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Quantitative description of ion exchange selectivity in non-ideal systems
Reactive Polymers, 1 (1983) 251-259
251
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
QUANTITATIVE DESCRIPTION OF ION EXCHANGE SELECTIVITY IN NON-IDEAL SYSTEMS VLADIMIR S. SOLDATOV and VALENTINA A. BICHKOVA
Institute of Physico- Organic Chemistry, B.S.S.R. A cadency of Sciences, 220603 Minsk ( U. S. S. R ) (Received June 29, 1982: accepted in revised form April 8, 1983)
Equations for ion exchange selectivity and resinate activity coefficients computed as a function of the ionic composition of resins have been derived. They are based on the assumption that the real non-ideal ion exchange process is a superposition of three ideal elementary processes related to the different types of exchange sites in resins. In the simplest non-ideal exchanger case, the equations derived for the computation of res'inate activity coefficients are comparable to Kielland's equation. The equations have been tested with nine ion exchange systems exhibiting strong deviations from ideal behaeiour (exchange of alkali metal ions for H + on highly crosslinked sulphonic resins). Correlation between the theory and experiment is good.
INTRODUCTION
A knowledge of the dependence of selectivity coefficients on the degree of ion exchange is not easily obtainable from theoretical models of ion exchange equilibria. It has been observed by many authors that the selectivity coefficient as a function of the ionic composition of the resin can behave in a variety of ways. Decrease in selectivity towards the preferably sorbed ion with increasing equivalent fraction of that ion occurs most often, while many cases where selectivity increases or passes through an extreme are also observed [1-3]. The numerous attempts to predict selectivity coefficients as a function of the ionic composition of ion exchangers have been summarized by Helfferich [4]. None of these, however, provides a theoretical approach of 016%6989/83/$03.00
general utility for predicting selectivity coefficients as a function of ion exchange resins. It is the aim of this paper to develop a simple equation of general use for describing the selectivity and activity coefficients computed as functions of the ionic composition of ion exchange resins. Recently a simple model of liquid ion exchangers has been developed by H6gfeldt and Soldatov [5]. Liquid ion exchangers are solutions of ionized, high molecular-weight organic substances dissolved in organic diluents immiscible with water. Molecules of the ionized substances are strongly associated in the organic solvents. According to the model proposed, the solutions of liquid ion exchangers containing counterions 1 and 2 are similar in properties to hypothetical solutions exhibiting only three kinds of aggregates of
q: 1983 Elsevier Science Publishers B.V.
252 the same size: two of them monoionic and the other mixed. It has been suggested that some properties of such solutions can be presented as a sum of the specific properties of the different aggregates. The following equation describing a property Y is based on this suggestion: r - - Yll.g 2 + 2yl2XlX 2 +Y22ff~,
(1)
where constants Yu and Y22 characterize the specific properties Y of type 1 and 2 monoionic associates, respectively; Yl2 is the value characterizing property Y in the mixed associate; and £~ and £2 represent equivalent fractions of 1 and 2 in the resin. It has been found [6,7] that eqn. (1) describes the selectivity coefficient, k, of a liquid-liquid ion exchange as a function of the liquid exchanger loading. In spite of the fact that the structure of liquid ion exchangers is obviously different from that of resins of the same chemical type, great similarity in their behaviour is observed [8]. The reasons for it are not yet understood. It will be shown in this paper that an equation of the type of eqn. (1) can be applied to resinous ion exchangers.
S E L E C T I V I T Y C O E F F I C I E N T S AS A FUNCTION OF THE IONIC COMPOSIT I O N OF R E S I N S Let us consider an ion exchange resin in equilibrium with an infinitely dilute solution containing monovalent ions of types 1 and 2. Assume that the free energy of the system can be defined as a sum of interactions of the nearest neighbour ions situated along the same hydrocarbon chain. Then each counterion has two neighbour sites, and three states can be imagined for each counterion, as follows: 111, 211(=112), 212, 222, 221(=122), and 121. Let us denote the chemical potentials related to these states, respectively, as ~l(.), t~l¢12), ~1(22), ~2(22), ~2(12), and /~2~ll)" The ions are
distributed between these states which can be considered as energetic levels of the exchange sites. Further, ion exchange is assumed to be ideal in each level, i.e., a relative equilibrium constant can be computed from ion concentrations for this level. Non-ideality of the total ion exchange is caused by an unequal loading of levels with the exchangeable ions at the given value of if,. When a mole of ion 1 is added to an infinitely large amount of resin with composition ffl, the change in Gibbs' free energy can be presented as a sum AG = ~1(22)AG1(22) -4--0~1(12)AG1(12) + al(11)AGl(ll),
(2)
where the symbols a are fractions of the mole of ions 1 added to the exchanger: a1(22 ) -~- O'l(12 ) -+- al(ll ) = l.
(3)
The value of AGl~:2 ) is related to the exchange of an equivalent of ion 2 by ion 1 in such a way that nearest neighbours of 1 are only ions of type 2: dGl(22) = fiq(22) dnl(22) +/-tl dn~ + ~2(22) dn2{22) + ~2 dn2,
(4)
where the symbols marked with a bar identify the resin phase and those without the bar the equilibrium solution; n corresponds to the number of moles. If dill(22 ) = - - d n l , dH2(22 ) =
-dn2,
(5)
dH1(22) = - dH2(22 ) , then dGl(22)=
(~1(22)-~1)
dnl(22)
- ( ~ 2 ( 2 2 ) - ~2) d~1(22).
