Kinetic mechanism limiting ion exchange rates in selective systems

Reactive Polymers, 7 (1988) 123-131 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

123

KINETIC M E C H A N I S M L I M I T I N G I O N E X C H A N G E R A T E S IN SELECTIVE S Y S T E M S * A.I. KALINITCHEV, E.V. KOLOTINSKAYA and T.D. SEMENOVSKAYA

Institute of Physical Chemistry, Academy of Sciences of the USSR Moscow, Lenin A venue 31 (U. S. S. R.)

(Received May 27, 1986; accepted in revised form January 26, 1987)

The possibility of identification of the kinetic mechanism is discussed for ion exchange systems in which counterions can exist in the resin phase in two states (free and bound). Selectivity within the resin bead is described by the relation between the free ions in the pore liquid and the bound ions fixed on the matrix. The dependences of the exchange rate in the bead on solution concentration and exchange isotherm shape (selectivity) are obtained for systems including complex forming, weakly dissociating and structurally inhomogeneous ion exchangers. This dependence is formally analogous to that for film-diffusion exchange kinetics. The relation between the mass transfer rate in the ion exchanger phase and in the solution film is analysed by use of experimental results on CI / Ag(S203) ~ and C I - / CNS - exchanges at various solution concentrations and with three types of resins of different porous structure (gel or macroporous).

INTRODUCTION The phenomenological theory of diffusion kinetic process for binary ion exchange, based on the N e r n s t - P l a n c k equation of the ion fluxes has been detailed by Helfferich [1] and Tunitsky and co-workers. [2,3]. Their theoretical considerations are valid for a variety of ion exchange systems including ions of different valences and free-acid and free-base forms of the exchangers, in which the exchange rate is controlled by the interdiffusion of counterions (B and A). However, the above approach based on a

* Paper presented at the 5th Symposium on Ion Exchange, Lake Balaton, Hungary, May 28-31, 1986. 0167-6989/88/$03.50

single diffusion mass transfer equation, where the ion fluxes are described by the N e r n s t Planck equations, becomes invalid for ion exchange involving a chemical reaction between the fixed groups (R) and counterions B and A or these counterions and co-ions (Y). For the theory of ion exchange kinetics in such ion exchangers the fundamental considerations have been put forward by Helfferich [4] who gives eleven typical examples with reactions such as dissociation, neutralization, hydrolysis and complex formation. According to [4] the effect of the chemical reaction on the mass transfer process is likely to lead to an appreciable decrease of the exchange rate (see also ref. [5]) and a change in the type of the dependence of the process on the solution concentration (C0).

,,v)1988 Elsevier Science Publishers B.V.

124 With few exceptions, discussed elsewhere [6] and also mentioned by Helfferich [4], the reaction rate is far higher than that of diffusion, so that the effect of the reaction on the kinetic process becomes evident through the equilibrium reaction parameters [7-14]. The purpose of this paper is to discuss ion exchange kinetics in selective systems using the kinetic model presented in refs. [8,9,11]. It is an attempt to demonstrate that phenomenological regularities and criteria describing intraparticle diffusional kinetics for conventional ion exchange are not applicable for selective systems. Ion exchange processes with chemical reactions have received much attention and, in the last decade, have been studied by many researchers [6,15-17]. Ion exchangers exhibiting exchange processes of this type are widely used for water treatment in the field of heat power engineering, extraction of metals from waste and natural waters [16] as well as in nuclear technology [6].

