398
J . Phys. Chem. 1992, 96, 398-403
Self-Diffusion in Electrolyte Solutions Using the Mean Spherical Approximation Olivier Bernard, Werner Kunz, Pierre Turq,* and Lesser Blumt Laboratoire d’Electrochimie Tour 74, UniversitE Pierre & Marie Curie, URA 430, 8 Rue Cuvier, 75005 Paris, France (Received: March 26, 1991; In Final Form: June 24, 1991)
We use the equations of the Fuoss-Onsager theory, which are solved by a Green’s function technique that yields explicit expressions in tems of the equilibrium pair correlation functions. The pair correlations needed to obtain these expressions are derived from either the hypernetted chain (HNC)or mean spherical (MSA) approximations. It is found that both yield rather good agreement with available experimentaldata for monovalent ions. The MSA has the advantage of yielding explicit expressions. In our treatment, the excluded volume of the ions is included in the corrections.
1. Introduction
Systems with long-range Coulomb interactions have been, for a long time, a challenging problem in statistical mechanics. Although the Debye-Hackel theory is the first major advance in our understanding of electrolytes, this theory cannot be applied to dense systems (concentrated solutions). The introduction of integral equations, most of them based on closures of the Omstein-Zernike (OZ) equation, and the availability of computer simulations have made possible further advances in this field. The systematic graphical analysis of the approximate closures of the OZl equation shows that the hypernetted chain (HNC) equation has the largest number of terms and therefore should be (and is!) the most accurate theory for electrolytes. This equation was first proposed in connection with electrolytes by E. Meeron.2 Numerical solution of the HNC by Rasaiah and Friedman3 showed that this approximation is very good for electrolytes. This fact has been confirmed by over 20 years of extensive use of this approximation. This is certainly the most accurate integral equation. However, the HNC has some limitations, mainly because it has to be solved numerically, and the highly nonlinear equations may not always converge. Alternative closures have been proposed, which correct thermodynamic inconsistencies4and the omission of higher graph^.^ A much simpler approximation is obtained from linear response theory. This is the mean spherical approximation (MSA), introduced by Percus, Lebowitz, and Ye~ick,~,’ originally as the embedding of the Debye-Huckel theory into the OZ equation. This theory was solved analytically by Waisman and Lebowitz* for the restricted case of equal-size ions and by Blum for the general case? The thermodynamic properties and pair correlation functions of the MSA are given in terms of a single parameter r, for any arbitrary mixture of hard-sphere ions. The expressions are formally identical to those of the Debye-Huckel theory.I0J’ The main feature of the MSA is its simple analytical form. However, it does yield poor pair distribution functions for highly charged systems at low concentrations. On the other hand, it leads to the exact Onsager bounds for the internal energy at infinite coupling and concentration.I2 The introduction of chemical association by Ebeling and c o - ~ o r k e r s lhas ~ ~ produced ’~ a much better theory. Again, if the chemical association is represented by a sticky potential, the same remarkable simplicity of the MSA is preserved.15 In recent years, integral equations were applied to study thermodynamic properties of electrolytes in various solvents. Phase behavior]’ and neutron scattering structure factorsIsJ9 were described by these methods. Self-consistent potentials were derived for the calculation of several different properties of the electrolytes.20 The O Z equation is
To whom correspondence should be addressed. ’ Permanent address: Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931-3343.
