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This article was downloaded by: [University of Arizona] On: 10 November 2012, At: 10:23 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Chemical Engineering Communications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcec20 THERMODYNAMICS OF CONCENTRATED ELECTROLYTE SOLUTIONS YUNDA LIU a b , ALLAN H. HARVEY a b & JOHN M. PRAUSNITZ a b a Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California, 94720 b Chemical Engineering Department, University of California, Berkeley, California, 94720 Version of record first published: 06 Apr 2007. To cite this article: YUNDA LIU, ALLAN H. HARVEY & JOHN M. PRAUSNITZ (1989): THERMODYNAMICS OF CONCENTRATED ELECTROLYTE SOLUTIONS, Chemical Engineering Communications, 77:1, 43-66 To link to this article: http://dx.doi.org/10.1080/00986448908940172 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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Page 1: Liu-Prausnitz (1989) Thermodynamics of Concentrated Electrolyte Solutions

This article was downloaded by: [University of Arizona]On: 10 November 2012, At: 10:23Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Chemical Engineering CommunicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcec20

THERMODYNAMICS OF CONCENTRATED ELECTROLYTESOLUTIONSYUNDA LIU a b , ALLAN H. HARVEY a b & JOHN M. PRAUSNITZ a ba Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley,California, 94720b Chemical Engineering Department, University of California, Berkeley, California, 94720Version of record first published: 06 Apr 2007.

To cite this article: YUNDA LIU, ALLAN H. HARVEY & JOHN M. PRAUSNITZ (1989): THERMODYNAMICS OF CONCENTRATEDELECTROLYTE SOLUTIONS, Chemical Engineering Communications, 77:1, 43-66

To link to this article: http://dx.doi.org/10.1080/00986448908940172

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

Page 2: Liu-Prausnitz (1989) Thermodynamics of Concentrated Electrolyte Solutions

Chem. Eng. Comm. 1989, Vol. 77, pp. 43-66Reprints available directly from the publisher.Photocopying permilled by license only.© 1989 Gordon and Breach Science Publishers S.A.Printed in the United States of America

THERMODYNAMICS OF CONCENTRATEDELECTROLYTE SOLUTIONS

YUNDA LIU,t ALLAN H. HARVEyt and JOHN M. PRAUSNITZ

Materials and Chemical Sciences DivisionLawrence Berkeley Laboratory

and

Chemical Engineering DepartmentUniversity of California

Berkeley, California 94720

(Received January 27, 1988; in final form August 2, 1988)

A new equation is derived for the activity coefficient of an electrolyte in a liquid solvent (such aswater) as a function of electrolyte concentration. This equation contains contributions fromlong-range electrostatic forces and from short-range attractive forces; it holds from high dilution to thesolubility limit. The new equation is based upon extended Oebye-Hiickel theory for long-rangeeffects and upon the local-composition concept for short-range effects. For a single-electrolytesolution, the new equation contains only one adjustable energy parameter and three other parametersthat are ion-specific, not electrolyte-specific. Therefore, extension to multicomponent electrolytesolutions follows without additional assumptions. Ion-specific parameters are reported for H+, K+,U+. Br" and CI- ions. Calculated and experimental activity coefficients are in excellent agreementfor aqueous solutions of HBr. HCl, KBr, KCI, LiBr and LiC!.KEYWORDS Thermodynamics Electrolytes Aqueous solutions Activity coefficients.

1. INTRODUCfION

In the last few decades, much progress has been made in representing thethermodynamic properties of electrolyte systems for engineering application.Chen et al. (1983), Pitzer (1981), Maurer (1983) and Renon (1986) give usefulreviews.

There are two fundamentally different methods for calculating vapor-liquidequilibria and related thermodynamic properties (Prausnitz et al., 1986). One isbased on activity coefficients for the liquid phase and fugacity coefficients for thevapor phase, and the other is based on fugacity coefficients for both phasesobtained from an equation of state. In spite of potential advantages provided byan equation of state, activity-coefficient models have attracted more attention forelectrolyte systems. In developing activity-coefficient models for electrolytesystems, the most popular and successful practice is to combine the electrostatic

t Department of Chemical Engineering Design. Chalmers University of Technology. S-41296Gothenburg, Sweden.

; Present address: Therrnophysics Division, NIST, Gaithersburg. MD 20899.

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44 Y. LID, A.H. HARVEY AND 1.M. PRAUSNITZ

theory of Debye and Hiickel with modifications of well-known methods fornon-electrolyte systems. (A different approach is that of Pailthorpe et al. (1984),who solve a Poisson-Boltzmann-like equation which is modified by a short-rangeion-solvent interaction). We follow the popular practice here but with someimportant modifications beyond those used by others.

Pitzer (1973; Pitzer and Mayorga, 1973) proposed a successful model based oncombination of an extended Debye-Hiickel form (which accounts for long-rangeforces) with a virial expansion (which accounts for short-range forces). Later,Cruz and Renon (1978), Chen et al. (1982) and Christensen et al. (1983) replacedthe virial expansion with various flexible local-composition models.

In the Cruz-Renon model, the excess Gibbs energy is assumed to be the sumof a local-composition term (NRTL equation), a Debye-Hiickel term, and aBorn contribution arising from the composition dependence of the dielectricconstant of the mixture. In the Chen model, the excess Gibbs energy is the sumof a NRTL term and a term of the Debye-Hiickel form. In Chen's model, twoassumptions are made: "local electroneutrality" and "like-ion repulsion".

Although both the Cruz-Renon model and the Chen model use the NRTLequation to account for short-range forces, there is an important differencebetween them. Cruz and Renon suggest that ionic species are completely solvatedby solvent molecules. Chen et al. note that that suggestion becomes unrealisticwhen applied to high concentrations, since the number of solvent molecules isinsufficient to solvate ions completely. In the Chen model, all ions are,effectively, completely surrounded by solvent molecules in the very dilute rangebut they are only partially surrounded by solvent molecules in the high­concentration range. That is the viewpoint adopted in this work.

