2120
J. A. Marinsky, M. M . Reddy, andS. A m d u r
(17) R. Schlogl, Z. Phys. Chem., 1, 305 (1954). (18) T. Teorell, Z. Elektrochem., 55, 460 (1951); Progr. Biophys. 610phys. Chem., 3,305 (1953); Discuss. FaradaySoc., 21,9 (1956). (19) H. Pleijel, 2.Phys. Chem., 72, 1 (1910). (20) F. Helfferich, "Ion Exchange," McGraw-Hill. New York, N. Y., 1962, Chapter 8, pp 357-359.
(21) P. Henderson, Z. Phys. Chem., 59, 118 (1907); 63, 325 (1908). (22) D. A. Maclnnes, "The Principles of Electrochemistry," Dover Publication, New York, N. Y., 1961, p 231. (23) S. W. Feldberg, "Electroanalytical Chemistry," A. J. Bard, Ed., Marcel Dekker, New York, N. Y . , 1969. (24) H. Cohen and S . W. Cooley, 6iophys. J., 5, 145 (1965).
Prediction of Ion-Exchange Selectivity' a J. A. Marinsky," M. M. Reddy, and S.
Amdur'b
Chemistry Department, State Universityof New York at Buffalo, Buffaio, New York 14214 (Received October 2, 1972; Revised Manuscript Received January 37, 1973)
Successful prediction of ion-exchange selectivity has been accomplished previously by using ion-exchange resin-phase activity coefficients calculated from polyelectrolyte osmotic coefficients with the GibbsDuhem equation. This approach has been further examined in this paper to establish the validity of assigning macroconcentration activity coefficients values to ions a t trace concentration. For this purpose ion-exchange selectivity reactions between a cross-linked ion-exchange resin and its polyelectrolyte analog have been studied. The reference state activity coefficient for each of the ions in this kind of experiment are the same in both phases, enabling direct calculation of the absolute value of the ion-exchange selectivity coefficient. Analysis of experimental ion-exchange selectivity data indicates that the trace ion activity coefficient is best expressed as a product of mean ion activity coefficient components. These components reflect electrostatic and solvation influences on the chemical potential of the ion in solution. The electrostatic contribution to the mean ion activity coefficient, ye/yecref), has been obtained by employing the experimentally observed limiting value of the polyelectrolyte osmotic coefficient, 41, with the Gibbs-Duhem equation. By using the electrostatic contribution term obtained from this treatment and the observed mean coefficient, y f l y f r e f , the solvation contribution to the mean activity coefficient, -ys/ys(ref),of the polyelectrolyte can be calculated over the concentration range of interest. Success of hybrid trace mean ion activity coefficients in the prediction of ion-exchange selectivity values is further demonstrated for ion-exchange reactions in simple electrolyte solutions.