(6)
Such a process occurs when an infinitely small increment of ion 1 is distributed between the ion exchanger saturated with ions 2 and electrolyte solution of type 2 at a concentration
253
C 2. If the standard states are chosen in such a way that /2,° =/~o at ff~ = x, = 1, then for the standard state, where AG, = AG[, we have: A G ~ ( 2 2 ) =/~1(22)-/~1 -o o =
-RTIn
AG R T - In k = In +In
k22.
(7)
Similarly, AG~(ll ) =
)=
- R T l n X - : 22 X 2 { x2 --,0)
= -RTln
I
(8)
and AG~(12 )=
-o o -o o = ]'/'1(12) - - I'Ll - - ~t2(12) + / - t 2
=-RTln
£1(12)x2X2(12)X1
RTlnk)2.
(9)
The counterions can be assumed to be statistically distributed between the levels since the difference in energy of an interionic interaction is much smaller than the absolute energy value of an energy level. Therefore, cq(22 ) is equal to the mole fraction X1(22) of ions 1 surrounded by ions 2 at each composition .~. The same holds true for a~(~2~ and al(l~: O~I(22) ~ X 1 ( 2 2 ) ' ~1(12) = X l ( 1 2 ) '
~.ll> =,G(~l).
(10)
The value of X1(22) is proportional to the probability that an ion of type 2 has type-2 neighbours in its vicinity. In this case, the probability is equal to x-22, since the sums of both probabilities and mole fractions are equal to
Yll =
1"
Og1(22) = X I ( 2 2 ) =
-2 x2.
(14)
In k~l,
Yl2 = In kl2, Y22 = In k22.
-AG~(ll
k22.~ 2 .
2 In kl2Xlx 2
Comparing eqn. (1) with eqn. (14), we find:
xl X 1 (a-~ ~ 0 )
= -RTln
kll.~ 2 +
(l 1)
Similarly, we have: a,(,,>=
(12)
Oq(12 ) = 2 . ~ 1 . ~ 2 .
(13)
It follows from eqns. (2), (7)-(9) and (1 1)-(13) that
(15)
The applicability of eqn. (14) to ion exchange resin equilibria has been examined in several systems exhibiting strong deviations from ideality. These systems include the N a + - H +, N H ~ - H + and N H ~ - N a + exchange on sulphonated polystyrene exchangers containing 12 and 25% by weight divinylbenzene ( D o w e x 50 WX12 and K U - 2 X 2 5 *). Equilibria of C s + - N H 2 , Cs + H + a n d L i + H + o n K R S - 2 5 * h a v e a l s o been studied. These equilibria, studied at a temperature of 25°C and at constant ionic strength of the equilibrium solution, have been reported earlier [9-11]. The values of In kjl, In k22 and In /£12 call be found by fitting the constants of eqn. (14) to the experimental data, In the case of liquid ion exchangers, constants In kit and In k22 were determined by extrapolating functions In k = f(Yj) at 21 ---, 1 and 21 ---, 0, or by measuring relative distribution coefficients at very low concentrations of one of the ions [6,7]. In the case of highly crosslinked resins, it is considered that the presence of small amounts of impurities or some irregular ion exchange sites can lead to a strong deviation of the experimental data at £ ~ ( 2 2 ) ~ 0 from the behaviour predicted by eqn. (14). Therefore, the values of log kll, log k22 and log k~2 have been computed by least squares fit of function S over the concentration range X~ = 0.1-0.9, provided that sum S reaches a
* Analogues
of D o w e x 50 W X 2 5 ( S o v i e t p r o d u c t i o n ) .
254 TABLE 1 Selectivity constants of elementary ion exchange processes. Logarithms of selectivity constants of elementary ion exchange processes used in eqns. (14) and (18). In right column, arithmetical mean of log k u and log k22 is given versus log k12 No.
Resins
Exchanged ions
log kll
log k22
log k12
{(log k,1 + log k22 )
1 2 3 4 5 6 7 8 9
Dowex 50 WX12 KU-2X25 Dowex 50 WX12 KU-2X25 Dowex 50 WX12 KU-2X25 KRS-25 KRS-25 KRS-25
NH~ -H + NH~ -H + Na+-H + Na + - H + NH~- - N a + NH~- - N a + Cs + - N H ~ Cs + - H + Li+-H +
0.092 - 0.060 0.024 - 0.276 0.090 0.155 - 0.046 - 0.503 -0.568
0.518 0.711 0.237 0.350 0.181 0.321 0.324 1.475 -0.015
0.338 0.399 0.343 0.459 0.082 0.157 0.067 0.505 -0.032
0.305 0.325 0.130 0.037 0.135 0.238 0.139 0.486 -0.291
-log -log
log k
log k 1.0
0.32 O. 20
0.6
0.16
0.4
0.12
0.2
0.08
0
0.04
-0.2
I 0.28
0.8
0.24 I
O. 20
~" \
I
RNH4
RH
0.6
\
0.4
\\
0.16
0.2
@\\o
\ \
0
-0.4
o
0.5
~,.
0.12
1.0
21 Fig. I. Logarithms of selectivity coefficient (right scale, dotted line), log k = f(xl), and activity coefficients, log e~ = f(xt), as functions of equivalent fraction of NH~- in resin phase of NH~- - H + system, Dowex 50 WXI2. @: Points on line log k = f(ffl), computed from experimental data. (D: Points on lines log ~ = f(£l), computed by graphical integration of eqn. (17). Curves computed from eqns. (14) and (18), using the constants given in Table 1.
