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Fluid Phase Equilibria 247 (2006) 1–7

A semi-empirical hydration model (SEHM) for describingaqueous electrolyte solutions

II. Mixing rules for multiple electrolyte systems at 25 ◦C

Efthymios Balomenos, Dimitrios Panias ∗, Ioannis PaspaliarisLaboratory of Metallurgy, School of Mining and Metallurgical Engineering, National Technical University of Athens,

Zografos Campus, 15780 Athens, Greece

Received 5 December 2005; received in revised form 1 June 2006; accepted 4 June 2006Available online 12 June 2006

bstract

Mixing rules for multiple electrolyte systems for the new semi-empirical hydration model (SEHM) are proposed. The SEHM model is tested ineveral ternary strong electrolyte systems and proves to provide accurate estimations of the solutions’ osmotic coefficients.

2006 Elsevier B.V. All rights reserved.

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eywords: Ionic activity coefficient; Osmotic coefficient; Ionic hydration; Mix

. Introduction

In part I of this series [1], a new semi empirical hydra-ion model (SEHM) for describing electrolyte solutions isroposed. The SEHM model, which is a combination of theitzer–Debye–Huckel theory with Schonert’s stepwise hydra-

ion model, needs one electrolyte regression parameter to accu-ately fit the mean ionic activity coefficient of any aqueous singletrong electrolyte system. The significant novelty of the SEHModel is its ability to estimate, in a semi-empirical fashion, the

olution’s water mole fraction, and thus theoretically predict theolution’s osmotic coefficient.

In the second part of this series the SEHM will be applied inernary strong electrolytes systems (solutions composed of threearts: water and two strong electrolytes). The ternary systemshat will be examined compose of both common ion systems andon-common ion ternary systems. The SEHM model’s abilityo predict the osmotic coefficient of these systems, without these of any mixing regression parameter, will not only strengthen

he basis of the model, but will establish it as a useful tool inhemical engineering.

∗ Corresponding author. Tel.: +30 210 772 2276; fax: +30 210 772 2168.E-mail address: panias@metal.ntua.gr (D. Panias).

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(b

378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2006.06.001

les

. Theory

The SEHM model for a single electrolyte system is presentedn detail in [1]. In summation the model states that in a system ofstrong electrolyte CzC

vCAzA

vA, a multi-step hydration equilibrium

xists in the form of:

(H2O)i−1 + H2O → p(H2O)i (1)

here p stands for the anion, the cation or any electrostatic ionair (including neutral species) which those two can form. Theseeries of reactions are characterized by two semi-empirical elec-rolyte parameters (nCA, kCA), which provide the overall hydra-ion number of the electrolyte (hCA) and the water mole fractionxW) of the solution through the following equations:

¯ CA = nCAxW

1 + xW(2)

W = Z

N(3)

= 1 − (vC + vA)mCAhCAMW (4)

= Z + (vC + vA)mCAMW (5)

kCA is taken to be 1 for all electrolytes [1], so it has been replacedy its numerical value in all equations).

2 hase

tt

l

wwm

l

Tieh

V

O

w

oasca

γ

Ste

φ

2

eD

efit

mt

x

Z

N

h

h

El(so

teatePa

y

y

O

es

X

Toaamath((

E. Balomenos et al. / Fluid P

The non-ideality of the solution is attributed to two factors:he long range electrostatic factor, which is described throughhe Pitzer–Debye–Huckel theory:

n γLR± = − κ2

6Dc

(2κ

(1 + κa)+ 1

aln(1 + κa)

)

+ πcα

3

(4α2 + κ4

(Dc(1 + κa))2

)(6)

here κ, D, c are given in detail [1] and the hydration factor,hich is described through the Schonert’s stepwise hydrationodel:

n γHYD± = nCA ln

(2

(1 + xW)

)− ln N (7)

he summation of (6) and (7) provide the logarithm of the meanonic activity coefficient of the electrolyte. The distance of clos-st approach in (6) is defined as twice the radius of the averageydration sphere (rh

