Extension of a two-structure model for electrolyte solutions to aqueous mixed electrolyte systems

J. inorg, nucl. Chem.. 1975. Vol. 37, pp 2503-2506. Pergamon Press. Printed in Great Britain

EXTENSION FOR

OF

ELECTROLYTE MIXED

A TWO-STRUCTURE

MODEL

SOLUTIONS;

AQUEOUS

ELECTROLYTE

TO

SYSTEMS*

M. H. LIETZKE and R. W. STOUGHTON Chemistry Division,Oak Ridge National Laboratory, Oak Ridge. TN 37830, U.S.A. (Received 17 January 1975~

Abstract--A two-structure model for electrolyte solutions has been extended to the prediction of the activity coefficient of each component in a mixed electrolyte solution. Activity coefficients predicted using this model agree as well with fitted-observed values for a number of mixtures as those predicted using a model which retains a Debye-Hiickel term at all ionic strengths. The advantages of the two-structure model include: (I) fewer adjustable parameters are needed per component electrolyte; and (2) the fact that two of the parameters describing the ionic strength dependence of the activity coefficientof the component electrolytes seem to have direct physical origin.

INTRODUCTION

IN A previous paper[l] we presented a model for electrolytic solutions which incorporated the ion atmosphere description of Debye and Htickel at low concentrations and a cell model at higher concentrations. Thus, an electrolytic solution of any particular concentration was treated as a mixture of these two models or structures, the contribution of each being weighted by a partition function, much as the sum of contributions from two significant structures has been used to describe the properties of liquids. A stochastic description of our two-structure model, as applied to solutions of a single electrolyte, is given by P ( c ) = P(O)f(c) + P(m)[l - f ( c ) ] ,

(I)

where P is the value of a given property of the solution at concentration c_, P(0) is the Debye-Hiickel description of that property, p(oc) is the cell model description and f(c) satisfies the conditions f(O) - I, f(oc) - O.

(2)

The symbols 0 and ~ in these expressions imply solute-solvent ratios of (): 1 and 1:0. The purpose of the present paper is to show how the two-structure concept can be extended to aqueous solutions containing more than one electrolyte. The property P considered will be the activity coefficient of each component in a solution containing more than one electrolyte. In implementing this extension use will be made of simple expressions we have previously developed for predicting such quantities for electrolyte mixtures. In dealing with the activity coefficients of the components of electrolyte mixtures we will retain the same formulations of P(0), P(~), and f ( c ) used in our previous communication. For P(0) we have P(O) = In v = - S",/-I/(I + 1.SX/I),

(3)

strength of the solution computed on a volume basis. For p(o:) we take p(~c) = In y = B I ~;~+ CI,

(4)

while for f(c) we again use the convenient exponential f ( c ) - e ''~

(5)

where a is a constant and /3 is the volume fraction of solute / 3 - Veil000

(6)

computed on the basis of the molar volume V_ of the dry salt. c is the concentration of solute in moles per liter: concentrations must be expressed in these units in our model since the significant variable is distance between ions. In the following sections we wilt demonstrate the simplicity that is achieved when our two-structure model is applied to the problem of predicting the activity coefficient of each component in an aqueous mixed electrolyte solution. The results obtained are entirely satisfactory for many practical applications where such quantities are needed. If higher accuracy is required, then experimental data on a particular mixture of interest can be used to refine the predicted values by adding terms to the model to account for ionic interactions in the solution. This contribution will be very small, however, since the predicted values are in most cases within one to two percent of the observed values. RESULTSOF THE CAI,CILATIONS When expressions (3), (4) and (5) are substituted into Eqn (I) and terms combined Eqn (7) is obtained for the logarithm of the activity coefficient ~' of a single electrolyte in solution. log y

= [ - S\/)/II

- 1.5\/I)-

BI ~:3- CI]e "~ + B I '/3 + (21

(7) where S_is the Debye-Hfickel coefficient and I is the ionic *Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation.

In this equation there are three adjustable parameters: a, _Band C_, whose values may be estimated by least squares using activity coefficient data from the literature[2].