0.08
-©. 2
o
-0.4 -0.6 0
0,5
Fig. 2. NH~- - H +, K U - 2 X 2 5 . See Fig. 1.
I,0
255 -lod (to
log k
1O . ~ 6
0.4
-log
(p
log k
0.12
0,2 ""<]~-@ - - -~- - -~--<]~ _ _@_ _ _(~_ .~9_ _(jr_ _<]L.
0.12
0.08
/ Ulo
O.Od
0.04
~
0
0.04
~
-0.2
~--°--~
~ m7_ m _ ~
~
-0.4
-0.2 o
0.5
1.0
fl t~
O
-0.4
0
0.5
Fig. 5. N H ~ - N a
1.0
+, Dowex 50 WX12. See Fig. 1.
log k
-log
Fig. 3. N a + - H +, Dowex 50 WXI2. See Fig. 1. 0.2
0 •08
-log T
1
0
o.36 [
1
}
log k
~1
o.8
~X~}{NH4
0.04
O
R};a
~
0
0.32
o,6
Ol
0.24
jlO
\
0.5
-0.2
~_
-0.4
1.0
Fig. 6. NH,~ - N a +, K U - 2 X 2 5 . See Fig. 1.
0.2
.~,
o
o
~-
-log
O, 20
O
0.20
I{H 0.16
log k
i\~ -0, 2
O. 16
-0.4
0.12
O. 12
0.4
"(]k
0
(:i
~ --<) "~
!/
0.2
-0.6
-0.8
0.04 0
-0.2
0.04
-0.4
0 -I .0
0
O. Ot~
0.5
1.0
-0.6
0 O
0.5
1.0
f~ Fig. 4. N a + - H +, K U - 2 X 2 5 . See Fig. 1.
Fig. 7. Cs + NH4 ~, KRS-25. See Fig. 1.
256
log k
log k
-log
O. 28
0.8
0.2
1.2
%
\ 0.24
0.7
\
O
1.0
\ -0.2
O° 20
\
0.~
.
\
xl.
0.8
\
0.16
\
0.5
-0.4
ELi
O.6
RIt
0.4
0.12
-0.6
0.08
-0.8
0.04
-1.0
0.4
\
0.3
0.2
\ \ \
0,2
o
o J -1.2
0.1
-0.2
0
1.0
0.5
i 1 -o.4 0
F i g . 8. C s + - H
0.5
+, K R S - 2 5 .
F i g . 9. L i + - H
+, K R S - 2 5 .
S e e F i g . 1.
1.0
most cases, good agreement is observed. Even in the case of the most non-ideal system (Na + - H + on K U - 2 X 2 5 ) , satisfactory agreement is obtained.
See F i g . 1.
minimum: ?t
RESINATE ACTIVITY COEFFICIENTS
S = ~ (log k ~ - log k)2 i=l
= ~ (log k i - log kll.~ 2 - 2 log kl2XlX 2 i=1
- l o g k22~22) 2 .
(16)
The experimental k~ values are related to different resin compositions and distributed regularly over the concentration range "~1 = 0.1-0.9; n is the number of experimental points (in our case, n = 9). The values of log kll , log k22 and log k m computed in this way are given in Table 1. Figures 1-9 compare the experimental points with those computed from eqn. (14). In
Equations given in Refs. [12-16] used for the calculation of activity coefficients contain the mathematically indefinite function In k = f(ffi). Depending on the above approach, all such equations are somewhat different from each other but all of them include, as the main part, the function In k = f(~,). We can therefore substitute this function, defined by eqn. (14), into eqn. (17): In ~ = - (1 - .~) Ink a +
£ In ka d ~ i,
(17)
x
where ka is an apparent equilibrium constant. This simplest equation is derived from the
257 assumption that a mixed swollen ion exchanger is a bicomponent system [12,13]i The result of the substitution is ln~, = _
k12 )
ln~-
- _ x~
2 {
kllk22
71/ln---TT-kl2 (18a)
In ,~:
= (kk__~12) 2 ln~kllk22 ff~. ,In 22. "~( + ~ k12
(18b)
The thermodynamic constant, K, can be computed as shown
InK;
lnk de =
ln(k k22k 2).
(19)
The particular case where In kl2 = :1 l n ( k l l k 2 2 )
(20)
deserves special attention. It corresponds to a linear dependence of I n k on ~, and is sometimes observed in ion exchange systems [17-18]. Under this condition: In kllk22 - 0
(21)
Equations (18a) and (18b) then become analogous to Kielland's equation: 1 (ln k " )Y 2 In @~ = ~ k2 2 ,
(22a)
In ~2 = ~ In
(22b)
2~, 22,
where expression (23) identifies Kielland's constant: 1
~ (ln~)
= constant.
(23)
It is also clear that In K = In
k12.
