Mean Spherical Model for Asymmetric
1311
Electrolytes
Acknowledgment. We wish to thank Professor T. Keii for helpful discussions.
References and Notes (1) F. Haber and J. Weiss, Proc. R. SOC.London, Ser. A , 147, 332 (1934). (2) N. Uri, Chem. Rev., 50, 375 (1952). (3) C. Walllng, Acc. Chem. Res., 8, 125 (1975). (4) W. T. Dixson and R. 0. C. Norman, J. Chem. SOC., 31 19 (1963). (5) E. Saito and B. H. Bielski, J . Am. Chem. Soc., 83, 4467 (1961). (6) Y. S.Chiang, J. Craddook, D. Micketvich, and J. Turkevich, J. Phys. Chem., 70, 3509 (1966). (7) F. Scillo, R. E. Florin, and L. A. Wall, J. Phys. Chem., 70, 47 (1966). (8) M. Setaka, Y. Kirino, T. Ozawa, and T. Kwan, J . Cafal., 15, 209 (1969). (9) J. Weiss, Trans. Faraday Soc., 31, 1547 (1935). (IO) J. Weiss, Adv. Catai., 4, 343 (1952).
(11) D. W. Mckee, J . Cafal., 14, 355 (1969). (12) D. C. Bond, "Catalysis by Metals", Academic Press, London, 1962, p 432. (13) S. Fukuzumi, Y. Ono, and T. Keii, Bull. Chem. SOC.Jpn., 46, 3353 (1973). (14) S. Fukuzumi, Y. Ono, and T. Keii, Inf. J. Chern. Kinet., 7, 535 (1975). (15) S.Fukuzumi and Y. Ono, J. Chem. Soc., Perkin Trans. 2, in press. (16) P. F. Knowles, J. F. Cibson, F. M. Pick, and R. C. Bray, Biochem. J.. 111. 53 (1969). (17) D.'Ballon, G: Palmer, and V. Massey, Biochem. Biophys. Res. Commun., 36, 898 (1969). (18) R. Nilson, F. M. Pick, and R. C. Brag, Biochem. Blophys. Acta, 192, 145 (1969). (19) J. M. McCord and I. Fridovich, J . Biol. Chem., 244, 6065 (1969). (20) M. G. Evans and N. Uri, Trans. Faraday Soc., 45, 224 (1949). (21) G. Czapski and L. M. Dorfman, J . Phys. Chem., 68, 1176 (1964). (22) J. Rabani and S. 0. Nielsen, J . Phys. Chem., 73, 3736 (1969). (23) D. Behar, C. Czapski, J. Rabani, L. M. Dorfman, and H. A. Schwarz, J . Phys. Chem., 74, 3212 (1970).
Mean Spherical Model for Asymmetric Electrolytes. 2. Thermodynamic Properties and the Pair Correlation Function L. Blum" Physics Department, College of Natural Sciences, University of Puerto Rico, Rio Piedras, Puerto Rico 00931
and J. S. H$ye Institutt for Teoretisk Fysikk, University of Trondheim, 7034 Trondheim NTH, Norway (Received December 28, 1976)
It is shown that the excess thermodynamic properties calculated from the mean spherical approximation (via the energy integrals) for the primitive model of ionic mixtures, that is, electrolytes and molten salts, can be expressed as functions of a single parameter r. The expressions are quite simple, and for what we call low concentrations (up to 1 N of a simple salt) reduce to the same relations that Debye found for the finite size ions, only that 2r takes the place of the Debye length. The interpretation that 2r is in fact the correct screening parameter for finite size ions is borne by the asymptotic form of the pair correlation function at small densities.
