ACTIVITY COEFFICIE~NTS OF ELECTROLYTES AT ISTERMEDIATE CONCENTRATIONS
2305
Activity Coefficients of Electrolytes at Intermediate Concentrations and the “Cube-Root” Law’
by J. E. Derinoyers Department of Chemistry, University
0.f
Sherbrooke, Sherbrooke, Quebec, Canaaa
and B. E. Conway Department of Chemistry, University of Ottawa, Ottawa, Ontario, Canada
(Received April
4, 1964)
The experimental mean molal activity coefficients of simple electrolytes at concentrations below l m are regarded as being separable into three main contributions arising from interionic coulombic forces, ion-solvent, and ion-cavity interactions. The ion-solvent contribution is calculatedL from the theory of Robinson and Stokes and previously deduced hydration parameters. The ion-cavity contribution is based on a mutual salting-out effect in which each ion is assumed to interact with the cavities in the solvent associated with the presence of all the other ions. The experimental molal activity coefficients are first corrected for hydration and the above mutual salting-out effects. The remaining coulombic contribution to the activity coefficients for most alkali halides can then be shown to follow the relation log f c =: a for c between 0.1 and 1 m. It is suggested that these linear plots can be interpreted in terms of a disordered lattice model.
Introduction The nature of deviations from ideal behavior in very dilute electrolyte solutions are now reasonably well understood. The limiting theory of Debye and HuckelZa for activity coefficients has been shown to be exact on statistical mechanical grounds by Kirkwood and Poirier2b and the limit of validity of the Debye-Huckel model (in distinction to that of their mathematical approximations) has been set by Frank and Thompson3 at about 0.001 m. The necessity of using an adjustable parameter, 8, the “distance of closest approach” in the Debye-Huckel theory, has been eliminated by Fuoss and Onsager4 when they showed, by integration of the Poisson-Boltzmann equation a t small and 1,arge distances from the ion using Bjerrum’s5parameter for the critical distance, that the activity coeecient -yh of a 1 : l electrolyte in dilute solution is independent of ion size. The prediction of activity coefficients a t intermediate concentrations is a much more diflicult problem. In addition to long-range coulombic fames, shorter range interactions associated with solvation and finite size
of the ions become increasingly important. Also, Frank and Thompson6 have suggested that the spatial distribution of ions varies with concentration; while a continuous ionic cloud model is correct for very dilute solutions, a disordered lattice model would be more satisfactory at intermediate and high concentrations and lead to a cube-root law in salt concentration for In yk which was demonstrated from ca. 0.001 to 0.1 M by these worker^.^,^ At present, the formulation of a theory capable of accounting quantitatively for the activity coefficients of electrolytes a t intermediate concentrations is presumptuous, and the aims of the (1) Presented at the Symposium on Solvation sponsored by the Chemical Institute of Canada, Calgary, Alberta, Canada, 1963.
(2) (a) P. Debye and E. Huckel, P h y s i k . Z . , 24, 185, 305 (1923); (b) J. G . Kirkwood and J. C. Poirier, J . P h y s . Chem., 5 8 , 591 (1954). (3) H . S. Frank and P. T . Thompson, J. Chem. Phys., 31, 1086 (1959). (4) R . M. Fuoss and L. Onsager, Proc. Natl. Acad. Sci. U . s., 47, 818
(1961). ( 5 ) N. Bjerrum, Danske Videnskab. Selskab, 7, No. 9 (1926). (6) H. S. Frank and P. T . Thompson, “The Structure of Electrolyte Solutions.” W. J. Hamer, Ed., John Wiley and Sons, Inc., New York, N . Y . , 1959, Chapter 8.
Volume 68, Number 8
August, 156.4
J. E. DESNOYERS AND B. E. CONWAY
2306
present paper will be restricted to an attempt to evaluate the contributions to the experimental activity coefficients of simple electrolytes in the range 0.1-1 m, which arise from the effects of hydration and inutual salting-out of ions and become significant in this range of concentrations. The resulting corrected activity coefficients should then reflect the nature of the distribution of ions in this concentration range and the presence of any other interactions neglected in this treatment.
