an analytic model for aqueous electrolyte solutions based on

Fluid Phase Equilibria, 39 (1988) 227-266 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

227

AN ANALYTIC MODEL FOR AQUEOUS ELECTROLYTE SOLUTIONS BASED ON FLUCTUATION SOLUTION THEORY

R.L. PERRY *, J.D. MASSIE * * and P.T. CUMMINGS

Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, VA 22901 (U.S.A.)

Received October 6, 1986; accepted in final form August 20, 1987)

ABSTRACT

Perry, R-L., Ma&e, J.D. and Cummings, P.T., 1988. An analytic model for aqueous electrolyte solutions based on fluctuation solution theory. Fluid Phase Equilibria, 39: 227-266.

A model for strong aqueous electrolyte solutions based on fluctuation solution theory is introduced. The key thermodynamic quantities in the model- direct correlation function integrals-are evaluated using results from the mean spherical approximation for mixtures of charged hard spheres for the electrostatic part, from hard sphere mixture theory for the excluded volume part and from corrections for infinite dilution properties obtained from experimental data. This model is compared to experimental data on aqueous NaCl and N&r solutions at room temperature from infinite dilution to 25%’ salt by weight at pressures of 1

Electrolyte solutions are ubiquitous both naturally occurring and industrial

and of fundamental importance in processes (such as natural biological

and batch biochemical processes, geochemistry, energy conversion, electrochemistry, corrosion and pollution) (Scrivner, 1984; Barry, 1985). Conse- quently, it is not surprising that the measurement of thermophysical and chemical properties and phase equilibria in electrolyte solutions and the development of theoretical tools to predict such properties continues to be an active area of research. A recent review of aqueous electrolyte solution thermodynamics relevant to industrial practice, including a compendium of available computerized data bases, is given by Davies (1985).

and 250 bar with very encouraging results.

INTRODUCTION

* Present address: Experimental Station/Bldg 323, Polymer Products Department, E. 1. DuPont de Nemours, Wilmington, DE, U.S.A. * * Present address: Goodyear Tire and Rubber Company, Akron, OH, U.S.A.

0378-3812/88/$03.50 0 1988 Elsevier Science Publishers B.V.

228

While the solution thermodynamics of non-electrolytes is, in many re- spects, a mature engineering discipline (Prausnitz, 1969; Walas, 1985), theoretical approaches to electrolyte solutions are less well developed. This state of affairs is a reflection of the strong interactions typically present in electrolyte solutions: charge-charge, charge-dipole and dipole-dipole interactions characterize the leading order ion-ion, ion-solvent and solvent-solvent intermolecular potentials, respectively. Since most common electrolyte solutions feature water as the solvent (or as one of the solvent species in a mixed solvent electrolyte solution), the interactions involving the dipolar solvent species will be very strong, mandating a modeling approach which can deal successfully with such interactions.

The classical prototypical model for electrolyte solutions is the Debye-Hiickel model introduced by Debye and Hiickel in 1923 (Debye and Hiickel, 1923). This model pictured the ions as point charges (i.e., with no excluded volume core regions} immersed in a dielectric continuum of dielectric constant 6. Thus, the interionic pair potentials are given by

e2zjzj u,j(r) = ~

EY

In this equation, e is the electronic charge and zi is the valence of the species i ion. If pi is the number density of ions of type i, the overall electroneutrality of the solution implies that

,,I

c Z,Pi = 0 r=l

Since the interionic potential is dominated by the Coulombic part at infinite dilution, the Debye-Hiickel theory yields exact infinite dilution limiting laws for the mean ionic activity coefficient.

Many semi-empirical theories of electrolyte solutions have evolved from the Debye-Hiickel theory and its limiting-law behavior, such as the models of Reilly and Wood (1969), Reilly et al. (1971) and Scatchard et al. (1970). Perhaps the most widely accepted such theory is that of Pitzer (Pitzer 1973; Pitzer and Mayorga, 1973, 1974; Pitzer and Kim. 1974). Pitzer’s model amounts to an extended Debye-Hiickel theory with second and third virial-like coefficients in the concentration expansion of the excess Gibbs free energy. The model is capable of correlating electrolyte data very well. However, in order to do so, it requires up to four temperature- and pressure-dependent parameters for each cation-anion pair. (As we shall see, the model proposed in this paper requires at most one state-independent parameter for each ion (not each ion pair).)

More recently, several attempts have been made to model electrolyte solutions using statistical mechanical methods. Fundamental modeling of

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electrolyte solutions has been reviewed by several authors (Friedman, 1961, 1971, 1981; Hafskjold and Stell, 1982) and we refer the reader to these reviews and Friedman’s monograph (Friedman. 1962) for more detailed discussions. From a statistical mechanical viewpoint, the first step in modeling an electrolyte solution is to specify the intermolecular potentials in- volved. There are thus two kinds of statistical mechanical models: Bom-Op- penheimer level models in which the solvent species as well as the ionic species appear explicitly in the solution and McMillan-Mayer level models in which the solvent species degree of freedom are integrated over yielding a continuum solvent approximation. Thus, for a Born-Oppenheimer level model, in addition to the interionic pair potentials one must specify the interactions between each of the ionic species and the solvent species and between solvent species molecules. In this case, the interionic potentials would not contain the solvent dielectric constant in contrast to the McMil- lan-Mayer level models (which clearly include the DebyeeHiickel model as a special case). Since most of the modeling approaches to electrolyte solutions have used the McMillan-Mayer approach, we review mechanics of such models in some detail.

The predominant McMillan-Mayer model is the so-called primitive model which consists of hard spheres containing point charges immersed in a continuum solvent of dielectric constant E. Thus, the interionic potentials are given by

Q(r) = cc r -=C ujj

e2zizj

6, r > ui, (3)

where

ui + a; 0,. = -

lJ 2 (4)

In eqn. (4), u, is the diameter of the species i ion. The primitive model, reviewed at length by Hafskjold and Stell (19X2), has been studied via computer stimulation, perturbation theory and integral equation approximations. It is this latter approach to the McMillan-Mayer modeling of electrolyte solutions that proves to be of considerable utility in developing practical expressions for thermodynamic properties and therefore demands further examination.

For a McMillan-Mayer level model of electrolyte solutions, we presume (although this is not a necessary restriction) that the ions are all spherically symmetric. Then, from a statistical mechanical perspective, an electrolyte solution containing m ionic species is described by the Omstein-Zernike

230

(OZ) equation for a mixture (Ornstein and Zernike, 1914)

where the integral is over all space, h,,( r12) and c,,( r12) are, respectively, the total and direct correlation functions describing the spatial distribution of and interaction between a species i ion located at 6 and a species j ion located at 5 and rlj = I< - 5 I. The total correlation function is simply related to the radial distribution function g,,( r12) which for spherically symmetric species i and j ions is proportional to the probability density of finding the ion centers separated by distance r12. In fact

g,,(r) = h,;(r) + 1 (6)

Equation (5) represents the definition of the direct correlation functions. As is made more explicit in the next section, for a system in which the

interactions between the ionic species consists only of spherically symmetric pair interactions U,,(Y), the knowledge of all the /Z,,(Y) or of the c,,(r) is sufficient to determine the molar Helmholtz free energy as a function of temperature T and pl,. . , p, which, as a fundamental equation of state, permits the calculation of all required thermodynamic properties. In order to calculate the correlation functions, one requires a so-called closure between the h,,(r) and the cij(r) which is independent of the OZ equation (eqn. (5)). Although the exact closure relation can be written down in terms of diagrammatic expansions, in practice approximations to the closure relation are used, Of most interest in this paper is the mean spherical approximation (MSA) (Lebowitz and Percus, 1966) given by

g,(r)=O=hii(r)= -1 where u,,(r)=00 (7a)

u,;(r) cij (r) = - k~ elsewhere

B

where k, is Boltzmann’s constant. Since

'ij(') ‘ij(‘)

+--asr-+m kJ

except at critical points, the MSA combines the exact statement of overlap (eqn. (7a)) with an approximate extrapolation to short ranges of the exact asymptotic behavior of the direct correlation function. Another typically more accurate but computationally much less convenient integral approximation which has been used for electrolyte solutions is the hypernetted-chain

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(HNC) approximation (whose genesis is described by Hansen and Mc- Donald (1976))

utj(r)

c,,b-) = g,,(4 - 1 - ln R&) - kT B

(9)

The solution of the OZ equation for the primitive and similar models for electrolyte solutions is reviewed by several authors (Friedman, 1971, 1981; Friedman and Dale, 1977; Blum, 1980). One key development in the integral equation approach is the analytic solution of the MSA for the primitive model by Blum (Blum, 1975; Blum and Hoye. 1977) which leads to the analytic expressions for the thermodynamic and structural properties of the primitive model given in the next section.

