solution, the concentrations of anionic and cationic species of that electrolyte in the liquid phase are determined by its solubility product: ln Ksp = νa ln(m a γ a ) + νc ln(m c γ c ) + n ln (aw ) .

(38)

The calculation of an electrolyte solubility product can therefore be based on the saturation molality of a pure solution of that electrolyte or the tabulated values of Gibbs free energies. Ideally, the solubility product should be the same no matter what the method of calculation. In reality, this is rarely the case. One of the solutions tested here is the solubility of some salts in binary electrolyte solutions: KCl–HCl–H2 O, KCl–NaCl–H2 O, NaCl–HCl–H2 O, NaCl–NaOH–H2 O, CsCl–KCl–H2 O, RbCl–NaCl–H2 O, CsBr–KBr–H2 O, Cs2 SO4 –CsCl–H2 O, NaCl–CaCl2 –H2 O and NaCl–MgCl2 –H2 O, at 298.15 K. Table 4 presents the used values of solubility product of salts in this calculation. These values are taken from literature [52]. The results of prediction related to the solubility of one electrolyte in the binary solutions are shown in Figs. 10–19. For comparison, we introduced the experimental data collected from literature [22, 51,52]. As shown in these figures, the new model gives a reasonable prediction for the solubilities for mixed salts with interactions parameters determined only from information on the properties of single-electrolyte aqueous solutions.

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

4. Conclusion

[39]

In this paper, a new model was developed for calculating the activity coefficient of electrolyte solutions. The two assumptions proposed by Chen and co-workers were used in the present work to reduce the parameters number involved in residual contribution of Uniquac model. The model has been tested on 86 aqueous electrolyte solutions at 298.15 K and results have been compared with those obtained from the Pitzer and electrolyteNRTL models. This new model requires more parameters than the electrolyte-NRTL model, but yields results compared to the Pitzer model. The modified electrolyte-Uniquac model led to the accurate calculation of individual activity coefficient of ion. This model can also be used to predict the excess properties and the salt solubility in aqueous multielectrolyte solutions with the interaction parameters obtained from binary data.

[40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

References [1] K.S. Pitzer, Phys. Chem. Earth 13–14 (1981) 249.

[51] [52]