Introduction In earlier studies293 we have used the Donnan membrane model of the ion-exchange process for the interpretation of cation-exchange selectivity data obtained with one of the exchanging ions present in trace quantity. The selectivity coefficient (i. e., the molality product ratio a t equilibrium for the Z I - , 22-valent cation exchange reaction) is expressed in this approach by
where K is the swelling pressure of the resin, VI and V2 are the partial molal volumes of the exchanging ions in the resin phase,4 and yl* and y2* are the mean molal activity coefficients of salts 1 and 2. Barred symbols denote the resin phase, 71 and 7 2 corresponding to the mean molal activity coefficients of the exchanging pair of ions in the gel. The charge of the coion of salts 1 and 2 is designated by 23. By dividing both sides of eq 1 by the mean molal activity coefficient ratio term for the two salts in the aqueous phase, a modified selectivity coefficient KlAC2is presented as a function of the activity coefficient The Journal of Physical Chemistry, Vol. 77, No. 17, 7973
ratio of the exchanging ions in the resin phase and the pressure-volume work of the exchange reaction. Evaluation of 72/71 and a(AV permitted computation of Kl& for comparison with the experimentally determined values. Resin-phase swelling pressure (a) was calculated from the osmotic coefficient data for the polyelectrolyte analog of the resin.5Jj Partial molal volumes of the ions in the resin phase were taken as the partial molal volumes of the ions in aqueous solution a t infinite dilution.' Resin-phase activity coefficients for the exchanging ions were determined by integration of the Gibbs-Duhem equation employing osmotic coefficients of the appropriate ion forms of the polyelectrolyte analog of the resin from a reference concentration of 0.02 m (0.01 m in the case of divalent salt forms) to the concentration of the resin phase. This integration gives 7.m/7refr the resin-phase activity coefficient at the concentralion of the ion exchanger divided by the activity coefficient a t the reference concentration. The computation of ( 7 2 / 7 l ) ( s l r e f / 7 2 r e f ) in place of 72/71 was necessary because it is not possible to anticipate the trend of osmotic coefficient values of polyelectrolytes in the dilute concentration region inaccessible to ac-
2129
Prediction of Ion-Exchange Selectivity
TABLE I: Computation of the Selectivity Coefficient for the Exchange of Trace Divalent Ion with Hydrogen Ion in Ion-Exchange Resin, Polyelectrolyte Analog Systemsa.b 6 H +- 2 ) / ( q H + - 2 ) r e f
% DVB
IResinl, m ~
1 2 4 8 12 16
~
1.012 1.604 2.80 4.50
5.61 6.64
6h2+)/(4MZ+)ref
(YMZ+)/ (YM2+)ref (rH+)*/(rH+)*ref
s,atm
AV, ml
e-rAv/RT
10.9 24.4 60 136 203 274
-17.1
1.003 1.007 1.018 1.04 1.06 1.08
K H E (calcd) ~ ~
KHExM (expt)
~~~~
0.243 0.257 0.348 0.558 0.798 1.108
1.61 1.61 1.61 1.61 1.61 1.61
a See paraqraph at end of text regarding supplementary material. 45Ca2+.
0.391 0.416 0.569 0.936 1.36 1.93
0.21 0.22 0.21 0.29 0.43 0.52
' Systems, hydrogen ion form resin (HR), 0.04 m polystyrenesulfonic acid (HPSS),
curate measurement. As a consequence any difference between the reference state activity coefficient of exchanging ions prevents a correct assignment to the K1*c2 values. This is particularly the case in unsymmetrical ion exchange (zl # 22). With such exchanging systems the utility of the method is limited to the prediction of trends in selectivity as a function of the concentration (cross-linking) of the ion in the gel phase. The predictive quality of the model so applied was good although trends in selectivity were exaggerated.2.3 In the case of symmetrical ion exchange selectivity coefficients as well as selectivity trends were predicted. Some divergence between experiment and calculation was noticeable with concentration change. Such divergence was attributed exclusively to neglect of interaction of the exchanging ions in the gel phase. An additional problem prevails, however, in proceeding with the above analysis of the selectivity coefficient. There is uncertainty in the appropriate assignment of the concentration equivalent of the trace ion. In the computation procedure it was assumed that the trace-ion behavior should be characterized