A. R. GUPTA
1152
A Thermodynamic Theory of Ion-Exchange Equilibria in Nonaqueous Solvents by A. R. Gupta Bhabha Atomic Research Centre, Chemistry Division, Trombay, Bombay -86, India
(Received November 30,1970)
Publication costs borne completely by The Journal of Physical Chemistry
Ion-exchange equilibria in nonaqueous and mixed solvents have been treated in a thermodynamically rigorous fashion. Using the concept of standard free energy of transfer of ions and resinate from the nonaqueousmixed solvent to water, the thermodynamic equilibrium constants for exchange equilibria in nonaqueousmixed solvents have been correlated with the equilibrium constants in aqueous systems. The free energies of transfer of the resinates have been formulated in terms of the free energies of swelling of resinates from dry monoionic forms of the resins to the equilibrium, a completely swollen state in the respective solvents. The equations developed for the mixed solvent systems have been applied to the available data in the literature on sodium-hydrogen exchange on Dowex 5OWX-8 in methanol-water mixtures. Ion-exchange equilibria in aqueous systems have been treated thermodynamically by considering them as reversible bimolecular reactions. In the earlier treatments on this basis, water activity terms and their variations were not included. lv2 Later this problem was treated by Hogfeldla Gaines and T h o r n a ~ ,and ~ Davidson and Argersinger6 independently. A good review of these approaches has been given by Holm.e The treatment of Gaines and Thomas is most thermodynamically rigorous. Using this approach as a model, the problem of ion exchange in mixed solvents has been treated' in a semiempirical manner. I n the present communication, the general problem of ion-exchange equilibria in nonaqueous solvents including mixed solvents has been considered in a thermodynamically rigorous fashion. An attempt has been made to treat all the solvent systems on the basis of a common set of standard states, so that a comparison among them is possible. Pure Nonaqueous Solvents. The problem of ionexchange equilibria in pure nonaqueous solvents, with reference to aqueous systems, is first considered. For simplicity, only uni-univalent exchanges have been treated. For the ion-exchange equilibrium
AR+B+=BR+A+
(A)
for every solvent, there exists an equilibrium constant
K,
=
defined with reference to the standard state of the monoionic form of the resinate in equilibrium with an infinitely dilute solution of the corresponding salt in that solvent. I n Gaines and Thomas' treatment of ionexchange equilibria, the equilibrium constant can be written in terms of the chemical potentials as
RT In K ,
= p o s ~ R-
~''BR
+
posg+
-
pos~+
(2)
A similar expression can be written for the aqueous system
RT In K ,
= p o W ~ R-
~'"BR
+
powBt
-
pow^+
(3)
Subtracting eq 3 from eq 2 Ks
RT In - = Kw (posB+
(~''AR
- p o W ~-~ (P'~BR )
- poWB+) -
- ~',BR)
- powAt)
AGt"(AR) - AGt"(BR)
+
=
+
AGt"(B+) - AG,"(A+)
(4)
The free energies of transfer are defined by the relation
aGt0(i) = post
- powa = RT In yl*
The individual equilibrium constants K , and K , are given by the following expression for moderately crosslinked resins (more than 6% DVB resins), where the molar volume terms can be neglected6
UBRaA +
-
aARaB +
where A and B are the univalent ions, R represents the resin matrix, a's denote the activities of the various components, and K , is the equilibrium constant in solvent s. The activities of the ions are defined with reference to the state of infinite dilution in the particular solvent, ie., lim y = 1, where y is the activity coeffic-co
cient on the molal scale. The resinate activities are The J O U Tof ~Physical Chemistry, Vol. 76, No. 8,1971
(1) E.Ekedehl, E.Hogfeld, and L. G. Sillen, Acta Chem. S c a d . , 4, 556,828 (1950). (2) W. J. Argersinger, Jr., A. W. Davidson, and 0. D . Bonner, Trans. Kana. Acad. Sci., 53,404 (1950). (3) E.Hogfeld, Ark. Kemi, 5, 147 (1950). (4) . . G. 1,. Gaines and H. C. Thomas, J. Chem. Phys., 21, 714 (1953). (5) A. W.Davidson and W. J. Argersinger, Jr., Ann. N . Y . Acad. Sci., 5'1, 105 (1953). (8) L.W.Holm, Ark. Kemi, 10, 151 (1958). (7) A. R. Gupta, J . Phys. Chem., 69,341 (1985).