THEORETICAL The methods of identifying the kinetic mechanism [1], effective in the case of nonselective exchange, are applicable only to a limited extent in selective systems. This conclusion is based on the results of the numerical analysis of the kinetic model in complex° forming ion exchangers obtained elsewhere [9-13]. In the papers [8,9,11,12] the phenomenological approach of thermodynamics of irreversible processes with Nernst-Planck equations for ion fluxes has been applied to the solution of the problem of intraparticle diffusional ion exchange with fast chemical reactions between counterions B and A and the fixed groups R of the ion exchanger. In the model considered, the slow and sole rate controlling step is diffusion within the exchanger particle. The following assumptions and restrictions are put forward: the

exchanger (resin) is regarded as a quasi-homogeneous phase; the exchange diffusion process is assumed to be isothermal; convective transport is neglected; sorption and desorption of solvent as well as swelling and shrinking of the exchanger are not considered; ion fluxes in the exchanger phase are described by the Nernst-Planck relations, and coupling of the fluxes other than by the electric diffusion potential is disregarded; the individual diffusion coefficients (Di) in these relations are assumed to be constant. These assumptions are considered satisfied in the majority of scientific works on ion exchange kinetics [1-8,14-17]. For a system with reactions, as discussed here, additional assumptions are made: the equilibrium dissociation coefficients KRi of the complexes Ri are assumed to be constant; local equilibrium of the association-dissociation reactions is assumed to be maintained at all times at any location in the resin. The last assumption is suitable in almost all cases as the vast majority of ionic reactions are very fast compared with diffusion in ion exchangers. It is believed that in this model [8,9,11,12] two states of counterions B and A can be recognized during the exchange RA + B RB + A in the resin phase: free ions B and A and ions B and A associated with fixed ion exchange groups R. Equilibrium is reached between the ions B and A contained in the pore liquid and the ions fixed on the matrix. Concentrations (C~, ai) of the free and bound ions (C~ and ai respectively) are linked by the dissociation-association equilibria. The dissociation constants KR, of the complexes Ri (i = B, A) are assumed to be independent of the degree of conversion (F). In the case of weakly dissociated components (when K R i / C o << 1) the KRA/KRB factor is responsible for the shape of the exchange isotherms (a i = f ( C A, C B, Cy)) in the resin phase. The exchange isotherm has "convex" shape if the resin is selective for the invading

125 ion B (KRA/KRB > 1), and "concave" if it is selective for the ion A being displaced (KRA/KRB < 1). The consideration of the two states, i.e., "free" and " b o u n d " , B and A ions is consistent with the qualitative "loose quasicrystal" molecular model [18]. In this model the free ions B and A are solely reponsible for mass transfer in the exchanger phase, rather than all counterions as suggested by the classical model [1,2]. Moreover, the equations of the model [8,9,11,12] include the co-ion concentration C v in the exchanger phase, in contrast to the conventional models. The model for complexing ion exchangers, as proposed in earlier work [8,9,11,12], describes a multicomponent system (n = 2). The effect of interference of components B and A on the interdiffusion process in the ion exchanger is allowed for by the interdependence of the fluxes 4 and ~ of the diffusing components i and j. The mathematical formulation of the kinetic process in this case consists of a system of two nonlinear diffusion equations in particular derivatives involving effective self-diffusion coefficients ( D , ) and the cross-interdiffusion coefficients (D~j) which depend on the concentration of the diffusing ions B, A and Y as well as on the equilibrium constants (KRA and KRB) characterizing the stability of the complexes RA and RB in the resin phase. A question of principle is that co-ion invasion into the resin phase is taken into consideration, so the effective diffusion coefficients D,I depend on the concentration Cy. The solution involves three ions (B, A and Y) so that the problem cannot be reduced to a single mass-transfer diffusion equation. A set of nonlinear differential mass balance equations was solved [9-11] by a numerical implicit finite-difference method for equally charged counterions B, A and a wide range of model parameters such as diffusion coefficients DB, D A and D r ; dissocia-