1. Introduction In a recent paper we found the general solution of the mean spherical approximation (MSA)' for a neutral mixture of charged hard spheres.' The unrestricted general solution is possible because in the MSA the properties and structure functions of the individual ions are scaled by a single parameter r, which has the dimensions of an inverse length. In our previous work, we gave rather complicated expressions for the thermodynamic properties of the mixture, involving integrals that had to be found numerically. However, for equal sizes, and/or low concentrations we found that the osmotic coefficient3(and also the activity coefficient4) could be integrated explicitly to yield a strikingly simple formula (in complete agreement with the Waisman-Lebowitz result'), which looked just like the Debye-finite size expression, but just with 2r in the place of the inverse Debye screening length.5 In the present work we give general expressions for the excess osmotic coefficient and excess activity coefficient of the general mixture. A formula for the excess free energy is also found, which is in agreement with an expression derived independently by Hiroike.6"1 All of these expressions consist of the sum of two terms: The first term is of the Debye-Huckel form, involving the parameter r, while the second depends also on the size differences of the ions, and vanishes when all the ions have the same diameter. The size difference term is analogous to the one
discussed by Stell and Lebowitz? and reflects the different shielding capability of the ions according to their size. Larger ions are less effective in their shielding ability than the smaller ones for the simple reason that their sheer size prevents them from coming close to the center ion. Indeed, (2I'-' is always larger than the Debye shielding length
The striking similarity of the MSA to the Debye-Huckel theory is also present in the binary correlation function. However here matters are more complex because the excluded volume effects should give charge oscillations in the ionic cloud, as predicted by the Stillinger-Lovett second moment condition' 471JOw drCzizjpipjgij(r)r4 =
id
-6C1 plzf2/ ~0
2
(1.2)
in a more classical language, the charge oscillation is reflecting the ion air formation that Onsager discussed many years ago.' Indeed, the hard core version of the Debye theory, which has been popular among electrochemists in its original form and in many improved versions, does not satisfy the second moment condition, while Groeneveld has shown" that the MSA and the hypernetted chain (HNC) approximations are among the theories that satisfy (1.2) exactly. Indeed, the work of Friedman and The Journal of Physical Chemistry, Vol. 81, No. 13. 1977
L. Blum and J. S. Hlye
1312
collaborators’’ has shown that the HNC12 equation is a very succesful one, since it does represent the results of computer experiments and real systems quite well. However it is not an easy theory to work with, and the calculations represent a substantial programming job. This explains why there is still so much interest in the improved versions of the Debye theory, which combines intuitive interpretation with simplicity in the calculation. We should mention in this connection the recent work of Olivares and McQuarrie,13Outhwaite,14and Pitzer15 in which different but ad hoc modifications of this theory are presented. However in the Debye theory, the ions of the ionic cloud are point charges, and this makes the theory inconsistent from the statistical point of view. Even worse, the thermodynamic limit does not exist for a system of point charges, without repulsion core. We believe that the new results presented here show that the MSA is the Debye-Huckel theory with the correct treatment of the exclusion core, not only by its formulation (Percus’), but also since it has a similar structure, that is the equations are formally the same, with the only difference that the ionic cloud is not calculated for point objects, but rather finite size charges. So we believe that the MSA represents the “middle of the road” alternative, and has simplicity and intuitive appeal, if perhaps not as accurate as the HNC. A short summary of the pertinent results of ref 2 are given in the next section, where we also discuss convenient ways to calculate the parameter r. In section 3 we give the main results of this work: quite simple formulas for the thermodynamic properties for arbitrary mixtures of hard ions in the MSA. Section 4 is devoted to the discussion of the pair correlation function. For the restricted primitive model, Hirata and Arakawa16 obtained the Laplace transform of this function and also a series for the function itself. We will get the general form of the Laplace transform of the pair correlation function for the arbitrary ionic mixture. This function is shown to be symmetric in a nontrivial calculation. Furthermore for the restricted primitive model the expression is in agreement with the previous work. In the low concentration regime, much of the complicated charge-hard core cross contribution vanishes and one recovers a simple formula which again has two terms: a hard core term, which is the PercusYevick pair correlation function for hard spheres, and a purely ionic term, which on Laplace inversion will yield the charge oscillation predicted by Stillinger and Lovett. For dilute solutions we give a more convenient expansion that starts with
the pair correlation function h,(r) = gij(r)- 1 (2.2) The hard core exclusion requires that for r IuLj
h,(r) = -1 (2.3) while for the direct correlation function the MSA condition is
cij(r)= -puij(r) for r > uLjhere uij =
(1/2)(Ui +
(2.4) Uj)
is the distance of closest approach. Furthermore, the electrostatic potential is
uij(r)= e 2 ~ i ~ j / ~ 0 r
(2.6) The solution of the Ornstein-Zernike equation of ref 2 was given in terms of the factor correlation function17 matrix Q ( r ) . For the general ionic mixture in the MSA has the following form Q i j ( r )= lim{(r - uii)Qij’+(1/2)(r- uij)2Qj”-ziuje-ctr} P+ 0
Xii < r < uii Qij(r)= lim(-ziuje-~‘}
~
ct+O
r > 0,;
Xji
(2.7) = ( 1 / 2 ) ( ~-j uZ)
(2.8) where Qcl’, Q,”, and a, are numeric coefficients that were found by solving a set of algebraic equations derived from the conditions (2.3) and (2.4); and also by considering the integral equation that defines the pair correlation function in terms of the matrix elements of Q,(r) Jii(r) = Qij(r) f
zk p k s hjk ”
dr, Jik(lr
- l-1 I ) Q k j ( r l )
(2.9)
where
(2.10)
Jij(r)= 2n4”drl r l h i i ( r l )
The coefficients of the factor correlation function are found to be 71 2r2 (2.11) 9.:= - u j j -uiuj(2> - -puiuj “ A 4A
+
”(
2n Qj”= ,[1 a. =
+ (2~j(n/2A)+ (1/2)ajPn]
(Y2
2 r ( 1 + roj)
2A
(2.12) (2.13)
which again is of the Debye-Huckel form, with 2 r as the shielding parameter. This result lends support to the interpretation of 2 r as the shielding parameter in the calculation of the thermodynamic properties. 2. Summary of Previous Results We will represent the actual electrolytic solution (or molten salt) by the primitive model, which consists of a mixture of charged hard spheres of species 1, . . i, ., n, with number density pL,diameter uc,and charge (actually electrovalence) 2,. The mixture is embedded in a continuum of dielectric constant to,and is electroneutral. In other words we require that
.