Several groups have attempted to use the analytic solution of the MSA for the primitive model as a key ingredient in developing practical models of electrolyte solutions. Triolo et al. (1976) modeled electrolyte solutions using the primitive model and regarded the diameter of one of the ions as adjustable. Trio10 et al. (1978) then considered concentration-dependent ionic diameters fitting the cationic diameter and its concentration derivative to experimental data on the osmotic and mean ionic activity coefficients. These authors also considered quite separately the effect of a concentration- dependent dielectric constant on solution properties using the primitive model analysis (Trio10 et al., 1977). Watanasiri et al. (1982) fitted experimental osmotic coefficients of several salts to the primitive model results adjusting both cation and anion diameters. These authors also considered temperature-dependent adjustable diameters. Landis (1985) used the MSA and the exponential (EXP) approximations (Andersen and Chandler, 1972) for the primitive model to mixed salt systems. The ionic diameters were regressed from experimental osmotic coefficient data. Planche and Renon (1981) introduced an extension to Blum’s analysis by considering the MSA for a mixture of ions whose intermolecular potentials consisted of hard core repulsions and Coulombic interactions (in common with the primitive model) as well as a temperature-dependent short-ranged interaction modeled as the derivative of a delta function. For each species, including the solvent, the short-ranged contribution introduces two adjustable parameters in addition to the ionic diameter (which is also considered to be adjustable). In a comprehensive study of 80 strong electrolyte solutions, Ball et al. (1985) found that the Planche and Renon model could successfully correlate the experimental data for solutions up to a molality of 6 M.

In this paper, we introduce a model which utilizes fluctuation solution theory (Kirkwood and Buff, 1951; O’Connell, 1971; O’Connell and De Gance, 1975; Perry et al., 1985) to express the thermodynamic properties in terms of direct correlation function integrals (defined in the next section)

232

and uses elements of Blum’s solution of the MSA for primitive model electrolytes to evaluate the electrostatic contributions to these quantities (as described in detail in the Description of Model section). As such, the model shares some general features with the models described above. In later sections we describe the analysis of the experimental data with which the predictions of the model are compared, and the comparison between theory and experiment.

FLUCTUATION SOLUTION THEORY

In this section, we review briefly the Kirkwood-Buff fluctuation theory of non-electrolyte solutions (Kirkwood and Buff, 1951) and its reformulation by O’Connell (O’Connell, 1971, 1981) in terms of volume integrals of the direct correlation function (defined below). The fluctuation solution theory is then defined for a single-salt/single-solvent electrolyte solution utilizing the recent results of O’Connell and co-workers (O’Connell and De Gance, 1975; Perry et al., 1980; Perry and O’Connell, 1984; Perry et al., 1985). Finally, we derive expressions for the volume integrals of the direct correlation functions implied by the Debye-Hiickel theory of electrolyte solutions (defined in the previous section) and the mean spherical approximation for the primitive model of electrolyte solutions, which represents a generalization of the Debye-Hickel theory since it includes hard core repulsion between the ions (Waisman and Lebowitz, 1972a, b). In each case, these results (which are used to define the model for aqueous electrolyte solutions described in the next section} are quoted without proof, the reader being referred to the relevant original reference for the derivation.

General formulation of fluctuation solution theoy

The original formulation of fluctuation solution theory was derived using the grand canonical partition function (McQuarrie, 1976) by Kirkwood and Buff (1951). As did Kirkwood and Buff, we consider an m-component mixture of non-dissociating non-reacting molecular species (i.e. a non-electrolyte solution). The 02 equation for such a system will be given by eqn. (5) if the molecules are all spherically symmetric; if this is not the case, an appropriate generalization to non-spherical interactions can be derived. The key results obtained by Kirkwood and Buff were expressions for fluctuations in the number of molecules of component i, N,, derivatives of the chemical potential p, with respect to yj at fixed volume and at fixed pressure, partial molar volumes, compressibihty and osmotic pressure in terms of Hi,, the

233

volume integral of the total correlation function between a type i and a type j molecule defined by

HLj = p J

A;;(;) dr’ (10)

In this equation, p is the total density of the solution

P’iPj i=l

and the integral is over all space. To represent the results as compactly as possible, we introduce the following notation: for a matrix 2, [Ziij is the i, j element of Z, ZT is the transpose of Z, (Z,,) is the matrix obtained from Z by deleting the ith row and the jth column, ]Z ] is the determinant of Z and Zt is the matrix of cofactors of Z (i.e. [Z?],, = (-)I+’ ](Z)i, 1). We use the symbols i, and 0, to indicate vectors of length n, in which each element is unity and zero, respectively. Using this notation, we can summarize the results of Kirkwood and Buff (1951) in the form given by O’Connell (1971)

iiXTD = 0: (12)

N, aPi i’,-l(Bji)t(Xjj)i,_, kBTaNj TpN = m

iT Bf Xi 7 7 rn.rnT, m

m [B+]jt pv;= c

i=I iLB+Xi,

1

w&J = i:B-‘Xi,

Nl lhr i:-l(B,ljt(Xjj)im-l pk,T q T,p, N

3 m.m+, Et’==, LB+] /cl

03)

(14)

(15)

06)

In these equations, (Ni} is the grand canonical ensemble average of the number N, of species i molecules

Sii is the Kronecker delta, CL, is the chemical potential of species i, [Xl,, = x,8,,, x, is the mole fraction of species i[ = (IV/N)], [B];; = Sj, +

234

xiHli, P is the pressure, Vi is the partial molar volume of species i and K~ is the isothermal compressibility,

1 av KT= ---

’ ap =,%-I.. .m 07)

In eqn. (16), ‘IT is the osmotic pressure of the solution over species 1 (i.e. assuming species 2,. . . , m are solutes and species 1 is the solvent). Note that eqn. (12) is the statement of the Gibbs-Duhem equation for the solution.

O’Connell (1971,198l) showed that these equations could be re-expressed in terms of direct correlation function integrals (DCFIs) defined by

Cij = p/cjj( r) dr’ (18)

since from the OZ equation

B=(I-xc)-’ 09)

where I is the m x m identity matrix and [Cl,, = Cil. Equations (13)~(16) then become

aPi Ni bTq TPN

= sij . I m.m+,

X,(1+ Ci,-Ckm_iXk(C,kt Cjk) +~~=~C;f-l-Uk-UI(ciikc~l-Ci~c~l)) -

1 - c;= J;llXk XlC,, W

(21)

(24

These equations are considerably simpler than their counterparts in terms of the total correlation function integrals. O’Connell (1981) obtained further

235

simplifications by combining several of these equations yielding

1 aP -_ kr3T aPi Tp 2 /3’ J=l

al-5 1 ‘ij ci,

k,T aPj Tp,+ = P, P

i ap 1 -- k,T aP T,x ,,.., x = PKTkBT

= f f x.x.(1 -c;,)

, m ;=I /cl ’ ’

(24)

(25)

(26)

(Note that in the thermodynamic expressions quoted above, the densities have been assumed to be number densities (i.e. measured in molecules per unit volume), but molar densities could easily be used (measured in moles per unit volume) by replacing k, by the gas constant R.) Thus, given equations for the C,j as functions of temperature and species densities, one can obtain all the thermodynamic properties of interest. By using models for the Clj, Mathias and O’Connell (1979, 1981) modeled liquid mixtures with supercritical components with considerable success. Since, in general, the asymptotic form of cij(r) is known (eqn. (8)) while the asymptotic form of hij( r) is not, the DCFI approach offers the possibility of direct modeling while minimizing the degree of semi-empiricism.

