Cmmmmiem®mm ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ c ~ c ~ c ~ Pergamon Chemical Engineerinq Scieme, Vol. 52, No, 14, pp. 2439 2442, 1997 c 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0009-2509(97)00047-X 00o9 2509/97 $1700 + 0.00 Dynamics of ion-exchange involving multivalent cations (Received 1 February 1996) Whenever a cation-exchange process involves multivalent counterions, the observed rates of diffusion within ion-ex- change resin particle are significantly lower than when both the counterions are monovalent. The reason for this differ- ence lies in the fact that the multivalent ions adsorb more strongly on the fixed sites of the resin. Hence in their pres- ence, the extent of dissociation of the fixed sites is reduced. Consequently, the pore walls are weakly charged and the electrical double layer contains a fewer number of the bal- ancing counterions. In our previous work on ion-exchange involving monovalent ions (Hasnat and Juvekar, 1996), we had shown that the diffusive flux in the pore of a resin particle is proportional to the concentration of counterions in the double layer. This explains why the rates of diffusion are lower in the presence of multivalent cations. In the present work we have developed and experi- mentally verified a modified version of our earlier model (Hasnat and Juvekar, 1996), which is now applied to quantify adsorption characteristics of multivalent ions. MODEL Consider a strong acid ion-exchange resin having m0no- valent fixed sites ~. Consider N particles of the resin in A-f0rm (i.e. as ~zAA) placed in a limited volume V of an aqueous solution of electrolyte BzxXzB. The i0n-exchange reaction can be written as ZAB + ZB~z,A ~ ZA~zBB + ZBA. (1) Solution Resin Resin Solution The total concentration of the fixed sites at any location in a resin particle equals the resin capacity, i.e. Q = ZA[~zAA] + ZB[gtztB ] + [~R,] (2) where ['.~] is the concentration of the ion-exchange sites in dissociated form. These sites impart electrical charge to the pore wall. We define, f,, the fraction of the total fixed sites in dissociated form as f, = [R~]/Q. (3) The value of f~ is dependent on the extent of adsorption of the counterions on the fixed sites. A multivalent cation adsorbs strongly because it binds with multiple fixed sites. We therefore expect f~ to be smaller in the presence of multivalent cations. We also expect, in view of high selectiv- ity of resin for multivalent cation, that the multivalent cation solely determines the value off,, even when its concentration in the solution is low. We can therefore treat f, as a charac- teristic constant of multivalent cations. We divide the space inside a pore into a central core region and an annular double-layer region (see Fig. 1). The area averaged continuity equation for ion i in the pore is written as c'~t? ~1 ~ 2 ~'tt][~z~i], <[q>=- ~[r <J,>]- i = A, B and X. (4) The area averaged concentration <[i]> and the flux (J~) are expressed as lI fl ] ([i]> = ~(Rp - 6d) [i]c + 27t _~ [i]¢g'd¢ (5) 2 c <Ji) = rc(Rp - ,Se) Ji + 2r~ J~(~¢d~" . (6) R p ~d In the above equations, the first term in the square bracket represents the contribution from the core region and the second, that from the double-layer region. The concentrations [i]¢ and [i]e can be related by the Boltzmann equation. Since the surface charge is low, we can linearize the Boltzmann equation and write [i]c (1 Z,3 d \ [i]¢= \ where ~b d is the electrical potential in the double layer and can be expressed by the Poisson equation: 1 a F a¢l "~ ~ ~zi[i]¢ (8) with boundary conditions: ¢ = R r - ha: i;~b--~d = 0 [9) a¢ ~b d 3Rp[~.] ( = Rp: t 10) ?¢ 2e, Co Combining eqs (7) and (8) and simplifying the resulting equation using electroneutrality condition in the core region, i.e. ~Z~[i]c=0 where i=A,B,X (11) i we get 1 a F aqS"7 ~2¢ Area integration of eq. (12) over the double-layer region yields fi " RTRZp[~] p_~ ~badd¢ = - 2~iZ2[i]c (13) in view of boundary conditions (9) and (10}. 2439