(24)
Figures 1 9 present activity coefficients 4' for the systems studied. The values c o m p u t e d from eqn. (18) and the constants given in the table are in good agreement with those found by graphical integration of eqn. (17), where
selectivity coefficients are used inslead of k~,. The neglect of activity coefficients of the aqueous electrolytes causes little error since all equilibria have been studied at constant ionic strength. It is interesting to note that some systems with strong deviations from ideality demonstrate regular behaviour and obey Kielland's e q u a t i o n (NH~- H + and C s + - H + on KRS-25). Activity coefficients for the L i ~ - H + and Na + H + systems can be described only by eqn. (19), while their absolute values are smaller than those in the previous case. It seems to be true that any ion exchange system involving a monofunctional ion exchanger with statistically distributed counterions can be satisfactorily described by the equations of type (14) and (18). This does not hold true for polyfunctional ion exchangers or for cases with a sieve effect where the exchange sites can be different relative to the same type of an exchangeable ion. In these cases, no simple statistical distribution of counterions occurs between the exchange sites. CONCLUSIONS The approach suggested in this paper allows ion exchange systems to be classified as follows: (1) Ideal systems, corresponding to a constant value of k and to the activity coefficients equal to 1. (2) Regular systems, corresponding to a linear function I n k = f(ff,) and to Kielland's equation for activity coefficients. (3) Irregular statistic systems, corresponding to an equation of the second power (eqn. (14)) for I n k = f(Y,) and to equations of the third power for activity coefficients. (4) Irregular nonstatistical systems requiring special treatment in any particular case. These are cases in which a statistic distribution of the counterions between exchange sites does not occur.
258
In conclusion, it is worth mentioning that eqn. (14) gives a compact and convenient way for tabulation and presentation of experimental data on ion exchange.
L I S T OF S Y M B O L S
Gibbs' free energy G k selectivity coefficient apparent equilibrium constant ka kn, k12, k22 selectivity constants of elementary ion exchange processes related to different types of exchange sites in resin hnumber of moles R gas constant T absolute temperature equivalent fraction of ion i Y some additive property of ion exchange system specific value of property Y reY lated to different states of exchangeable ions mole fraction of ions i entering o¢i a certain energy level of resin at given £i value mole fraction of ions i sur~i(ij) rounded by ions i and j chemical potential resinate activity coefficient Note: barred symbols refer to resin phase; unbarred symbols to equilibrium solution.
REFERENCES 1 G.E. Myers and G.E. Boyd, Thermodynamic calculations of cation exchange, J. Phys. Chem., 60 (1956) 521. 2 O.D. Bonner and V. Rhett, Equilibrium studies of siever-sodium-hydrogen system on Dowex 50, J. Phys. Chem., 57 (1953) 254. 3 V.S. Soldatov, Simple Ion-Exchange Equilibria (in Russian), Nauka i Tekhnika publ., Minsk, 1972. 4 F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962, pp. 5-249.
5 E. H6gfeldt and V.S. Soldatov, On the properties of solid and liquid ion exchangers. VII. A simple model for formation of mixed micelles applied to salts of dinonylnaphtalene sulfonic acid, J. Inorg. Nucl. Chem., 41 (1979) 575. 6 Z.I. Kuvaeva, V.S. Soldatov, F. Fredlund and E. HOgfeldt, On the properties of solid and liquid ion exchangers. VI. Application of a simple model to some ion exchange reactions on dinonylnaphtalene sulphonic acid in different solvents and for different cations with heptane as solvent, J. Inorg. Nucl. Chem., 40 (1978) 103. 7 V.S. Soldatov and A.V. Mikulich, Parallel investigation of ion exchange selectivity of methylammonium ions by a H-ion on liquid and polymer sulphonic cation exchanger, J. Phys. Chem. (U.S.S.R.), 53 (1979) 1279. 8 V.S. Soldatov and E. HOgfeldt, On the properties of solid and liquid ion exchangers. I. Selectivity studies on dinonylnaphtalene sulphonic acid dissolved in heptane. Comparison between liquid and solid cation exchangers, Ion Exch. Membr., 2 (1974) 13. 9 V.S. Soldatov and V.A. Bichkova, Experimental investigation of the N H ~ - - N a + - H ÷ ion exchange equilibrium on a sulfostyrene ion exchanger, J. Phys. Chem. (U.S.S.R,), 45 (1971) 1242. 10 V.A. Bichkova and V.S. Soldatov, Ion-exchange equilibrium in the N H ~ - - N a + - H + system on KU-2X25 resin, Izv. Akad. Nauk B.S.S.R., Ser. Khim. Nauk, 1 (1973) 66. 11 V.A. Bichkova and V.S. Soldatov, Mutual influence of ions in three-ion NH~- - I + - H + systems on highly cross-linked sulphostyrene ion exchangers, Izv. Akad. Nauk B.S.S.R., Ser. Khim. Nauk, 2 (1977) 17. 12 E. H6gfeldt, On ion-exchange equilibria. II. Activities of the components in ion exchangers, Ark. Kemi, 5 (1952) 147. 13 A.W. Davidson and W.J. Argersinger, Equilibrium constants of cation exchange processes, Ann. N.Y. Acad. Sci., 57 (1953) 105. 14 G. Gaines and H.C. Thomas, Adsorption studies on clay minerals. II. A formulation of the thermodynamics of exchange adsorption, J. Chem. Phys., 21 (1953) 714. 15 A.M. Tolmachev and V.I. Gorshkov, Some thermodynamics problems of ion exchange, J. Phys. Chem. (U.S.S.R.), 40 (1966) 1924. 16 V.S. Soldatov, On thermodynamics of ion-exchange equilibria. Calculation of thermodynamic quantities of ion-exchange equilibria, J. Phys. Chem. (U.S.S.R.), 46 (1972) 434, 1078. 17 R.M. Barrer and J.D. Falconer, Ion exchange in feldspathoids as a solid-state reaction, Proc. Roy. Soc. Ser. A, 236 (1956) 227.
259 18 A.V. Mikulich and V.S. Soldatov, Comparable studies of ion-exchange selectivity of ethylammonium ions by an H-ion on liquid and polymer sulfonic
cation exchangers, Izv. Akad. Nauk B.S.S.R., Set. Khim. Nauk, 2 (1979) 39.