Fluctuation theory of electrolyte solutions

A single-solvent (1) single-salt (2) electrolyte solution has two components, the solvent and the salt. In strong (completely dissociating) electrolyte solutions, three species are present in solution: the solvent (1) molecules and the positive ( +) and negative ( - ) ions. If the valence of the positive and negative ions is V+ and V_ respectively, then the salt completely dissociates according to

A,+B,_ + v+A’+ + v-B= (27)

where A’- and B’- are the positive and negative ions, respectively, and Y, and v_ are stoichiometric coefficients. Charge neutrality demands that

v+z+ + v-z_ = 0 128)

Clearly, the density of positive and negative ions is given by

P+ = V+Pz, P- = V-P2 (29)

For an electrolyte solution, it is clear that the number of components (c) differs from the number of species (m) and that because of stoichiometric

236

(or, equivalently, electroneutrality) constraints not all of the species densities can be varied independently. Thus, we must distinguish between several definitions of mole fraction as follows: the species mole fraction on a total species basis is given by

4 N+ N- x1=-,x+=-,x_=-

N N N , N=N,+N++N_

The component mole fraction on a species basis is defined to be

hi x0,= N’ i=l,2 (31) where Noi is the number of molecules of component i. Finally, the component mole fractions on a component basis are given by

y,=+, No= i Noi 0 1=1

(32)

Thus we see that Cx, = 1 and Cy, = 1 while xo2 = x+/v., = x-/v . The Kirkwood-Buff and O’Connell equations given in the previous

section are not directly applicable to electrolyte solutions, essentially because of the electroneutrality condition (eqn. (28)) which imposes stoichiometric constraints on the number of moles of ionic species (O’Connell and De Gance, 1975; Perry et al., 1985). Equation (28) leads directly to the Stillinger-Lovett zeroth moment condition (Stillinger and Lovett, 1968) which implies that

B=I+XH

is a singular matrix. The singularity of B can be derived from the stronger statement (eqn. (8)) which, for electrolyte solutions, implies

'ij('> aizje2

-+ -- asr+co cr (33)

where E is the solvent dielectric constant for a McMillan-Mayer model and is unity for a Born-Oppenheimer model. This implies that the [Cl,, associ- ated with ion-ion interactions are infinite. A more precise statement of the singularity in the [C],j-and another way to see that eqns. (20)-(23) are not directly applicable-comes from eqn. (8) which implies that for direct correlation functions involving ion-ion interactions the three-dimensional Fourier transform of cjj( r)

tij(k) = /efz.Gij(r) d7-, 4~~~~2 -$ + O(k”) as k --) 0

Since from their definition eqn. (18) the DCFI are given by

cjj = p2ij(o)

(34

237

and since from eqn. (33) it is clear that ?jj( k) + 00 as k + 0 we conclude that for ion-ion interactions the [Cl, are infinite.

Several authors have addressed the question of how to modify the Kirkwood-Buff equations for electrolyte solutions and the appropriate modification to eqns. (20)-(26) expressing thermodynamic properties in terms of the direct correlation function integrals. The most general analysis has been performed by O’Connell and co-workers (Perry et al., 1985) who derived expressions for thermodynamics analogous to eqns. (20)-(23) and applicable to electrolyte solutions containing an arbitrary number of solvent species and an arbitrary number of ionic species. This analysis was an outgrowth of a related study of chemically reactive mixtures (Perry et al., 1980), which like electrolyte solutions are subject to stoichiometric constraints. The results of Perry et al. are given in terms of short-ranged direct correlation functions ci”j ( r ) defined by

ci”(r) =c;,(r) - y&f B

(35)

Clearly, for ion-ion interactions C:.(Y) is short ranged; for all other interactions, one or both of z, will be zero implying that ct ( Y) = C,,(Y), which is also short ranged. Thus for all the interactions, the DCFIs

C$-=p/cp,(r) dr’ (36)

are well-defined. For a single-solvent/single-salt electrolyte solution, the general expressions of Perry et al. (1985) reduce to

1 i3P 1 =- k,T + T,N PKTRT

=x&(1- CPJ + 2~X,,X,*(l- CPJ

1 8 lny+ ~

PK&rJ’ 8x02 rpx = vx&[(l- Cfi)(l - c&) - (1 - ce,‘] (38)

. 1 01 _

1 aP VI2 -- = ~ = V(XOi(l - cp-J + V”02jl - G)l W 8Poz ~,p,, KTkBT

1 i3P G -- = ~ = [x0$ - cg + “X,,(l - G)l k,T %%i T.p,,, KTkBT

(39)

(40)

Equations (37)-(40) are written on a component basis where component 1 is

238

the solvent and component 2 is the salt, v,, is the molar volume of the salt, y+ is its mean ionic activity coefficient defined by

y* = ( u”;v’_)l’y (41)

pot, po2 are the component densities and v = v+ + VL There are six species DCFIs (Cf,, Cf+, Cp-, CT,, Ct_, Co-) and only

three component DCFIs (Cp,, Cfz, C,“,). These quantities are related by (Perry et al., 1985)

cp, = (v+cp+ + v-c;-)/+ (42)

c2”2 = (V:CS+ + 2u+u_c:_ + ylCO)/Y (43)

Equation (38) for In y+ is singular near x,,* + 0 so we use it in the form

xii2 d Iny ,

PK&J ax02 T,p,x,, = vx&x;{“[(l - Cf,)(l - C-j!) - (1 - cpJ2] (44)

The solvent activity coefficient can be derived from eqn. (38) and the Gibbs-Duhem equation, which for the single-salt/single-solvent system under consideration in this paper can be written as (O’Connell and De Gance, 1975; Perry et al., 1985)

d lw, ‘01 dx,,

T,P

+ VXmdx,, =

T,P

(45)

In summary, given models for the C$, which in turn have their origin in the C$, a, p=1, + -, eqns. (37)-(40) and (45) provides routes to the thermodynamic properties of interest. Alternatively, one can invert these relations to find expressions for the C,; from experimental data, namely

where

(46)

(47)

(49)

and voz, POT are the molar volumes of the salt at xo2 and at infinite dilution,

239

respectively (Perry, 1980). The DCFIs at infinite dilution of the salt are given by

lim (1 - Cf,) = (1 - Cf2)1 = $ x02 l o 1

lim xhi2(1 -C&) =0 %Z’O

(50)

(51)

(52)

where pi is the isothermal compressibility of the pure solvent. The limiting slope of (1 - Cp,), C:,’ is given by

(53)

where @,” is the apparent molar compressibility at infinite dilution. Harned and Owen (1958) define the apparent molar volume, @, and the apparent molar compressibility, OK by

(55)

where v and Vi are the molar volumes of the solution and the pure solvent, respectively.

This summarizes the major thermodynamic results which are required in the subsequent analysis of our model for electrolyte solutions.