We develop a new activity-coefficient model similar to that of Chen but basedon a more reasonable combination of Debye-Hiickel theory and the local­composition concept.

2. THERMODYNAMIC BACKGROUND

We seek an expression for the excess Gibbs energy. For simplicity, we considerhere aqueous single-electrolyte systems. Multi-electrolyte systems can be handledin the same way.

Throughout this paper, the concentration scale is the true mole fraction xdefined as

(2-1)

where nk is the number of moles of species k per unit volume; nr is the total molenumber of all species per unit volume, i.e., the true mole density of theelectrolyte solution:

Here the sum is over all ionic and molecular species. Equation (2-1) is based onthe assumption that all electrolytes are completely dissociated.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 45

Consider a salt with the formula Cw,A wo where We and Wa are stoichiometriccoefficients. The dissolution equation is written

Cw,A wo~ weC + waA

Dissociation is described by the equation of equilibrium

(2-2)

We relate the chemical potential of an ion to its true mole fraction through theactivity coefficient

Ile = Il~ + RT In YeXe

Ila = Il~ + RT In YaXa

(2-3a)

(2-3b)

where Ye and Ya are the individual ionic activity coefficients of cations and anions,respectively. R is the gas constant, and superscript a denotes the standard state.

To obtain an expression for Ilea' we write

and we define

We then obtain

Y = (yw,ywo)I/(Wc+WD)±,CQ c a

(2-4)

(2-5)

(2-6)

(2-7a)

For a real, aqueous, single-electrolyte system, the Gibbs energy, G(<cal), isgiven by

(2-8)

where nea is the number of moles of electrolyte ca in the liquid phase. Thechemical potential of water is related to its true mole fraction through its activitycoefficient by

Ilw = Il':" + RT In YwXw (2-3c)

For an ideal, aqueous, single-electrolyte system, the Gibbs energy, G(lde,,')' isgiven by

where

and

Il'w = Il':.- + RT In Xw

Il~ = Il~ + RT In Xc

u; = Il~ + RT In X a

(2-9)

(2-lOa)

(2-lOb)

(2-lOc)

(2-7b)

Comparing (2-9) with (2-8), we see that GOdc,,') is equivalent to G(<cal) when allactivity coefficients become unity.

For water, we choose the pure liquid as the reference state where Ilw = Il':". Itfollows from Eq. (2-3c) that

Yw~ I as Xw~ 1 (2-lla)

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46 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

(2-12)

The reference state for the cation is a hypothetical ideal dilute solution of unitconcentration; we follow a similar convention for the anion. Therefore:

y;-l, y:-l, and Y:.ca-l as xc-O,xa-O, andx±.ca-O (2-11b)

As shown by (2-11a) and (2-11b), the activity coefficients of water and soluteare not normalized in the same way. Equations (2-11a) and (2-11b) give theunsymmetric convention; we have used an asterisk to denote the normalizationgiven in (2-11b).

The unsymmetric excess Gibbs energy, CE', is defined by

C E ' - C* C*- (rcal) - (ideal)

=n.R'T In Yw + n.R'T In y; + naRT In y:=nwRT In Yw + nea(we + wa)RT In Y:.ea

3. NEW MODEL

For mixtures of liquid nonelectrolytes, local-composition models (Wilson, 1964;Renon and Prausnitz, 1968; Abrams and Prausnitz, 1975) often representactivity-coefficient data well. But if a local-composition model is used to accountfor the contribution from short-range forces, we expect that the long-rangeelectrostatic forces must influence the local composition. In contrast to previousmodels, our model takes into account the effect of long-range electrostatic forceson local composition.

Let us arbitrarily choose an ion in the solution which, for simplicity, containsonly a single electrolyte. The local composition in the immediate vicinity of thatcentral ion is determined by the overall composition of the mixture, by the sizesof the species in the solution and by the energies of interaction between thecentral ion and all species (ions or uncharged molecules) in the first coordinationshell. Figure I shows a cation C, surrounded by nearest neighbors that form a firstcoordination shell. The distance between the center of C, and the outer boundaryof the first coordination shell is r~. Some ions, such as C, and A j , are located

outer boundary of the firstI coordination shell

O/ '-D-' " 0 6) 0r \

/ rc

\ 0 00\ 100-'0 / 0 \J.J 00 , ",--_/

FIGURE 1 Long-range and short-range interaction in a single-electrolyte system.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 47

(3-2)In Y~.ea = In Y;a.LR + In Y;a.SR

outside the first coordination shell, i.e., their distances from cation C are largerthan r~. It is assumed that these ions interact only electrostatically with C. Theinteraction energy between central cation C, and anion Ai in the first coordinationshell is given by Eea, which includes contributions from both short-range andelectrostatic forces.

Since Eea includes an electrostatic contribution, the unsymmetric excess Gibbsenergy, CE", is given by the sum of two contributions

CE" = Cf~ + Cf~ (3-1)

where Cf~ is the contribution from long-range electrostatic interaction forcesbetween each central ion and all ions outside the first coordination shell; C§~ isthe contribution from both the short-range interaction forces of all kinds and thelong-range electrostatic forces between each central ion and all ions inside thefirst coordination shell.

For Cf~, we use a modified Debye-Huckel expression. For Cf~, we use alocal-composition expression similar to those proposed by Renon and Prausnitz(1969) and Chen et al. (1982). Through the usual thermodynamic relations, weobtain

where Y~.ea is the unsymmetric mean activity coefficient of the electrolyte (ca) ona true mole fraction scale.

4. LONG-RANGE INTERACTION CONTRIBUTION

The long-range interaction contribution is represented by a modified Debye­Hiickel expression. The modified Debye-Hiickel expression is based on classicalPoisson-Boltzmann theory and the local-composition concept.