i pizj = 0
i =1
(2.1)
We recall, furthermore, the MSA boundary conditions for The Journal of Physical Chemistry, Vol. 81, No. 13, 1977
(2.17) (2.18) finally the parameter I? is determined by solving the algebraic equation3
the degree of this equation depends on the number of ions involved: for a binary salt it is of the 6th degree and has
1313
Mean Spherical Model for Asymmetric Electrolytes
to be solved numerically. From all the six solutions we have to select the physically acceptable, which is positive and tends to the Debye inverse length K~ from below for infinite dilution < 217 -+x 0 (2.20) The numerical solution can be obtained from either the Newton-Raphson formula or even by simple iteration of some guessed initial value of r. For 1-1 electrolytes a very good initial value is obtained from the inverse screening length obtained from the Waisman-Lebowitz solution but using the average diameter (2.21) 0 = 61/60 as the hard core diameter. That is, the initial value is
2 r 0 = [- 1 + (I t 2 ~ ~ i /ir ~ ) ~ / ~ ]
(2.22)
to conclude this section we should mention that the coefficients QV’ and Q,” are simply related to physically interesting quantities. It is easy to see from (2.9) that the contact value of the pair correlation function is given by g{(ojj) = Q ~ ’ / ( ~ T c J ~ )
(2.23)
while, differentiating (2.9) twice we also obtain
Qj”/27r= 1- h $pkJ21 - s F ~ j [ ( Q l ” ) 2 - (Qlor’)*1
(3.3)
this result is obtained from ref 18 together with eq 2.23 and 2.25. The superscript zero means that we are taking the values for the discharged case. Some algebra shows that (3.3),together with (2.11), (2.12), and (2.13) yields the surprisingly simple formula PAA = PAE ry3.n (3.4)
+
This same relation was obtained recently in an independent investigation by Hiroike.‘ Next we compute the excess osmotic coefficient from the thermodynamic relation
(3.5)
which together with
where AI’ is the excess pressure, and /3 is the Boltzmann thermal factor (l/kBT). For this quantity we have from ref 18
where’ (see also ref 2 for notation) 6, - (pipj)”*~ij(O)= Z:&zik(O)$k(o) k
(2.26)
yields the simple formula for the inverse compressibility which, after some algebra, leads to
X-l
(2.27)
(3.7)
3. Thermodynamic Properties One of the convenient features of the MSA is that it gives explicit, yet general relations for the thermodynamic properties, such as the osmotic and activity coefficients, derived via the internal energy. We recall first that since the MSA is not the exact solution of the statistical mechanical problem of the neutral mixture of hard spheres, it will be inconsistent, in the sense that the same quantity, for example, the pressure, when computed in different ways, say, the virial, compressibility, or energy formulas, will yield different numerical answers. From past experience and comparison to computer experiments it is expected that the best results will be those of the internal energ path. It has been recently shown by HBye and Stell’ that the virial pressure equals the energy pressure when the second-order corrections from the graphical expansion to the MSA are taken into account. This result, general for the MSA, is also the basis for the calculations of the thermodynamic properties of this section. Let us first recall that the excess internal energy is given by’s3
Finally, we get the activity coefficient from the thermodynamic formula AA p=-A$ + A Iny+ (3.8)
l
60
and the previous results (3.4) and (3.7). The activity coefficient is then
(3.9) where AI3 is given by (3.1). We observe that all of the above expressions for the excess thermodynamic properties in the MSA consist of the sum of two terms, one of which is exactly of the Debye-Huckel form, but with 2 r as the shielding parameter, while the other contains the parameter p,, defined by eq 2.14. For low numerical densities (still a respectable 1 N of a 1-1 electrolyte) this parameter is quite small and can be neglected. In this case, we obtain
A lnyk =-71 + -f2Pn2 2A
t
(3.10)
A @ = -r3/3nfo
C? r
2
pi
4nf0 i 1 +
~ i 2
roi
(3.11)