DCFIs from Debye- H iickel theory

The Debye-Hiickel theory of electrolyte solutions was discussed in the Introduction as a McMillan-Mayer model in which the ions are modeled as point ions. Thus, the Debye-Hiickel theory gives rise to thermodynamic expressions which are valid in the limit of low salt concentration where short-range parts of the interionic potentials become negligible. In the limit of very low concentration, results of the Debye-Hiickel theory are known collectively as the Debye-Hiickel limiting law expressions since they become exact in this limit. They are derived from the Debye-Hiickel expressions for the excess Hehnholtz free energy

AA = - n3k,T

12~ N;“q3’*Vp;{

240

where NA is Avagadro’s number, q is CS=l~,z,2( s being the number of ionic species) and (Y is given by

Equation (56) can be used to derive the expression for the activity coefficient since

where the superscript el indicates the electrostatic contribution to y. Using eqns. (46)-(48) we obtain the following expressions for the DCFIs

cODH = _ 11 16~ (21,, - 31:)

CODu = _ 3a3( P0243%)1’2P11 12 167~~

C ODH _ a3

22 167rv2

In eqns. (59)-(61)

/ J)lnc 1 alne 1

aP T, POT PlKl ap j-

(59)

(60)

(61)

(64

DCFIs from the mean spherical approximation

The MSA for an arbitrary mixture of charged hard spheres was solved analytically by Blum (1975) generalizing the MSA solution by Waisman and Lebowitz (1972a, b) for the restricted primitive model (equivalent binary mixture of equal diameter charged hard spheres). Blum used the Baxter factorization technique (Baxter, 1968, 1970). The details of the analytic solution will not be repeated here. The solution reduces to solving a single non-linear equation for a parameter F which is a generalization of the Debye-Hiickel screening length (i.e. it reduces to this quantity in the limit

241

of zero ionic diameter). The excess internal and Hel~oltz free energies, AE and AA respectively, and the mean ionic activity coefficient are given by

(44)

AA=AE+g (65)

In y+= [

tx= PAE - $‘,? 1 /VP,, W

where Sk, A and P, are variabIes in the MSA analytic solution defined by

A = 1 - $ &;u; (67) 1

As in the case of the electrostatic terms for the DCFIs based on the ~eby~-nickel theory, a simiIar de~vation was carried out for the MS.4 terms yielding

(73)

(74)

243

perturbation term accounting for electrostatic effects. The hard sphere terms, C$ , is given by the Camahan and Starling equation of state for a mixture of hard spheres (see the Appendix), the correction term, C$, includes experimental, infinite-dilution solution properties, and the electrostatic term, Cup, must be formulated so as to approach the correct Debye-Hiickel limit at low salt concentrations.

The original model of Perry was modified by Reid (1982) by adding a second virial coefficient term to the solvent-solvent direct correlation function integral and by adding a term of first order in salt density to the solvent-salt DCFI term from the MSA. This model, which required the fitting of five parameters from experimental data, was in good agreement for the solvent-~solvent and solvent-salt DCFIs but fit poorly for the salt-salt DCFI for the NaCl-H,O system at 25 ‘C.

The use of the early forms of the model as a tool for predicting thermodynamic properties for an aqueous, strong electrolyte system were limited because they required extensive volumetric and activity data to fit the model parameters. In order to make the model more general in its applicability, a method to reduce the dependence on fitted parameters was sought. By modifying the original model of Perry (19&O), Moffitt (1983) was able to reduce the number of fitted parameters to two; the hard sphere diameter of water and a correlation factor. Because the electrostatic portions of the solvent-solvent and solvent-salt direct correlation function integrals are of lesser importance than in the salt-salt case, for the solvent-solvent and solvent-salt DCFIs, Moffitt used the Debye-Hiickel limiting forms (eqns. (59) and (60)) to account for the electrostatic portion. However, for the salt-salt DCFI, where electrostatic effects are more significant, Moffitt used the DCFI (eqn. (75)) obtained from the analytic solution of the MSA for the primitive model electrolyte. Moffitt used the Pauling crystal radii for the ions but permitted the water molecule diameter to be adjustable. He introduced a second adjustable parameter, which he called the correlation factor, that varied between 0 and 1 and was a multiplicative factor which scaled the size of the MSA electrostatic contribution.

The MSA and the Debye-Htickel theory do not approach the correct limits for the DCFI at infinite dilution (with the exception of the salt-salt DCFI) and the hard sphere portion is not very good at low salt concentrations. Thus, Perry (1980) imposed the infinite dilution solvent-solvent and solvent-salt DCFIs on the hard sphere reference state in addition to the electrostatic portions to force the DCFIs to the correct limits as the salt concentration approaches zero.

The initial model proposed by Perry and modified by Moffitt is the basis of this work. However, two major modifications have been employed. First, Moffitt used a correlation which was unrelated to the DCFIs to estimate the

244

solution density, whereas in this work the solution density is obtained from the proposed solution theory by numerically integrating the density derivative of pressure, which is in terms of the density-dependent DCFIs, from a reference state to a final state (Mathias and O’Connell, 1981). Secondly, the correlation factor was dropped as an artificiality. (Indeed, several comparisons with experimental results showed that its optimal value is near unity, corresponding to the inclusion of the full MSA contribution to the salt-salt DCFI.) These modifications lead to the development of a one parameter model in which the hard sphere diameter of the water molecule is the only adjustable parameter. As described in the Results section, this model yields acceptably accurate predictions for activity coefficients and for osmotic pressure of the electrolyte solutions studied to date. The greatest accuracy is obtained when both the water and the ionic diameters are permitted to be adjustable, and results from this model are also reported in the Results section. In the remainder of this section, we describe the model for the DCFI in more detail.

Following Perry (1980) and Moffitt (19X3), for a single-solvent (l)/single-salt (2) system, the model equations for the solvent-solvent, solvent-salt. salt-salt DCFIs based on a component mole fraction are as

l-C,,= &- I 1

(1 - C;!)] + (1 - c;;) - CgH

1 ~ c,, = (1 ~ c*2)hs ~ cgsA

(77)

(78)

(79) The hard sphere reference DCFIs are expressed on a species basis in the Appendix which in turn are related to a component basis by (see the Fluctuation Theory of Electrolyte Solutions section)

which gives

c:; = (V+C,+ + Y_C1&V (81)

c~;=(I:C+++2u+V_C__+VZC~~)/V~ (82)

for a single-solvent/single-salt system. The solvent-solvent expression (C::)

245

is unaffected so it is the same for both species and component. The superscripts hs, hso, and hs’o refer to hard sphere portions of the model given in the Appendix and (a:, @,” refer to the standard state (i.e. infinite dilution of the solute) apparent molar volume and compressibility respectively (Redlich and Rosenfeld, 1931). The terms in brackets for the solvent-solvent and solvent-salt DCFls are the C$ terms. Because the hard sphere portion of the model is not very good as the salt concentration approaches zero, we force the model to fit the experimental values at infinite dilution. This accounts for the (1 - C:;“) and ~/(~,K,RT) terms in the solvent-solvent DCFI and for the (1 - C,“,““) and @~/(vK~RT) terms in the solvent-salt DCFI. The slope of (1 - C:,“) is not very good at infinite dilution so we force it to agree with experimental values at infinite dilution which account for the yz{ [v/( pl~l RT)] - [ a,“/( KURT)] - Cf,“‘“) term.

To calculate the terms given in eqns. (77) through (79), we require expressions for rci, K~, @:, a,“, (a~~/aP),, (d ln c/ap), and (a’ ln r/aP2)r, the last three terms appearing in the Debye-Hiickel and MSA contributions (see eqns. (62) and (63)). Pure water volumetric and compressibility data were obtained from an equation of state for water given by Chen et al. (1977)

where rcl, IC,” and V,, VI0 are the secant bulk moduli and molar volumes of pure water at the system pressure P and at the reference pressure PO, respectively, and A,,, and B, are temperature dependent parameters. Using this equation of state, the molar volume, V,, and compressibility, K~, for pure water are

v

1

= p [KP + (A, - w + 4J21 ’ (K,O+A,P+B,,,P~]

K; - B,P2

(K:+A,P+B,P~)~ I

(84)

(85)

B,P(K~+A,P+B,P~)+(A,+~B,P)(Ic;-B,P2)

(IC; + A,P + BwP’)3 1 @6)

Equations (54) and (55) for the apparent molar volume QO and the apparent molar compressibility QK may be expressed in half powers of salt

246

concentration in the limiting law regime by the following (Harned and Owen, 1958; Redlich and Rosenfeld, 1931) _

a’, = @; + SJ?“~ + b,,C (87) (88)

I I 2

_ 6 a’ln c

T aP

(89)

(90)

II (91) T

S, and S, are the Debye-Hiickel limiting slopes for the corresponding apparent molar volume and compressibility, C is the molar concentration of the salt

(92) and cbf and a,” are the standard state apparent molar quantities. The constants b, and b, are empirical coefficients used to extend the apparent molar quantities over the salt composition range of interest. The parameters @z, b,, 0: and b, were determined by Moffitt (1983) by least-squares fitting of eqns. (87) and (88) to experimental data.