CA), which in turn is given by

hCA = 4πNA

3(rh

CA)3 = 1

2(V c

C + V cA) + hCAVW (8)

r:

a

2= rh

CA = (0.5((rcC)3 + (rc

A)3) + hCA(rW)3)1/3

(9)

here from [1] Vw = 0.0180/dw, dw being the water density.It is useful to notice that both Eqs. (6) and (7) depend solely

n electrolyte parameters (nCA, rhCA). Through the SEHM model

nionic and cationic contributions are averaged out and con-idered equal. Thus the anionic (γA) and cationic (γC) activityoefficients are also equal. From the definition of the mean ionicctivity coefficient:

± = (γvCC γvA

A )1/(vC+vA)=(γvC+vAi )1/(vC+vA) = γi, i=C, A

(10)

o Eqs. (6) and (7) yield the ionic activity coefficients (γ i) ofhe electrolyte as well the mean ionic activity coefficient of thelectrolyte.

Finally the solution’s osmotic coefficient is given by

= − ln xW

1000VWc− κ3

6Dc(1 + κa)

+ πac

3

(2a2 + κ4

(Dc(1 + κa))2

)(11)

.1. Ternary systems with common ion

The first case to be considered is that of a system of two stronglectrolytes with a common ion, e.g. the system of CzC

v AzAv and

C A1zDvD

AzAvA2

. Knowing the parameters nCA, nDA for the two singlelectrolyte systems, the fist step for describing the system isnding a way to express the hydration of the mixed system. If

he molality of the first electrolyte is mCA and of the second is

r

Es

Equilibria 247 (2006) 1–7

DA, then the hydration of the system is described by applyinghe same reasoning as in a single electrolyte system [2]:

W = Z

N(12)

= 1 − ((vC + vA1)mCAhCA + (vD + vA2)mDAhDA)MW

(13)

= Z + ((vC + vA1)mCA + (vD + vA2)mDA)MW (14)

¯ CA = nCAxW

1 + xW(15a)

¯ DA = nDAxW

1 + xW(15b)

qs. (12)–(15) compose a system of three independent non-inear equations (Eqs. (13) and (14) can be combined in Eq.12)) with three unknowns (xW, hCA, hDA). The solution of thisystem provides the solution’s water mole fraction and the twoverall electrolyte hydration numbers.

The second step to describing the system would be to expresshe Eqs. (6), (7) and (9) for it. These equations however expresslectrolyte specific properties of a single electrolyte system. Inmixed system appropriate mixing rules are required, in order

o include the contributions from both electrolytes in a singlequation. The mixing rules used here, are those provided byatwardhan and Kumar [3,4], which assign to each electrolyten ionic strength factor yij:

CA = 0.5(mCz2C + mAz2

A)

Im(16a)

DA = 0.5(mDz2D + mAz2

A)

Im(16b)

bviously yCA + yDA = 1.These ionic strength factors can be used to provide the prop-

rty Xmix of the mixed system, from the properties of the con-tituting single electrolyte systems, by writing:

mix = yCAXCA + yDAXDA (17)

hese mixing rules are in compliance with the basic principlef the SEHM model, which treats anions and cations in anverage way, as if the electrolyte solution consisted of oneverage sized particle, hydrated an average hCA times. Whenixing two single electrolyte systems there would be two such

veraged sized particles (hydrated hCA and hDA times) that needo be combined in to a new average sized particle. The averageydration sphere radius of this “averagely mixed” particlerh

mix) will be produced from the combination of Eqs. (9) and17):

hmix = yCArh

CA + yDArhDA = amix

2(18)

q. (18) yields the distance of closest approach for the mixedystem, and by returning to Eqs. (6) and (10) for any ion (i=C,

hase

D

l

Ts

l

l

A

l

l

Efi(t

l

FitusemE

φ

2

D

tui

pc

t

x

Z

N

h

A

x

Z

N

h

Ep(ttb

oof

Y

w(1

X

Rt

Z

N

x

Tw

E. Balomenos et al. / Fluid P

, A) in the system:

n γLRi = − κ2

6Dc

(2κ

1 + κamix+ 1

amixln(1 + κamix)