2503

2504

M. H. LIETZKEand R. W. STOUGHTON

When this is done for a variety of electrolytes of different valence types it is observed that the value of a correlates with valence type, i.e. the value of a increases when plotted against the product of the valence of the cation and anion Z , Z . Values of a above 1.0 for 1-I electrolytes, 1.2 for 2-1 and 1-2 electrolytes, and 1.4 for 3-1 electrolytes did not give significantly better fits of the activity coefficient data and hence a was fixed at these values for the electrolytes of corresponding valence types. Values of _a below the above values resulted in significantly poorer fits of the data. As shown in our previous paper[l] the values of B_also correlate with valence type and indeed, as a consequence of our model, vary as

Z+Z-[2(v+ + v_)l(v+Z? + v / 2)]l/3, In this expression v+ and v are the number of cations and anions, respectively, into which the electrolyte dissociates. It was found that for 1-1 electrolytes the values of B estimated by least squares scattered around a mean value of -0.617483, for 2-1 and 1-2 electrolytes around a mean value of -1.11332, while for 3-1 electrolytes the mean value was - 1.52854. Accordingly, the values of B were fixed at these mean values for the electrolytes of corresponding valence type. Hence, the final least squares fits for each single electrolyte solution involved only the estimation of the parameter _Cin Eqn (7). As mentioned in our previous communication[l] the term CI in Eqn (7) is a catch all for everything except short-range interaction of ions which are in adjacent and nearby cells in our model. Salts of other valence types were also included in our survey. Values of the _aand B_parameters for these types are as follows: for 2-2 electrolytes the value of a_was 1.6 with the mean value of _B -2.60762; for 3-2 electrolytes the value of _a was 2.0, while the mean value of B_ was -3.03534. No further mention will be made of these electrolytes, however, since activity coefficient data on mixtures involving these salts were not available. Values of the parameters of Eqn (7) for the electrolytes which are components of the mixtures considered in this paper are shown in Table 1. When Eqn (7) is used to represent the ionic strength dependence of the activity coefficient of a single electrolyte in solution, it can readily be shown [1] that in the range of moderate ionic strength the Debye-Hiickel term contributes very little to the total value of the activity coefficient. This is in sharp contrast to the Table 1. Values of the parameters of Eqn (7) for a number of electrolytes Substance

a

B -0.617483

C

HCI

1.0

0.426114

LICl

1.0

-0.617%83

0.378973

NaCI

1.0

-0.517~83

0.207801

KCI

1.0

-0.517483

0.14q451

RbCI

1.0

-0.617483

0.12766~

CsCI

1.0

-0.617483

0.0599~23

MECI 2

1.2

-1.11332

0.362553

CaCI 2

1.2

-1.11332

0.316898

SrCI 2

1.2

-1.11332

0.325674

BaC12

1.2

-1.11332

0.230943

LaC13

1.4

-1.52954

0.290468

situation where a conventional expression consisting of a Debye-Hfickel term plus a power series in the ionic strength is used, such as Eqn (8). logy =-S~¢/-I/(I+A~v/-I)+BI+CI2+DI3+ ...

(8)

In this case, even at high ionic strength, the Debye-Hiickel term contributes a large fraction of the total activity coefficient value, since this term has not been phased out by a partition function as in the two-structure model. Moreover, even when the _Aparameter in Eqn (8) is fixed at some definite value, values of the other parameters scatter widely for different electrolytes and hence for moderate concentrations at least two, and often three or four, adjustable parameters are needed per electrolyte. Hence, a considerable simplification results when Eqn (7) is used, since, as shown in Table 1, only one adjustable parameter is needed per electrolyte. We have previously demonstrated[l] that the ionic strength dependence of the activity coefficient of single electrolytes in solution can be about as well expressed by Eqn (7) as by Eqn (8). Thus it seems reasonable that a model for predicting the activity coefficient of each component in a mixed electrolyte solution which is based on the two structure concept should be simpler, in the sense that it would involve fewer adjustable parameters, than a model in which the Debye-Hfickel term is retained even at concentrations so high that the linearized Poisson-Boltzmann equation is no longer valid. Several different models, some very elaborate, have been proposed for predicting the activity coefficients of the components of electrolyte mixtures. We have proposed [3] a particularly simple equation for predicting the activity coefficient of a component of an electrolyte mixture from the activity coefficients of the individual components in pure solution. This equation is 1

1

iz÷z_l log w, = ~

log v,°

where 71 is the mean ionic activity coefficient of the ith component in a mixture containing n electrolytes; the superscript 0 indicates activity coefficient values in the pure state at the total ionic strength of the mixture; Z ÷ and Z - represent the charges on the cation and anion, respectively; and _Fi is the ionic strength fraction I~/I of the ]th component in the solution. The charges have been included so that the logarithms of all the activity coefficients approach the same limiting slope at low concentrations, Since activity coefficient values predicted using Eqn (9) are about as good as those predicted using the more elaborate models [3], we have used the form of this equation as a basis for combining the parameters of Eqn (7) for each component electrolyte into a two-structure model for electrolyte mixtures. The resulting expression for activity coefficients is log Yli