Owen et al. (1961) fitted the dielectric constant of pure water (25’ C) as a function of pressure to the equation

6 = exp[ A, + B,P + CeP2] (93)

where the parameters A,, Be and C, are given as polynomials of temperature. Using this expression, the following pressure derivatives of the dielectric constant can easily be obtained

a In c ~ =B,+2C,P

ap T

a*‘nE =2c

ap2 T e (95)

The partial molar volume, Fo2, is derived using the relationship (Kortiim,

1965)

v,, = g 02 T.P&:,,

= *‘, +y2$ 2

(96)

247

to obtain

To, = [ a),” + sJY2 + b,C ] + $ s,c l’* + b,,C

1 + 10-V [ $,c”* + b,C]

where

(97)

The isothermal compressibility, K~, is given by the expression (Harned and Owen, 1958)

1 av “= -vaP T=

YF!‘k + Y, v, ‘5

V (99

To calculate the various solution properties, we must determine the density of the solution. From the fluctuation solution theory presented in this section, we have

where c is the number of components in the solution. Because the DCFIs are functions of the density, we must numerically integrate this equation from the reference state to the final state (where Pr is known) in order to determine the solution density. Integrating eqn. (100) yields (Mathias and O’Connell, 1979, 1981)

where

x0/3 = 1 X&P' + (X&P/ - x&qPr t /P )I p=p’+ (pf-pyt

002) (103)

0 I t I 1, f refers to the final state, and r refers to the reference state. The reference state, which is at infinite dilution of the solute, is defined as

x,& = 1.0, x& = 0.0, pr = p1 = l/u,, P’ = 1.0 bar (104)

The numerical integration was performed by utilizing a 20-point Gauss-Legendre quadrature. Once the solution density was obtained the DCFIs and thus the activity and osmotic coefficients were calculated.

248

ANALYSIS OF EXPERIMENTAL SOLUTION PROPERTIES

The experimental expressions for the solvent and solute activity coefficients for an isothermal system at pressure P were obtained from the Gibbs-Duhem equation and the definition of the chemical potential

PY ln Y, I T,p,_x = ln Yi I T,PO,_X + poRT 1 I dP

r,x W)

The pressure correction, the integral term on the right-hand side, was obtained by numerically integrating from the reference pressure P” to the required pressure P at constant temperature and composition. Thus we can obtain the following for the solute and solvent activity coefficients

ln Y f I T,p,_x = ln Y* I T,PO,_X + J qo2-@,0) (lP

po vRT TX ?_

In 71 1 T,P,_X = In 71 I T,P’, x + J

f+%-%)Y2 dP

pu YlRT TX

3_

006)

(107)

Using the conditions for osmotic equilibrium, the molar osmotic coefficient @, is related to the solvent activity y1 by (Hamer and Wu, 1972)

Rearranging this equation yields

ln y1 I T,PD,_X =f!$i@-ln ”

i 1 Y, + VY2

(108)

009)

The mean ionic activity coefficient at the reference state was obtained from an empirical equation for the molal mean ionic activity coefficient for various electrolytes given in Hamer and Wu (1972)

log r,(T, _X, P”) = -Iz+z_ IA”K

l+B*yTi;;; +p*I,+c*I;+D*I;

+E*14 + F*15 M m

where A* = 0.5108 at 25°C

I, = fCm,.z,2

(110)

(W

W) y*lOOO

m=- YF,

249

Z,, the molal ionic strength, reduces to the molality, m, for a solution of monovalent ions, A* is the Debye-Hiickel constant, and B *, ,3 *, C*, D*, E *, F* are fitted constants, which are characteristic for each salt, tabulated by Hamer and Wu (1972). The first term on the right-hand side of eqn. (110) is the extended Debye-Hiickel limiting law expression for the mean ionic activity coefficient based on the ionic strength.

The molal mean ionic activity coefficients were converted to a component mole fraction basis by the expression

(113)

The experimental expression for the osmotic coefficient, obtained from Hamer and Wu (1972), is given by the following

@=l-- I + B*G) - 2 ln(1 + B*h) - l/(1 + B*J;;;)]

-1 2p*m _ 3C*m2 _ iD*m’ I (114)

where all variables have been previously defined. It is assumed that the osmotic coefficient is independent of the pressure since the system is dominated by water and the partial molar volume of water changes very little over the pressure range considered.

Thus, from the experimental volumetric data and from parameters tabulated by Hamer and Wu (1972): we are able to calculate the osmotic coefficient and the solvent and mean ionic activity coefficients of the electrolyte solutions studied by these authors.

RESULTS

In comparing the predictions of the theory presented in The Description of Model section with experimental data on electrolyte solutions, two approaches were adopted. In the first case, the ionic radii were set to their Pauling crystallographic values and the diameter of the water molecule was the only adjustable parameter. The model was compared to experimental data on NaCl and NaBr aqueous electrolyte solutions at 25 o C and one bar and the water diameter determined by least-squares fit to these data. The volumetric properties of the salt solutions were obtained from the data of Gibson and Loeffler (1948). They tabulated the specific volumes of the NaCl-H,O and NaBr-H,O systems for compositions ranging from 5 to 25 wt. % salt, for pressures from 1 to 1000 bar absolute and for temperatures

250

TABLE 1

Hard sphere diameters for the water molecule and the ionic species determined by least-squares fit to experimental data on NaCl and NaBr aqueous electrolyte solutions

Species

Hz0 2.5868 2.0502 Nat 1.90 2.9631 Cl- 3.60 3.3802 BI- 3.90 3.9971

a Only water diameter fitted; h all diameters fitted.

from 25 o to 85 O C. The resulting water molecule diameter is given in Table 1, along with the Pauling radii for the ions, Using these values, the osmotic coefficient, mean ionic activity and solvent activity were calculated and are displayed in Figs. l-3 for NaCl at one bar. Comparison between the predicted and experimental quantities is evidently quite good at atmospheric pressure; although results are not given here, the quality of the theoretical predictions decreases with increasing pressure.

The second approach adopted is to consider the diameters of the ions as well as the water molecule to be adjustable. Using the experimental data at 1

020, , , I , I , I , I , I [

/ - H,O-NoCI at 25’C and Ibor

0.10 -

Fig. 1. The log of the mean activity coefficient as a function of salt mole fraction for an aqueous solution of NaCl at 25 ’ C and 1 bar. The circles represent the experimental data (see Analysis section of text); the dashed curve shows the theoretical results obtained when only the water diameter is fitted to the experimental data while the solid curve is obtained by fitting the ionic diameters as well (see Results section and Table 1). The results shown for NaCl-H,O in Figs. l-3 range from infinite dilution to 25% NaCl by weight (which is close to saturation for this solution).

251

Fig. 2. The log of the water activity coefficient as a function of salt mole fraction for an aqueous solution of NaCl at 25’ C and 1 bar. The symbols have the same significance as in Fig. 1.

bar the diameters of the water molecule and the Na+, Cl- and Br- ions were determined by least-squares fitting to the experimental data. The diameters thus obtained are reported in Table 1. The Cl- and Br- diameters are not that different from the Pauling values. However, the Na+ diameter is considerably larger than its Pauling value and the water diameter considerably smaller than the value found previously. The increase in cation diameter is consistent with the findings of Triolo et al. (1976) who fitted experi-

H,O-N&I at 25+C and I bar

0.00 O-02 004 O-06 %at

Fig. 3. The osmotic coefficient as a function of salt mole fraction for an aqueous solution of NaCl at 25°C and 1 bar. The symbols have the same significance as in Fig. 1.

252

H,O-NaBr of 2YC and I bar

-025 -

Fig. 4. The log of the mean activity coefficient as a function of salt mole fraction for an aqueous solution of NaBr at 25’C and 1 bar. The symbols have the same significance as in Fig. 1. The results shown for NaBr-H,O in Figs. 4-6 range from infinite dilution to 25% NaBr by weight.

mental osmotic coefficients to the MSA solution to determine effective hard sphere diameters. These authors conjectured that the larger effective diameter could be attributed to solvation effects. We are unable to draw the same conclusion from this work due to the limited nature of the comparison of the theory with experiment. Further comparisons, which are underway? will reveal whether the larger cation diameter is a consistent feature of the model and therefore attributable to solvation. One intriguing observation for which we do not have any physical interpretation is that the increase in the diameter of the Nat ion is almost exactly equal to the effective radius of the water molecule.