Both of these relations were derived previously from approximate relations that neglected P, from the start.334 The Journal of Physical Chemistry, Voi. 81, No. 13, 1977
1314
L. Blum and J. S.Hdye
In this case the integration of (3.2) is especially simple, since all the thermodynamic quantities depend on the single parameter a 2 p , instead of a' and p separately. As was already mentioned in the Introduction (3.10) and (3.11) are very similar to the Debye-Huckel formulas (for the primitive model in which the ion has a finite size) A $ = - zo3/24=bo
1
(3.12) x 03
A In y+ = -8 ~ 6 01 t
(3.13)
all that simplelgand an explicit, closed formula for gij(r) cannot be obtained. Numerically, however, the Laplace inversion of (4.1) should be rather simple, since the Laplace transform can be converted to a Fourier transform, for which very efficient and fast programs for calculating the inverse exist. To compute 6iJ.s)we first take the derivative of (2.9) with respect to r, and the take the Laplace transform. A simple calculation leads to
UXO
with the only difference that the inverse screening length is now 2I'. In the next section we will show that this is due to the fact that in the MSA and for relatively low concentrations, 2I' is the inverse shielding length in the Debye sense, that is, the decay rate of the correlation function, and therefore it is only natural that all the thermodynamic quantities are scaled by this length. The surprising simplicity of the above results is in sharp contrast with the rather involved manipulations that are necessary to derive them. There is also no rigorous proof that the scaling relation is correct as conjectured in ref 2. Therefore it is quite important to have independent tests of the solution: One of these tests is the fact that the pair correlation function gij(r) is symmetric, as will be shown in the next section. This result is nontrivial and depends quite heavily on all the details of the analytic solution. A further, perhaps more reliable, check is possible using the general results of ref 18, in which relations that do not depend on the details of the solution were derived. The only relations needed were the Ornstein-Zernike equation and eq 2.3 and 2.4. The first of these relations is
where
(4.3)
(4.4) where +,(a) functions
are slightly modified incomplete gamma
p l ( u h )= ( 1 - s(sk - e-suh)/s2 v z ( ( 3 h ) = ( 1 - s ( s k + s z 0 k 2 / 2- f?-s0k)/s3
(4.5)
(4.6)
From (2.11), (2.12), and (2.13) we readily obtain N
p h Q h j ( s =) e-shjh[Skj- p h ( a k + OjPk + ajrk)]
-
+ A In yi:dto
(3.14) Here, we have checked the numerical derivatives against (3.11). the analytical expressions of hE (3.1) and A In We found agreement within the precision of the computer. Another check can be obtained from the relation (anA/api)P,pj+i= P i (3.15) = A E do
in which the right-hand side is calculated by numerical differentiation of (3.3)'' while the left-hand side is obtained from the analytic solution of the MSA. This test also shows perfect agreement. A further test of the solution was to obtain the dense point limit of Waisman and Lebowitz.' Again here, there was complete agreement. As an interesting mathematical sideline, we note that in the cases were the temperature 6 or the interaction strength 01 (which is dependent on the dielectric constant to) are not the fixed quantities, it will be unnecessary to solve (2.19) to obtain .'I All we need to do is to compute a, and therefore as a function of.'I In fact, everything else depends only on r (and of course on the individual zi,q,and pi). Thus, r plays the crucial role of the universal scaling parameter. 4. The Pair Correlation Function A quantity of major interest in the study of ionic solutions and molten salts is the pair correlation function g&). In this section we will derive a general formula for the Laplace transform
Eij(s)= J i d r e-srrgij(r)
(4.1) As is known, in fluid theory, even for the simple models, the analytic structure of G&s) in the complex plane is not The Journal of Physical Chemistry, Vol. 81,No. 13, 1977