where cij(r)is the direct correlation function, and is defined by this relation, and hi,(r) = gi,(r) - 1 where gij(r) is the pair distribution function. This equation represents merely a definition of cii(r)and needs to be supplemented by another relation between the correlation functions. In the HNC we have
[
5j(r) gi,(r) = exp -&BT
+ hij(r)- cij(r)
where
V j ( r ) = v j C b ( r )+
v;(r)
(3)
where vi,Cb(r) is the Coulomb pair potential of an ionic pair in a continuum solvent of dielectric constant e (4)
with K / ( r ) , the hard-sphere potential
where uijis a distance parameter, usually the sum of the ionic radii of ions i and j . We use ei = eZi for the product of the elementary charge by the charge number of the ion under consideration. kB is Boltzmann’s constant, and T i s the temperature. In the MSA, we approximate cij(r)and gij(r)by the superposition of Coulomb and hard-sphere contributions: if r < uij gij(r) = 0 (6) Viyb(r) cij(r)= - - if r> uij (7) kBT Transport properties are less understood from the theoretical point of view. (1) Morita, T.; Hiroike, K. Prog. Theor. Phys. 1961, 25, 537. (2) Meeron, E. J . Chem. Phys 1957, 28, 630. (3) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968, 48, 2742. (4) Zerah, G.; Hansen, J. P. J . Chem. Phys. 1986, 84, 2336. (5) Rossky, P. J.; Dudowicz, J. B.; Tembe, B. L.; Friedman, H. L.J. Chem. Phys. 1980, 73, 3372. (6) Percus, J. K.;Yevick, G. J. Phys. Rev. 1964, 136, B290. (7) Percus, J. K.; Lebowitz, J. L. Phys. Rev. 1966, 144, 251. (8) Waisman, E.; Lebowitz, J. L. J . Chem. Phys. 1972, 56, 3086; 1972, 3093. (9) Blum, L. Mol. Phys. 1975, 30, 1529. (10) Blum, L.;Heye, J. S. J . Phys. Chem. 1977, 81, 1311. (11) Hiroike, K . Mol. Phys. 1977, 33, 1195. (12) Rosenfeld, Y.;Blum, L. J. Chem. Phys. 1986, 85, 1556. (13) Ebeling, W.; Grigo. M. J . Sol. Chem. 1982, 11, 15 1. (14) Kremp, D.; Ebeling, W.; Krienke, H.; Sindig, R. J. Star. Phys. 1983, 33, 99. (15) Herrera-Pacheco, J. N.; Blum, L. J. Chem. Phys. 1991, in press. (16) Rasaiah, J. C. Inr. J . Thermodyn. 1990, 1 1 , 1. (17) BeIloni, L. Chem. Phys. 1985, 99, 43. (18) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (19) Wu, C.-F.; Chen, S.-H. J. Chem. Phys. 1987, 87, 6199. (20) Kunz, W.; Calmettes, P.; Turq, P. J . Chem. Phys. 1990, 92, 2367.
0022-36S4/92/2096-398$03.00/00 1992 American Chemical Society
Self-Diffusion in Electrolyte Solutions One basic difficulty is that the solvent-averaged Hamiltonian (McMillan Mayer level as indicated in ref 21) does not include the solute-solvent interactions in a direct way. It is therefore impossible to predict the transport properties of the solute at infinite dilution in the solvent from first principles. However, the solutesolvent collisions can be mimicked by friction forces which balance the thermal motion of the solute particles. This can be done by several types of Brownian approximations, either by computer simulation or by semianalytical theories: Brownian dynamics simulations can be done either at the Langevin leve121or at the Fokker-Planck-Smoluchowsky l e ~ e l ? Z ~ ~ Most of the theoretical treatments 14,24-27start from the Fokker-Planck-Smoluchowsky level. Hydrodynamic continuity equations have been used to extend Debye and Hiickel equilibrium theory to transport processes by Onsager and c o - w o r k e r ~ . ~ ~ - ~ ~ Improved transport coefficients can be obtained using more accurate correlation functions, such as those obtained from the HNC a p p r o x i m a t i ~ n . ' ~ * ~ ~ . ~ ~ It is possible to use directly in the Fuoss-Onsager theory more accurate pair distribution functions, using a Green's function technique, with appropriate boundary conditions. This is done in the present paper. The results are compared to other work in the l i t e r a t ~ r e . ~ ~ . ~ ~ We study the corrections of a given property from its value at infinite dilution. For self-diffusion this variation involves only the so-calledrelaxation effect, which gives the return to equilibrium in the linear response regime after a perturbation by an external force. In computer simulation the self-diffusion coefficient is simply related to the velocity autocorrelation function.21 Selfdiffusion coefficients are the simplest transport coefficients, because they involve only one type of particle. Extensive sets of experimental data on them can be found in ref 33. In the Fuoss-Onsager theory there are different laws which describe the selfdiffusion coefficient as a function of concentration: the limiting laws, for low concentrations, and the extended laws, for higher concentration^.