251
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
QUANTITATIVE DESCRIPTION OF ION EXCHANGE SELECTIVITY IN NON-IDEAL SYSTEMS VLADIMIR S. SOLDATOV and VALENTINA A. BICHKOVA
Institute of Physico- Organic Chemistry, B.S.S.R. A cadency of Sciences, 220603 Minsk ( U. S. S. R ) (Received June 29, 1982: accepted in revised form April 8, 1983)
Equations for ion exchange selectivity and resinate activity coefficients computed as a function of the ionic composition of resins have been derived. They are based on the assumption that the real non-ideal ion exchange process is a superposition of three ideal elementary processes related to the different types of exchange sites in resins. In the simplest non-ideal exchanger case, the equations derived for the computation of res'inate activity coefficients are comparable to Kielland's equation. The equations have been tested with nine ion exchange systems exhibiting strong deviations from ideal behaeiour (exchange of alkali metal ions for H + on highly crosslinked sulphonic resins). Correlation between the theory and experiment is good.
INTRODUCTION
A knowledge of the dependence of selectivity coefficients on the degree of ion exchange is not easily obtainable from theoretical models of ion exchange equilibria. It has been observed by many authors that the selectivity coefficient as a function of the ionic composition of the resin can behave in a variety of ways. Decrease in selectivity towards the preferably sorbed ion with increasing equivalent fraction of that ion occurs most often, while many cases where selectivity increases or passes through an extreme are also observed [1-3]. The numerous attempts to predict selectivity coefficients as a function of the ionic composition of ion exchangers have been summarized by Helfferich [4]. None of these, however, provides a theoretical approach of 016%6989/83/$03.00
general utility for predicting selectivity coefficients as a function of ion exchange resins. It is the aim of this paper to develop a simple equation of general use for describing the selectivity and activity coefficients computed as functions of the ionic composition of ion exchange resins. Recently a simple model of liquid ion exchangers has been developed by H6gfeldt and Soldatov [5]. Liquid ion exchangers are solutions of ionized, high molecular-weight organic substances dissolved in organic diluents immiscible with water. Molecules of the ionized substances are strongly associated in the organic solvents. According to the model proposed, the solutions of liquid ion exchangers containing counterions 1 and 2 are similar in properties to hypothetical solutions exhibiting only three kinds of aggregates of
q: 1983 Elsevier Science Publishers B.V.
252 the same size: two of them monoionic and the other mixed. It has been suggested that some properties of such solutions can be presented as a sum of the specific properties of the different aggregates. The following equation describing a property Y is based on this suggestion: r - - Yll.g 2 + 2yl2XlX 2 +Y22ff~,
(1)
where constants Yu and Y22 characterize the specific properties Y of type 1 and 2 monoionic associates, respectively; Yl2 is the value characterizing property Y in the mixed associate; and £~ and £2 represent equivalent fractions of 1 and 2 in the resin. It has been found [6,7] that eqn. (1) describes the selectivity coefficient, k, of a liquid-liquid ion exchange as a function of the liquid exchanger loading. In spite of the fact that the structure of liquid ion exchangers is obviously different from that of resins of the same chemical type, great similarity in their behaviour is observed [8]. The reasons for it are not yet understood. It will be shown in this paper that an equation of the type of eqn. (1) can be applied to resinous ion exchangers.
S E L E C T I V I T Y C O E F F I C I E N T S AS A FUNCTION OF THE IONIC COMPOSIT I O N OF R E S I N S Let us consider an ion exchange resin in equilibrium with an infinitely dilute solution containing monovalent ions of types 1 and 2. Assume that the free energy of the system can be defined as a sum of interactions of the nearest neighbour ions situated along the same hydrocarbon chain. Then each counterion has two neighbour sites, and three states can be imagined for each counterion, as follows: 111, 211(=112), 212, 222, 221(=122), and 121. Let us denote the chemical potentials related to these states, respectively, as ~l(.), t~l¢12), ~1(22), ~2(22), ~2(12), and /~2~ll)" The ions are
distributed between these states which can be considered as energetic levels of the exchange sites. Further, ion exchange is assumed to be ideal in each level, i.e., a relative equilibrium constant can be computed from ion concentrations for this level. Non-ideality of the total ion exchange is caused by an unequal loading of levels with the exchangeable ions at the given value of if,. When a mole of ion 1 is added to an infinitely large amount of resin with composition ffl, the change in Gibbs' free energy can be presented as a sum AG = ~1(22)AG1(22) -4--0~1(12)AG1(12) + al(11)AGl(ll),
(2)
where the symbols a are fractions of the mole of ions 1 added to the exchanger: a1(22 ) -~- O'l(12 ) -+- al(ll ) = l.
(3)
The value of AGl~:2 ) is related to the exchange of an equivalent of ion 2 by ion 1 in such a way that nearest neighbours of 1 are only ions of type 2: dGl(22) = fiq(22) dnl(22) +/-tl dn~ + ~2(22) dn2{22) + ~2 dn2,
(4)
where the symbols marked with a bar identify the resin phase and those without the bar the equilibrium solution; n corresponds to the number of moles. If dill(22 ) = - - d n l , dH2(22 ) =
-dn2,
(5)
dH1(22) = - dH2(22 ) , then dGl(22)=
(~1(22)-~1)
dnl(22)
- ( ~ 2 ( 2 2 ) - ~2) d~1(22).