= [--

Si ~/-I/(1 "+ 1"5~/-I) + B J ''3 + CJ,

B i J 1/3 -

Cii[]e-<"t~)i] (10)

where logy~j refers to the logarithm of the activity coefficient of the ith component in a mixture where the remaining electrolytes are indexed j; _S~ is the DebyeHiickel limiting slope appropriate for component i; and the remaining coefficients (a/3)ij, B_~jand _C~jare weighted

2505

Extension of a two-structure modelfor electrolyte solutions averages computed from the corresponding terms in Eqn (7) for each component as follows. (a/3)~, = (a/3)~ +~,, ~/2[(a[3)j-(a[3)~]

(11)

Table 3. Standard deviations (~)a between fitted-observed and predicted values of the logarithmof the activity coefficientof each componentin a number of solutionscontainingthree electrolytes Mixture °A

J

B,~ = B, llZ~Z I, + ~ F,12[B, IIZ.Z Ij-B, IIZ÷Z I,]

I, + ~

c,, = c./Iz-z

~/'..lC/Iz+z

IJ - C, ltZ+Z 1,1.

HCI-NaC1-MgCI 2

TS 0 . 0 1 7 3 DH

HCI-KC1 CaCI 2

TS

(12)

(13) HCI-MgCI2-CaC/2

In these latter expressions the sum over j is taken for all the electrolytes in the solution except that indexed i. Each t3~ and L3jin Eqn (11) represents the volume fraction of the respective solute i or i in the solution, while F_~ has the same significance as in Eqn (9). Using Eqn (10) and the parameters in Table 1 we have predicted the activity coefficient of each component in a number of electrolyte mixtures which have been investigated experimentally[4]. The predictions were made for solutions of total ionic strengths one and two with varying ratios of the components, a total of six solutions in each case. The standard deviations over the range of these six solutions between fitted-observed values of the logarithm of the activity coefficient of each component in a series of two-electrolyte mixtures and values predicted by Eqn (I0) are shown in the rows labelled TS in Table 2. For comparison standard deviations for activity coefficient values predicted by Eqn (9) with the activity coefficient of each component represented by Eqn (8) with four adjustable parameters (A__,_B,_C, _D) are shown in the rows labelled DH in Table 2. As shown previously[3] these latter values are representative of the standard deviations observed when any of the predictive methods which retain the Debye-Htickel term at all concentrations are used. The overall (over all eighteen mixtures listed in Table 2) mean deviation between observed-fitted values of the logarithm of the activity coefficient of the first-listed compound in each mixture and predicted values was 0.011 in the case of the two-structure model (hereafter referred to as Model TS) and the same in the case of the model Table 2. Standard deviations (o')" between fitted-observed and predicted values of the logarithmof the activity coefficientof each component in a number of solutionscontainingtwo electrolytes Mixture

~A

cB

LdCI-CsCI

TS 0.0153 DH

.0226

,0193

NaCI-KOI

TS

.O047

.0048

DH

.0026

.0036

NaCI-CsCI

TS

°0063

.0108

DH

.0049

.0111

KCi-CsC1

TS

.018~

.0219

DH

.0148

.0258

T8

.0069

.0045

DH

o0121

.0099

TS

.0092

.0193

DH

.0080

.0144

TS

.0058

.0123

CsC1-BaCI 2

NaCI-MgCI 2

KCI-CaC12

0.0185

DH

.010~

.0129

MgC12-CaCI 2 TS

.0073

.0043

DH

.0081

.0071

TS

.0103

.0060

DH

.0065

.0063

HCI-LiCI

Mixture

oA

HCI-NaC1

TS 0,0125 DH

,0085

HC1-KC1

TS

.6061

DH

.0024

HC1-RbC1

TS

,022B

DH

.0212

HC1-CsCI

TS

,0379

DH

,0361

TS

,0826

DH

.0036

TS

.0076

HCI-MgCI 2

HCI-CaC12

DH

.O072

HCI-S~Cl 2

TS

,0083

DH

,0082

HCI-BaC12

TS

.0080

DH

,0082

HCI-LaC13

TS

.0998

DH

,0110

~B

0.01,1 .oo8~ .0o~2 .0036 •o2o2 .0220 .0263 .0335 .0079 .0009 .o16~ .o19o .o188 .ol,2 .o123 .oi77 .0,61 .0820

aThe subscripts A and B refer to the compound in the mlxtur~ listed first and second, respectively.