Theore tical 1. Introduction. If we restrict the present study to solutions below 1 m in concentration, it is reasonable to assume that the experimental molal act,ivity coefficient y is separable into three main component terms (expressed as rational activity coefficients7) log y
=
log fc
+ log + log fh
fso
(1)
associated with coulombic, hydration, and mutual salting-out interactions. The coulombic contribution f c to the rational activity coefficient is the most important term in dilute solutions since it depends on long-range electrostatic forces. The hydration contribution fh effectively takes into account the number of water molecules which are removed from their solvent role due to hydration of the ions and this term is equivalent to Huckel’s “cp” corrections arising from the mutual salting-out effect associated with hydration also discussed by B ~ t l e r . The ~ term fso is an extra salting-out term which allows for the fact that other ions distributed about a given ion occupy volume in the polar solvent medium but have less polarization than an equal volume of solvent in the field of the reference ion and therefore tend to be excluded froin regions near the reference ions. This last term is closely linked with fc in that both these coefficients deal with the distribution of particles around a reference hydrated ion. The coulombic term gives the distribution of ions taken as point charges while log fso gives the distribution of ions taken as uncharged spheres, i.e., as cavities of a medium of different polarizability from that of the solvent. In reality, we have a distribution of charged spheres, and consequently the above two terms can be regarded as additive only if log fso is corrected for the effects of coulombic forces and log fc for mutual salting-out effects. 2 . Hydyation Contribution. The relation between hydration parameters and activity coefficients has been dealt with principally by Robinson and Stokes7 using mole fraction statistics and by Glueckauf lo using volume fraction statistics. It has been difficult up to now to The JournaE of Physical Chemistry
choose between these two treatments since in both cases the hydration numbers and radii were treated as adjustable parameters which effectively took account of all other interactions except the Debye-Huckel electrostatic contribution. The hydration numbers obtained for individual ions are not additive and do not agree well with other valuesll obtained from partial molal volumes. In addition, the above treatments are based on the assumption that the coulombic contribution to the activity coefficient is given correctly by Debye and Huckel’s theory for concentrations up to 4 m which is que~tionable.~Despite the fact that the equations of Glueckauf and Robinson and Stokes cannot be used reliably to obtain primary hydration parameters12 from activity data, these equations may still give the correct relation between activity and hydration data provided that knownbvaluesfor hydration numbers and radii are used. Hildebrai~dl~ has presented arguments that mole fraction statistics are preferable to volume fraction ones when the dissolved electrolyte ions are spherical. It will also ‘be shown below that better results are obtained with the theory of Robinson and Stokes than with that of Glueckauf. The relation derived by Robinson and Stokes will therefore be used to correct for the hydration contribution to log y, vix. logfh
0.018hmb, - log [l - 0.018(h - ~ ) m ] ( 2 ) 2.303
= -~
where m is the molality, h the hydration number, the osmotic coefficient, and v the total number of gram-ions arising from dissociation of 1 g.-mole of the electrolyte. The osmotic coefficient can be obtained experimentally and is known for most simple electrolytes as a function of m.7 The exact value of the hydration number to be used in a theoretical expression such as that above is always difficult to choose since hydration numbers depend to some extent on the property investigated. However, we have shown elsewhere’l that primary
+
(7) Cf. R . A . Robinson and R . H. Stokes, “Electrolyte Solutions,” Butterworth and Co. Ltd.. London, 1959, Chapter 9 . For a summary of Stokes and Robinson’s numerical data, see B. E. Conway, “Electrochemical Data,” Elsevier Publishing Co., Amsterdam, 1952. (8) W. Hackel, Physik. 2 , 26, 93 (1925). (9) J. A. V. Butler, J . Phys. Chem., 33, 1015 (1929). (10) E. Glueckauf, Trans. Faruday SOC.,51, 1235 (1955). (11) J. E . Desnoyers and B. E. Conway, Can. J . Chem., in press. (12) B. E. Conway and J. 0’31.Bockris, “Lfodern Aspects of Electrochemistry,” Vol. I, J. 0’34. Bockris and B. E. Conway, Ed., Butterworth and Co. Ltd., London, 1954, Chapter 2 ; J. 0’31. Bockris, Quart. Reo. (London), 3 , 173 (1949). (13) K. Shinoda and J. H . Hildebrand, J . Phys. Chem., 61, 789 (1957).