(6)
Such a process occurs when an infinitely small increment of ion 1 is distributed between the ion exchanger saturated with ions 2 and electrolyte solution of type 2 at a concentration
253
C 2. If the standard states are chosen in such a way that /2,° =/~o at ff~ = x, = 1, then for the standard state, where AG, = AG[, we have: A G ~ ( 2 2 ) =/~1(22)-/~1 -o o =
-RTIn
AG R T - In k = In +In
k22.
(7)
Similarly, AG~(ll ) =
)=
- R T l n X - : 22 X 2 { x2 --,0)
= -RTln
I
(8)
and AG~(12 )=
-o o -o o = ]'/'1(12) - - I'Ll - - ~t2(12) + / - t 2
=-RTln
£1(12)x2X2(12)X1
RTlnk)2.
(9)
The counterions can be assumed to be statistically distributed between the levels since the difference in energy of an interionic interaction is much smaller than the absolute energy value of an energy level. Therefore, cq(22 ) is equal to the mole fraction X1(22) of ions 1 surrounded by ions 2 at each composition .~. The same holds true for a~(~2~ and al(l~: O~I(22) ~ X 1 ( 2 2 ) ' ~1(12) = X l ( 1 2 ) '
~.ll> =,G(~l).
(10)
The value of X1(22) is proportional to the probability that an ion of type 2 has type-2 neighbours in its vicinity. In this case, the probability is equal to x-22, since the sums of both probabilities and mole fractions are equal to
Yll =
1"
Og1(22) = X I ( 2 2 ) =
-2 x2.
(14)
In k~l,
Yl2 = In kl2, Y22 = In k22.
-AG~(ll
k22.~ 2 .
2 In kl2Xlx 2
Comparing eqn. (1) with eqn. (14), we find:
xl X 1 (a-~ ~ 0 )
= -RTln
kll.~ 2 +
(l 1)
Similarly, we have: a,(,,>=
(12)
Oq(12 ) = 2 . ~ 1 . ~ 2 .
(13)
It follows from eqns. (2), (7)-(9) and (1 1)-(13) that
(15)
The applicability of eqn. (14) to ion exchange resin equilibria has been examined in several systems exhibiting strong deviations from ideality. These systems include the N a + - H +, N H ~ - H + and N H ~ - N a + exchange on sulphonated polystyrene exchangers containing 12 and 25% by weight divinylbenzene ( D o w e x 50 WX12 and K U - 2 X 2 5 *). Equilibria of C s + - N H 2 , Cs + H + a n d L i + H + o n K R S - 2 5 * h a v e a l s o been studied. These equilibria, studied at a temperature of 25°C and at constant ionic strength of the equilibrium solution, have been reported earlier [9-11]. The values of In kjl, In k22 and In /£12 call be found by fitting the constants of eqn. (14) to the experimental data, In the case of liquid ion exchangers, constants In kit and In k22 were determined by extrapolating functions In k = f(Yj) at 21 ---, 1 and 21 ---, 0, or by measuring relative distribution coefficients at very low concentrations of one of the ions [6,7]. In the case of highly crosslinked resins, it is considered that the presence of small amounts of impurities or some irregular ion exchange sites can lead to a strong deviation of the experimental data at £ ~ ( 2 2 ) ~ 0 from the behaviour predicted by eqn. (14). Therefore, the values of log kll, log k22 and log k~2 have been computed by least squares fit of function S over the concentration range X~ = 0.1-0.9, provided that sum S reaches a
* Analogues
of D o w e x 50 W X 2 5 ( S o v i e t p r o d u c t i o n ) .
254 TABLE 1 Selectivity constants of elementary ion exchange processes. Logarithms of selectivity constants of elementary ion exchange processes used in eqns. (14) and (18). In right column, arithmetical mean of log k u and log k22 is given versus log k12 No.
Resins
Exchanged ions
log kll
log k22
log k12
{(log k,1 + log k22 )
1 2 3 4 5 6 7 8 9
Dowex 50 WX12 KU-2X25 Dowex 50 WX12 KU-2X25 Dowex 50 WX12 KU-2X25 KRS-25 KRS-25 KRS-25
NH~ -H + NH~ -H + Na+-H + Na + - H + NH~- - N a + NH~- - N a + Cs + - N H ~ Cs + - H + Li+-H +
0.092 - 0.060 0.024 - 0.276 0.090 0.155 - 0.046 - 0.503 -0.568
0.518 0.711 0.237 0.350 0.181 0.321 0.324 1.475 -0.015
0.338 0.399 0.343 0.459 0.082 0.157 0.067 0.505 -0.032
0.305 0.325 0.130 0.037 0.135 0.238 0.139 0.486 -0.291
-log -log
log k
log k 1.0
0.32 O. 20
0.6
0.16
0.4
0.12
0.2
0.08
0
0.04
-0.2
I 0.28
0.8
0.24 I
O. 20
~" \
I
RNH4
RH
0.6
\
0.4
\\
0.16
0.2
@\\o
\ \
0
-0.4
o
0.5
~,.
0.12
1.0
21 Fig. I. Logarithms of selectivity coefficient (right scale, dotted line), log k = f(xl), and activity coefficients, log e~ = f(xt), as functions of equivalent fraction of NH~- in resin phase of NH~- - H + system, Dowex 50 WXI2. @: Points on line log k = f(ffl), computed from experimental data. (D: Points on lines log ~ = f(£l), computed by graphical integration of eqn. (17). Curves computed from eqns. (14) and (18), using the constants given in Table 1.