tion constants KR, and solution concentration Co. The problem was confined to exchange between an ion exchanger initially containing ion A as the only counterion and a solution of constant concentration containing ion B as the only counterion ("infinite solution volume" condition). It follows from our computer calculations that the type of concentration profiles of the ions in the resin bead depends on two factors: the diffusivity ratio DA/D B and a selectivity factor KRA/KRB. As distinct on the one hand from nonlinear physical sorption kinetics, for which the concentration profiles in the sorbent bead, are dictated by the isotherm shape and on the other hand from conventional ion exchange kinetics, for which they are dictated by the diffusivity factor (DA/DB) [1], the diffusion process of ion exchange accompanied by association is characterized in the discussed model by the joint action of the two factors: DA/D B and KRA/KRB. It appears that system behaviour is strongly affected by the mobility of the preferred ion (i.e., D R if KRA/KRB > 1, and D A if KRA/ KRB < 1). Furthermore, in contrast to the situation in nonreacting systems, selective sorption of the invading ion is not a sufficient condition for development of sharp profiles in the resin bead, and the concerted action of diffusivity and selectivity factors has to be considered. Formation of a sharp concentration profile is typical if both the ion being displaced is much more mobile (DA>~ DB) and the exchange isotherm in the resin is convex (KRA > KRB ). Comparison of the results of the numerical calculations reveals a very interesting feature of the kinetic process under study, complicated by ion association in the exchanger bead. Kinetic curves F(t) calculated for the convex isotherm and the relation D A >~ D B are described by the equation of diffusion into a spherical bead with constant effective diffusivity Deff. In other words the kinetic

126

curves F(t) coincide with those for Deft = const, if a sharp concentration profile appears in the bead, and do not if the exchange zone in the bead is spread out. It should also be noted that a linear dependence of conversion F o n (f)1/2, characterizing intraparticle exchange kinetics for nonselective processes, can be observed for selective ion exchangers in the case of a convex isotherm only (when KRA/KRB > 1) [9-13]. However even in this case the effective diffusivity depends on the selectivity factor. Theoretical studies confirmed by experiments have shown the interdiffusion rate in the exchanger phase to depend on the external solution concentration (Co) and the exchange isotherm shape (KRA/KRB [9--13]. These dependences are formally similar to those observed in film diffusion kinetics, as follows from a comparison of curves 1 and 2 in Fig. 1. These curves are the calculated exchange rates for the case of a concave isotherm [10,11,13] and for film-diffusion control [1]. In selective systems, the application of the Helfferich-Tunitsky criterion [1,3] may result in erroneous estimates of the contributions of film and intraparticle diffusion. This is because the ion diffusion coefficients in the resin phase that are included in this criterion can undergo very sharp variations with the change of exchange direction. These variations are not taken into account in the theoretical expressions. Moreover the criterion neither includes the influence of the equilibrium parameters characterizing the selectivity in the resin phase nor accounts for co-ion invasion. The effective diffusivity of the kinetic process may be quite different if these factors were taken into consideration. A deviation from the ideal exchange kinetic dependences caused by selectivity action can occur in any ion exchange system in which ions in the resin phase can exist in two different states: i.e., relatively free and bound, as is assumed in our model [8,9,11,12]. This is true

I0 ._o

1.0

f

g ~ 10_1

10 - 2

I 50 Separation factorp

I 100 OC~

Fig. 1. Half exchange time as a function of separation factor aAB for film (1) and intraparticle (2) diffusion control: (1) results of calculation using the relation [1] to.5(ro6C/DC)=O.167+O.O67aA; (2) the results of computer simulation of ion exchange kinetics accompanied by complex formation [10,13].

for complex-forming, weakly dissociating, chemically and structurally inhomogeneous ion exchangers. The theoretical and experimental features of ion exchange kinetics in complex forming and weakly dissociating ion exchangers are discussed elsewhere [8-13] with use of the above-mentioned model. The effect of selectivity on the ion exchange rate is discussed here based on experimental data for structurally inhomogeneous ion exchangers. In this work the experimental results on desorption kinetics of silver thiosulphate complex from anion exchangers with different internal porous structures were analysed. This complex, preferred by the anion exchanger, is displaced in the experiments by sodium cloride solutions of different concentrations (Co). This particular system gives an opportunity to observe the exchange rate over a wide range of exchange isotherm parameters since, in accordance with the electroselectivity condition, the shape of the isotherm for exchange of a trivalent for a monovalent ion depends on the solution concentration. Sorption and desorption of Ag(S203)~--

127

ions have been studied previously [19-21] with the aim of extracting silver from various solutions.