The radii reported in Table I were used to calculate the osmotic coefficient, mean ionic coefficient and water activity coefficient at 1, 250, 500, 750 and 1000 bar. The results at 1 bar are compared with experiment in Figs. 1-3 for NaCl and Figs. 4-6 for NaBr; the results at this pressure and 250 bar are tabulated in Tables 2-5. (Higher pressure calculations are reported by Massie (1986) as are the values of the quantities a,“, b,, a,” and bk.)

The results at 250 bar are obtained using the diameters obtained by fitting at the experimental data at 1 bar and performing the integration in eqn. (101) from P’ = 1 bar to Pf = 250 bar, thus using the infinite dilution parts of the DCFIs calculated for one bar. (One could also perform these calculations using P’ = 250 bar so that the infinite dilution corrections in the DCFI are calculated at the higher pressure. This would have required the use of experimental data at the elevated pressure.) Consequently, the results

253

o-5 1 , 8 , 1 , 1 , 1 , !

H,O-NaBr ot 25°C and I bar _

. _

-2,5- 8 I 8 ’ n ’ ’ ’ ’ ’ 000 0.01 002 O-03 004 005 DO6

x San

Fig. 5. The log of the water activity coefficient as a function of salt mole fraction for an aqueous solution of NaBr at 25OC and 1 bar. The symbols have the same significance as in Fig. 1.

at higher pressure represent predictions of the model obtained with no experimental information at the elevated pressure. It is clear that the accuracy of the model deteriorates as the pressure is increased. Nevertheless, it is equally clear that the model is capable of representing the experimental data at one atmosphere extremely well with a small number of adjustable parameters and is acceptably accurate up to quite high pressures (the error

Fig. 6. The osmotic coefficient as a function of salt mole fraction for an aqueous solution of NaBr at 25 o C and 1 bar. The symbols have the same significance as in Fig. 1.

254

TABLE 2

Comparison of theoretical and experimental values of the mean ionic activity coefficient, osmotic coefficient and water activity coefficient of aqueous solutions of NaCl at 1 bar absolute and 25 ’ C

NaCl-H,O(P=l bar, T= 25°C)

YSdI y* @ Yw

Experi- Theory % error Experi- Theory % error Experi- Theory ment ment ment

0.0025 0.7611 0.7569 0.56 0.928 0.923 0.0050 0.7206 0.7135 0.98 0.921 0.915 0.0075 0.7006 0.6928 1.11 0.920 0.916 0.0100 0.6895 0.6820 1.08 0.922 0.919 0.0125 0.6834 0.6767 0.99 0.926 0.924 0.0150 0.6807 0.6749 0.84 0.930 0.930 0.0175 0.6804 0.6758 0.68 0.935 0.937 0.0200 0.6819 0.6785 0.51 0.941 0.944 0.0225 0.6850 0.6827 0.33 0.947 0.951 0.0250 0.6893 0.6882 0.16 0.954 0.959 0.0275 0.6948 0.6948 0.00 0.961 0.967 0.0300 0.7013 0.7024 -0.15 0.968 0.976 0.0325 0.7088 0.7108 - 0.29 0.976 0.984 0.0350 0.7171 0.7201 - 0.41 0.984 0.993 0.0375 0.7263 0.7301 - 0.52 0.993 1.002 0.0400 0.7363 0.7408 - 0.61 1.002 1.011 0.0425 0.7470 0.7522 - 0.69 1.011 1.020 0.0450 0.7586 0.7643 - 0.75 1.020 1.030 0.0475 0.7710 0.7770 - 0.79 1.030 1.039 0.0500 0.7841 0.7904 - 0.81 1.040 1.049 0.0525 0.7980 0.8045 - 0.81 1.050 1.059 0.0550 0.8127 0.8192 - 0.80 1.061 1.069 0.0575 0.8282 0.8346 - 0.77 1.072 1.079 0.0600 0.8445 0.8506 - 0.72 1.083 1.089 0.0625 0.8616 0.8673 -0.66 1.094 1.100 0.0650 0.8796 0.8847 - 0.58 1.105 1.110 0.0675 0.8984 0.9028 ~ 0.48 1.117 1.121 0.0700 0.9181 0.9215 - 0.37 1.129 1.131 0.0725 0.9387 0.9410 -0.24 1.141 1.142 0.0750 0.9602 0.9612 - 0.10 1.153 1.153 0.0775 0.9827 0.9821 0.05 1.166 1.164 0.0800 1.0061 1.0038 0.22 1.178 1.175 0.0825 1.0304 1.0263 0.40 1.191 1.186 0.0850 1.0558 1.0495 0.60 1.204 1.197 0.0875 1.0822 1.0736 0.80 1.217 1.208 0.0900 1.1097 1.0984 1.02 1.230 1,219 0.0925 1.1382 1.1241 1.24 1.243 1.231

0.57 1.000 1.000 0.61 1.001 1.001 0.51 1 .OOl 1.001 0.36 1.001 1.001 0.19 1.002 1.002 0.02 1.002 1.002

-0.15 1.002 1.002 -0.30 1.002 1.001 -0.44 1.001 1.001 -0.57 1.001 1.001 -0.67 1.001 1.000 - 0.76 1.000 1.000 -0.83 0.999 0.999 -0.88 0.999 0.998 - 0.92 0.998 0.997 - 0.94 0.997 0.996 - 0.94 0.995 0.994 -0.93 0.994 0.993 -0.91 0.992 0.991 -0.88 0.991 0.990 -0.83 0.989 0.988 -0.77 0.987 0.986 - 0.70 0.985 0.984 -0.62 0.982 0.981 -0.53 0.980 0.979 - 0.43 0.977 0.976 -0.33 0.974 0.973 - 0.21 0.971 0.970 - 0.09 0.967 0.967

0.03 0.964 0.964 0.16 0.960 0.961 0.29 0.956 0.957 0.43 0.952 0.953 0.57 0.948 0.949 0.71 0.944 0.945 0.86 0.939 0.941 1 .oo 0.934 0.937

255

TABLE 3

Comparison of theoretical and experimental values of the mean ionic activity coefficient,

osmotic coefficient and water activity coefficient of aqueous solutions of NaCl at 250 bar

absolute and 25 o C

NaCI-H,O (P = 250 bar, T = 25 ’ C)