J.l N.C. V o l 37. No. 12--G

aB

°C

0.0284

O,OlOg

.0177

.0271

.0111

.0084

.0114

.0~0~

DH

.009~

.0133

.0721

TS

.0018

.0289

.0287

DH

.015B

.0287

,0302

aThe subserip%s A~ B and C r e f e r to the compound in the mixture l i s t e d first, second, and third, respectively.

retaining the Debye-Hiickel term (hereafter refered to as Model DH). In the case of the second-listed compound the values of the overall mean deviations were 0.015 for Model TS and 0.016 for Model DH. When both components in all the mixtures were considered simultaneously the values of the overall mean deviations were 0.013 and 0.014, respectively, for the two models. Hence, there is no significant difference between the models in their ability to predict the activity coefficient values of each component in the mixtures over the range of ionic strength investigated. Using both Models TS and DH predictions were also made of the activity coefficient of each component in three different mixtures each containing three electrolytes. The standard deviations between observed-fitted and predicted values of the logarithms of the activity coefficients of each component over the same range used for the two-electrolyte systems are shown in Table 3. As before, the rows labelled TS refer to the two-structure model, while the rows labelled DH refer to Model DH. Except in two cases, the standard deviations are slightly lower for Model TS than for Model DH. The overall mean deviation is 0.0207--0.014 for Model TS and 0.0251 :~ 0.018 for Model DH. Hence, there is again no significant difference in the ability of the two models to predict the activity coefficient of each component in these mixtures, although in general, the standard deviations in the case of the two-structure model are slightly lower. CONCLUSIONS The results of the calculations indicate that activity coefficient values predicted for the components of a series of electrolyte mixtures by the two-structure Eqn (10) are as close to fitted-observed values as those predicted using a model which retains the Debye-HiJckel term for each electrolyte at all concentrations. As shown previously activity coefficient values predicted using this latter model are consistent with values predicted by several other models, all of which retain the Debye-Hiickel term at all concentrations. An advantage of the two-structure representation of electrolyte mixtures lies in the fact that only one adjustable parameter is needed for each component electrolyte, since the two remaining parameters in Eqn (7) may be fixed at the same value for each electrolyte of a given valence type. In contrast, models which retain the Debye-Hfickel term at all values of the ionic strength require more parameters to represent equally well the activity coefficient behavior of each component electrolyre in a solution containing several electrolytes. Another feature of the two-structure model should also be noted. When Eqn (8) is used to fit activity coefficient

2506

M.H. LIETZKEand R. W. STOUGIffON

data at concentrations above the range of high dilutions, the coefficients B_, _C, etc. are empirical constants, devoid of physical significance. The values of _B, for example, scatter widely for different electrolytes even of the same valence type. The source of this empiricism lies in the retention of the term in _p~2,which has its origin in the continuous space charge of the Debye-Hiickel model, and which must lose all physical significance when the concentration is so high that the linearized PoissonBoltzmann equation is no longer valid. In contrast, the values of the _aand _Bparameters in the two-structure Eqn (7) are very nearly the same for electrolytes of the same valence type. This result suggests that these are coefficients with direct physical origin, in the sense that the Debye-Hiickel coefficient _Sis derivable from a model and fundamental principles. Granting this premise, the twostructure Eqn (10) is thus verified as a more valid description of the properties of electrolyte mixtures over

the entire concentration range than a model which retains the Debye-Hiickel term at all concentrations. Acknowledgements--The authors wish to express their sincere appreciation to WilliamJ. Rogers of Angelo State University for performing the computer calculations in connection with this project. REFERENCES

1. M. H. Lietzke, R. W. Stoughton and R. M. Fuoss, Proc. Nat. Acad. Sci. 59, 39 (1968). 2. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd Edn revised, Butterworths, London (1965), Appendix 8. 3. M. H. Lietzke and R. W. Stoughton, J. Solution Chem. 1,299 (1972). 4. Activity coefficientdata on the salt-salt mixtures were taken from the compilationof R. M. Rush, ORNL-4402 (Apr. 1969), where the original references are given. Activity coefficient data on the acid-salt mixtures are from our own work.