1153
A THERMODYNAMIC THEORY OF ION-EXCHANGE EQUILIBRIA ~ N B =R1)
lnK, = L'1nK.dNsR-k
S
UW~N= B 1) R
a w = ~ ( N B R1)=
UW(NBR=O)
+S
a ~ ~=( N B = R0)
log K , - log K, =
AG+,"(B+) - AGt"(A+) -
2.303RT
a. = 1 (NBR= 0 )
~ N B =R 0)
n, d In a,
-
n, d In a, n,
d In a, (5)
where K, is the selectivity coeffcient (KD) for reaction A corrected for the solution phase activity coefficients (KD) being defined by
S S S
+ nw d log a, + n, d log a,
~,=O(NBR=O)
aw = ~ ( N B =O) R
aw=O(NBR =0)
a 8 = ~ ( N B R1) =
u,=O(NBR=~)
S
n, d log a,
-
aW= ~ ( N B= R1)
NBRmA+ KD = NARmB+
u~=O(NBR= 1)
~
where N's are the equivalent fractions of the resin components, the m's are the molalities of the ions in the outside solution, and n, is the number of moles of water associated with 1 eq of the exchanger. A similar expression can be written for K , in terms of solvent activity terms a, and n,, the number of moles of solvent associated with 1 eq of the exchanger. The various terms in eq 5 are, in principle, accessible to experimental d e t e r m i n a t i ~ n . ~The ~ ~ left-hand side of eq 4 can, therefore, be evaluated from experimental data. The quantities AGtC for the ions are well defined and have been reported for many ions in different so1vents.10-12 The AGto terms for the resinates have not yet been defined in terms of experimentally obtainable quantities. These can be best understood in terms of the free energy changes occurring in the ~ w e l l i n g ' ~ *of' ~the ion exchangers in the two solvents. For this purpose, the reference state of dry monoionic form of the resin is chosen for the resins. For the solvent vapor, the vapor over the pure solvent is taken as the reference state. When the resin in ionic form i is in equilibrium with pure water, the free energy of swelling is given byl3,l4
where a, and n, have their usual meaning. Similarly the free energy of swelling of resin iR in the solvent s is
As Wp,,(iR) and sqsw(iR)are measured with reference to the same state of dry monoionic form of the resin, their difference gives the change in free energy in going from the equilibrium state of resin iR in solvents s to the equilibrium state of resin iR in water, i e . , the free energy differences in the standard states of the resinates in the two solvents, as used in Gaines and Thomas treatment. Therefore sp,w(iR) - "v,,(iR)
AGot(iR)
(8) Thus AGt"(iR) can be evaluated in terms of spgW(iR) and wpsw(iR)with the help of eq 6 and 7. Substituting for these terms in eq 4,we get =
n, d log a,
(9)
Mixed Solvents. The case when one of the solvents is water, has been experimentally investigated by many workers.15 The problem has also been treated theoretically in a semiempirical manner.' As the problem of ion exchange in mixed solvents is an important one it will now be treated in a more rigorous fashion. The quantities pertaining to the mixed solvent containing an organic solvent and water will be denoted by the superscript "mix." Rewriting eq 4 for the mixed solvent
log Kmix - log K w =
{ AmixGt0(AR) -
AmixGto(BR)
+
AmixGto(B +) - AmixGto(A+)] /2.303RT
(10)
Now Kmix represents the ion-exchange equilibrium constant referred to the standard states in the mixed solvent having N H ~ O mole fraction of water and N, mole fraction of the nonaqueous solvent. As has been pointed out earlier,' Gaines and Thomas equations can be used for evaluating equilibrium constants in mixed solvents. AmixGto(i) terms have definite thermodynamic significance and have been determined for many ions in mixed solvents." The free energy of transfer of the pure resinates from the mixed solvent to water, i e . , AmiXGto(iR) , needs further analysis. Again for the resins, the reference state of the dry, monoionic form of the resin is chosen and for the component solvents, the pure solvents are taken as the reference (8) M. R. Ghate, A. R.Gupta, and J. Shankar, I n d i a n J. Chem., 4, 64 (1966). (9) M. R. Ghate, A. R. Gupta, and J. Shankar, ibid., 4, 353 (1966). (10) Roger G. Bates in "The Chemistry of Nonaqueous Solvents," J. J. Lagowski, Ed., Academic Press, New York, N. Y., 1966, pp 97-127. (11) D.Feakins in "Physico-chemical Processes in Mixed Aqueous Solvents," F. Franks, Ed., Heinemann Educational Books Ltd., London, 1967,pp 71-89. (12) H . Strehlow in "The Chemistry of Nonaqueous Solvents," J. J. Lagowski, Ed., Academic Press, New York, N. Y.,1966,p 152. (13) G. E. Boyd and B. A. Soldano, 2.Elektroehem., 57, 162 (1953). (14) G. V. Samsonov and V. A. Pasechnik, Russ. Chem. Rev., 38 (7), 547 (1969). (15) M . R.Ghate, A. R.Gupta, and J. Shankar, I n d i a n J. Chem., 3, 286 (1965); M. R. Ghate, A. R. Gupta, and J. Shankar, ibid., 5, (1967); M. R. Ghate, A. R. Gupta, and J. Shankar, ibid., 6, 98 (1968). For other references see ref 7.