at the concentration of the dominant ion. However, the water activity of the system controlled by the dominant ion is different from that of the trace ion a t that concentration and this is a complication of the system that may not be adequately treated. Ion-exchange reactions between polyelectrolyte solutions and cross-linked polyelectrolyte ion-exchange resins were analyzed to determine the appropriate expression for the concentration equivalent of the trace ion. These experiments are definitive because the reference state activity coefficient in each phase is the same and cancels in the selectivity coefficient expression. Ion-exchange selectivity coefficients correctly predicted for the above systems should be directly comparable with experimental observation. Analysis of the results of these ion-exchange studies by using the thermodynamic method employed in our earlier publications2J leads to unsatisfactory agreement between predicted and experimental selectivity values as shown in Table I. This discordance must be the result of inappropriate choice of the concentration equivalent of the trace ion in the polyelectrolyte solution and in the cross-linked gel. The activity coefficient of an ion A at trace-concentration levels in a solution containing ion B at molality m l differs from the activity coefficient of ion A at molality ml as a consequence of different solvent activities in each solution. For simple electrolyte solutions the mean activity coefficient of the ions in solution can be factored into terms
representing electrostatic and solvational components.s-ll The electrostatic contribution to the mean ion activity coefficient in solution is calculated by employing a semiempirical or limiting law. By analogy it may be appropriate to calculate similarly the electrostatic and solvational contributions to the trace ion mean activity coefficient for the polyelectrolyte solutions and the cross-linked polyelectrolyte gels examined in this study. The polyelectrolyte solution osmotic coefficient, 4, approaches a limiting value at low solution concentrations, &. This limiting value reflects only electrostatic interactions between the polyion and the counterions in solution.12 Electrostatic contributions to the ionic activity coefficient of the polyelectrolyte solution can be obtained by using the limiting value of the polyelectrolyte osmotic coefficient, 41,with the Gibbs-Duhem equation r m
where Y~ is the electrostatic contribution to the mean ionic activity coefficient at concentration m and Te(ref) is the electrostatic contribution to the mean ionic activity coefficient a t the reference state concentration mref. Equation 2 integrates directly giving In
Ye/Ye(ref)
=
($1
- 1)In m/mret
(3)
The assumption that $1 is invariant up to concentrations as high as 6 m is implicit in the integration of eq 2. Support for the validity of this estimate comes from the observation that the osmotic coefficient of the Cs+ ion form of PSS increases very slowly, indeed, over this concentration range.13 Because of the lower charge density of Cs+, ion-solvent interaction is minimal and electrostatic interaction between counterion and charged polymer matrix is apparently the dominant factor in determining the colligative properties of this polyelectrolyte. The constancy of 4 with concentration that is observed is thus justification of use of $1 in eq 3 to compute In Ye/Tref. A plot of log ye/Te(ref) and log y i / y * r e f US. log m for polystyrenesulfonic acid is shown in Figure 1. The log Tf/y*ref term is obtained by employing the experimental 4 values for the pure polyelectrolyte in the integrated form of the Gibbs-Duhem equation.2,3-5-6Below a polyelectrolyte concentration of 0.1 m the electrostatic and mean ionic activity coefficients are essentially the same. Above 0.1 m these terms diverge with increasing concentration. This divergence is attributed to solvation effects.6 The solvation contribution to the mean ionic activity coefficient is given by The Journal of Physical Chemistry, Vol. 77, No. 17, 1973
21 30
J. A. Marinsky, M. M. Reddy, and S. Amdur
HPSS -
Y . 10-
0.M
I d.020.03
1 0.05
I I 0.080.1
I
I
I
0.2
0.3
0.5
I
1
2
0.61
3
5
8
log m
Figure 2. Plot of y s vs. log
Figure 1. Plot of log yf/yfrefvs. log m.
rpm.
