J. A. MARINSKY
1572
Prediction of Ion-Exchange Selectivity
by J. A. Marinsky Department of Chemistry, State University of New York at Buffalo,Buffalo,New York
(Received March 7, 1966)
An equation introduced for the anticipation of ion-exchange selectivity by Gregor, who employed the model of an elastic matrix to describe ion-exchange equilibria, has been reexamined. The polyelectrolyte theory of FUOSS, Katchalsky, and Lifson, as adapted by Gregor and Kagawa to counterions of different sizes, has been used according to the method suggested by Feitelson to estimate their activity coefficient ratio in the exchanger. Hydration parameters obtained via the polyelectrolyte model appear to enhance the quantitative estimate of this ratio and of swelling pressure effects as well.
Introduction The prediction of ion-exchange equilibria, a problem of considerable importance, has been extensively investigated for many years. The numerous attempts that have been employed to resolve this problem may be classified into two categories. In the first category, rigorous thermodynamic treatment that requires no model and no assumptions with respect to the mechanism of the phenomenon is employed. The second approach consists of the introduction of models with particular properties resembling those of the ion exchanger to permit the derivation of equations which reflect the action of various physical forces in the exchange process. A rather complete discussion of the spectrum of theoretical approaches and models has been presented by Helfferich.‘ One of the simplest and most useful quantitative treatments of ion-exchange equilibria employs the model of an elastic matrix that was introduced by Gregor.2 This model leads to the relation for the molal selectivity coefficient,K
71 Y2 In K12 = In : In Yz
+
Y1
+ (F - .)(IT RT
- 72)
(1)
where subscripts 1 and 2 refer to the counterions of uni-univalent electrolytes 1 and 2 with a common byion, the terms are the partial molal volumes of the respective ions, and F - n is the resin-swelling pressure. Bars are placed above the appropriate symbols to designate the resin phase. Gregor in his earliest application of the model assumed that the system behaved ideally except for The Journal of Physical Chemistry
solvation and attempted to correlate selectivity on the basis of swelling pressure and ionic size3s4alone. In anion exchange, where the selectivity could not be explained by a swelling-pressure effect, Gregor and coworkers postulated ion-pair f ~ r m a t i o n . ~Their distinction between free solvent and solvation shells and associated and free counterions was arbitrary and employment of one or both approaches is drastically deprived of thermodynamic rigor as a consequence. Qualitative rather than quantitative analysis of the ion-exchange phenomenon was the result. The swelling-pressure term of eq 1 becomes relatively unimportant when, in an attempt for greater thermodynamic rigor, the poorly defined solvation effects are not specifically considered. The effects are instead reflected in the activity coefficients of the ions in the resin phase. Myers and Boyds and Boyd, Lindenbaum, and Myers’ avoided completely the use of empirical relations in their computation of selectivity coefficients by using exact thermodynamic equations to evaluate the three right-hand members of eq 1. The evaluation of the equations, however, (1) F. Helfferich, “Ion-Exchange,” McGraw-Hill Book Co., Inc., New York, N. Y.,1962. (2) H. P. Gregor, J . Am. Chem. SOC.,70, 1293 (1948); 73, 642 (1951). (3) H. P.Gregor and J. I. Bregman, J . Colloid Sci., 6 , 323 (1951). (4) H.P. Gregor and M. Frederick, Ann. N . Y . Acad. Sci., 57, 87 (1953). (5) H. P. Gregor, J. Belle, and R. A . Marcus, J. Am. Chem. SOC., 77, 2713 (1955). (6) G. E. Myers and G. E. Boyd, J . Phys. Chem., 6 0 , 521 (1956). (7) G. E. Boyd, S. Lindenbaum, and G. E. Meyers, ibid., 65, 577 (1961).