1-0,1 5eC

10 4

k

EXPERIMENTAL Kinetic determinations were performed [22] using a flow cell with a thin layer of ion exchanger. The concentration variation of ions desorbed by the flowing solution was determined from UV detector readings [22]. The ion exchanger was presaturated under static conditions w i t h Ag($203)32 - ions sythesized by stirring of dilute silver nitrate and sodium thiosulphate solutions. The NaC1 desorbate concentration was measured within a range of 0.1-2 N. Sodium chloride solutions are the most convenient as they have low optical density in this concentration range. A noticeable decomposition of the thiosulphate with formation of silver chloride will take place if the chloride ion concentration exceeds 2 N [22]. The linear flow rate through the ion exchanger layer was maintained constant at c, = 5 cm min 1; the ion exchanger sample weight was 0.01-0.005 g. Under these conditions the dimensionless concentration equiv. fraction of Ag(S203) 3- ions in the eluate did not exceed 2 × 10 3 (relative to the total solution concentration). Table 1 shows the specific volume (V) and the volume capacity (C) values for the strong-base resin AV-17 with different porosities. In addition, the C N S - / C 1 - ion exchange rate for equally charged ions was measured by the same technique as described above for TABLE 1 Specific volume (V) and volume capacity (C) for anion exchangers Ion exchanger

p (cm3 g 1)

C" (mg-eq cm 3)

AV-17-8 AV-17-8P AV-17-20P

1.3 2.9 1.8

2.5 1.2 1.1

10 3 "7. 0

10 2 2

8 101

1.0 h

I

I

1.0

1 2.0

M NaCI

Fig. 2. Dependence of desorption rate of different ions with NaC1 concentration variation in eluent: 1, 2. 3--experimental data on desorption of Ag(S203)3 ions from anion exchangers AV-17-8 (1), AV-17-20P (2) and AV-17-8P (3); 4, 5--experimental data on desorption of CNS ions from anion exchangers AV-17-8 (4) and AV-17-20 P (5); 3', 6, 7, 8--results of calculation for film diffusion kinetics in anion exchangers AV-17-8 (6), AV-17-8P (7, 3') and AV-17-20P (8).

the Ag(S203) 3 /C1 exchange. The time (t0.1) for 10% conversion (instead of the widely used half time to.5 value) was used for description of the kinetic rate (Fig. 2) because of the slowness of the process.

RESULTS AND DISCUSSION Figure 2 (curves 1-3) presents the dependence of the desorption rate of Ag(S203) ~ions on the NaCI solution concentration for anion exchangers of the same chemical nature but with a different pore structure. According to Fig. 2 (curves 2 and 3) the dependence of to., on the solution concentration (NaC1) is stronger for resin with higher water content. This conclusion becomes evident from the

128

comparison of the kinetic curves (Fig. 2, curves 2, 3) for anion exchangers AV-17-8P and AV-17-20P, which have the same volume capacity (C) but different water contents (Table 1). The augmentation of time-concentration dependence caused by electrolyte penetration into the pores of exchanger has been repeatedly noted not only for macroporous but also for gel ion exchangers [23,24]. Moreover the experiments discussed are characterized by the fact that for the same concentration range (Fig. 2) the rate variation is wider for exchange of ions of different charges (curves 1-3) than for that of equally charged ions (Fig. 2, curves 4, 5). The separation factor strongly increases with decreasing solution concentration in displacement of trivalent by monovalent ions. (Table 2). Thus, the rate variations (Fig. 2) are associated not only with the concentration factor ( C / C o ) , but also with the separation factor (aA). While transport of the silver thiosulfate complex anion in the bead is conclusively established as predominantly rate-controlling, the results on C1-/Ag($203)32 - exchange do not entirely exclude a contribution from film diffusion. Curves 6-8 in Fig. 2 represent the theoretical rates calculated with the equation for film diffusion kinetics [1]