(4.7)
where
where p&a) is the Laplace transform of the square wave pulse cpo(o) = Joudre-sr =
(1- e-su)/s
(4.11)
The matrix defined by (4.7) is of the so-called Jacobi type, and has an inverse that can be found explicitly for any size of the matrix. When this inverse is replaced into eq 4.2, we obtain after some rather laborious algebra (see Appendix A for details)
+ AEij(s)
(4.12)
here, the first term gives essentially the Percus-Yevick, pure hard sphere pair distribution, the second gives the MSA pure excess electrostatic contribution, and the third one, the cross interactions of the hard core and the charges. The terms in (4.12) are defined as follows:
1315
Mean Spherical Model for Asymmetric Electrolytes
DT = DOD,
+ ADT
(4.13)
(4.25) and, using (2.13) and (4.15), we obtain in this regime N
N
Gij(s)= Gi:(s) (4.14)
+
= zizja2/[4n(i roj)(l
(4.15)
+ ruj)]
(4.27)
The Laplace inverse of the excess electrostatic part can be obtained either by expanding the denominator of (4.26)
(4.28)
(4.16)
and obtain the zone expansion of the pair distribution function16
(4.17)
0
Agij(r)= 2 Agij(rn) rn =O
(4.18)
2r2 [sin r ( r - uij) r - cos p ( r - oU)]e-r(r-aij) (4.29) uij < r < min ( o i j ul) This expansion is rather inconvenient for low concentrations, because then the shielding length is much larger than the hard core diameter, and very many terms of (4.29) are needed. These terms become increasingly complicated, as can be appreciated. A more sensible expansion for the low concentration regime is obtained using the fact that for this case it is true that A g i j ( ' ) ( r= ) -A,-
2r P, = - C p , u p l CY2 1
(4.19) (4.20) (4.21)
With these definitions, we write first the "discharged" pair correlation function
+
on2
i
(4.30) and therefore we expand
- uj))
(4.22)
while the cross contribution to the pair correlation function is
The Laplace inversion of this series is quite straightforward, since, apart from some discontinuities, it has only one pole at s = -2r. Using (4.11) we have for the first few terms
gij(r) =gijo(r) - (Aij/r){ q ( r - 0ij)e-2r(r - oij)
- ~2r2 ~ P l a 1 Z / ~ - u i j d r l r lv(ol e - 2+r ruij ~-r
+ rl) + . with
(4.24) It is easy to see that all the Ps vanish for equal size ions, and that they are quite small for low ionic concentration (pa3 0.1). Therefore for not too concentrated ionic solutions, they can be neglected. We obtain
-
a}
(4.32)
where gi,'(r) is the Percus-Yevick, hard core pair correlation function, and ~ ( x is) the Heaviside function. This approximation for the pair correlation function is quite appealing. Due to its simplicity, it can be used to compute higher order graph contributions beyond the ring diagrams which yield the Debye-Huckel approximation, with relatively little effort? and indeed, we plan to do such a calculation in the near future, and expect to get results considerably more accurate than the classic ones because of the optimization of the MSA.20 Finally, we should The Journal of Physical Chemistry, Voi. 81, No. 13, 1977
1810
L. Blum and J. S.H@ye
remark that to the first approximation in the ionic strength (4.32) is just of the same form as the classical Debye result
6g..(r) I/ = A..e-2r(r-uij)/r 11 r > uji
(4.33)
be symmetric constitutes an argument for the correctness of the symmetrization conjecture used in this work. One of the crucial relations used derives from (2.13), and reads
-2sr2
but with the different shielding length 2r.
Acknowledgment. We are indebted to Professor K. Hiroike for sending us a copy of his work prior to publication.
CYz
( i f=)f[2(bne^)+ s(d"e") t P , ( 1 - a"b^)]
similarly, we also need
sqi?J
Appendix A. Derivation of the Pair Correlation Function The solution of the system of linear equations (4.2) requires the inversion of a matrix with elements
-2sr2
M. 11 = &ij
where the symbols are defined above and in section 2. Repeated used of these relations shows that the pair correlation function is indeed symmetric:
[aij
- ~iGij(s)l
(A.1)
We first observe that M is a matrix of the form
a2
(cf ) =
.-.