T h e Journal of P h y s k a l Chemistry, Vol. 76, No. 8,lQ7l
A. R. GUPTA
1154 state. Let a‘, and a‘, denote the activity of solvent “s” and water in the mixture, respectively, and n, and n, the corresponding moles of solvent s and water absorbed by the resin iR when in equilibrium with the mixed solvent. The free energy of swelling, i.e., the change in the free energy of the system when dry resin iR comes to equilibrium with the mixed solvent, can be expressed as mix
a!w,a’B (PSW
=
-RT
aw,as) = 0
n, d In a,
The corresponding free energy change in the readjustment of the water content of the resinate is
-RT
l::::,=o n, d In a, =
RT
+
saw=l,a’=o
aw-l,ag=O(N~~=O)
S S
Uw=
The total free energy change in the second step, thus, becomes
l,as=O(NBR=1)
=O
n, d In a,
a)w,a‘s
Therefore
+
+
miXqsw(iR) = ~ ( 1 ) ( ~ ( 2= ) w ~ s w ( i R ) ~ ( 2 ) (11) The free energy of transfer of resin iR from the mixed solvent to water is defined by - Wq,,(iR) AmixGto(iR)= miXqsw(iR)
+
=
Wqsw(iR) ~ ( 2 ) wcpsw(iR)=
4‘ =)
n, d In a, The Journal of Physical Chemistry, Vol. 76, N o . 8,1971
(12)
n, d log a, -
S
~’W,U’MBR = 1)
n, d log a,
(13)
A comparison of eq 9 for pure nonaqueous systems with eq 13 for mixed solvents shows that the two types of systems have different integration limits for the various integrals. This basic difference has some far reaching consequences. The integration limits in eq 9 imply that the absorption isotherms of resinates, needed for the evaluation of integrals, should be obtained by equilibrating the resinates with solutions in pure solvents, having different solvent activities. On the other hand, the absorption isotherms needed for the evaluation of various integrals in eq 13, should be obtained by equilibrating the resinates with mixed solvents, having different activities of the two components. It is interesting to compare eq 13 with the semiempirical equation derived by the author earlier. For this purpose it is better to rearrange the equation to give log K , in terms of log Kmix. From the definition of free energy of transfer of an ion from solvent s to water
- powt = AGto(i) = RT In y**
where y* is interpreted as an activity coefficient related to the free energy of transfer, AGto(i). Writing AmixGto(i)in terms of y * in eq 13 and rearranging log K , = log Kmix
= La,
1)
u~=~,u,=O(NBR=~)
pas$
n, d In a,
a’w,a’,
RT
+ n, d log a, + n, d log a,
aw = I,U,=O(NBR=O)
U’wta’a(NBRc
a‘WP’B
aw = l,as=O
S
U‘~,U’,(NBR=O)
~’w,u’,(NBR=O)
(n,, n,) d In (awJa,)
This free energy change can be described in terms of the free energy changes in the following two steps. 1. Dry resin iR (one equivalent) absorbs nowmoles of water at water activity a, = 1, i.e., pure water. 2. Organic solvent is gradually added maintaining the equilibrium between solvent and resinate, until the requisite composition of the solvent mixture (aw, a,) is reached. During this process resin absorbs n, moles of the solvent and water attains its new equilibrium value of n, moles. As discussed earlier, the free energy in step 1 is given by ,qSw (eq 6). I n the second step, the free energy changes pertaining to the solvent and water can be treated separately as shown below. I n the absorption of the solvent, the free energy change is
-RTS
Finally, putting the values of AmixGto(iR)in eq 10, we get
+ log Y + other terms *A+ y*B+
~
(14)
Here log K , and log Kmix are given by expression 5 for the two solvents involved, i.e., water and the mixture. I n the earlier treatment y* values were incorporated in the first integral in the expression for log Kmix. Therefore, the first four integrals in eq 10 of ref 7 are equivalent to log Kmix log (Y*A+/Y*B+) of eq 14. Further comparison of these two equations reveals that the remaining two integrals of the former are the same as the two integrals involving nW and a, terms in the latter. Thus, eq 14 differs from the eq 10 (ref 7) in having two additional integrals involving n, and a, terms. In these mixed solvent systems, where the ion exchangers do not absorb one of the components (like the organic component in water-organic solvent mixtures), i.e., n, is equal to zero, eq 14 reduces to eq