TABLE 2 Dependence of separation factor on input concentration

Co Separation factor a~

(IzAI =3, IzBI =1, K~'=3 C'= 1.2 mg-eq, cm -3)

104

CO, g - e q . l - a

0.1

49

13

0.2

3

1

0.5 1.0 2.0

one of the anion exchangers (AV-17-8P) with only the assumption that all three AV-17 samples have the same selectivity (a~) for Ag(S203) 3- ions. It follows from the calculations with eqn. (1) that the film-controlled exchange rate (Fig. 2, curves 6-8) is considerably higher than that observed in the experiments with all of the ion exchanger samples.

O.B 1

0.6

ln(1 - F ) + (1 - 1 / a A ) F 0.4

= - 3DCt/(roSC~t # )

(1)

The following values were used in the calculations: D = 0 . 5 × 1 0 -5 cm 2 s -1 ro = 0 . 0 4 cm, v=5 cm min -a, 8 = 0 . 2 r o / ( 1 - 7 0 r 0 v ) = 0.007. The separation factor is assumed to vary in accordance with the expression [1]

I

2 t x

~

0.21

3

.

(2) O

~ 500

Table 2 presents the separation factors calculated with eqn. (2) for C A / C o = 0.01. The equilibrium coefficient K# was determined from independent experiments. The equilibrium characteristics were calculated for

1000 Time, s

Fig. 3. Experimental kinetic curves for desorption of Ag(S203) 3- ions from AV-17-8P at different NaC1 solution concentrations 2 N (1), 0.5 N (2) and 0.1 N (3).

129

In addition, an attempt was made to obtain information on the role of film diffusion analysing the shape of the initial part of the kinetic curves (Fig. 3) according to the conventional approach [1]. Unfortunately, theoretical kinetic curves are not available for the complex ion exchange system considered with ZA 4= ZB' ~A 4= 1, D A :¢: D B. The reason is that the programming of the finite-difference scheme for the model [8,9,11,12] is essentially hindered because an explicit mathematical formulation of the isotherm dependence a, = f , ( C A, CB) is impossible for exchange of ions of different valences. However, it is possible to demonstrate how the kinetic curves will change (Fig. 4) with the change of the exchange mechanism and selectivity by using relations that are applicable for equally charged counterions with equal diffusivities. Two possible cases may be assumed: either intraparticle diffusion over the whole concentration range, or film diffusion for low input concentration and transition to intraparticle diffusion for high input concentration. The auA value varies from 100 (at Co = 0.1 N) to 1 (at C0 = 2N). It follows from Fig. 4 that in both cases the hypothetical deviations of the kinetic curves are so small, especially for the initial state, that for the system of interest the identification of the rate-controlling step by use of the shape of the initial

1

2

©.4

0,2

1 O

--

i ~,

2 1'O

1'5

I

Dt/4 Dtc/%6?

Fig. 4. Comparison of shape of theoretical curves for film (1) [1] and intraparticle (2) [10,13] diffusion kinetics for separation factor aA = 100.

part of the kinetic curve may not be reliable. Besides, the experimental kinetic curves are slightly distorted because of a peculiarity of the experimental technique. * The discussion of the experimentally observed kinetic dependences has led to the conclusion that the exchange rate is essentially controlled by intraparticle diffusion. According to the model [8,9,11,12] the influence of selectivity on the exchange rate is due to the presence of ions in "free" and " b o u n d " states in the resin bead. Selectivity inside the bead is described by the relation between the free ions in the pore liquid and ions fixed on the matrix (KRA/KRu factor) [8-13]. If local equilibrium is established instantaneously, the ionic composition of parallel diffusion fluxes in the bead pores (i.e., free ions) and gel phase of the bead (i.e., bound ions) is determined in particular by the selectivity factor - - the shape of the exchange isotherm. If the mobility of the ions in the pores is much higher than that in the gel portions of the resin bead, then, in terms of the "loose