[(s
+ t2-36e^)+
(de^)
where we are using the notation
gj = 1
A
ai
= pjaj
A
A
cj = pjpj
dj = uj
A
h
e 1. = p rYi .
f j
= (5
By simple substitution we can check that the inverse of this matrix, M-l, has elements 1 M ~ T '= ~ j +j ----{ijij[(l - 2)(1e^f) - ( c f ) ( e d ) l AA
AA
DT
2r2s
+ ( c f ) ( a d ) ] - --J[ U j U j + U j U j ] [ ( c ^ f ^ ) ( l - 8) A A
A A
+ &2j[(c"S)(1- Gf) + ( b e ) ( c f ) l + u j G [ ( o e ) ( l- G2)+ ( b c ) ( d e ) l + c7:d;.[(1 AA
A
A
Ah
A A
-
A h
A h
A h
u ~ ) (-I if)- ( a f ) ( b e ) ]+ ^cjXj[[iad)(l-if) AA
AA
+ (if)(d^e")] + ij3[(d"i)(l -;?I+ ) @)(S2)] + 2&1- &(l - 2 )- ( a d ) ( b c ) ] + & 6 j [ ( $ ) ( l - 22) + (^u;z)(2p)+ 2Jj[(2f)(l-;b) AA
+
A h
(mhl}
(-4.4)
where
(1- ;8)(1 - &?)(I - i f )- ( a d ) ( b c ) ( l - ^e?) A A
DT =
A h
AA
- ( c f ) ( e d ) ( l -3 )- ( a f ) ( b e ) ( l -2) AA
-
M
A A
(2)(6i)(Gf)- (;?)(%)(2;)
(A.5)
and we used the notation n
A A
(a"b") = 2=4 c: qq
(A.6)
throughout. Replacing this result into (4.2), and making use of (A.3), (A.4), (429, (4.9), and (4.10) yields an explicit, but quite complex expression for Glj(s). This expression is not symmetric as required by simple reciprocity. The fact that after a nontrivial algebraic cglculation, which makes full use of the solution of ref 2, GJs) turns out to
The Journal of Physical Chemistry, Vol. 81, No. 13, 1997
and replacing (A.3), (4.8-lo), and (2.13) into this yields after some more algebra, eq 4.12.
References and Notes (1) J. K. Percus and G. Yevick, Phys. Rev., 136, 8290 (1964); J. L. Lebowitz and J. K. Percus, ibid., 144, 251 (1966); E. Waisman and J. L. Lebowitz, J . Chem. Phys., 56, 3086, 3093 (1972). (2) L. Blum, Mol. Phys., 30, 1529 (1975). (3) R. Triolo, J. R. Grigera, and L. Blum, J. Phys. Chem., 80, 1858 (1976). (4) J. C. Rasaiah, D. N. Card, and J. P. Valleau, J . Chem. Phys., 56, 248 (1972). (5) J. E. Mayer, J. Chem. Phys., 18, 1426 (1950). (6) K. Hiroike, Mol. Phys., in press. (7) G. Stell and J. L. Lebowitz, J. Chem. Phys., 49, 3706 (1968). (8) F. H. Stillinger and R. Lovett, J . Chem. Phys., 48, 3858 (1968). (9) L. Onsager, Chem. Rev., 13, 73 (1933). (10) J. Groeneveld, unpublished. (11) J. C. Rasaiah and H. L. Friedman, J. Chem. Phys., 48, 2742 (1968); 50, 3965 (1969). (12) A. R. Allnatt, Mol. Phys., 8, 533 (1964). (13) W. Olivares and D. McQuarrie, Biophys. J., 15, 143 (1975). (14) C. W. Outhwaite, "Equilibrium Theory of Electrolyte Solutions" in "Statistical Mechanics' , Vol. 11, The Chemical Society, London, 1975. (15) K. S. Pitzer, J. Chem. Phys., 77, 268 (1973). (16) F. Hirata and K. Harakawa, Bull. Chem. SOC.Jpn., 48, 2139 (1975). (17) R. J. Baxter, J . Chem. Phys., 52, 4559 (1970). (18) J. S. Hb/e and G. Stell, "Thermodynamics of the MSA for Simple Fluids", SUSB Report No. 287, Stony Brook, 1976. (19) M. S. Wertheim, J. Math. Phys., 5 , 643 (1964). (20) H. C. Andersen and D. Chandler, J. Chem. Phys., 57, 1818 (1972); D. Chandler and H. C. Andersen, bid., 57, 1930 (1972). (21) While this work was in preparation we received a preprint from Professor K. Hiroike in which, among other results, an expression for the free energy was given.