+
A THERMODYNAMIC THEORY OF ION-EXCHANGE EQUILIBRIA 10 (ref 7). Therefore, the latter equation is strictly applicable only to those systems where ion exchangers completely exclude the nonaqueous component. On the other hand, eq 14 is a general one and applicable to all mixed solvent systems. This equation can be used for computing log Kmixfrom log K , and other parameters which can be determined by independent experiments. For this purpose, further simplification of eq 14 can be achieved for solutions of low ionic strength ( 6 0.1 M ) . Under these conditions, as has been shown p r e v i o u ~ l y , the ~ - ~last three integrals in the right-hand side of eq 5 defining log K, or log Kmixare unimportant and log K , or log Kmixis given by the simple expression
sb
log K, (or log Kmix) =
log "K, (or log miXKa) CWBR
Using this simplification one can rewrite eq 14 for computing log Kmixas follows log miXKa, CWBR =
log K, - log - Y*B+
s
aW= alw a, = a ' B ( N B R = 0)
n, d log a,
a,=O ( N B R = O )
+
aW=l
aB=O
L-
aw = I ( N B R w
=o)
- U'W(NBR = O)
n, d log a,
+
a8=afB aw = a',,.
S
a8
= a',(NBR = 1)
a, = O ( N B R = 1 )
n, d log a, -
uw=l a,=O
lUw=
1 ( N B R = 1)
w-a'w(NBR=l)
n, d log a,
(15)
a, = alg
Application of Ep 16. In the numerous studies reported in the literature on ion exchange in mixed solvents, the swelling data needed for the evaluation of various integrals in eq 15 have not been recorded. However, Starobinets, Novitskaya, and Sevostyanova16 have recently published their results on Na/H exchange on Dowex 50 WXs in methanol-water mixtures, which includes the swelling data on sodium and hydrogen forms of the exchanger in the various solvent mixtures. These workers used expt 12 of ref 7 (valid for ionic strength 6 0.1 M ) for calculating log K m i x . These calculated value of log Kmixdiffered from the experimentally observed values by an order of magnitude. ~ a + )in They had ignored the 2 log ( m i X y ~ + / m i x yterm their computation, which is not permissible as this term has significant values in methanol-water mixtures. All the same, their work emphasizes the impor-
1155
tance of integrals involving n, and a, terms. Their data have been used for computing log Kmix by use of eq 15. As they have reported the experimental values of the integra1.f; log m i x KdNm, ~ the same has been computed here, i e . , log miXyiterm has been taken on the right-hand side of eq 15. The activity coefficients of NaCl and HC1 in methanol-water mixtures a t an ionic strength of 0.1 M (mixyiterms) have been obtained from the data of Aker10f.l~ As the value of the activity coefficients of the NaC1 were not reported at this molality, they were obtained by interpolation using the data at 0.05, 0.2, and 0.5 M solutions. The values of the integrals involving water activity terms have been tabulated by Starobinets, et al.,16 in their paper. The values of y*H+ and y * ~ ~ for+ various methanol-water mixtures have been reported by Aker10f.l~ The same values have been used here and were apparently used by the Russian workers (this could be checked by an analysis of their data). The evaluation of integrals involving a, and n, poses a problem as the limits of integration are from a, = 0 (log a, = - a ) to a, = a',. This difficulty was circumvented by extrapolating the plots of n, vs. log a, to values of n, = 0. These plots were constructed using the reported values of