* This shape distortion is due to post flow cell desorbate dispersion. Fig. 4 presents the experimental dispersion of thiosulphate complex microquantity in the device when the flow cell does not contain an ion exchanger layer. The initial part of the corresponding kinetic curve (2, Fig. 5) is concave. The experimental kinetic curve that allowed for post flow-cell dispersion is not as steep as the imaginary theoretical kinetic curve for desorption without allowance for this dispersion. As a result, desorbate was initially registered in the experiment 15 s earlier than it would be if this dispersion was not a factor. The error in the time determination of desorbate appearing may be neglected for slow processes at low eluent concentration. This error at high eluent concentration was evaluated and corresponding corrections were made. However, as mentioned above, t[~e shape of the kinetic curve is distorted. Therefore, no definite conclusion about the rate-determining step using the shape of the initial part of the kinetic curve is possible. That is why the conclusion about prevalence of intraparticle diffusion in the kinetic process is made as a result of the discussion presented above.

130

1.0

1.O

._o

o~ 0

O 1

2

Time, rnin

Fig. 5. Dispersion of microquantity of thiosulphate complex: 1 change of Ag(S203)~- ions concentrations in eluate at 5 cm min -1 flow velocity; 2 dispersion kinetic curve. quasi-crystal" hypothesis, the kinetic system is analogous to ion exchange accompanied by association of counterions with fixed exchange groups. Therefore it is believed that the kinetic regularities for macroporous ion exchangers are similar to those for weakly dissociating or complex-forming exchangers. It is still impossible to make a quantitative comparison of the experimental and calculated kinetic dependences for macroporous exchangers because of the absence of sufficiently accurate results for equilibrium and kinetic parameters that are necessary for computer calculations. However, it follows from the presented results that the kinetic dependences (i.e., concentration and selectivity effects) are analogous in complex forming and macroporous ion exchangers. The electrolyte invading the resin bead can also be regarded as a "stagnant mobile phase" (a term from the field of liquid chromatography). It is important in practice that the exchange process can be strongly retarded in macroporous ion exchangers in the case of a concave isotherm because of the absence of automatic removal of the displaced ions (by convective flow) from the "stagnant phase" of the resin bead.

CONCLUSION The analysis of experimental results on C1-/Ag(S203) ~- and C 1 - / C N S - exchanges

at various solution concentrations and with three gel-type resins of different pore structures has led to the conclusion that the exchange rate is essentially controlled by intraparticle but not film diffusion. For selective systems it has been shown that the effect of isotherm shape (selectivity factor) on the exchange rate as well as on the development of the concentration profiles may be interpreted within the framework of the model proposed elsewhere [8,9,11,12] for complex-forming ion exchangers. The combination of ion binding in the resin and co-ions invasion produces effects that are expected to be analogous for complex-forming and macroporous ion exchangers.

REFERENCES 1 F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962. 20.P. Fedoseeva, E.P. Cherneva and N.N. Tunitsky, The Research of Kinetics of Ion Exchange. II, III, Zh. Phisich. Khim., 33 (1959) 936, 1140. (in Russian). 3 B.P. Nikolsky and P.G. Romankov (Eds.), Ionites in Chemical Technology, Chimia, Moscow, 1983 (in Russian). 4 F. Helfferich, Ion exchange kinetics. V. Ion exchange accompanied by reactions, J. Phys. Chem., 69 (1965), 1178. 5 A. Schwarz, J.A. Marinsky and K.S. Spiegler, Self exchange measurements in chelating ion exchange resins, J. Phys. Chem., 68 (4) (1964) 916. 6 M.K. Streat, Kinetics of Slow Diffusion Species in Ion Exchangers, Reactive Polymers, 2 (1984) 79. 7 U.S. Ilnitskii. Diffusion ion fluxes in ion exchange processes with formation of weak-dissociating compounds in Solution phase, Zh. Phisich. Khim., 50 (1976) 2132 (in Russian). 8 A.I. Kalinitchev, T.D. Semenovskaya, E.V. Kolotinskaya, A.Ya. Pronin, and K.V. Chmutov, investigation into the kinetics of ion exchange processes accompanied by complex formation, J. Inorg. Nucl. Chem., 43 (1981) 787. 9 K.V. Chmutov, A.I. Kalinitchev, and T.D. Semenovskaya, an approximate solution of equations of ion exchange kinetics in complex forming ion exchangers, Dokl. Akad. Nauk SSSR, 239 (1978) 650 (in Russian).