We consider first the non-ideal behavior of the cation. We consider twocontributions: first, the long-range contribution from electrostatic interaction withions outside the coordination shell; second, the contribution from electrostaticinteraction with ions inside the first coordination shell plus the contribution frominteraction with all species in the first coordination shell due to short-rangeforces. Then we may write the chemical potential of one mole of cations, u., as

(4-1)

(4-2)

/-Ie = /-I~ + RT In Xc + RT In Y;.LR + RT In Y;.SR

From Appendix A, we obtain

• vee2

[( VaXaZe )eXP[K(r~- rJl ]In~u= ~+ -~

. 2DrekT X a+ xwCwe.ae (1 + Kr~)

where Y;' LR is the individual activity coefficient of the cation arising from thelong-range contribution; Ve and Va are the algebraic valences of the cation andanion, respectively; k is Boltzmann's constant; e is the protonic charge; D is thedielectric constant; K is the inverse Debye length defined by (A-to); z, is thecoordination number of the cation; r; is the radius of the cation; r~ is the outerradius of the coordination shell; Cwe.ae is defined by (B-7).

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Page 7: Liu-Prausnitz (1989) Thermodynamics of Concentrated Electrolyte Solutions

48 Y. LIU, A.H. HARVEY AND .r.M. PRAUSNITZ

Similarly for anions, we obtain

• vae2

[( VcXcZa )exP[K(r~-ra)]In Ya LR = Va +--'---"---"---. 2DrakT ,Xc + xwGwa,ca (1 + Kr~)

From (2-5), (4-2) and (4-3), we have

• 'e2

{WcVc[( VaXaXc )exP[K(r~- rc)]In YcaLR= -- vc + - -----==----=---=---, 2DkT(ta; + wa) rc Xa+ xwGwc,ac (1 + Kr~)

WaVa[( VeXcZa )exP[K(r~ - ra)]+-- v +-----''---''---''---'0 a Xc + xwGwo.ca (1 + Kr~)

(4-3)

(4-4)

(5-2)

Equation (4-4) is a significant modification of the Debye-Hiickel equation.Equation (4-4) gives only the contribution due to electrostatic forces betweeneach central ion and the ions outside its first coordination shell, while the originalDebye-Hiickel equation gives the contribution due to electrostatic forcesbetween each central ion and all other ions in the solution.

5. SHORT-RANGE INTERACTION CONTRIBUTION

The local-composition concept is used to account for contributions fromshort-range forces of all kinds, including the electrostatic interaction forcesbetween each central ion and all ions inside its first coordination shell.

Our local-composition expression differs from that of Chen in three ways:First, our local-composition expression is based on a previous derivation of the

three-parameter Wilson equation (Renon and Prausnitz, 1969). We apply thelocal-composition concept to the excess enthalpy H'fR; we then integrate theGibbs-Helmholtz equation

(5-1)

to obtain G'fR' We use the boundary condition G'fR =0 when I/T =O.Second, we abandon the second assumption of the Chen model wherein the

distribution of cations and anions around a central solvent molecule is such thatthe net local ionic charge is zero. While this assumption may hold for theconcentration range near the fused pure electrolyte, we see no reason to assume"local electroneutrality" for the dilute and intermediate concentration range.

Third, as suggested by Guggenheim's quasi-chemical lattice theory, we use aclassical definition of the local composition; in our work the definition of the localcomposition is

Xij Xi exp( - £ii/kT)-=Xki Xk exp(-£k/kT)

where Xii is the local mole fraction of i in the first coordination shell surrounding

t Energy parameter gij is the product of Eij and Avogadro's number.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 49

species j and elj is the characteristic interaction energy between one nearest­neighbor i-j pair. t Equation (5-2) is similar to that used by Chen except thatChen introduces into the argument of the exponential an arbitrary nonrandom­ness factor.

From Appendix B, we obtain

• 1[ (XwGcw.ww )InYc,sR=-Z- z., G G -xw-Gcw.ww+1X w +xc cw,ww +xa QW,WW

(5-3)

X w- Gaw. ww+ 1)

(5-4)

From (2-5), we obtain

In Y:a,SR1

Finally, from (3-2), we have the complete expression for the mean ionic activitycoefficient of an electrolyte in an aqueous single-electrolyte system. Further, withthe Gibbs-Duhem equation, we can obtain the activity coefficient or osmoticcoefficient of water in the solution.

The sum of Eqs. (4-4) and (5-5) gives In Y~,ca' To use these equations werequire the following parameters:

1. radii rc and ra for the cation and anion, respectively. Anionic radius ra comesfrom crystal data. In this work, we consider only two anions; for CI- theradius is 0.181 and for Br" it is 0.196 nanometers. From fitting binary data, weobtain five sets of cationic radii, shown in Table II. Each set corresponds to aseries of fixed coordination numbers, shown in Table I.

2. energy parameters characterizing the interaction between two species. Forexample, gcw refers to the interaction between cation (C) and water (W).However, these parameters always appear as differences according to

Gji.kl = exp[ -(gjl - gki)/RT]

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50 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

TABLE I

The Influence of Coordination Number on the Simultaneous Fit of Mean-Ionic-Activity-CoefficientData for Aqueous Electrolytes at 25°C

Set Zw 1 11+ ZK+ ZLi-+ ZOr- ZO- °lnY±.c<I

[I] 4.5 4.5 4.5 4.5 4.5 4.5 0.017/21 10.0 10.0 10.0 10.0 10.0 10.0 0.01813] 4.5 4.0 4.0 5.0 5.0 4.0 0.01814] 4.5 4.0 4.0 6.0 7.0 3.0 0.Ql5151 4.5 4.0 4.0 6.0 7.0 3.0 0.016

Maximum molalities are 11.0(HBr). 16.0(HCI). 5.5(KBr). 5.0(KCI). 20.0(LiBr) and 19.2(LiCI).A=1.5 for all electrolytes except HBr where A=1.70.In Set151. the true density is fixed at 60 mol/L.