0.08
-©. 2
o
-0.4 -0.6 0
0,5
Fig. 2. NH~- - H +, K U - 2 X 2 5 . See Fig. 1.
I,0
255 -lod (to
log k
1O . ~ 6
0.4
-log
(p
log k
0.12
0,2 ""<]~-@ - - -~- - -~--<]~ _ _@_ _ _(~_ .~9_ _(jr_ _<]L.
0.12
0.08
/ Ulo
O.Od
0.04
~
0
0.04
~
-0.2
~--°--~
~ m7_ m _ ~
~
-0.4
-0.2 o
0.5
1.0
fl t~
O
-0.4
0
0.5
Fig. 5. N H ~ - N a
1.0
+, Dowex 50 WX12. See Fig. 1.
log k
-log
Fig. 3. N a + - H +, Dowex 50 WXI2. See Fig. 1. 0.2
0 •08
-log T
1
0
o.36 [
1
}
log k
~1
o.8
~X~}{NH4
0.04
O
R};a
~
0
0.32
o,6
Ol
0.24
jlO
\
0.5
-0.2
~_
-0.4
1.0
Fig. 6. NH,~ - N a +, K U - 2 X 2 5 . See Fig. 1.
0.2
.~,
o
o
~-
-log
O, 20
O
0.20
I{H 0.16
log k
i\~ -0, 2
O. 16
-0.4
0.12
O. 12
0.4
"(]k
0
(:i
~ --<) "~
!/
0.2
-0.6
-0.8
0.04 0
-0.2
0.04
-0.4
0 -I .0
0
O. Ot~
0.5
1.0
-0.6
0 O
0.5
1.0
f~ Fig. 4. N a + - H +, K U - 2 X 2 5 . See Fig. 1.
Fig. 7. Cs + NH4 ~, KRS-25. See Fig. 1.
256
log k
log k
-log
O. 28
0.8
0.2
1.2
%
\ 0.24
0.7
\
O
1.0
\ -0.2
O° 20
\
0.~
.
\
xl.
0.8
\
0.16
\
0.5
-0.4
ELi
O.6
RIt
0.4
0.12
-0.6
0.08
-0.8
0.04
-1.0
0.4
\
0.3
0.2
\ \ \
0,2
o
o J -1.2
0.1
-0.2
0
1.0
0.5
i 1 -o.4 0
F i g . 8. C s + - H
0.5
+, K R S - 2 5 .
F i g . 9. L i + - H
+, K R S - 2 5 .
S e e F i g . 1.
1.0
most cases, good agreement is observed. Even in the case of the most non-ideal system (Na + - H + on K U - 2 X 2 5 ) , satisfactory agreement is obtained.
See F i g . 1.
minimum: ?t
RESINATE ACTIVITY COEFFICIENTS
S = ~ (log k ~ - log k)2 i=l
= ~ (log k i - log kll.~ 2 - 2 log kl2XlX 2 i=1
- l o g k22~22) 2 .
(16)
The experimental k~ values are related to different resin compositions and distributed regularly over the concentration range "~1 = 0.1-0.9; n is the number of experimental points (in our case, n = 9). The values of log kll , log k22 and log k m computed in this way are given in Table 1. Figures 1-9 compare the experimental points with those computed from eqn. (14). In
Equations given in Refs. [12-16] used for the calculation of activity coefficients contain the mathematically indefinite function In k = f(ffi). Depending on the above approach, all such equations are somewhat different from each other but all of them include, as the main part, the function In k = f(~,). We can therefore substitute this function, defined by eqn. (14), into eqn. (17): In ~ = - (1 - .~) Ink a +
£ In ka d ~ i,
(17)
x
where ka is an apparent equilibrium constant. This simplest equation is derived from the
257 assumption that a mixed swollen ion exchanger is a bicomponent system [12,13]i The result of the substitution is ln~, = _
k12 )
ln~-
- _ x~
2 {
kllk22
71/ln---TT-kl2 (18a)
In ,~:
= (kk__~12) 2 ln~kllk22 ff~. ,In 22. "~( + ~ k12
(18b)
The thermodynamic constant, K, can be computed as shown
InK;
lnk de =
ln(k k22k 2).
(19)
The particular case where In kl2 = :1 l n ( k l l k 2 2 )
(20)
deserves special attention. It corresponds to a linear dependence of I n k on ~, and is sometimes observed in ion exchange systems [17-18]. Under this condition: In kllk22 - 0
(21)
Equations (18a) and (18b) then become analogous to Kielland's equation: 1 (ln k " )Y 2 In @~ = ~ k2 2 ,
(22a)
In ~2 = ~ In
(22b)
2~, 22,
where expression (23) identifies Kielland's constant: 1
~ (ln~)
= constant.
(23)
It is also clear that In K = In
k12.