131 10 T.D. Semenovskaya, A.I. Kalinitchev and E.V. Kolotinskaya, The rate of ion interdiffusion in complex-forming ionites for various convex and concave isotherms, in: M. Streat, and D. Naden (Eds.), Ion Exchange Technology, Ellis Horwood, Chichester, 1984, p. 257-265. 11 A.I. Kalinitchev, E.V. Kolotinskaya and T.D. Semenovskaya, ion exchange kinetics accompanied by complex formation in the Ion Exchanger, Theor. Osn. Khim. Technol., 17 (1983) 313 (in Russian). 12 A.I. Kalinitchev, E.V. Kolotinskaya and T.D. Semenovskaya, computerized analyses of the diffusion processes in complexing ionites, J. Chromatogr., 243 (1982) 17. 13 A.I. Kalinitchev, E.V. Kolotinskaya and T.D. Semenovskaya, analyses of diffusion processes in complex forming ion exchangers., I, II, Zh. Phisich. Khim., 58 (1984) 2807, 2811 (in Russian). 14 Y.L. Hwang, and F. Helfferich, Generalized model for multispecies ion exchange kinetics including fast reversible reaction, Reactive Polymers, 5 (1987) 237. 15 L. Liberti and F. Helfferich, (Eds.), Mass Transfer and Kinetics of Ion Exchange (Hague: Martinus Nijhoff, Alphen aan den Rhijn, 1983). 16 F. Helfferich, Ion Exchange Kinetics-Evolution of a Theory, ibid., pp. 157-180. 17 F. Helfferich, L. Liberti, D. Petruzzelli, and R. Passino, Anion Exchange Kinetics in Resins of High Selectivity, Isr. J. Chem., 26 (1985) 1.

18 N.I. Nikolaev, (Ed.), Diffusion Process in Ionites, NIITEChIM, Moscow, 1973, (in Russian). 19 A.E. Agadzhanyan, K.A. Ter-Arakelyan and G.G. Babayan, Some regularities of silver thiosulphate complex sorption on anion exchanger AM-2B, Arm. Khim. Zh., 35 (1982) 151. 20 B.N. Laskorin, G.I. Sakovnikov, Ya.P. Novikov, A.P. Lebedeva, and A.N. Eryomkina, Au and accompanying admixtures desorption from anion exchanger with porous structure, Zh. Prikl. Khim., 47 (1974) 254 (in Russian). 21 V.A. Davankov, V.M. Laufer, and k.B. Zubakova, On the question of silver thiosulphate complex ions elution after their sorption. In: Theory and Practice of Sorption Processes, 15, VGU, Voronezh, 1982, p. 180 (in Russian). 22 A.I. Kalinitchev, E.V. Kolotinskaya, T.D. Semenovskaya, and V.T. Avgul, relationship between the mass-exchange rate in an ion exchanger and in solution for selective systems in the case of a concave exchange isotherm, J. Chromatog., 364 (1986) 119. 23 D. Richman, and H. Thomas, Self-diffusion of sodium ion in a cation-exchange resin, J. Phys. Chem., 60 (1956) 237. 24 N.I. Nikolaev, M.D. Kalinina, and G.G. Tchuveleva, concentration action of electrolyte solution on counterions diffusion in cation exchanger, Zh. Phisich. Khim., 44 (1970) 3110 (in Russian).