TABLE 11

Effective Radii for Cations (A) from Simultaneous Fit of Mean-Ionic-Activity-Coefficient Data forAqueous Electrolytes at 25°C

Set H+ K+ u+

[I] 0.6991 0.6753 0.769812] 0.6582 0.5542 0.708413] 0.6835 0.6458 0.744714] 0.5924 0.6154 0.6956151 0.5241 1.0301 0.5909

All sets as in Table I.

where subscript (i) denotes the species at the center surrounded by acoordination shell containing species j and/or k (If i stands for water, thenspecies i may also be in the first coordination shell.).

3. the dielectric constant as discussed in Section 6.

To illustrate. consider an aqueous solution of LiCI. We need radii for Li" andCl". The dielectric constant is calculated as shown in Section 6. Further, werequire guCl, guw and gClw' The first of these must be obtained from experimentaldata for LiCI solution but the others could be obtained from experimental datafor other salts where lithium is the cation or where chloride is the anion.

6. THE DIELECTRIC CONSTANT

Although there has been some experimental study of the concentration depend­ence of the dielectric constant of electrolyte solutions, few attempts have beenmade to take it into account when modeling deviations from ideality forelectrolyte solutions. Triolo et al. (1976) tried to improve the primitive model

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 51

TABLE III

Numerical Values of the Constants in Eq. (6-1) for the Dielectric Constant

8, = 7.62571Hz = 244.0038 3 = -140.5698 4 = 27.78418 5 ~ -96.28058 6 = 41.7909B; = -10.20998 8 = -45.20598 9 = 84.63958 10 = -35.8644

within the framework of the Mean Spherical Approximation (MSA), using eithera two-parameter, density-dependent dielectric constant or an adjustable (but notdensity-dependent) dielectric constant up to 2 m. But the calculated values of thedielectric constant are not realistic, since they are greater than that for purewater. In the concentration range to 6 m, an extension of Giese's equation (Gieseet al., 1970) was used by Ball et al. (1985) for the MSA model of electrolytesolutions. In the Cruz-Renon model, the expression of Pottel (1973) was used toestimate the variation of the dielectric constant with concentration.

Based on the assumption that ions are dielectric holes, we assume that at highelectrolyte concentrations, water becomes steam-like. Therefore, we use here anexpression for the dielectric constant based on an equation of Uematsu andFranck (1980) that represents the experimental dielectric constant of gaseous andliquid water as a function of temperature and density. In this work, we changedensity dependence to apparent-mole-fraction dependence. When calculating theapparent mole fraction, an electrolyte is considered as a molecular component; inother words, for calculating the dielectric constant, the electrolyte is consideredto be undissociated.

The calculated dielectric constant D is a function of temperature andelectrolyte concentration but it is the same for all electrolytes in water:

D = 1 + (B,/1;.)(1 - xca ) + (B2/1;. + B3 + B,1;.)(1- xca ) 2(6-1)

+ (8511;. + B61;. + 87 T~)(1 - X ca)3 + (B8IT~ + B9/1;. + 8 10)(1 - xca ) '

where Xca is the apparent mole fraction of electrolyte ca; 1;. = T /298.15 with TinKelvins; Table III gives constants 8 as reported by Uematsu and Franck.

7. IONIC RADII

In expressions of the Debye-Hiickel form, it is usually assumed that all ions havethe same size. This assumption is reasonable only when electrolyte solutions aredilute, where the distances between ions are so large that the influence of ionicsize can be neglected. There is much evidence that, for models covering the high

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52 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

concentration range, results are sensitive to the values chosen for ionic radii (e.g.Scatchard, 1961).

It has been customary to use crystal ionic radii for ionic radii in solution.Several sets of crystal radii are available; the most widely used are due toGoldschmidt (1926), Pauling (1927), and Gourary and Adrian (1960). In thepresent model, for anions, we choose Goldschmidt's radii. For cations, we chooseradii fit from activity-coefficient data. However, to obtain radii from activity­coefficient data, it is necessary to assign values to the coordination numbers.

Table II shows some results of the fit. For potassium ion, the values of 'K arenot all realistic, perhaps because there is some distortion in the simultaneous fitwhen the hydrogen ion is included. Based on complete dissociation, the fit iscalculated as though the hydrogen ion were a bare (not hydrated) proton. Whenwe fit KBr or KCI separately, we can obtain much more realistic results for 'K'

These results show that special care must be exercised when the cation is H+which is probably better represented by H30+.

8. DATA CORRELATION AND DISCUSSION

To obtain binary parameters and cationic radii, we use experimental mean-ionic­activity-coefficient data (Hamer and Wu, 1972) for six aqueous single-electrolytesystems (HBr, HCl, KBr, KCI, LiBr, LiCI). [The Hamer-Wu activity coefficientsare based on a molality concentration scale; the conversion between this and ourtrue-mole-fraction scale can be found for example in Chen et al. (1982).] We usethese data to obtain 11 energy parameters and 3 cationic radii; they are obtainedby least-square analysis on deviations between calculated and experimentalquantities:

s = '" (In y •. calc - In y •. cXP.) 2Y L..J ±,ca,l ±.ca,/

i(8-1)

These six electrolytes were chosen because first, the activity coefficient curves forLiBr and LiCI are different in type from those for KBr and KCI as shown inFigures 2 and 3; second, there is a large difference in the maximum molalities forthese two types. Therefore, fitting data for these four systems to the presentmodel provides a stringent test. Third, aqueous HBr and HCI data were used toinvestigate the behavior of H+ in the present model.

The classical Debye-Hiickel expression uses the density of water instead ofthat of the solution. In the high-concentration range, this approximation maycause large deviations. To investigate this effect, we initially chose a version ofClarke's density model (Aspen Technology, Inc., 1984) with parameters deter­mined from three sources of experimental density data (Haase et al.• 1965;Lengyel et aI., 1964; Bogatykh and Evrovich, 1965) to calculate the densities ofthese six aqueous electrolytes. We found, however, that the true mole density,nT, does not vary significantly among the six electrolytes or with electrolyteconcentrations. If we fix the value of the true mole density as 60 moles per liter,the deviations are within ±5% for concentrations to 20 m. It is therefore not

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 53

1.00~ ,

'c.D.