^^-^'^^^-^^ In section 2, we present the general solution scheme for the evaluation of the relaxation effect using the Green's function method. Section 3 is devoted to the explicit solution of the first-order equation. The higher order terms are examined in the section 4. The results for both the MSA and HNC approximations are compared to experimental data in section 5 . 2. General Theory The Fuoss-Onsager treatment is a linear response mean field approximation, in which the total pair distribution functions h,(r,t) (equilibium hi)'(r) and the nonequilibrium hi'(r,t)) are related to the total potential qi(r,t) (equilibrium qi1(r) and the none(21) Turq, P.; Lantelme, F.; Friedman, H. L.J. Chem. Phys. 1977, 66, 3039. (22) Ermak, D. L.; McCammon, J. A. J . Chem. Phys. 1978, 69, 1352. (23) Friedman, H. L.; Raineri, F. 0.;Wood, M. D. Chem. Scr. 1989,29A, 49. (24) Mou, C. Y.; Thacher, T. S.;Lin, J.-I. Chem. Phys. 1983, 79, 957. (25) Thacher, T. S.; Lin, J.-I.; Mou, C. Y. J . Chem. Phys. 1984,81,2053. (26) Altenberger, A. R.; Friedman, H. L.J . Chem. Phys. 1983,78,4162. Zhong, E. C.; Friedman, H. L. J . Phys. Chem. 1988, 92, 1685. (27) Vizcarra-Rendon, A.; Ruiz-Estrada, H.; Klein, R. J . Chem. Phys. 1987, 86, 2976. (28) Onsager, L.; Fuoss, R. M. J . Phys. Chem. 1932, 36, 2689. (29) Onsager, L. Ann. N.Y.Acad. Sci. 1945, 46, 241. (30) Onsager, L.;Kim, S. K. J . Phys. Chem. 1957, 61, 215. (31) Engel-Herbert, C.; Kremp, D.; TBwe, J. J . Solution Chem. 1990, 19, 225. (32) Scherwinski, K.; WPchter, U.; Sindig, R. Wiss. Z . WPU. Rostock 1989, 38, 43. (33) Mills, R.; Lobo,V. M. M. Selj-Diffusion in Electrolyte Solutions; Physical Sciences Data 36; Elsevier: Amsterdam, 1989. (34) Turq, P. Chem. Phys. Let?. 1972, I S , 579. (35) Micheletti, C.; Turq, P. J . Chem. SOC.,Faraday Truns. 2 1976, 73, 743. (36) Murphy, T. J.; Cohen, E. G. D. J . Chem. Phys. 1970, 53, 2173. (37) RCibois, P. M. Electrolyte Theory; Harper & Row: London, 1968.
The Journal of Physical Chemistry, Vol. 96, No. I , 1992 399 quilibrium q;(r,t)) by means of the Poisson equation. In the linear response theory, the averaged densities and potentials are expressed as the sum of an equilibrium part and a perturbed part that is proportional to the external perturbation.
h,W) = h;(r> + h/(r,r)
(8)
+ *;(r,r)
(9)
qi(r,t) = * p ( r )
The Poisson equation is
The corresponding equilibrium and nonequilibrium averaged densities Ay(r.(r,t)= pipjgij(rJ)
(1 1)
satisfy the hydrodynamic continuity equation. The continuity equation is
5
where is the velocity of ion j in the vicinity of an ion i a n d h j =j&r2,t). The pair distribution function gij(r,t)is related to hij(rJ) by gj, = 1
+ hi,
(13)
We consider a solution of a simple salt. Since we are interested in experiments with tracers, we will consider three ionic components: the two ionic species 2 and 3 of the salt and the tracer 1, in very small quantity, which is an isotope of either 2 or 3. The velocity ijlof an ion of species 1 in the vicinity of an ion of species j is given by zj1
=
+
W1(KJ1
- kBTv lnfj,)
(14)
where the diffusion coefficient Di of the ith ion is related to its generalized mobility ai by the relation Di = oikBT. In the self-diffusion experiment, the average relative velocity ZlSof the solvent with respect to the ion of species 1 is zero. Following O n ~ a g e r ,the ~ ~hydrodynamic ,~~ corrections to the ion velocity are cero for the self-diffusion to lowest order. The force Kjl acting on an ion of species 1 in the neighborhood of species j is
where il,the acting (diffusion) force on the tracer ion 1, is given by kl = -kB7V log C,, where C, is the concentration of the tracer ion 1. 6kl is the relaxation force. This nonideal contribution due to the relaxation of the ionic atmosphere has been calculated by using a Fokker-Planck equation following Resiboi~.~'Falkenhagen and ~ o - w o r k e r used s ~ ~ a similar generalization of Onsager's expressions for a system of charged hard spheres:
qjis the potential due to the ionic atmosphere of ion j acting on ion 1. In the stationary regime the left-hand side of eq 12 is zero. Without external force, the potentials of mean force are related to the pair distribution function by the equilibrium relation
which also displays the symmetry of the correlation function. In (38) Falkenhagen, H.; Ebeling, W.; Kraeft. W. D. Transport Properties of Dilute Solutions. In Ionic Interactions; Petrucci, S., Ed.; Academic Press: New York, 1971; Vol. 1 .