TABLE 11: Reanalysis of Selectivity Coefficient Predictions for the Exchange of Trace Divalent Ion with Hydrogen Ion-Exchange Resin, Polyelectrolyte Analog Systems
--
System: HR, 0.04m HPSS, 45Ca2+;(Yca2+/Yca2+ref)/r(YH+)/(YH+ref))2 = 1.61
1 2 4 8 12
0.00723 0.00592 0.00705 0.0157 0.0299
0.0325 0.0215 0.0132 0.0088 0.0073
1.40 1.93 3.43 8.3 16.5
1.003 1.007 1.018 I .04 1.06
0.256 0.232 0.254 0.354 0.424
0.21 0.22 0.21 0.29 0.43
System: HR, 0.01 m HPSS, s5Zn2+(60C02++, 63Ni2+);(YZnl+/YZnref)/~(YH+)/(YH+ref)l'= 0.615
2 4 8 12 16
0.00723 0.00592 0.00705 0.0157 0.0299 0.0576
1 2 4
0.00723 0.00592 0.00705
1 2 4
0.00723 0.00592 0.00705
1 2 4
0.00723 0.00592 0.00705
1
1S I 2.13 4.85 18.0 54 94
0.0345 0.0223 0.0139 0.00926 0.0076 0.0066
1.004 1.009 1.023 1.053 1.08 1.11
0.05(0.048,O.052) 0.039(0.043,0.048) 0.035(0.041,0.045) 0.030 0.026 0.027
0.0851 0.0777 0.0655 0.0613 0.0486 0.0633
System: HR, 0.01 m HPSS, 45Ca2+;(YCa2+/YCa2+ref)/((YH+)/(yH+ref)~'= 0.615
0.0325 0.0215 0.0132
1.40 1.93 3.43
1.003 1.007 1.018
0.0953 0.0886 0.0977
0.055 0.053 0.053
System: HR, 0.01 m HPSS, 90Sr2+;(YSr2+/YSr2+ref)/(YH+/YH+ref)2= 0.615
0.031 0.0208 0.0127
1.27 1.63 3.0
System: HR, 0.01 m HPSS,
[HR],m
0.112 0.109 0.116
0.053 0.057 0.064
= 0.615 0.045 0.039 0.039
1ogCd2++; (YCdZ+/YCdZ+ref)/(YH+/YH+ref)' 1.53 2.25 4.55
0.033 0.022 0.0135 (qH+)
[HPSS],m
1.003 1.008 1.019
'/ ( T H +) 2ref
0.0884 0.0747 0.0724
1.004 1.009 1.022 (?'M2+)/(YM2+
)ref
[ ( 4 ~ ' + ) / ( 4 ~ ' + ) r e ~ l e ( ~ ~ ~ + )(YH+)'/ s (YH+)'ref
e-a
*'
Hi
KHExCo(CalCd)
KHExco(expt)
System: HR (8% DVB), HPSS, 6oCo2t
2.41 X 2.42 X 4.61 X lo-' 5.82 X 1.06 X IO-' 0.91 X lo-' 1.628 X IO-' 1.666 X 10-1
4.425 4.44 4.44 4.45 4.46 4.46 4.49 4.49
0.0925 0.0925 0.0925 0.0925 0.0925 0.0925 0.0925 0.0925
T h e J o u r n a l of P h y s i c a l C h e m i s t r y , Vol. 77, No. 17, 1973
1.18 1.18 1.65 1.93 2.98 2.57 3.84 3.81
1.053
1.14 X 1.14 X 1.61 X 1.98 X 2.90 X 2.51 X 3.74 x 3.72 X
lo-* IO-' lo-' 10 -I 10-1 10-1 lo-'
9.74 x 10-2 8.97 X 1.61 X 10-1 1.88 X lo-' 3.6 X IO-' 2.86 X 10 -1 4.14 X 10-I 4.12 X 10-I
21 31
Prediction of Ion-ExchangeSelectivity COMPARISON OF COMPUTED AND EXPERIMENTAL VALUES OF K x
COMPARISON OF COMPUTED AND EXPERIMENTAL VALUES OF ,K,
SYSTEM: Sr(II) R.X% DVB, 0.1m Sr (ClO,),dn)
COMP.(METHOD I)
0.7
0.6-
% I'
2.0
\ -----e
0
SI%
P
s
X
0.5-
d SYSTEM: Ca (II)R.X% DV8.O.lm Ca(CIO,),
0.4 -
M'UI)
----I I.5
-
0.3
0.2-
c
co'
0.1
L
0.0
K -vr
A
1.0
2.0 3.0 iii,f+t(RESIN MOLALITY)
Figure 3. Comparison of computed KAC:divalent-divalent exchange.