Ow 0.50 E- b. e'0

E- '0. .:b -;

0, • o'lJ

/00, ~.PI Col~d

I ',",0'

UCI 0

~ KCI ° P

"y± 10 f-,

0

°,

,0',0-

,0-

I~.a -0.5

, , ,

0 0.05 0./0 0.15 0.20 0.25Mole Fraction of Ions

FIGURE 2 Mean ionic activity coefficient and water activity for aqueous lithium chloride andpotassium chloride at 25'C.

surprising that the fit of activity coefficients using the fixed molar density is almostas good as that using measured values of the true density. [See case [5] in Table I.]

For the system H20-LiBr at 25°C, Figure 4 shows the modified Debye­Huckel contribution and the local-composition contribution to the mean ionicactivity coefficient. Unlike the original Debye-Huckel expression, our long-rangecontribution to In Y;a (Eq. (4-4» becomes positive at high ion concentrations.This occurs because, at high concentrations, the average number of counterions

11

10",

}.o,P',,,

[J'

,d'

,c,d'

0"

O',a"O-'

I ~foo-~'O~'~::-~.~'.J..L....L..........J..L...........J'0.5.~o 0.05 0.10 0.15 0.20 0.25

Mole Fruction of Ions

E..pl cereeLiS' 0

roo b:- 1<8r

Ow 0.50

FIGURE 3 Mean ionic activity coefficient and water activity for aqueous lithium bromide andpotassium bromide at 25°C.

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54 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

o E_porimentQI- Calculoted- - - Modified O-H Contribution

Local-CompositionContribution

0.200.150.05 0.10

II

II

//

//

/../(, ./

/

,/

oMole Fraction of Ions

FIGURE 4 Contributions to mean ionic activity coefficient for aqueous lithium bromide at 25°C.

in the coordination shell becomes large enough that the solution outside thecoordination shell has a charge of the same sign as that of the central ion.

To obtain energy parameters from experimental activity-coefficient data, it isnecessary to specify coordination numbers as shown in Table I. The last columnin Table I gives the standard deviation in fitting experimental activity coefficientsY~,ca to the new equation. For all five sets, the goodness-of-fit is about the same.

The size constant A is used to determine the radius of the coordination shellshown in Figure 1: r~ = r; + Ara • (Similarly, r~ = ra + ArC")

Table IV gives energy parameters gjilR at 25°C upon setting gww = O.Table V gives a comparison between the present model and that of Chen.

Unfortunately, the comparison is not as strict as we might wish because resultsfrom the Chen model are obtained from his paper (Chen et al., 1982), which uses

TABLE IV

Interaction Energy Parameters, gjJ R (K), from Simultaneous Fit of Mean-Ionic-Acitivity-CoefficientData for Aqueous Electrolytes at 25°C. upon Selling gww= 0

Br- CI- H2O

Set[4] Setl4\' Set[4] Set[41' Setl4] Set[41'

H+ -113.86 186.34 14.0 -59.75 -113.33 -137.12K+ -203.85 -190.34 -195.82 -175.19 -287.89 -270.69Li+ -260.34 -185.66 -274.89 -190.28 -494.51 -443.50H,o 315.38 325.13 296.69 316.13 0.00 0,00

Set[4] as in Table I.Set[41' is the same as Set[4] except that maximum molalities are 6.0(HBr), 6,0(HCI), 5.5(KBr),

5,0(KCI), 6.0(LiBr) and 6.0(LiCI).

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 55

TABLE V

Comparison Between the-Present Model and the Chen Model (Results from Fit of Mean-Ionic­Activity-Coefficient Data of Aqueous Electrolytes at 25°C)

Set[4] Set[41' Chen Model

°lnY:t,..<Imaxm °In Y±,<"a max.m °In Y±,ca maxm

HBr 0.013 11.0 0.005 6.0 0.015 3.0HCI 0.023 16.0 0.003 6.0 0.035 6.0KBr 0.005 5.5 0.004 5.5 0.004 5.5KCI 0.007 5.0 0.005 5.0 0.003 4.5LiBr 0.012 20.0 0.004 6.0 0.050 6.0LiCI 0.029 19.2 0.004 6.0 0.040 6.0

Set[41 and Set[41' as in Tables I and IV.

data sources different from ours. Nevertheless, for most systems studied here, thepresent model gives a much better representation of the experimental data.

With a numerical integration of the Gibbs-Duhem equation, we obtain wateractivities at 25°C for aqueous solutions of the six electrolytes, shown in Figures2,3 and 5. Table VI gives the standard deviations of these calculated wateractivities.

9. CHARACTERISTICS OF PARAMETERS

Whereas the parameters in the Chen model are salt-specific, those in thepresent model are ion-specific. There is only one adjustable parameter, gca, for

FIGURE 5 Mean ionic activity coefficient and water activity for aqueous hydrogen chloride andhydrogen bromide at 25°C.

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56 Y. LIU, A.H. HARVEY AND 1.M. PRAUSNITZ

TABLE VI

Standard Deviations of Water Activities at 25'C for Aqueous Solutions 'of Six Electrolytes. WaterActivities Calculated Using Gibbs-Duhem Equation and Mean Activity Coefficients for the

Electrolytes

Set[4]

G•• rnax.m

HBr 0.0066 11.0HCI 0.0103 16.0KBr 0.0007 5.5KCI 0.0012 5.0LiBr 0.0061 20.0LiCI 0.0143 19.2

Set[4] as in Tables I and IV.

