NOTES
They showed that one can, however, fit the data by assuming higher order (1:2) complexes to be formed in addition to the 1 : l complexes. Hayman,13 however, cast doubt on the physical significance of the equilibrium constants derived in this fashion. Gardner, et a1.,l7 found K values dependent on the wave length for the association of substituted naphthylenes with picric acid. They interpreted this in terms of contact charge-transfer interactionsIs and solvent perturbation^,'^ rather than by assuming the presence of higher order associates. Hayman13 has discussed the interpretation of spectrophotometric data in some detail and has shown that complementary data from other sources are needed to distinguish between the various possibilities. In any case, it is important to note that the amines suspected to experience much steric hindrance to H bonding, that is di-p-tolylamine and di-1,l-naphthylamine, seem most susceptible to the above-mentioned complications, in accordance with Gardner’s findings. Obviously, the present results mean that much care is necessary in the selection of model compounds for the study of structural effects on H-bonding equilibria.
Acknowledgment. The authors are indebted to Dr. E. Loeser and Mr. R. Basalay for their preparation of compounds needed in this work.
339
water activity, where apparently anomalous effects appear, an investigation along the lines here suggested may be in order. For an ion-exchange reaction between A“’ and B+ZBin an exchanger containing a definite number of fixed charges which also imbibes anion, say X-zx, one may write an equilibrium constant expression for the cation exchange
I
Kc-fAzB fBZA
In this expression the m’s and y’s are written for molalities and activity coefficients in the solution phase. The ratio of the ion activity coefficients is thermodynamically determinate
The composition of the exchanger phase is expressed SO that
and Henry C. Thomas
that is, the N’s are given as fractions of the exchange capacity in equivalents ; “exchange capacity” here refers to the total fixed charge, exclusive of imbibed anion. One equivalent of exchanger therefore conNx equivalents of cation. tains 1 The activity coefficients j A and f B refer to the neutral combinations of cation, exchanger, and imbibed anion and are thermodynamically defined quantities for the exchanger phase. When desired, numerical values of these activity coefficients may be obtained from integrations of the Gibbs-Duhem equation of the exchanger phase, provided that sufficient experimental data are available. The Gibbs-Duhem equation for one equivalent of exchanger containing n A = N A / Z A , nB = N B / Z B , and nX = N X / Z Xmoles of the various ions and ns moles of water is
Department of Chemistry, University of North Carolina, Chapel Hill, Y o r t h Carolina (Received J u l y 15, 1964)
n.4 d In fANA
(17) P. D Gardner, R. L. Brandon, N. J. Nix, and I. Y . Chang, J . Am. Chsm. Soc., 81, 3413 (1959). (18) L. E. Orgel and R. S. Mulliken, ibid., 79, 4839 (1957). 119) N. S.Bayliss and C. J. Breckenridge, ibid., 77, 3957 (1955).
The Effect of Water Activity on Ion-Exchange Selectivity
by H. Laudelout Catholic University of Louvain, Heverlee, Belgium
+
+ nB d In fBNB -/nx d In f x N x
It appears that a purely thermodynamic relation between the water content of an ion exchanger and its selectivity has not been exploited in the past. This relation predicts the change in selectivity with water activity from the change in water content with composition. Since considerable interest attaches to ionexchange behavior for concentrated solutions, with low
+ ne d In a,
=
0
Here a, is written for the activity of the imbibed water, which we may take to be always equal to the activity of the water in the equilibrating solution, All quantities in the equation will then refer to an exchanger in equilibrium with a given solution. The quantity f x has the nature of an individual ion activity coefficient. Its use is forced upon us by our choice of Volume 69, N u m b m 1
January 1965
NOTES
340
means for expressing the composition of the exchanger; it is not, however, susceptible of determination from exchange data and drops out of the final expressions for the determination of f~ and f ~ .In the following argument we need only its existence, and our result in no way depends upon a knowledge of its value. The Gibbs-Duhem equation may then be written
N A d In ANA)'^
+ N B d In G~BNB)'~ +
From the expression for the true equilibrium constant
K we get d ln K ,
+ d 1nfAzB= d 1nfBzA
(4)
Using this relation to eliminate f~ from (3), separating the various activity coefficients from the concentration variables, and making a series of replacements such as
N B d In K O = d(NB In K,)
+ Nx) d InfA
Z B ( ~
=
d(zB(1
-
In K OdNB
N x ) 1nfA) ZB
lnfx dNx
we arrive at the expression
ZAZBns d In
U,
(5)
The quantity on the left of ( 5 ) is a perfect differential; so, then, must be the quantity on the right. Part of the necessary and sufficient condition for this is the relation we seek =
-)
-zAzB( bns bNB bns
=
+~AzB(-)
a..Nx
~ N aA. . N x
(6)
since, also, a t constant NX we have dNB = -dNA. Expression 6 in essence predicts the change in the selectivity coefficient with total molality of the equilibrating solution, which determines the solvent activity, from a knowledge of the change in water content with composition. As remarked above, this result in no way depends on a knowledge of fx. It The Journal of Physlcal Chemistry
might be suggested at once that the simplest case of application of (6) will be to an exchanger with no imbibed anion, Le., for NX constant and equal to zero. In this case the formula can be derived directly without the necessity of introducing the anion activity coefficient at any stage. Zeolites form a class of cation exchangers which imbibe little or no anion and so offer an opportunity for the simplest application of (6). A study of the consequences of our expression 6 might be made in the following manner. The selectivity coefficient K O would be obtained from a series of chromatographic experiments in the usual fashion. At the same time, the free volume of the chromatographic column would be determined by a measurement of its anion content, say by displacing chloride with nitrate, or less ambiguously by displacing chloride with an identical solution containing tagged chloride ion. Such a procedure would deJine the region of cation uptake as that region free of anion. If this definition coincides closely with that given by a free volume determined simply by weight increase, the zeolite will thus be shown not to take up anion. The water content of the zeolite would then be determined by a measurement of the volume of the column accessible to Hz018. The difference between this volume and that accessible to anion measures ns. In order to obtain sufficiently large effects, it would be necessary to compare values of K , for dilute solutions with those obtained at high concentrations of electrolyte, selecting solutions for which sufficient vapor pressure data are available to make possible the calculation of a,. Such a study would make possible an assessment of the extent to which sorbed ions are associated with a constant amount of water. This extreme of simdicitv (which is scarcely to be expected) would be represented by
that is, a linear formula for the variable part of the water content, implying fixed ionic hydration. This would result in the following requirement for a pair of singly charged ions
N s A - nsB
(8)
Thus one would expect a linear variation of In K , with In us,provided that the water content of the pure ionic forms changes inappreciably with water activity as is undoubtedly true for the rigid zeolitic crystals. This relation implies that one could predict the change in In K O with In a, from a pair of measurements of
NOTES
34 1
water content for the monoion zeolites. It would be of interest to see how closely this simple relation approximates to the facts.