Y&It Y f @ YW

Experi- Theory ‘% error Experi- Theory % error Experi- Theory

ment ment ment

0.0025 0.7619 0.7477 1.87

0.0050 0.928

0.7217 0.7021 2.72 0.921

0.0075 0.7019 0.6796 3.18 0.920

0.0100 0.6910 0.6672 3.45 0.922

0.0125 0.6852 0.6603 3.63 0.926

0.0150 0.6827 0.6571 3.75 0.930

0.0175 0.6825 0.6564 3.83 0.935

0.0200 0.6843 0.6576 3.90 0.941

0.0225 0.6875 0.6603 3.96 0.947

0.0250 0.6921 0.6642 4.02 0.954

0.0275 0.6917 0.6692 4.08 0.961

0.0300 0.7044 0.6752 4.15 0.968

0.0325 0.7120 0.6820 4.22 0.976

0.0350 0.7206 0.6896 4.30 0.984

0.0375 0.7299 0.6979 4.38 0.993

0.0400 0.740 1 0.7070 4.48 1.002

0.0425 0.7511 0.7167 4.58 1.011

0.0450 0.7629 0.7271 4.69 1.020

0.0475 0.7754 0.7381 4.82 1.030

0.0500 0.7888 0.7497 4.95 1.040

0.0525 0.8029 0.7620 5.09 1.050

0.0550 0.8179 0.7750 5.25 1.061

0.0575 0.8336 0.7885 5.41 1.072

0.0600 0.8502 0.8028 5.58 1.083

0.0625 0.8676 0.8176 5.76 1.094

0.0650 0.8858 0.8332 5.94 1.105

0.0675 0.9049 0.8494 6.14 1.117

0.0700 0.9249 0.8663 6.34 1.129

0.0725 0.9458 0.8839 6.55 1.141

0.0750 0.9676 0.9022 6.76 1.153

0.0775 0.9904 0.9213 6.98 1.166

0.0800 1.0141 0.9411 7.20 1.178

0.0825 1.0388 0.9617 7.43 1.191

0.0850 1.0646 0.9830 7.66 1.204

0.0875 1.0914 1.0052 7.89 1.217

0.0900 1.1192 1 SO283 8.12 1.230

0.0925 1.1482 1.0522 8.36 1.243

1.000 0.911

0.919

1.13

0.96

1.001 1 .ooo

1.001

0.910 1.13 1.001 1.001

0.912 1.08 1.001 1.002

0.916 1.00 1.002 1.002

0.921 0.92 1.002 1.002

0.927 0.85 1.002 1.002

0.934 0.78 1.002 1.002

0.940 0.73 1.002 1.002

0.947 0.68 1.001 1.001

0.955 0.65 1.001 1.001

0.962 0.64 1.000 1.000

0.970 0.63 1 .ooo 1 .oMl

0.978 0.64 0.999 0.999

0.986 0.66 0.998 0.998

0.995 0.69 0.997 0.997

1.004 0.72 0.996 0.996

1.012 0.77 0.994 0.995

1.022 0.83 0.993 0.993

1.031 0.89 0.991 0.992

1.040 0.96 0.989 0.990

1.050 1.04 0.987 0.988

1.060 1.12 0.985 0.986

1.070 1.20 0.983 0.984

1.080 1.30 0.980 0.981

1.090 1.39 0.977 0.979

1.100 1.49 0.974 0.976

1.111 1.59 0.971 0.973

1.122 1.69 0.968 0.970

1.132 1.80 0.965 0.967

1.143 1.90 0.961 0.964

1.154 2.00 0.957 0.960

1.166 2.11 0.953 0.957

1.177 2.21 0.949 0.953

1.189 2.31 0.945 0.949

1.200 2.41 0.940 0.945

1.212 2.50 0.935 0.940

256

TABLE 4

Comparison of theoretical and experimental values of the mean ionic activity coefficient, osmotic coefficient and water activity coefficient of aqueous solutions of NaBr at 1 bar absolute and 25 o C

NaBr-H,O (P =l bar, T= 25°C)

Experi- Theory % error Experi- Theory W error Experi- Theory ment ment ment

0.0025 0.7669 0.0050 0.7307 0.0075 0.7148 0.0100 0.7077 0.0125 0.7057 0.0150 0.7071 0.0175 0.7111 0.0200 0.7172 0.0225 0.7249 0.0250 0.7341 0.0275 0.7447 0.0300 0.7566 0.0325 0.7696 0.0350 0.7838 0.0375 0.7992 0.0400 0.8156 0.0425 0.8332

0.0450 0.8519 0.0475 0.8717 0.0500 0.8926 0.0525 0.9147 0.0550 0.9379

0.7680 -0.14 0.7311 - 0.06 0.7158 -0.14 0.7097 - 0.28 0.7087 - 0.43 0.7111 - 0.56 0.7159 -0.66 0.7225 - 0.74 0.7306 - 0.78

0.7399 - 0.79 0.7504 ~ 0.76 0.7618 - 0.69 0.7742 - 0.60 0.7875 - 0.47 0.8016 -0.31 0.8166 -0.12 0.8324 0.10

0.8489 0.35 0.8663 0.62 0.8845 0.91 0.9035 1.22 0.9234 1.55

0.932 0.929 0.27 1.000 1 .ooo 0.928 0.926 0.19 1.001 1.001 0.930 0.929 0.04 1.001 1.001 0.935 0.935 -0.10 1.001 1.001 0.941 0.943 - 0.21 1.001 1.001 0.948 0.951 - 0.30 1.001 1.001 0.956 0.960 - 0.35 1.001 1.001 0.965 0.969 -0.37 1.001 1 .ooo 0.975 0.978 - 0.36 1.000 1 .ooo 0.985 0.988 -0.31 1.000 0.999 0.995 0.998 ~ 0.24 0.999 0.999 1.006 1.007 -0.14 0.998 0.998 1.017 1.018 - 0.01 0.997 0.997 1.029 1.028 0.14 0.995 0.995 1.041 1.038 0.31 0.994 0.994 1.054 1.049 0.50 0.992 0.993 1.067 1.059 0.71 0.990 0.991 1.081 1.070 0.94 0.988 0.989 1.094 1.081 1.19 0.986 0.987 1.108 1.092 1.45 0.984 0.985 1.123 1.104 1.72 0.981 0.983 1.138 1.115 2.01 0.978 0.981

in the computed properties is less than 10% for pressures up to and exceeding 250 bar).

The error at the higher pressure can be characterized as the model predicting a larger pressure effect on the activity coefficients than is ob- served experimentally. In view of the calculational procedures, there are three possible sources for this discrepancy: failure to adopt pressure dependent (equivalent, density dependent) hard sphere diameters; the use of P’ = 1 bar (and therefore the low pressure infinite dilution corrections to the DCFIs) rather than P’ = 250 bar; and increasing inaccuracy with increasing pressure in the approximations adopted. To determine which of these factors is dominant is beyond the scope of the current paper and will be the subject of future research.

257

TABLE 5

Comparison of theoretical and experimental values of the mean ionic activity coefficient, osmotic coefficient and water activity coeffrclent of aqueous solutions of NaBr at 250 bar absolute and 25 o C

NaBr-H,O (P = 250 bar, T = 25°C)

Y+ @ Yw

Experi- Theory B error Experi- Theory % error Experi- Theory ment ment ment

0.0025 0.7676 0.7587 1.16 0.932 0.925 0.0050 0.7316 0.7193 1.68 0.928 0.921 0.0075 0.7158 0.7018 1.94 0.930 0.923 0.0100 0.7088 0.6938 2.11 0.935 0.928 0.0125 0.7069 0.6910 2.25 0.941 0.935 0.0150 0.7084 0.6915 2.39 0.948 0.942 0.0175 0.7125 0.6944 2.54 0.956 0.950 0.0200 0.7186 0.6992 2.71 0.965 0.958 0.0225 0.7265 0.7054 2.90 0.975 0.966 0.0250 0.7358 0.7128 3.12 0.985 0.975 0.0275 0.7464 0.7213 3.36 0.995 0.984 0.0300 0.7584 0.7308 3.63 1.006 0.993 0.0325 0.7715 0.7413 3.91 1.017 1.002 0.0350 0.7858 0.7526 4.22 1.029 1.012 0.0375 0.8012 0.7648 4.55 1.041 1.022 0.0400 0.8177 0.7777 4.89 1.054 1.032 0.0425 0.8353 0.7915 5.25 1.067 1.042 0.0450 0.8541 0.8061 5.62 1.081 1.052 0.0475 0.8740 0.8214 6.01 1.094 1.063 0.0500 0.8950 0.8376 6.41 1.108 1.074 0.0525 0.9171 0.8546 6.82 1.123 1.085

0.66 0.71 0.69 0.66 0.64 0.66

0.70 0.77 0.87 0.99 1.14 1.30 1.48 1.68 1.89 2.12 2.35 2.60 2.86 3.12 3.40

1.000 1 .ooo 1.001 1 .OOl 1.001 1 .OOl 1.001 1 .OOl 1.001 1 .OOl 1.001 1.001

1.001 1 .OOl 1.001 1.001 1.000 1.001 I .ooo 1 .ooo 0.999 0.999 0.998 0.999 0.997 0.998 0.996 0.997 0.994 0.995 0.992 0.994 0.991 0.993 0.988 0.991 0.986 0.989 0.984 0.987 0.981 0.985

DISCUSSION

In this paper, we have introduced a model for electrolyte solutions which utilizes the analytic solution of the mean spherical approximation for the primitive model of electrolytes and the Debye-Hiickel theory to calculate the electrostatic portions of the DCFIs required to calculate thermodynamic properties of interest (such as the osmotic coefficient and the activity coefficients). The volumetric portions of the DCFTs are obtained from hard sphere mixture theory and all of the DCFI are corrected to ensure the correct infinite dilution behavior. The resulting model of electrolyte solutions was compared with experimental results on NaCl and NaBr and found to be capable of representing the atmospheric data very well and capable of accurate predictions up to quite high pressures.