'" [(a CDI, _ aexP)/aexp]2LJ W,f W,/ "',I

i

Y(n -p -1)

where n is the number of data points and p is the number of adjustable parameters. In above cases,p ~o.

each electrolyte; other fitted parameters (gcw, g.w, 'c) are common to allelectrolyte systems containing the same cation and/or the same anion. In thepresent model, all interaction-energy parameters appear as differences, i.e.,gji - gki> therefore, we can arbitrarily fix one interaction energy parameter; wechoose gww = O. All other interaction-energy parameters are relative values.

In view of the semi-empirical nature of our model, we cannot assign clearphysical significance to the parameters. We may, however, notice some reg­ularities. The present model suggests that the attraction energies between variouspairs are in the order: gaw > gca > gcw (Table IV). Also, from Table IV, we seesome reasonable attraction-energy sequences: gKB, > gLiB,; gKCI > gLiCO andgKw > gLiw'

Table VII indicates the effect of temperature on the fit of experimental data(Caramazza, 1960) with our (temperature-independent) parameters. Table VII

TABLE VII

Effect of Temperature on Standard Deviation of Mean Ionic Activity Coefficient for AqueousPotassium Chloride Using Temperature-Independent Parameters Obtained at 25'C

T ('C) 0 18 25 35 50

G 0.056 0.017 0.005 0.037 0.088

Maximum molality is 4.0 m.Parameters are those in Set[4] at 298.15 K.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 57

shows that for a good fit the parameters must be somewhat dependent ontemperature but we have not investigated that dependence here.

10. ION ASSOCIATION

Electrolytes may dissociate partially or completely in solution. With nsmgtemperature, the dielectric constant decreases and association increases. Withincreasing electrolyte concentration, association rises due to a mass-action effectand a lower dielectric constant.

Molecular-thermodynamic models for electrolyte systems can be divided intotwo limiting categories: one is based on the assumption of complete dissociationof electrolytes and the other on chemical equilibria to describe the extent of ionassociation. Although the chemical-association models may be regarded as morerealistic and flexible (Helgeson, 1981), they have serious disadvantages (Pitzerand Mayorga, 1974), especially upon extension to mixed-electrolyte systems.

The complete-ionization assumption has been used consistently in previouswork based on the local-composition concept. It has therefore become customaryto ascribe inadequacies of local-composition correlations to ion association.However, it appears that the local-composition models can generally representthe contribution due to the short-range forces even when there is some ionassociation.

For nonelectrolyte systems, the UNIFAC model (Fredenslund et al., 1977),based on the local-composition concept and the group-contribution method,often provides a good approximation. If a molecule can be divided into groups,we can similarly divide the ion pair into ionic "groups" and then use the "groups"in our expression representing the contribution from the short-range forces.Therefore, ion association mainly affects the expression representing the con­tribution from the long-range forces. In the present model, although assuming

wcter 01 center

anion at center cotion 01 ceere-

FIGURE 6 The three types of cells in a single-electrolyte system.

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58 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

complete dissociation, the modified Debye-Hiickel expression accounts indirectlyfor ion association, as indicated in Appendix A.. In effect, the present model makes a detour around the problem of ionassociation. The Cruz-Renon model uses a known dissociation constant and sixparameters to describe the vapor-liquid equilibrium of the HCI-H20 system at25°C for acid composition from infinite dilution to 18 m. By contrast, as shown inFigure 3, the present model uses only one specific adjustable parameter inaddition to two common parameters and one effective cationic radius. Thepresent model represents the mean ionic activity coefficient of the HCI-H20system at 25°C for concentrations from infinite dilution to 16 m. For this system,the specific parameter is gHel and the two common parameters are gHw and gClw'

11. A THEORETICAL CONSIDERATION

We recognize that there is some inconsistency in using a long-range electrostaticcontribution (derived in a McMillan-Mayer framework) in conjunction with anexcess Gibbs energy model which implies a Lewis-Randall framework. This issuehas recently been examined by Cardoso and O'Connell (1987); they find that theinconsistency has no effect on the thermodynamics of single-solvent systems.However, extension of our model (and all similar models) to mixed-solventsystems requires special care; Cardoso and O'Connell give the proper procedure.

12. CONCLUSION

A new molecular-thermodynamic model has been proposed for aqueouselectrolyte solutions. It goes beyond previous models because it is based on asemi-theoretical combination of Debye-Hiickel theory and the local-compositionconcept and because it is designed to represent the properties of a wide variety ofelectrolyte systems over the entire range of electrolyte concentration. Thus far,application of the new model indicates encouraging results. However, the resultsshown here are clearly preliminary because it is not yet clear that the ion-specificparameters reported here may be used for other salts. Much further study isrequired before we can claim success for a model for electrolyte solutions thatuses primarily ion-specific (instead of salt-specific) interaction parameters.

ACKNOWLEDGEMENTS

This work was supported by the Director, Office of Energy Research, Office ofBasic Energy Sciences, Chemical Sciences Division of the U.S. Department ofEnergy under Contract No. DE-AC03-76SF00098. Y. Liu is a recipient of a grantfrom The National Swedish Board for Technical Development. For helpfuldiscussions, the authors are grateful to Prof. U. Gren, M. Wimby and Dr. C.-c.Chen.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 59

NOMENCLATURE

D dielectric constant

G Gibbs energy

Gji.ki quantity defined by (B-7)H enthalpy

No Avogadro's number

R gas constant

T absolute temperature (K)

e protonic charge

Bn interaction energy parameter of j-i pairs

k Boltzmann constant

m molalityn, molar density of species i (except in Appendix A, where n, denotes

number density)

r the distance from the ion which is the origin of the coordinates

rJra ) radius of cations (anions)

r' outer radius of the first coordination shell

Xi true mole fraction of species i based on all species (molecular and ionic)

Xji local mole fraction of species j where species i is the center

z coordination number

Greek Letters

y activity coefficient on true mole-fraction scale

Eji interaction energy for one j-i pair

K inverse Debye length defined by (A-lO)

A size constant in (A-4)

Ilk chemical potential of species k

Ve> Va algebraic valences of cations and anions, respectively

p; charge density at distance r from the cation which is the origin of thecoordinates

a standard deviation

1jJ the electric potential

We> W a numbers of cations and anions, respectively, produced by the completedissociation of one electrolyte molecule

Superscripts

E excess properties

ideal solution

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60

o•00

LRSR

Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

standard state

unsymmetric convention

infinite dilution

long-range

short-range

Subscripts

a anion

c cation

ca electrolyte ca

m molar quantity

w water

APPENDIX A

DERIVATION OF THE MODIFIEDHOCKEL EXPRESSION

DEBYE-

(A-I)

Based on Poisson's equation of electrostatic theory, the Boltzmann distributionlaw and the local-composition concept, the modified Debye-Hiickel expression isto account for the long-range electrostatic-interaction contribution to the excessGibbs energy.