X C
' '0
4.0
P
'r
and experimental values of
(4) where ys is the solvation contribution to the mean ion activity coefficient at concentration m and Yscref) is the solvation contribution to the mean ion activity coefficient at the reference concentration. For polystyrenesulfonic acid and all other polyelectrolyte solutions in this study osmotic coefficient and mean ionic activity coefficient data obtained for pure polyelectrolyte solutions with eq 3 and 4 enable calculation of the electrostatic and solvation contributions to the mean ion activity coefficient over the range of solution concentration of interest (Figures 1and 2). In the calciulation of an ion-exchange selectivity coefficient (when one ion is at a macroconcentration level and the other is a t a trace concentration level) the activity of the macrocomponent is that associated with the pure substance in solution. Thus, the activity coefficient in the selectivity calculation is the mean ionic activity coefficient determined from the experimental osmotic coefficients of the pure polyelectrolytes with the Gibbs-Duhem equation. However, for the ion a t trace concentrations the activity coefficient of trace ion A is the product of two terms: (1) the electrostatic contribution to the mean ion activity coefficient of a solution containing pure A at the concentration of the macroion and (2) the solvation contribution to the mean ion activity coefficient of a solution containing pure A a t the water activity of the macroion solution. Thus
(Y+IY
+ref)
=
s
Y
(relYe(,,n)(rslY,(,eo)
trace ion A ion A a t macro ion concn concn m , m,, solvent activity aHZO
(5)
ion A a t solvent activity aH,O
Results of selectivity calculations employing this procedure are compared with experimental values in Table 11. Agreement between computation and experiment is considerably improved by this approach. When the polyelectrolyte Concentration does not fall below the concentration of the reference solution which represents the lower concentration limit of the osmotic measurements
0.1'
1.0 iii,ddRESIN
20 30 MOLALITY)
Figure 4. Comparison of computed KAC:divalent-divalent exchange
COMPARISON
1.0
40
and experimental values of
OF COMPUTED
2.0
3.0
4.0
i i i z p ( R E S I N MOLALITY)
Figure 5. Comparison of computed KAC: divalent-divalent exchange.
and experimental values of
made in this laboratory agreement between experiment and computation is excellent. At lower polyelectrolyte concentrations the computed K E is~ somewhat larger than the experimental value. In all cases, however, trends in selectivity are anticipated with accuracy. The discrepancy between predicted and observed selectivities when the experimental polyelectrolyte concentration is less than the reference concentration is probably a consequence of error introduced by the linear extrapolation of the osmotic coefficient data for the HPSS to lower The Journal of Physical Chemistry, Vol. 77, No. 17, 1973
21 32
J. A.
""I
COMPARISON OF COMPUTED AN0 EXPERIMENTAL VALUES OF K,, SYSTEM: M (lI)R.X% DVB.
/ /
/' COMP
1.0U
GU0.5-
V.V V.V(
&S
0.4. 0.2-
" "
0o1
-.-
* 1.0
EXPA --*-,^----v
2.0
3.0
to analysis by the new method (method 2 in Figures 3-6) of computation as the other exchanging pairs of ions studied. This result combined with the demonstrated fact that the absolute value of the selectivity coefficient of two ions exchanging between a PSS resin and its linear polyelectrolyte analog can be evaluakd with accuracy when the polyelectrolyte concentration does not fall below the reference molality attests to the validity of the modified approach that has been developed to compute relative mean molal activity coefficient values in the resin phase. The tendency for divergence in predicted selectivity trends from observation with increasing cross-linking of resin is undoubtedly a consequence of neglect of ion-ion interaction. A study of three component mixtures is planned, e.g., MI (11)-Mz( 11)-PS S and Hz0, isopiestically to evaluate the extent of such interaction in an attempt to corroborate the above statement. In addition, the predictive quality of this improved method of selectivity analysis is being examined for the pairs of ions studied here over the complete composition range of these pairs of ions in the resin.