258

Future research on this model will involve determining the effective diameters for a large number of cations and anions, evaluating the predictive power of the model for electrolytes composed of anion-cation pairs not used in the determination of the ionic diameters and extending the model beyond the single-salt/single-solvent case studied in this paper. In this regard, the multiple salt extension is straightforward as the MSA solution is for an arbitrary number of ionic species. The extension to the more interest- ing (and industrially relevant) case of mixed solvent electrolytes is more difficult as it depends on the development of theories for solvent mixture properties (such as the dielectric constant) which represent major problems in their own right.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support for this research by the National Science Foundation through grant CPE-8307280. The authors are indebted to Professor J. O’Connell for invaluable advice and encouragement in relation to the research presented in this paper.

APPENDIX

Hard sphere DCFIs from the Munsoori-Carnahan-Starling-Leland equation of state

In this Appendix, we provide expressions for the direct correlation function integrals obtained from the Mansoori-Carnahan-Starling-Leland equation of state for a hard sphere mixture (Mansoori et al., 1971). The chemical potential for species i is given by

kT =I,!$ -ln(l-<,> + s 3[,5 3410,

qtnt +Fx+ l-Ej

+ 9&,’ + 3 52Qi 2

i I[ ln(l t3)

- +

& 3

s,’ -

2(1 - ‘$)’ t3 20 - 4,)’ I

E 20, 3 -- i I[ 2 ln(1 -

53

t,) + 53fI:3)] 3

(Al)

where Ai = [h2/(2am,kBT)] II2 is the thermal de Broglie wavelength, 41”’ is the partition function for the internal degrees of freedom (McQuarrie, 1976) and &, k = 0, 1, 2, 3 is defined by

& = t t piok 642) r=l

Combining eqn. (25) with the chemical potential expression given in eqn.

259

(Al), we obtain the following equation for the DCFls

c;:.”

50

( ui + UJ + 3&( Di0J2( a; + a,)

(1 - 5,) (1 - G3)’

+ 3u,uj<,[ (‘i + ‘I)‘+ ‘1’j] + (“;u~)3SlJ + 9(u;u~EZ)’ + S*C"iuj12

0 43)" (1-t,>" (1 4,)'

X

i

6 + &(9t3 - 15) (aj + e&(6 + S,[U, - 151)

53 - tf

+ uiuj6:(6 +[3[t3(26-1453) -2’1)

a1 - 53) 1 + 6t,(g~j)~ ln(l -E,)(E3- [~;+u,]~~+o;u,CZG’) t: (A3)

where the subscripts i and j refer to the solvent (l), positive (2) and negative (3) salt ions and C,;: = Cjz hs. These species expressions are used in eqn. (80) to determine the DCFIs on a component basis.

With the exception of the salt-salt DCFI, the DCFIs obtained from the MSA and Debye-Hiickel theories do not approach the correct limits at infinite dilution and the hard sphere portion is not very good at low salt concentrations. Therefore, the infinite dilution limits for the solvent-solvent and solvent-salt DCFIs were superimposed on the hard sphere reference state to force the DCFIs to the correct limits as the salt concentration approaches zero. For the solvent-solvent DCFI, the hard sphere limits are

lim Ch” = 11

;l”” = c;;[ -8 + #(1+ 12%$(1 - 12E!))] C

Y,PO (1 - (iI4

(A4)

where

and

:1” lim -!z&

.R’O 2

k=O, 1,2,3 (A5)

260

where

For the solvent-salt DCFI, the hard sphere limit is

(A7)

LIST OF SYMBOLS

Helmhol tz free energy fitted constant in eqn. (93) fitted constant in eqn. (83) Debye-Hiickel constant in eqn. (110) parameters in the expansion of apparent molar properties defined in eqns. (87) and (88) fitted constant in eqn. (93) fitted constant in eqn. (83) fitted constant in eqn. (110) number of component in solution molar concentration of salt fitted constant in eqn. (93) fitted constant in eqn. (110) direct correlation function between species i and j molecules/ ions three-dimensional Fourier transform of ciJ (r ) volume integral of PC;,(~) short-range part of the direct correlation function between species i and j molecules/ ions volume integral of pc,y,(r) infinite dilution correction to C,, C,, obtained from Debye-Hiickel theory electrostatic contribution to Ci, C;, for a mixture of hard spheres specific contributions to C;:” defined in the Appendix electrostatic part of C,) obtained for a mixture of charged hard spheres using the mean spherical approximation

261

D*

; E* F*

g;,(r)

h,j(r)

m m

mi N

NA N, No, P

PO

C,

4 r

r ‘J

s

s,, Sk, s,

fitted constant in eqn. (110) electronic charge internal energy fitted constant in eqn. (110) fitted constant in eqn. (110) radial distribution function between species i and j molecules/ions total correlation function between species i and j molecules/ ions volume integral of ph,,

vector of length n in which each element is unity molal ionic strength Boltzmann’s constant parameters in Debye-Hiickel form of direct correlation function integrals defined in eqns. (62) and (63) number of components in mixture molality of salt in solution in eqns. (112) and (114) molality of species i

total number of moles in mixture Avagadro’s number number of molecules of i in mixture on species basis number of molecules of i in mixture on component basis pressure reference pressure parameter in mean spherical approximation for direct correlation function integral defined in eqn. (69)

cS&J,Z12 intermolecular separation position of molecule/ion i in space gas constant distance between centers of species i and species j molecules/ ions number of ionic species in solution parameters in the expansion of apparent molar properties defined in eqns. (87)-(91) absolute temperature pair potential between species i and j molecules/ions partial molar volume of i on species basis partial molar volume of i on component basis partial molar volume of salt at infinite dilution molar volume of pure solvent at pressure P

molar volume of pure solvent at pressure PO

262

Greek letters

ff

B P* 4, A

UT

species mole fraction of i on species basis component mole fraction of i on species basis final and reference component mole fraction of i in eqn. (101) component mole fraction of i on component basis valence of species i ion generic symbol for a matrix i, j element of matrix Z transpose of matrix Z matrix obtained from Z by deleting i th row and j th column determinant of matrix Z matrix of cofactors of Z

parameter in the Debye-Hiickel expression for excess Helm- holtz free energy defined in eqn. (57)

VW fitted constant in eqn. (110) Kronecker delta parameter in mean spherical approximation for direct correlation functional integral defined in eqn. (67) excess Helmholtz free energy excess Helmholtz free energy dielectric constant molar osmotic coefficient apparent molar compressibility apparent molar compressibility at infinite dilution apparent molar volume apparent molar volume at infinite dilution activity coefficient of i

mean activity coefficient of salt electrostatic contribution to mean activity coefficient of salt parameter in mean spherical approximation for direct correlation function integral defined in eqn. (72) isothermal compressibility solvent isothermal compressibility at pressure P solvent isothermal compressibility at pressure P” thermal de Broglie wavelength chemical potential of species i

stoichiometric coefficients in salt dissociation equilibrium

v, + v- osmotic pressure

263

Xok

T

parameter in mean spherical approximation for direct correlation function integral defined in eqn. (71) total density of mixture final and reference total densities of solution in eqn. (104) density of species i density of water density of component i diameter of species i molecule/ion distance of closest approach of species i molecule/ion to species j molecule/ion parameter in the hard sphere direct correlation functions defined in eqn. (A7) dimensionless density parameters in the hard sphere direct correlation functions defined in eqns. (A2) and (15) parameter in mean spherical approximation for direct correlation function integral defined in eqn. (70) parameter in mean spherical approximation for direct correlation function integral defined in equation (68)

Other symbols

on vector of length n in which each element is zero

C> grand canonical ensemble average of enclosed quantity

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an analytic model for aqueous electrolyte solutions based on

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