400
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
the nonequilibrium case, the nonequilibrium qjare related to the nonequilibrium h i by mean of the Poisson equation (10). It should be noticed that the Poisson equation is used in this made1 as a closure equation only for the perturbed parts of the potential and of the pair distribution functions, since the unperturbed parts can be provided by any modern equilibrium statistical theory such as H N C or MSA. Because there are three ionic species, the continuity equation is a set of three simultaneous partial differential equations: VrfI2v',2 + vs21v'21 = 0 vlf13v'13
+ vlf31v'31 =
V2f325'2 + V3fZ'i;z' = 0
The continuity equations can be written in the form of an inhomogeneous differential equation ( v 2 - %dI2)h(r) = F(?,i)
(27)
where F ( i , i ) is the driving force and corresponds to the right-hand side of the continuity equation (12). The Green's function of eq 27 is
(18) (19)
and we get the solution h(r) by convoluting with F ( i , i ) : h(r) =
(20)
The perturbation due to the presence of the tracer 1 will only have a negligible effect on the interaction of species 2 and 3. Therefore, eq 20 can be simplified using the equilibrium distribution functions of 2 and 3. Since the concentration of tracer 1 is very small, we can neglect the perturbed potentials \k2' and *3'.
Rearranging eq 12 for the stationary case, we get dg2I O klwl- kT(w, w2)Ah2,' g210e2w2A@l'= dx -e2w2V\kl'-Vg210+ (elwlA\k: + e2w2A!VI0)h2,'+ (elw,V*20 e2w2V0,0).Vh21'- bklw1-Vg210(21)
+
Bernard et al.
+
+
A similar equation is obtained for the distribution functions between particles l and 3 by replacing the index 2 by 3. In this equation, the left-hand side corresponds to the limiting law, Le., the equation obtained by substituting the Debye-Htickel potential and by keeping only the first terms in dC. The right-hand side corresponds to terms of higher order in C, where C is the salt concentration. The procedure of the solution is as follows: Replacing the \ko and gi; in the left-hand side of the continuity equation by their equilibrium H N C or MSA functions, we get the first-order solution of the continuity equation (i.e., without right-hand side). This solution will be the Green's function either in 0'or in h,' needed to solve the continuity equation to second order. Using the first-order solution, we evaluate the effect of the second-order terms (right-hand side of the fully expanded continuity equation), by convolution with the Green's f ~ n c t i o n . ' ~ After summation of eq 21 and the corresponding one for particle interactions 1-3, we get the first- and second-order contributions
s
di, G ( J i- ?,I) F ( i 1 , i )
(29)
Expanding F(r'l,i) in spherical harmonics, we get
F ( i l , i ) = b ( r l , k ) Pi cos y
(30)
/=O
z
where y is the angle between iland and PIcos y is the Legendre polynomial of degree 1. Integrating over the angles yields r12i/(Kdlrl)f/(rl,k) +
h(r) = - K d I e p / aselk/(Kdlr)Jrdrl /=0
i/(Kdlr)
1-
drl r12k/(Kd,rl)fi(ri~k))(31)
where 8 is the angle between r' and and P,(t) are the Legendre polynomials. We use the modified Bessel functions i,(r) =
71, (-1)n
+I
dt aZlPn(t)
k,(z) =
lm dt e-"P,,(t)
Pn(t) =
-
(33)
with 1 P ( t 2 - 1)"
2"n!
(34)
dtn
Similarly, the potential is of the form V*(r) = Q(?,Z) Using again the Green's function method
(35)
where
'1
After performing angular integration, we obtain *(r) =
-x- 21 + 1 /=o
'Os
- J rdr, r12rl/q/(rl,k)+ #+I
\
where XI' and XI" are defined by and for the perturbed part XI' of the distribution functions and
hjI" denotes the second-order nonequilibrium distribution function. is defined by
Kd12
xI'= 2 1 ' e. ~ ~ The term 6kl' is discussed below (cf. eq 46).