For a charge distribution possessing spherical symmetry about the origin,Poisson's equation has the form

..!-!!.-(r2 d1J!) = _ 4n pr2dr dr D

where 1J! is the potential at a point where the charge density is p; r is the distancefrom the origin.

If a particular ion is chosen as the origin of coordinates and no external forcesact on the ions, the time-average distribution of charges about that ion hasspherical symmetry. Equation (A-I) is therefore taken to apply to the time­average values of 1J! and p at r.

Since the electrolyte solution as a whole is electrically neutral, we have

(A-2)

where nj denotes the number of ions j per unit volume, (i.e., the bulkconcentration) and the summation is over all ionic species in the solution.

We select a cation as the center of coordinates. According to the local­composition concept, we consider a cell which consists of the central cation andits first coordination shell. The condition of electrical neutrality tells us that the

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 61

net charge in the solution outside this cell must be equal but of opposite sign tothat inside the cell. Thus (A-2) may be replaced byr4:rrr2Pedr + Vee + Naevae = 0 (A-3)

where r~ is the outer radius of the central cation cell. This outer radius is given by

(A-4a)

where A is a size constant to account for the effect due to the outside shape of thefirst coordination shell. This size constant is in the range 1 to 2.

Similarly(A-4b)

Nae is the time-average number of anions in the first coordination shell of the cellhaving a cation at the center; it is given by

(A-5)

where z; is the coordination number of the cation and X ae is the local molefraction of anions around the central cation:

(A-6)

(A-7)

Equation (A-6) is discussed further in Appendix B.Equation (A-3) is related to the 'like-ion repulsion' assumption (Chen et al.,

1982), which states that the local composition of cations around cations is zero,and similarly for anions. This assumption is equivalent to assuming that repulsiveforces between ions of like charge are extremely large; Equation (A-3) does notinclude the term NeeVee because, X ee = O.

According to the Boltzmann distribution law, the average local number of ionsj per unit volume, n;, at distance r, is

n; = nj exp( - V:ie)

The product VFIPe is the electrical potential energy of ion j when the coordinatesystem is centered on the cation. The product kT is its thermal energy.

For the volume element considered, the net charge density is obtained bysumming over all ionic species

(A-8a)

Upon expanding the exponentials in (A-8a), we obtain

(A-8b)

The first term on the right of (A-8b) vanishes by the condition of electrical

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62 Y. LIU, A.H. HARVEY AND 1.M. PRAUSNITZ

(A-8c)

neutrality. (For electrolytes with symmetric valence, the third term alsoishes.) If VjetJIe« kT, only the second term (linear in tJle) is appreciable:

2 2'" njvje tJle

p; = - L. kT}

van-

This approximate form is consistent with the superposiuon principle ofelectrostatics, which says that the potential due to two sets of charges in specifiedpositions is the sum of the potentials due to each set taken separately. Equation(A-8c) is strictly valid only when the potential energy, VjetJIe. of the ion j is smallcompared to its thermal energy, kT. Although this may well be true for themajority of the ions j in the solution, it is questionable for those ions which areclose to the central cation. But those ions are not of concern to us because thepotential calculated here does not include the ions in the first coordination shell.

Substituting (A-8c) into the Poisson Eq. (A-I), we have

!.- !!.-(r2d tJIe) = K2",r2dr dr 't'e (A-9)

where K is defined by

K2= 4Jt'e22: njv;/DkTj

(A-lO)

(A-ll)

Equation (A-9) is a linear second-order differential equation relating w; to r.To solve (A-9), we use two physical conditions: first, that the potential mustremain finite at large r; second, that electroneutrality (A-3) must hold in theentire solution. Finally, we obtain

(VaXaIe )eexP[K(r~-r)lltJI = v + - ---',.-'-"'-"--:::-~

c c Xu + xwGwc.ac D (1 + Kr~) r

This fundamental expression is valid only in the region r 2: r~. For the case r = reowe have to add the potential, tJle,FCS' due to the ions in the first coordinationshell.

For an isolated cation in a medium of dielectric constant D, the potential tJI~ atdistance r is given by

(A-12)

By the principle of linear superposition of potential fields, for r = r-, we have

w: + tJle,FCS = tJI; + tJI~ + tJle,FCS (A-l3)

(A-14)

where tJI; is the potential due to all the ions outside the first coordination shell.Substituting (A-ll) and (A-12) into (A-l3), we obtain

'( VaXaZe) e exp[K(r~-re)1 VeetJI = v + - ---'--"----'---=--'-'-"c c Xu + xwGwc,ac Dr; (1 + Kr~) Dr;

Within r < re , no other ions can penetrate and tJI; is therefore constant for allr < re and equal to its value at r = r-. Thus (A-14) accounts for the contribution ofall ions outside the first coordination shell on the potential of the central cation.

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 63

The change of electrical energy of the central cation, due to interaction withions outside the first coordination shell, is equal to the product of its charge, Vee,

and the potential, 1jJ;. However, if we apply the argument to every ion in thesolution, we count each ion twice, once as the central ion and once as part of thesurrounding ions. Thus the change of the electrical energy for one mole ofcations, !illc . LR' is given by

!illc • L R = RT In Y;.LR = ~o vce1jJ; (A-IS)

where No is Avogadro's number.Substituting (A-14) into (A-IS), we obtain (4-2). Similarly, we obtain (4-3).