COMP
40 4.0
TM++ (RESIN MOLALITY)
Figure 6. Comparison of computed and experimental values of KAC: univalent-divalent exchange.
concentrations. Nagasawa, e t al.,14 have shown that at concentrations below 0.02 m, the reference molality of HPSS, its osmotic coefficient deviates increasingly from the linearly extrapolated value. As a consequence, the value of Y H + / Y H + r e f at 0.01 and lower is probably larger than calculated. Since this term is squared the value of ((YMZ + / y M z + ref )/ ( Y H + /YH + r e f ) l2 may be significantly smaller than calculated to account for the difference between the predicted and observed KEx values a t these low polyeleetrolyte concentrations.
Predictions Based on Modified Selectivity Computation Method To examine further the utility of this approach to the prediction of ion-exchange selectivity in general all of the systems investigated in the earlier papers have been reanalyzed. Prediction and experiment are plotted for each of these systems in Figures 3-6,15 the ratio between the observed and predicted value a t 1%cross-linking being used to normalize the computed points. In a number of instances the predictions based upon the earlier computational approach (method 1) are included in these figures to provide a comparison of their respective predictive quality. From these graphical representations it is seen that predicted trends in selectivity though still somewhat exaggerated in the higher cross-linked resins are more consistently in better agreement with experimental observation than by the earlier approach. In addition those exchanging pairs of ions which were most troublesome to analyze earlier,3 e.g., the Sr(I1)-MT(II), Ca(I1)-MT(II), M(I1)-SrT(II), M(I1)-CaT(II), and hl(II)-NaT systems, are as susceptible
The Journal of Physical Chemistry, Vol. 77, No. 17, 1973
Marinsky, M . M. Reddy, andS. Amdur
Acknowledgment. The authors wish to express their appreciation to the United States Atomic Energy Commission for financial support through Contract No. AT(301)-2269. Supplementary Material Available. Three additional figures to illustrate the utility of the computation, seven additional exchange systems, will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 x 148 mm, 20x reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 Sixteenth St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-73-2128. References and Notes (a) Presented in part at the Charles D. Coryell Memorial Symposium during the National Meeting of the American Chemical Society, Boston, Mass., April 1972. (b) Present address, Soreq Nuclear Research Centre, Israel Atomic Energy Commission, Yavne, Israel. M. M. Reddy and J. A. Marinsky. J. Macromol. Sci. Phys., B5(1), 135 (1971). M. M. Reddy, J. A. Marinsky, and S. Amdur, J. Amer. Chem. SOC., 94, 4087 (1972). The 7r ( A V )term, because of its relative unimportance, was neglected in ref 2 and 3. M . M. Reddy and J. A. Marinsky, J. Phys. Chem., 74,3884 (1970). M. M. Reddy, J. A. Marinsky, and A. Sarkar, J . Phys. Chem., 74, 3891 (1970). P. Mukerjee, J. Phys. Chem., 65, 740 (1961). R. H. Stokes and R. A. Robinson, J. Amer. Chem. SOC.. 70, 1870 (1948). E. Glueckauf, Trans. Faraday Soc., 51, 1235 (1955). R. A. Robinson and R . H. Stokes, "Electolyte Solutions," 2nd ed, Revised, Butterworths, London, 1959, Chapter 9. E. Glueckauf in "The Structure of Electrolyte Solutions," W. Hammer, Ed., Wiley, New York, N. Y., 1959, pp 97-1 12. G. S. Manning, J. Chem, Phys., 51, 924, 3249 (1969). B. Soldano and Q. V . Larson. J. Amer. Chem. SOC., 77, 133 (1955). A. Takahashi, N. Kato, and M. Nagasawa. J. Phys. Chem., 74, 944 (1970). See paragraph at end of text regarding supplementary material.