)

+ �cαmix

3

(4α2 + κ4

(Dc(1 + κamix))2

)(19)

he ionic hydration coefficient in the single electrolyte CzCvC

AzAvA1

ystem will be:

n γHYDC = nCA ln

(2

1 + xW

)− ln N (20a)

n γHYDA = nCA ln

(2

1 + xW

)− ln N (20b)

nd for the single electrolyte DzDvD

AzAvA2

system:

n γHYDD = nDA ln

(2

1 + xW

)− ln N (21a)

n γHYDA = nDA ln

(2

1 + xW

)− ln N (21b)

qs. (20a) and (21a) can be used to describe the hydration coef-cients of the cations C and D in the mixed system, while Eqs.20b) and (21b) need to be combined through Eq. (17) in ordero produce the hydration coefficients of the common ion A:

n γHYDA = (yCAnCA + yDAnDA) ln

(2

1 + xW

)− ln N (22)

inally the third step in describing the mixed system is produc-ng an observable property of the mixed system and comparinghe SEHM’s results with experimental data. The propertysed for this purpose is the osmotic coefficient of the mixedystem, which can be measured directly (activity coefficients oflectrolytes in mixed electrolytes systems generally cannot beeasured directly) and in the SEHM model is calculated usingqs. (11) and (18):

= − ln xW

1000VWc− κ3

6Dc(1 + κamix)

+ πamixc

3

(2a2

mix + κ4

(Dc(1 + κamix))2

)(23)

.2. Ternary systems without a common ion

In a system of two strong electrolytes CzCvC

AzAvA

andzDvD

BzBvB

one cannot use the same mixing rules developed inhe previous section, since the definition of the system is notnequivocal. Indeed one can define the same exact system byntroducing the two electrolytes CzC

v′CBzB

v′B

and DzDv′

DAzA

v′A

at appro-

riate molalities. If the first system CzCvC

AzAvA

and DzDvD

BzBvB

isharacterized as I and the system CzC′ BzB′ and DzD′ AzA′ as II

vC vB vD vAhen Eqs. (12)–(15) are written for system I:

WI = ZI

NI(24)

r

Ul

Equilibria 247 (2006) 1–7 3

I = 1 − ((vC + vA)mCAhCA + (vD + vB)mDBhDB)MW

(25)

I = ZI + ((vC + vA)mCA + (vD + vB)mDB)MW (26)

¯ CA = nCAxWI

1 + xWI

, hDB = nDBxWI

1 + xWI

(27a,b)

nd for the system II:

WII = ZII

NII(28)

II = 1 − ((vC + vB)mCBhCB + (vD + vA)mDAhDA)MW

(29)

II = ZII + ((vC + vB)mCB + (vD + vA)mDA)MW (30)

¯ CB = nCBxWII

1 + xWII

, hDA = nDAxWII

1 + xWII

(31a,b)

qs. (24)–(31) provide two systems of 3 × 3 inde-endent non-linear equations, with unknown variablesxWI , hCA, hDB), (xWII , hCB, hDA). In order for the two sys-ems to be thermodynamically equivalent one must demandhat xWI = xWII , but with this restraint the combined systemecomes mathematically insolvable.

To overcome this problem, another set of mixing rules devel-ped by Patwardhan and Kumar [5] is introduced. If i is anyne of the two cations and j any one of the two anions, then theractional contribution factor Yij of the ij electrolyte is:

ij =(

0.5miz2i

Im

)(mj|zj|CHm

)+(

0.5mjz2j

Im

)(mi|zi|CHm

)(32)

here CHm is the system’s charge balanceCHm = mC|zC| + mD|zD| = mA|zA| + mB|zB|) and

∑i

∑jYij =

. The X property of the mixed system is then derived as

mix =∑

i

∑j

YijXij (33)

eturning to the system at hand, the solution’s water mole frac-ion is now defined as

= (YCA + YDB)ZI + (YCB + YDA)ZII (34)

= (YCA + YDB)NI + (YCB + YDA)NII (35)