DERIVATION

APPENDIX B

OF THE LOCAL-COMPOSITIONEXPRESSION

While the derivation that follows may be generalized, the derivation given here isbased on a single, completely dissociated aqueous electrolyte.

It is assumed that in the solution there are three types of cells as shown inFigure 6. One type consists of a central water molecule with water molecules w,anions a, and cations c in the first coordination shell. The other two, based on the"like-ion repulsion" assumption, have either an anion or cation at the center anda first coordination shell consisting of water molecules and oppositely-chargedions, but no ions of like charge (i.e., Xcc = X aa = 0). The like-ion repulsionassumption follows from assuming that the values of repulsive interactionenergies between ions of like charge are much greater than any other interactionenergies between those ions.

According to the two-fluid theory (Scott, 1956), we have

E = N (x e(w) + x e(c) + x e(a» (B-1)mow c a

where Em is the interaction energy of one mole of solution; e(w) is the interactionenergy of one cell where water is the center, with similar definitions for e(c)

and e(a).

The local mole fractions are related to one another by

X w w + X cw + X aw = 1 (central-water cell)

X wc + Xac = 1 (central-cation cell)

X wa + Xca = 1 (central-anion cell)

For each type of cell, we have

(B-2a)

(B-2b)

(B-2c)

(B-3a)

(B-3b)

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64 Y. LIU, A.H. HARVEY AND J.M. PRAUSNITZ

(e) _~( )e. - 2 XwaEwa + XcaEca (B-3c)

where Eji is the interaction energy of one j-i pair (Eji = Eii)'

For one mole of pure water or pure, completely dissociated electrolyte, wehave

where

Zce.: ='2 gac

ZaEm,ca ='2 gcu

(B-4a)

(B-4b)

(B-4c)

gji = NoEji

The excess interaction energy of one mole of solution, E;" is defined by

(B-5)

Since we are concerned with liquids at low pressures, following Renon andPrausnitz (1968), we use the approximation

H;, = E;,where H;, is the excess enthalpy of one mole of solution.

We now defineGi l .ki = exp[ -(gil - gkl)1RT]

Upon combining (B-2) and (6-2), we have

(B-6)

(B-7)

XiGiw,ww(i=w,c,a) (B-8a)

«o.:(i = w, a) (B-8b)

(B-8c)

To obtain an expression for the excess molar Gibbs energy, G;" we use theGibbs-Helmholtz equation

B(G;,IT) = HEB(IIT) m

Assuming gil independent of temperature, we integrate (B-9), giving

G;, 1[RT = -:2 ZwXw In(xw+ xcGcw.ww + xaGaw,ww)

(B-9)

(B-IO)

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THERMODYNAMICS OF ELECTROLYTE SOLUTIONS 65

At the low limit of the integration, we use 1fT =O. Therefore, G;;,/T =0 atthis limit.

Through the thermodynamic relation

(dnTG;;' ) =RTln Y

an; T,P,nj#-i "

finally, we obtain

(B-11a)

(B-llb)

Equations (B-lla) and (B-llb) give symmetrically normalized, individual ionicactivity coefficients. For consistency with the unsymmetrically normalized mod­ified Debye-Hiickel expression, we renormalize the activity coefficients using therelations

In Y:.SR = In Yc.SR - In Y~SR

In Y:.SR = In Ya.SR - In Y:;'.SR

where (00) denotes infinite dilution.From (B-12a) and (B-12b), we obtain Eqns (5-3) and (5-4).

LITERATURE C[TED

(B-12a)

(B-12b)

Abrams, 0.5., and J.M. Prausnitz, AIChE J., 21, 116 (1975).Aspen Technology, Inc., ASPEN PLUS Electrolytes Manual, Cambridge, Mass. (1984).Ball, F.-X., H. Planche, W. Furst, and H. Renon, AIChE J., 31, 1233 (1985).Bogatykh, S.A., and 1.0. Evrovich, Zh. Prikl. Khirn., 38,945 (1965).Caramazza, R., Gaz, Chim. Ital., 90, 1721 (1960).Cardoso, MJ.E. de M., and J.P. O'Connell, Fluid Phase Equilibria, 33, 315 (1987).Chen, C.-C., H.1. Britt, J.F. Boston, and L.B. Evans, AIChE J., 28,588 (1982)Chen, C.-c., H.1. Britt, J.F. Boston, and W.M. Clarke, AIChE Symp. Ser., No. 229, 126 (1983).Christensen, C.; B. Sander, A. Fredenslund, and P. Rasmussen, Fluid Phase Equilibria, 13, 297

(1983).Cruz, J.-L., and H. Renon, AIChE J., 24,817 (1978).Fredenslund, A., J. Gmehling, and P. Rasmussen, "Vapor-Liquid Equilibria Using UN[FAC",

Elsevier, (1977).Giese, K., V. Kaatz, and R. Pottel, J. Phy". Chem., 74,3718 (1970).Goldschmidt, V.M., Skr. Norske Vid. Akad. Oslo, Math. Nat. KI., No.2, 1 (1926).Gourary, B.S., and F.J. Adrian, Solid State Phys., 10, 127 (1960).Haase, R., D.-F. Sanermann, and K.-H. Ducker, Z. Phys. Chern. N.F., 47,224 (1965).Hamer, W.J., and Y.-c. Wu, J. Phys. Chern. Ref. Data, 1, 1047 (1972).Helgeson, H.C., Phys. Chern. Earth, 13-14, 133 (1981).Lengyel,S., J. Tamas, J. Giber, and J. Holderith, Acta Chim. Hung., 40, 125 (1964).Maurer, G., Fluid Phase Equilibria, 13,269 (1983).

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66 Y. LIU, A.H. HARVEY AND 1.M. PRAUSNITZ

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