This formal expression is directly useful when H N C pair correlations are used. The MSA pair correlations will yield an explicit
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 401
Self-Diffusion in Electrolyte Solutions lT-----l
where Gj1(Kdl) = x,>jIo(r)e-Kdlr dr
1.004
0.0
0.2
0.4
0.6
OS
1.0
(48)
The integrals Gjl(Kdl)can be evaluated using the Laplace transform of rhjlo(r):
1
sqrt[ NaCI] Figure 1. Self-diffusion coefficients of Nat in aqueous NaCl solutions as a function of the square root of molarity: (X) experimental values taken from ref 33; (-*-) Onsager's limiting law; (---) theoretical values predicted by using static correlation functions obtained by HNC; (-) theoretical values predicted by the analytical MSA expressions; theoretical values adjusted to experimental data, taken from ref 3 1.
fiil(s) = lmrhjlo(r)e-" 0 dr
(49)
In the MSA we havelo
(.e.)
expression of the second-order terms. Similarly, from the perturbed part of the potential we get where
and 4 r 2 N a2Cpi i
(1
2;
+ rni)*
(53)
The electrostatic part of Gjl(Kdl) is given by Gjlcl(K,jl)= fijle'(S Kdl) (54) The hard-sphere contribution of G.,(Kdl)can be evaluated by the Laplace transformation of the hard-sphere part of rhi? within the Percus-Yevick a p p r o x i m a t i ~ n . ~ ~ . ~ ~ G,lHs(Kd,) = (Laplace of [rgjlHS(r);S = Kdl]) e-Kdlajl y ( 1 + Kdlcjl) (55)
hjlf = hiIf.cos 8
3. First-Order Theory We have for the self-diffusion coefficient
Kdi
4. Second-Order Theory Using the same method and after considerable algebra, we get
where (43) is the first-order nonideal contribution and bkl"
-
(44) kl is the second-order term. These nonideal contributions due to the relaxation of the ionic atmosphere have been calculated by using eq 16. For the first-order term, we have
+
6kl' = - ~ j ~ m V ( V , i C bVl/)hjlf d7
(56) where djl(r) is defined by dil(r) = - I m F j l ( r ) d r and Fil(r) by
(45)
Using the fact that hi,' must be equal to zero for r < ujl(hardsphere potential), we obtain after angular integration
Fil(r)kl=
(46) The first term of eq 46 can be evaluated by using eq 39 and the second term by using eqs 40 and 41. If, in eqs 45 and 46 we replace hjlf by hjlN,we will obtain 6klff. More explicitly, the first-order term is
where
epj
(
(57)
dhjIo d$If
-+ w I + oj dr dr
e
dhjlf d y 1 0 w16kl' dhjIo h , l r V 2 ~ l+o -- (58) dr dr (aI+ w j ) dr Wjlo = -ksT log (gilo) = +ej'Plo = +el*:
(59)
(39) Lebowitz, J. L. Phys. Rev. 1964, 133, A895. (40) Sanchez-Castro, C.; Blum, L.J . Phys. Chem. 1989, 93, 7478.
402 The Journal of Physical Chemistry, Vol. 96, No. I, 1992
Bernard et al. J~llrdjl(r)e-Kdir dr. We obtain
0 ** ll
0.80 4
0.0
0.2
0.6
0.4
0.8
1.0
I
sqrt[LiCI] Figure 2. Self-diffusion coefficients of Li+ in aqueous LiCl solutions as a function of the square root of molarity: ( X and +) two sets of experimental values taken from ref 33; (-.-) Onsager's limiting law; (---)
theoretical values predicted by using static correlation functions obtained by HNC; (-) theoretical values predicted by the analytical MSA expressions.