n, for the hydrogen and sodium form of resinates for various water-methanol mixtures. The activity of methanol in these solvent compositions was obtained from the data of Butler, et aZ.18 The values of the two integrals involving n, and a,, were computed by graphical integration of the plots of n, us. log a, within the appropriate limits for the various mixtures. The relevant data are shown in Table I. All the quantities are thus available for the computation of log Kmixby use of eq 15. The calculated values of log Kmixand the corresponding observed values16are also given in Table I. The agreement between the values of log Kmixcomputed on the basis of present thermodynamic approach and the observed ones, at low values of N , , is more than satisfactory. At larger values of N,, the computed values show larger deviations from the observed ones. Considering the uncertainties in the values of various quantities used in this computation, this is not surprising. The difficulties in the experimental determination of y * or AG," (i) and uncertainties involved in their reported values, have been pointed out by Strehlowlgin a general way. The uncertainty in the value of y* increases with its intrinsic value. As y * for both HC1 and NaCl increases with methanol content, the reported values of y" at high values of N , are unreliable to the same extent. Thus, the larger deviations in the computed values of (16) G.L. Starobinets, L. V. Novitskaya, and L. I. Sevostyanova, Zh. Fiz. Khim., 42 (5),1098 (1968). (17) G.Akerlof, J . Amer. Chem. Soc., 52, 2353 (1930). (18) J.A. V.Butler, D. V. Thomson, and W. H. Mailennan, J . Chem. Soc., 674 (1933). (19) Reference 12,pp 145,146. The Journal of Physical Chemistry, Vol. 76,No. 8,1971
A. R.GUPTA
1156 Table I: Evaluation of Methanol Activity Integrals in Eq 15 for Na/H Exchange on Dowex 5OWX-8 and Computed and Observed Values of Log Kmix
Na
a8
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.148 0,282 0.393 0.476 0.555 0,636 0,740
a
~B(HR)~
1.2 1.8 2.4 2.8 3.2 3.4 3.6
n8(NaR)b
Diff erenoea in the value of integral
0.66 1.10 1.54 1.76 2.00 2.07 2.07
-0.093 -0.256 -0.374 -0.459 -0.532 -0.605 -0.680
YNaCl
0.735 0.675 0.625 0.57 0.518 0,455 0.415
log Kmix (es 16)
log Kmix'
mix+fHCl
0.768 0.735 0.705 0.676 0.653 0.62 0.585
0.46 0.64 0.84 0.97 1.30 1.56 1.69
0.36 0.62 0.78 0.89 0.98 1.07 1.16
(obsd)
Values tabulated are for
-S
a w = a'w,
a'a(NNaR
0)
+S
aw3a'w, aa3a's(NNaR
n,
d log as
a,=O(NNaR=O)
aw=l
b
mix
=S
1)
ns d log as
as = O ( N N ~ R= 1) aw-1
Taken from ref 16.
log K m i x compared to the observed ones may arise due to these uncertainties in the values, and the data in Table I can be taken in a general way as an experimental verification of eq 15. In conclusion, it should be emphasized that eq 9 and 13 are exact equations and have been derived thermodynamically in a rigorous fashion. Equation 9 relates the ion exchange in a pure nonaqueous solvent to ion exchange in aqueous medium. Equation 13 does the same correlation for ion exchange in mixed solvent systems, one solvent being water. Here for convenience the aqueous system has been chosen as the reference
The Journal of Physical Chemistry, Vol. 76, No. 8, 2971
system, primarily because most of the thermodynamic data, like standard free energy of transfer of an ion from one solvent to another, are available with reference to aqueous medium. I n principle, any solvent could have been chosen as the reference solvent. These equations thermodynamically describe the ion-exchange systems in any solvent, pure or mixed, in terms of the ion exchange in aqueous or any other chosen medium.
Aclcnowbdgment. The author expresses his sincere thanks to Dr. J. Shankar for his encouragement and keen interest durin.gthe course of this investigation.