W = Z

N(36)

he average hydration sphere radius of the mixed system (rhmix)

ill be produced from the combination of Eqs. (9) and (33):

hmix = yCArh

CA + yDArhDA + yCBrh

CB + yDBrhDB = amix

2(37)

sing the distance of closest approach from Eq. (37) theong range activity coefficient for any ion of the system is

4 E. Balomenos et al. / Fluid Phase Equilibria 247 (2006) 1–7

Figs. 1–6. Model results for various ternary electrolyte solutions with common ions. Concentration is measured in total molal ionic strength, yB is the ionic strengthfactor of the second electrolyte in the title, experimental data are represented with hollow shapes and model results with smaller black shapes and dotted lines.

hase Equilibria 247 (2006) 1–7 5

a

l

Fa(

l

l

l

l

Fi

φ

3

tcfimtir

fF

TAe

E

CCL

σ

Table 2Summary of model results for ternary systems with common ions

System σ =√

1n

∑n

i=1(yexpi − ymodel

i )2

Reference

KCl–CsCl 5.24E−03 [7]LiCl–CsCl 8.42E−02 [7]LiCl–KCl 3.56E−02 [8]LiCl–NaCl 1.22E−02 [8]LiCl–LiNO3 9.71E−02 [8]MgSO4–MgCl2 3.50E− 01 [9]NaCl–NaBr 1.56E−03 [10]NaCl–CsCl 2.50E−02 [11]NaCl–KCl 1.61E−02 [12]NaCl–MgCl2 5.59E−02 [13]NaCl–Na2SO4 7.81E−02 [9]MgCl2–NaCl 2.56E−02 [9]Na2SO4–MgSO4 1.43E−01 [9]

Table 3Summary of model results for ternary systems without common ions

System σ =√

1n

∑n

i=1(yexpi − ymodel

i )2

(xWI − xWII ) Reference

CsCl–Na2SO4 9.788E−03 6.595E−03 [14]LiCl–Na2SO4 3.667E−02 4.229E−03 [14]KCl–NaBr 2.030E−02 4.744E−03 [10]Na2SO4–MgCl2 2.160E−01 3.375E−02 [15]NaCl–MgSO4 1.569E−01 2.609E−03 [15]

Tf

4

stcttXitci

E. Balomenos et al. / Fluid P

gain defined:

n γLRi = − κ2

6Dc

(2κ

1 + κamix+ 1

amixln(1 + κamix)

)

+ πcαmix

3

(4α2 + κ4

(Dc(1 + κamix))2

)(38)

ollowing the reasoning of the previous section, the hydrationctivity coefficients for this system are defined based on Eq.33):

n γHYDC = (YCAnCA + YCBnCB) ln

(2

1 + xW

)− (YCA + YCB) ln N (39a)

n γHYDD = (YDAnDA + YDBnDB) ln

(2

1 + xW

)− (YDA + YDB) ln N (39b)

n γHYDA = (YCAnCA + YDAnDA) ln

(2

1 + xW

)− (YCA + YDA) ln N (39c)

n γHYDB = (YCBnCB + YDBnDB) ln

(2

1 + xW

)− (YCB + YDB) ln N (39d)

inally the observable osmotic coefficient of the mixed systems:

= − ln xW

1000VWc− κ3

6Dc(1 + κamix)

+ πamixc

3

(2a2

mix + κ4

(Dc(1 + κamix))2

)(40)

. Results

The mixing rules developed in the previous section wereested in several ternary strong electrolyte systems. Osmoticoefficient data from thirteen systems with common ions andve without common ions were compared with the SEHModel. The SEHM’s nCA values are taken from [1] and addi-

ional values for some electrolytes are given in Table 1, theonic radii data are taken from Marcus [6] and in all cases

w = (3VW/4�NA)1/3 = 1.9277 × 10−10 m at 25 ◦C.