Some Thermodynamic Aspects of Ion-Exchange Equilibria i n Mixed Solvents
by A. R. Gupta Atomic Energy Establishment, Trombay Chemistry Division, Bombay 88, India (Received June 2i'* 1564)
When ion-exchange equilibrium is regarded as the equilibrium sta,te of the reversible bimolecular reaction ZBAR
+ Z*BfZBe Z B A + ~+* ZABR
(1)
where ZA and ZB are the valencies of ions A and B, respectively, and R refers to the univalent functional group of the ion exchanger, the equilibrium constant K is given by
Ekedahl, Hogfeld, and SillBn and, independently, Argersinger, Davidson, and Bonner2 have considered the activities of the components AR and B R as a whole, rather than the single ion activities in exchanger phase; in other words, the exchanger phase has been regarded as a solid solution of the components AR and BR. I n the earlier treatments, solvent absorbed by the exchanger was completely neglected, but in later analysis of this problem this has been taken into considerati0n.~-5 Gaines and Thomas4 have given a thermodynamically rigorous treatment of this equilibrium and have defined the standard state for the exchanger phase as that of the monoionic form of the exchanger in equilibrium with an infinitely dilute solution of the corresponding salt, ie., pure water. This choice of the standard state implies that it is independent of the outside solution. The equation for the thermodynamic equilibrium of the uni-univalent exchange system In water (subsequently we will be considering only unlunivalent exchanges-an extension to other cases 1s obvious), derived on this basis, using the GibbsDuhem equation and the law of mass action, is
where K. is the selectivity coefficient ( K D )for reaction 1 corrected for the solution phase activity coefficients ( K D being defined by KD = (NBRmA+)/(NARmB+), where N ' s are the mole fractions of the resin components and m's are molalities of the ions in the outside solution), VB and VA are the equivalent volumes of pure B and A forms of the exchanger, 7 is the molar volume of the water vapor, and nw is the number of moles of water associated with one equivalent of the exchanger. The first integral takes into consideration the dependence of selectivity coefficients on resin composition. The other three integrals refer to the variations in the water sorption by the ion exchanger. The second integral implies the integration of the equivalent moisture, nw, of the ion exchanger over the two extremes of resin composition, a t a constant ionic strength, m. The second and third integral rationalize the standard state of the ion exchanger from that of the monoionic form of the exchanger in equilibrium with an m molar solution of the corresponding salt to that of the monoionic form of the exchanger in equilibrium with pure water (an infinitely dilute solution of the corresponding salt). The resin phase activity coefficients are implicitly taken care of in this equation (for a fuller discussion of this subject, see ref. 6). When ion-exchange equilibria in mixed solvents are to be considered on this basis, many difficulties arise in obtaining a thermodynamically rigorous solution of the problem. The addition of another solvent introduces another term in the Gibbs-Duhem equation. The integration to be performed on isothermal surfaces (see ref. 4) to solve the resulting equations for the exchanger phase activity coefficients becomes quite complicated. Another difficulty lies in the choice of the standard states. For these reasons, a mathematically rigorous solution will not be attempted here, but the problem will be considered on general physicochemical principles. Equation 3, derived for the aqueous systems by Gaines and Thomas, will be taken as the model, and then the various terms will be corrected, using the appropriate activity coefficients. The final results thus will be derived on a semiempirical basis, but will be thermodynamically correct as will be pointed out a t appropriate places. (1) E. Ekedahl, E. Hogfeld, and L. G. SillBn, Acta Chem. Scand., 4 , 556, 828 (1950). (2) W. J. Argersinger, Jr., A. W. Davidson, and 0. D. Bonner, Trans. Kansas Acad. Sci., 53, 404 (1950).
(3) E. Hogfeld, Arkiv Kemi, 5, 147 (1952). (4) G. L. Gaines and H. C. Thomas, J . Chem. Phys.. 21, 714 (1953). (5) A. W. Davidson and W. J. Argersinger, Jr., Ann. S. Y . Acad. Sci., 57, 105 (1953). (6) L. W. Holm, Arkiv Kemi, 10, 151 (1956)
Volume 69, Number I
January 1565