ACTIVITY COEFFICIENTS OF ELECTROLYTES AT h T E R M E D I A 4 T ECOXCENTRATIONS
2307
hydration datal2 obtained from partial molal volume attraction of a charged ion for polar solvent molecules measurements are probably minimum values relevant rather than for the less polarizable cavities associated to many of the physic0 chemical properties of ions with the presence of the other ions. In considering (e.g., salting-out of nonelectrolytes, l4compressibilities) the number dn; of ions in an element of volume of in ionic solutions while values from mobility data are thickness dr at distance r from a reference hydrated maximum values. These two limiting values for ion j, t,he effects of long-range coulombic forces and hydration parameters (primary hydration numbers or mutual salting-out will be taken into account by the primary hydration shell volumes) agree very well with relation the hydration data necessary for the interpretation of dni = Co exp - __ 2ielC.j 4nr2dr (3) measured salting-out constants for nonelectrolytes kT kT and theoretical dielectric saturation effects near i o n ~ ~ , ' ~ and p o l y i o ~ i s . ~It~ is therefore reasonable to assume where Cois the average number of ions per unit volume, that the hydration numbers required in eq. 2 should lC.j the potential of the reference hydrated ion, and Au not be smaller that those obtained from partial molal the, "salting-out energy" associated with the intervolumes. The values of h and r h , the radii of hydrated action of the field of the reference ion j with the polarizions, are reproduced in Table I for alkali and alkaline able cavity of the ion i. "Salting-out energy" has been earth halides. Utilization of these data should give previously derived for the problem of salting-out of the minimum influence of hydration effects on activity nonelectrolytes by ions.I4 It can be extended to coefficients. mutual salting-out between ions by replacing the partial molal volume of the nonelectrolyte by that of each ion P,, and neglecting the polarizability of the Table I : Hydration Radii and Numbers Calculated cavity occupied by ion i. The energy leading to the from Partial RiIolal Volumes at 25"" tendency for mutual salting-out is then
[
Ion
Li Na +
K+ Rb' cs
+
Mg2+ a
'h,
A.
2.18 2.61 2.36 2.30 2.27 2.95
h
2.6 4.0 2.6 2.1 1.6 6.6
Ion
CaZ+ Sr2+ Ba2+
c1-
IBr-
Th,
A.
2.90 3.20 3.45 2.21 2.16 2.23
@]
h
6.1 7.9 9.0 1.1 0 0.8
From ref. 11; cf. also ref. 12.
3. Mutual Saltinpout. Ions in solution will attract or repel each other on account of the coulombic forces acting between their charges. Superimposed on these electrostatic iiiterac tions ill be a certain amount of electrochemical exclusion between all ions as a result of mutual salting-out between ions of finite sizes2b discussed above. This mutual salting-out interaction occurs beyond the primary hydration shell and is to be distinguished from the hydration salting-out effect associated with diminished activity of free solvent water. This latter contribution to activity coefficients is already accounted for in eq. 2 and in the Huckel term in the extended Debye-Huckel equation in which the linear empirical term in concentration, introduced in the latter equation for high concentrations, implicitly includes both effects. The mutual sal ting-out contribution associated with ion cavities may be calculated by assuming that the only role of a dissolved ion without its charge is to occupy space. There will then be a preferential
(4) where E is the relevant dielectric constant of water,14 e the electronic charge, and N Avogadro's number. The distribution of ions dn?, in the case for which only long-range interionic coulombic forces are considered (Au = 0), is given by
We may now treat the problem of the mutual salting-out effect in an analogous way to that involved in the case of nonelectrolyte^.^ A solution element of volume dv near the reference ion j can be regarded as containing a real concentration n,/dv of cavities (as defined in the sense above) of the other ions i, when the mutual salting-out effect is operating (Au finite). In the absence of such salting-out effect an "ideal" concentration n,0/dv of such cavities would have arisen. The ratio of these two quantities gives a measure of the activity coefficient ( c j . nonelectrolyte case) contribution arising from the mutual salting-out between the central ion j and all the other distributed ions is9 That is (14) B. E. Conway, J. E. Desnoyers, and C. A . Smith, Phil. Trans. Roy. Soc. (London), in press.
Volume 68, A'umber 8 August, 196.4
J. E. DESSOYERS AND B. E. CONWAY
2308
1000 _1 -- Ncj fji
The solution of this equation for all the possible combinations of j and i will give the mutual salting-out contribution to the rational activity coefficient corrected for the influence of long-range coulombic forces in the ion distribution. An exact solution of eq. 6 is impossible but an approximate solution can be obtained if the following assumptions are made. Both Au/kT and zie$,/kT can be considered small compared with unity, so that the exponentials may be expanded as in the DebyeHuckel theory in a power series retaining the first term only. Also, since $ j appears only as a correction factor, it may be replaced by the Debye-Huckel potential even though it is known that the validity of this potential is doubtful a t such high concentrations. Equation 6 thus becomes
(xje)2
Pi
100Oxixj
2ekTN r h ( j ) 2NI 1000 lO0OZiXj N c j 2NI I---
-
(xje)2cj Pi I IOOOE~TT h ( j ) ZI - XiZjCj
(11)
Taking logarithms and using the approximation In (1 x) = zwhen x