The standard deviation of experimental osmotic coefficientsrom SEHM model results are tabulated in Tables 2 and 3. Inigs. 1–8 some of the results are presented graphically as well.

able 1dditional model results, fitting up to molalities shown in the third column,

xperimental data taken from [20]

lectrolyte nCA Fit up to m Max m σ Fitting γ± σ Evaluating ϕ

sCl 1.1323 6.0 4.11E−02 9.91E−03s2SO4 0.8804 1.8 1.08E−02 5.46E−02i2SO4 2.8383 3.0 3.29E−02 7.38E−02

=√

1n

∑n

i=1(yexpi − ymodel

i )2.

Ttr[tHtctwtsMnT

he third column indicates the mean absolute difference between the water moleractions of Eqs. (19) and (23).

. Discussion

The mixing rules of the SEHM model tested well in all theystem examined. Exceptions are the systems that containedhe MgSO4 electrolyte, but the deviations in those system arelearly attributed to the failure of the SEHM model to predicthe osmotic coefficient of the single electrolyte MgSO4 sys-em [1]. A more detailed investigation in the single electrolyte+x2 (SO4)x systems shows that while the SEHM fits the activ-

ty coefficients in every one of these system extremely well,he predictions for the osmotic coefficient are mediocre in theases of the univalent cations X: Na, Li, Cs and rather poorn the cases of the bivalent cations X: Mg, Cu, Zn (Fig. 9).his behavior can only be attributed to a poor estimation of

he systems’ water mole fraction caused by the fact that theeaction H2O + SO4

− = HSO4− + OH− (log K298 = −12.0043,

16]) was ignored. In the case of the bivalent cationic saltshe prediction is further worsen by the fact that the reaction

2O + X2+ = H+ + XOH+ was also ignored. In the case of Mg2+

he first hydrolysis reaction is characterized by an equilibriumonstant of log K298 = −11.44, [17]. Also for the MgSO4 sys-em, the possible formation of the complex ion MgHSO4

+

as ignored, an ion which if formed would cause a shift inhe H2O + SO4

− = HSO4− + OH− equilibrium towards the con-

umption of more water molecules (solubility studies of thegSO4–H2SO4 at high temperatures have shown that the domi-

ant ion in these systems is indeed the complex MgHSO4+ [18]).

hus the failure of the SEHM model to predict the osmotic coef-

6 E. Balomenos et al. / Fluid Phase Equilibria 247 (2006) 1–7

F mmos esentel

fitniee

S

l

wofmtN

Fiw

dspceeemdmp

igs. 7 and 8. Model results for various ternary electrolyte solutions without cotrength factor of the second electrolyte in the title, experimental data are reprines.

cient of the MgSO4 system is clearly attributed to the fact thathe complex chemical equilibria of the system were ignored. TheCA value for the MgSO4, mathematically is able to fit the activ-ty coefficient data, but physically it cannot account for weaklectrolytes or for complex ion associations between differentlectrolytes.

As far as the mixing rules are concerned, for an ion i theEHM model results in the general equation:

n γi = DH(I) +∑

j

εijf (I) (41)

here DH(I) denotes a Debye–Huckel term of some form, j ionsf the opposite charge, εij is an electrolyte specific parameter and

(I) is a function of ionic strength. Eq. (41) is consistent with

any successful semi-empirical models like the SIT-Theory,he Bromley Equation, the Pitzer ion interaction model and theRTL model [19].

ig. 9. The osmotic coefficients of single electrolyte X+x2 (SO4)x systems. Exper-

mental data from [20] are represented with hollow shapes and SEHM resultsith smaller black shapes and dotted lines.

tsdptoass

aw

LcDhImMNrVx

n ions. Concentration is measured in total molal ionic strength, yB is the ionicd with hollow shapes and model results with smaller black shapes and dotted

In the ternary systems without common ions the averageifference between xWI and xWII (indicated in Table 3) isurprisingly small, indicating perhaps that while the SEHM’sarameters are not ion specific [1], their thermodynamicombination yields almost ion specific results. Indeed an exactlectrolyte theory would deepen solely on ion specific param-ters, with clear physical meanings. The SEHM model requireslectrolyte specific parameters, simply because it lacks in ther-odynamic detail. The fact that ion-equivalent ternary systems

erived from different electrolytes, produce almost equal waterole fractions, proves that the SEHM is treading on the right

ath.The Semi-Empirical Hydration Model (SEHM) is thus true

o its name, since it describes the non-ideality of an electrolyteolution in a semi empirical way, using fitting parameters toescribe multi step hydration equilibria, but at the same timeroducing physical significant properties of the system, likehe mole fraction of the solvent or the osmotic coefficientf the solution. It is thus a model best suited for modelingnd studying the behavior of electrolyte systems rather thenimply predicting the activity coefficients of the ions in theystem.