(24)
Figures 1 9 present activity coefficients 4' for the systems studied. The values c o m p u t e d from eqn. (18) and the constants given in the table are in good agreement with those found by graphical integration of eqn. (17), where
selectivity coefficients are used inslead of k~,. The neglect of activity coefficients of the aqueous electrolytes causes little error since all equilibria have been studied at constant ionic strength. It is interesting to note that some systems with strong deviations from ideality demonstrate regular behaviour and obey Kielland's e q u a t i o n (NH~- H + and C s + - H + on KRS-25). Activity coefficients for the L i ~ - H + and Na + H + systems can be described only by eqn. (19), while their absolute values are smaller than those in the previous case. It seems to be true that any ion exchange system involving a monofunctional ion exchanger with statistically distributed counterions can be satisfactorily described by the equations of type (14) and (18). This does not hold true for polyfunctional ion exchangers or for cases with a sieve effect where the exchange sites can be different relative to the same type of an exchangeable ion. In these cases, no simple statistical distribution of counterions occurs between the exchange sites. CONCLUSIONS The approach suggested in this paper allows ion exchange systems to be classified as follows: (1) Ideal systems, corresponding to a constant value of k and to the activity coefficients equal to 1. (2) Regular systems, corresponding to a linear function I n k = f(ff,) and to Kielland's equation for activity coefficients. (3) Irregular statistic systems, corresponding to an equation of the second power (eqn. (14)) for I n k = f(Y,) and to equations of the third power for activity coefficients. (4) Irregular nonstatistical systems requiring special treatment in any particular case. These are cases in which a statistic distribution of the counterions between exchange sites does not occur.
258
In conclusion, it is worth mentioning that eqn. (14) gives a compact and convenient way for tabulation and presentation of experimental data on ion exchange.
L I S T OF S Y M B O L S
Gibbs' free energy G k selectivity coefficient apparent equilibrium constant ka kn, k12, k22 selectivity constants of elementary ion exchange processes related to different types of exchange sites in resin hnumber of moles R gas constant T absolute temperature equivalent fraction of ion i Y some additive property of ion exchange system specific value of property Y reY lated to different states of exchangeable ions mole fraction of ions i entering o¢i a certain energy level of resin at given £i value mole fraction of ions i sur~i(ij) rounded by ions i and j chemical potential resinate activity coefficient Note: barred symbols refer to resin phase; unbarred symbols to equilibrium solution.
REFERENCES 1 G.E. Myers and G.E. Boyd, Thermodynamic calculations of cation exchange, J. Phys. Chem., 60 (1956) 521. 2 O.D. Bonner and V. Rhett, Equilibrium studies of siever-sodium-hydrogen system on Dowex 50, J. Phys. Chem., 57 (1953) 254. 3 V.S. Soldatov, Simple Ion-Exchange Equilibria (in Russian), Nauka i Tekhnika publ., Minsk, 1972. 4 F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962, pp. 5-249.
5 E. H6gfeldt and V.S. Soldatov, On the properties of solid and liquid ion exchangers. VII. A simple model for formation of mixed micelles applied to salts of dinonylnaphtalene sulfonic acid, J. Inorg. Nucl. Chem., 41 (1979) 575. 6 Z.I. Kuvaeva, V.S. Soldatov, F. Fredlund and E. HOgfeldt, On the properties of solid and liquid ion exchangers. VI. Application of a simple model to some ion exchange reactions on dinonylnaphtalene sulphonic acid in different solvents and for different cations with heptane as solvent, J. Inorg. Nucl. Chem., 40 (1978) 103. 7 V.S. Soldatov and A.V. Mikulich, Parallel investigation of ion exchange selectivity of methylammonium ions by a H-ion on liquid and polymer sulphonic cation exchanger, J. Phys. Chem. (U.S.S.R.), 53 (1979) 1279. 8 V.S. Soldatov and E. HOgfeldt, On the properties of solid and liquid ion exchangers. I. Selectivity studies on dinonylnaphtalene sulphonic acid dissolved in heptane. Comparison between liquid and solid cation exchangers, Ion Exch. Membr., 2 (1974) 13. 9 V.S. Soldatov and V.A. Bichkova, Experimental investigation of the N H ~ - - N a + - H ÷ ion exchange equilibrium on a sulfostyrene ion exchanger, J. Phys. Chem. (U.S.S.R,), 45 (1971) 1242. 10 V.A. Bichkova and V.S. Soldatov, Ion-exchange equilibrium in the N H ~ - - N a + - H + system on KU-2X25 resin, Izv. Akad. Nauk B.S.S.R., Ser. Khim. Nauk, 1 (1973) 66. 11 V.A. Bichkova and V.S. Soldatov, Mutual influence of ions in three-ion NH~- - I + - H + systems on highly cross-linked sulphostyrene ion exchangers, Izv. Akad. Nauk B.S.S.R., Ser. Khim. Nauk, 2 (1977) 17. 12 E. H6gfeldt, On ion-exchange equilibria. II. Activities of the components in ion exchangers, Ark. Kemi, 5 (1952) 147. 13 A.W. Davidson and W.J. Argersinger, Equilibrium constants of cation exchange processes, Ann. N.Y. Acad. Sci., 57 (1953) 105. 14 G. Gaines and H.C. Thomas, Adsorption studies on clay minerals. II. A formulation of the thermodynamics of exchange adsorption, J. Chem. Phys., 21 (1953) 714. 15 A.M. Tolmachev and V.I. Gorshkov, Some thermodynamics problems of ion exchange, J. Phys. Chem. (U.S.S.R.), 40 (1966) 1924. 16 V.S. Soldatov, On thermodynamics of ion-exchange equilibria. Calculation of thermodynamic quantities of ion-exchange equilibria, J. Phys. Chem. (U.S.S.R.), 46 (1972) 434, 1078. 17 R.M. Barrer and J.D. Falconer, Ion exchange in feldspathoids as a solid-state reaction, Proc. Roy. Soc. Ser. A, 236 (1956) 227.
259 18 A.V. Mikulich and V.S. Soldatov, Comparable studies of ion-exchange selectivity of ethylammonium ions by an H-ion on liquid and polymer sulfonic
cation exchangers, Izv. Akad. Nauk B.S.S.R., Set. Khim. Nauk, 2 (1979) 39.