PREDICTION OF ION-EXCHANGE SELECTIVITY
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electrostatic interactions in ion-exchange gels should requires a large number of measurements, and the be sought to explain ion-exchange selectivity, emmethod, although theoretically important, is not ployed the polyelectrolyte theory of FUOSS, Katchalsky, practical for predicting ion-exchange equilibria. and Lifson15 for this purpose with some success. He Glueckauf,* on the other hand, has been able to use limited his examination to highly swollen ionized gels, the osmotic coefficient measurements of weakly crosshowever, justifying his use of polyelectrolyte theory linked (0.5% DVB) exchangers in the pure salt forms by pointing out the similarity between the linear to determine the activity coefficient ratios of the ions analog and the highly swollen cross-linked gel. We in the resin phase and thereby predict ion-exchange have since shown that the profitable use of polyelecequilibria. Harned’s ruleg was employed to estimate trolyte theory for the interpretation of ion-exchange the activity coefficients of the resinates in the mixed phenomenals can be extended to the more concentrated form. highly cross-linked ion-exchange gels. In addition Glueckauf’s approach has been modified by Soldano, we have demonstrated that the unique patterns of et u Z . , ~ ~ ’who ~ employed the Gibbs-Duhem equation to physical-chemical behavior that are observed with determine ion-activity coefficients. The activity colinear polyelectrolyte systems are to be expected in efficients of the heteroionic forms of the resin were also the highly cross-linked systems as weIl.l7 For excomputed by Harned’s rule.s The amalgamation of ample, the observation in numerous investigations of polyion and simple counterion interactions, implicit ionized polyelectrolyte-simple salt mixtures that the in their approach, leads to coefficients that are different colligative properties of the pure components are from those that are generally observed in aqueous solualso applies to cross-linked ionized polytions. The Harned coefficients are much too large, electrolytes (ion-exchange resins)-simple salt mixand the thermodynamic requirement that their sum ture~.~’ be a constantla is not obeyed.” The complete ionModel. The gel, in the theory of FUOSS, Katchalsky, exchange isotherm is predicted by them, however, and Lifson15 a6 modified by Kagawa and Gregor,lB from water vapor sorption isotherms of the pure salt is pictured to consist of a parallel arrangement of forms of the resin and two equilibrium measurements cylindrical polyion rods carrying evenly distributed with one component present in trace quantity. fixed charges with mobile counterions of finite radius The treatment by Gregor of the problem of ion-exdistributed in accordance with the electrostatic field change selectivity appears to be most practical and it is in the vicinity of the polyion strands. At the midpoint, the objective of this article to provide an appropriate modification of his approach. For this purpose we have adapted Feitelson’s14 application of the FUOSS, (8) E. Glueckauf, Proc. Roy. SOC.(London), A214, 207 (1952). (9) H. 8. Harned and B. B. Owen, “The Physical Chemistry of Katchalsky, and L i f ~ o n ’polyelectrolyte ~ theory as Electrolyte Solutions,” 3rd ed, Reinhold Publishing Corp., New modified by Gregor and Kagawa16to evaluate the first York, N. Y., 1958,p 600 and Appendix A. term of eq 1, In (TI/%), which was neglected by Gregor (10) B. A. Soldano and Q. V. Larson, J. Am. Chem. SOC.,77, 1331 (1955). and co-workers. Our employment of the third term of (11) B.A. Soldano and D. Chesnut, ibid., 77, 1334 (1955). eq 1, (i? - T)(~I- 72)/RT, is less arbitrary than (12) B. A. Soldano, Q. V. Larson, and G . E. Myers, ibid., 77, 1339 Gregor’s, hydration parameters for the estimate of ion (1956). volumes being derived from our interpretation” of (13) E.Glueckauf, H.A. C. McKay, and A. R. Mathieson, J . Chem. SOC.,5299 (1948). the observed variation of the osmotic properties of the (14) J. Feitelson, J . Phya. Chem., 66, 1295 (1962). essentially unrestrained ( kT. I t is quite conceivable, for example, that the ability of an ion to fit in this highly ordered solvent region will have a noticeable effect on (27) D. Whitney and R. Diamond, Inorg. Chem., 2 , 1284 (1963); J . Inorg. Nucl. Chem., 27, 219 (1965). (28) R. Diamond and D. Whitney, "Ion-Exchange," Vol. I, J. A.
Marinsky, Ed., Marcel Dekker, Inc., New York, N.Y.,1966,Chapter 7. (29) H. s. Frank and W. Y. Wen, D i S C U S 8 i O n S Faraday soc., 24, 133 (1957). (30) R. L.Kay and D. F. Evans, J. Phys. Chsm., 70, 2326 (1966).
Volume 71, Number 6 M a y 1987
J. A. MARINSKY
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its distribution in the high-potential region in the neighborhood of the polyion strand. One can also expect that hydrogen-bonding effects will be operative as well in the case of the NH4+ and H+ ions to favor selective behavior. Two opposing factors apparently operate in determining the arrangement of ion species in the ordered solvent. A large disruptive ion which effectively disturbs the solvent structure operates to oppose the preferred arrangement of a second ion which for the benefit of this argument will be considered to be weakly ordering in its interaction with solvent (e.g., Cs+ and E(+). On this basis there will be a predilection toward disorder, and the order-disturbing ion will assume a preferred distribution in the region bounded by bz and the radial dimension at which e$ = kT. If, however, the second ion is strongly ordering (e.g., Hf), it must overcome the opposing disordering effect thereby leading to its preferred distribution. The relative magnitude of A characterizing each ion is thus qualitatively explained. This type of behavior should be concentration independent as well since the highly ordered region near each polyion strand is essentially invariant. If we examine the relative magnitude of A parameters for the various systems, we can assign a characteristic A value to each counterion species. Such an analysis results in the values listed for each ion in Table V.
The Journal of Physical Chemistry
Table V : Value of A Parameter Characterizing Univalent Ion
Ion
K+ "4
+
cs
+
Na +
H+ TMA Li +
Bare ionic radiue X 108, om
A
1.33 1.48 1.69 0.97
-0.078 -0.128 -0.128
0.33 +
2.44 0.60
0.0
-0.325 -0.383 -0.412
It seems a reasonable postulate that the A term originates as a result of the neglect in the model of counterion selectivity arising from the different characteristic assembly behavior of each ion in the highly ordered water structure that exists in the high-potential region of each polyion strand. The anomalous behavior of Li is not easily resolved. There is no immediately obvious explanation for the observed reduction (Bm, = +0.05mm) of the preferred arrangement of Li+ ion ( A = -0.412) in the highly ordered zone of the polyelectrolyte as its concentration, on a monomer basis, increases. Acknowledgment. Financial support through Contract No. AT(30-1)-2269 with the U. s. Atomic Energy Commission is gratefully acknowledged.