The next step for the development of the SEHM model is itspplication at higher temperatures as well as in systems witheak electrolytes.

ist of symbolssummation of the system’s ionic molar concentrationD = 4π1000 NA

i hydration number of specie isolution’s ionic strength

i molal concentration of specie iW molar mass of the solvent (kg/moles water)

A Avogadro’s constant

i radius of specie ii molar volume of specie ii mole fraction of specie i

hase

yYz

Gα

γ

κ

ν

φ

SACmW±

Sch

A

wN

R

[

[[[[[

[

[

E. Balomenos et al. / Fluid P

ij ionic strength factor of electrolyte ijij fractional contribution factor of electrolyte ij, Eq. (19)i valance of the i-type ions

reek symbolsdistance of closest approach

i activity coefficient of specie iPitzer–Debye–Huckel reciprocal length [1]

i Grions of i-ions produced per mole of electrolytesolution’s omotic coefficient

ubscripts,B quantity referring to an anion,D quantity referring to a cation

quantity in molal concentration scalequantity referring to the solvent (water)mean ionic quantity

uperscriptsionic quantity (non-hydrated)hydrated quantity

cknowledgments

This work was carried out under the “HERAKLITUS” projecthich is co-funded by the European Social Fund (75%) andational Resources (25%).

[[

[

Equilibria 247 (2006) 1–7 7

eferences

[1] E. Balomenos, D. Panias, I. Paspaliaris, Fluid Phase Equilib. 243 (2006)29–37.

[2] H. Schonert, Ber. Bunsenges Phys. Chem. 94 (1990) 664–670.[3] V.S. Patwardhan, A. Kumar, AlChE J. 32 (1986) 1419–1428.[4] V.S. Patwardhan, A. Kumar, AlChE J. 32 (1986) 1429–1438.[5] V.S. Patwardhan, A. Kumar, AlChE J. 39 (1993) 711–714.[6] Y. Marcus, J. Chem. Soc. Faraday Trans. I 87 (1991) 2995–

2999.[7] R.A. Robinson, Trans. Faraday Soc. 49 (1953) 1147–1149.[8] R.A. Robinson, C.K. Lim, Trans. Faraday Soc. 49 (1953) 1144–

1147.[9] Y.C. Wu, R.M. Rush, G. Scatchard, J. Phys. Chem. 72 (1968) 4048–

4053.10] A.K. Convington, T.H. Lilley, R.A. Robinson, J. Phys. Chem. 72 (1968)

2759–2763.11] R.A. Robinson, J. Am. Chem. Soc. 74 (1952) 6035–6036.12] R.A. Robinson, J. Phys. Chem. 65 (1961) 662–667.13] R.F. Platford, J. Phys. Chem. 72 (1968) 4053–4057.14] R.A. Robinson, J. Sol. Chem. 1 (1972) 71–76.15] Y.C. Wu, R.M. Rush, G. Scatchard, J. Phys. Chem. 73 (1969) 2047–

2050.16] Outokumpu Research Oy, HSC Chemistry Ver 3.0, http://www.

outokumpu.fi/hsc, Finland.17] C.F. Baes, R.E. Mesmer, The Hydrolysis of Cations, John Wiley & Sons

Ltd., New York, 1976.

18] H. Liu, V.G. Papangelakis, Hydrometallurgy 79 (2005) 48–61.19] E. Balomenos, D. Panias, I. Paspaliaris, Miner. Process. Extr. Metall. Rev.

27 (2006) 1–60.20] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, second ed., Butter-

worths, London, 1959.