In order to evaluate analytically the integral .f~llrdjl(r)e-Kdlr dr, we use the explicit value of the first-order terms, as given by the M S A expressions. A simplified MSA expression of hjlo(r)is used to approximate the functions hi:(r), q I ' ( r ) ,and X l ( r ) a n a l y t i ~ a l l y . ' ~ ~ ~ ~ el eJe-x(rn,i)
hjlo(r)
-
tkBT(1
+ r U & i + ruj)(i - r2a2)r
(60)
where
where
and
Y
5. Results and Discussion
The explicit expression for hjl' is
The above expressions are easy to evaluate, using either numerical expressions for the equilibrium correlation functions (HNC) or the analytical MSA functions. In all cases, our first-order formal expressions were equivalent to those of Lin et al.,24*25 who expressed the relaxation correction in Fourier space instead of coordinate space
where Using this result, we evaluate the second-order contribution to
403
J. Phys. Chem. 1992,96,403407 2.10%
1
The contribution of the second-order terms is small and diminishes the relaxation field, which causes an increase of the values of the self-diffusion coefficient as compared to the first-order electrostatic non-hard-sphere correction. We show an experimental curve (Figure 1) for a system studied by Kremp et al.?l Na+ in NaCl solutions. For all systems (Figures 1-3) our calculated values corres nd to the crystallographic radii: Na+, r = 0.98 A; C1-, r = 1.81 Li+, r = 0.78 A; K+, r = 1.33 A; Br-, r = 1.96 A. By analogy with the MSA case, we used a charged hard-spheres model potential in our HNC calculations. Clearly, there is little difference between the HNC and MSA results. The HNC is the more accurate theory, and one would normally expect it to perform better than the MSA. The fact that it is even marginally worse in our work may be due to larger error cancellation in the MSA. This also happens in the theory of charged interfaces. Overall, however, we think that this good agreement is due to the fact that the transport coefficient studied in this paper depends on the long-range part of the correlation function and is relatively insensitive to the short-range part, where the discrepancies between the MSA and the HNC are more severe. The good agreement between our treatment and the experimental values, even without introducing adjustable parameters, at least for simple systems, as well as its consistency with other available theoretical approaches, gives us confidence for its application to more complicated situations such as electrolyte or polyelectrolyte mixtures and associated electrolytes.
E
1304 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
I
sqrt[KBrl Figure 3. Self-diffusion coefficients of Br- in aqueous KBr solutions as a function of the square root of molarity: (+) experimental values taken Onsager’s limiting law; (-- -) theoretical values prefrom ref 33; dicted by using static correlation functions obtained by HNC; (-) theoretical values predicted by the analytical MSA expressions. (-a-)
These expressions can be compared to ours (eqs 47 and 48). The second term of the right-hand side of eq 47 is missing in the preceding equations because Lin et al. do not consider hard-sphere potentials in the expression of the relaxation correction 45, and for this reason they neglect excluded volume effects. This term is small for the ions we studied. In our expression the first term in the right-hand side differs only of the preceding expression by the function io(Kdlgjl).Lin et al. do not use the fact that hjl’ must be equal to zero for r C ajl in eq 45. As a consequence, their integral begins from zero instead of r = ujl. The term i O ( K d l g j l ) is 1 in the Lin et al. expression.
Acknowledgment. L.B. acknowledges with gratitude the hospitality of the Universitt de Paris and partial support through Grants NSF-CHE-89-01597 and EPSCOR RII 86-10677.
Thermodynamic Properties of Aqueous Mixtures of HCI and CoCI, at Different Temperatures. Application of Pltzer’s Formalism Rabindra N. Roy,* C. Porter Moore, Monica N. White, Lakshmi N. Roy, Kathleen M. Vogel, Department of Chemistry, Drury College, Springj?eld, Missouri 65802
David A. Johnson, Department of Chemistry, Spring Arbor College, Spring Arbor, Michigan 49283
and Frank J. Miller0 Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida 33149 (Received: May 16, 1991)
Emf measurements were made on the cell without liquid junction: Pt,H2(g,latm)lHCl(ml),CoClz(mz)(AgC1,Ag (A). Activity coefficients of HCI and CoClz for the system H+-Cd+-Cl--HzO have been investigated at temperatures ranging from 278.15 to 318.15 K at 10 K intervals and at constant total ionic strengths from 0.1 to 4.0 mol kg-I. The results have been interpreted using Harned’s empirical equations and the comprehensive treatment of Pitzer, including the contributions of higher order electrostatic terms. The Pitzer mixing parameters %H,Co and $H,C~,CIas well as the temperature derivatives of the mixing coefficients have been well represented by linear equations. These results may be used to calculate the relative apparent molal enthalpy for H+, Co2+,and CI- aqueous solutions. The activity coefficients and excess enthalpies for HC1 and CoCI2 are reported at 298.15 K.
Introduction There has been a renewed interest in the thermodynamic properties of aqueous mixed electrolyte solutions, particularly in applications involving seawater desalination, geothermal energy, chemical oceanography, hydrometallurgy, and pulp and paper chemistry. Such applications require accurate values of the thermodynamic properties of aqueous mixed electrolyte solutions.
Activity coefficients of all components of such mixtures are of primary importance in describing accurately the thermodynamic behavior of mixed electrolyte solutions. Very precise values of activity coefficients in such systems can be determined using isopiestic vapor pressure methods and emf techniques. The emf method, which gives the activity coefficient of at least one of the solutes directly, was chosen to determine
0022-3654/92/2096-403%03.00/0 0 1992 American Chemical Society