G. Maurer, Fluid Phase Equilib. 13 (1983) 269. H. Renon, Fluid Phase Equilib. 30 (1986) 181. L.A. Bromley, AIChE J. 19 (1973) 313. P. Debye, E. Hückel, Phys. Z. 24 (1923) 185. E.A. Guggenheim, Philos. Mag. (Ser. 7) 19 (1935) 588. E.A. Guggenheim, J.C. Turgeon, Trans. Faraday Sot. 51 (1935) 747. J.N. Bronstedt, J. Amer. Chem. Sot. 44 (1922) 938. K.S. Pitzer, J. Phys. Chem. 77 (1973) 268. KS. Pitzer, J. Phys. Chem. 77 (1973) 2300. J.L. Cruz, H. Renon, AIChE J. 24 (1978) 817. F.X. Ball, W. Fiirst, H. Renon, AIChE J. 31 (1985) 392. C.C. Chen, H.I. Britt, J.F. Boston, L.B. Evans, AIChE J. 28 (1982) 588. C.C. Chen, L.B. Evans, AIChE J. 32 (1986) 444. A. Haghtalab, J.H. Vera, AIChE J. 34 (1988) 803. Y. Liu, A.H. Harvey, J.M. Prausnitz, Chem. Eng. Commun. 77 (1989) 43. H. Renon, J.M. Prausnitz, AIChE J. 15 (1969) 785. Bo Sander, Aage Fredenslund, Peter Rasmussen, Chem. Eng. Sci. 41 (5) (1986) 1171. Eugénia A. Macedo, Per Skovborg, Peter Rasmussen, Chem. Eng. Sci. 45 (1990) 5. D.S. Abrams, J.M Prausnitz, AIChE J. 21 (1) (1975) 116. W.J. Hamer, Y.-C. Wu, J. Phys. Chem. Ref. Data 1 (1972) 1047. J.F. Zemaitis, D.M. Clark, M. Rafal, N.C. Scrivner, Handbook of Aqueous Electrolyte Thermodynamics, DIPPR, New York, 1986. John G. Albright, Joseph A. Rard, Samuel Serna, Erin E. Summers, Michelle C. Yang, J. Chem. Thermodynam. 32 (2000) 1447. R.A. Robinson, R.H. Stokes, Electrolyte Solutions, 2nd revised edition, Butterworths, London, 1965. M. EL Guendouzi, A. Mounir, A. Dinane, J. Chem. Thermodynam. 35 (2003) 209. D.W. Marquardt, J. Soc. Indus. Appl. Math. 11 (2) (1963) 431. Cheng-long Lin, Liang-sun Lee, Fluid Phase Equilib. 205 (2003) 69. Vahid Taghikhani, Hamid Modarress, Juan H. Vera, Fluid Phase Equilib. 166 (1999) 67. Eva Rodil, J.H. Vera, Fluid Phase Equilib. 187–188 (2001) 15. Eva Rodil, J.H. Vera, Fluid Phase Equilib. 205 (2003) 115. R.G. Bates, B.R. Staples, R.A. Robinson, Anal. Chem. 42 (1970) 867. R.H. Stokes, R.A. Robinson, J. Amer. Chem. Soc. 70 (1948) 1870. Mohamed El Guendouzi, Asmaa Benbiyi, Rachid Azougen, Abderrahim Dinane, Comput. Coupling Phase Diagrams Thermochem. 28 (2004) 435. Colin J. Downes, Kenneth S. Pitzer, J. Solution Chem. 5 (6) (1976) 1989. C.W. Childs, C.J. Downes, R.F. Platford, J. Solution Chem. 3 (2) (1974) 139. Mohamed El Guendouzi, Asmaa Benbiyi, Abderrahim Dinane, Rachid Azougen, Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 213. Abdelfetah Mounir, Mohamed El Guendouzi, Abderrahim Dinane, J. Solution Chem. 31 (10) (2002) 793. Mohamed El Guendouzi, Abdelfetah Mounir, Abderrahim Dinane, Fluid Phase Equilib. 202 (2002) 221. Abdelfetah Mounir, Mohamed El Guendouzi, Abderrahim Dinane, Fluid Phase Equilib. 201 (2002) 233. Mohamed El Guendouzi, Asmaa Benbiyi, Abderrahim Dinane, Rachid Azougen, Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 375. Abderrahim Dinane, Mohamed El Guendouzi, Abdelfetah Mounir, J. Chem. Thermodynam. 34 (2002) 423. Mohamed El Guendouzi, Asmaa Benbiyi, Abderrahim Dinane, Rachid Azougen, Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 97. Abderrahim Dinane, Abdelfetah Mounir, Fluid Phase Equilib. 206 (2003) 13. Mohamed El Guendouzi, Rachid Azougen, Abdelfetah Mounir, Asmaa Benbiyi, Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 409. Abdelfetah Mounir, Mohamed El Guendouzi, Abderrahim Dinane, J. Chem. Thermodynam. 34 (2002) 1329. Mohamed El Guendouzi, Rachid Azougen, Asmaa Benbiyi, Abderrahim Dinane, Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 329. Abderrahim Dinane, Abdelfetah Mounir, J. Solution Chem. 32 (5) (2003) 395. Mohamed El Guendouzi, Abderrahim Dinane, Abdelfetah Mounir, J. Solution Chem. 31 (2) (2002) 119. Abderrahim Dinane, Mohamed El Guendouzi, Abdelfetah Mounir, J. Chem. Thermodynam. 34 (2002) 783. Mohamed EL Guendouzi, Mohamed Marouani, Fluid Phase Equilib. 216 (2004) 229. Bin Hu, Pengsheng Song, Yahong Li, The Open Thermodynam. J. 1 (2007) 1. Bin Hu, Pengsheng Song, Yahong Li, Wu Li, Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 541.