1178
F. HELFFERICH
Ion-Exchange Kinetics.
V.
Ion Exchange Accompanied by Reactions
by F. Helfferich Shell Development Company, Emerwille, California (Receiaed September 29, 1966)
A theoretical analysis of various ion-exchange processes involving ionic reactions such as neutralization and coniplex formation is presented. The derived rate laws differ appreciably from those for ordinary ion exchange in the absence of reactions. Co-ion diffusion can affect the rate or even be the sole rate-controlling step. Although still much in need of verification by systematic experimental studies, the proposed rate laws are in agreement with earlier experimental observations by other authors.
Introduction I t has long been known that ion-exchange rates are controlled by diffusion.’I2 Quantitative theories based 011 diffusion e q ~ a t i o n s l -are ~ now well confirmed for simple systems. All previous theories, however, are based on the premise that the exchanging ions retain their identity, that is, are not consumed by accompanying chemical reactions. Obviously, this underlying assumption is not admissible if the ions, in the course of ion exchange, undergo neutralization, association, or complex-formation reactions. The purpose of this communication is to investigate, from a theoretical viewpoint, the effects of accompanying reactions on the kinetics of ion exchange. 1v2,5-13
Types of Reactions Eleven characteristic examples of ion-exchange processes involving reactions of the ions are compiled in Table I. Chemically, the reactions would best be classified as neutralizations, hydrolyses, and complex formations. For a quantitative ‘treatment, however, it is more convenient to classify the processes by types as follows. In processes of type I, the counterions released by the ion exchanger are consumed by reactions with the co-ions from the solution (reactions 1 to 4 in Table I). In those of type 11, the counterions originating from the solution are consumed by reactions with the fixed ionic groups of the ion exchanger (reactions 5 to 7). In those of type 111, undissociated fixed ionogenic groups of the ion exchanger are ionized by reactions with the co-ions from the solution (reactions 8 to 10). In those of type IV, fixed ionogenic groups are converted from one undisThe Journal of Physical Chemistry
sociated form to another by reactions with the counterions from the solution (reaction 11).
Theory The theoretical treatment will be based on the following siniplifying assumptions commonly used in theories of ion-exchange kinetics. It is assumed that the systems are isothermal, that the ion-exchanger particles are spherical, uniform, and of equal size, and that the individual diffusion coefficients of the ions are constant. Furthermore, swelling changes of the ion exchanger, activity-coefficient gradients, and coupling effects other than by electric fields are neglected. Also, the treatment remains restricted to systems in which the rates of the chemical reactions are fast enough not to limit the exchange rates. Of course, equations based on these assumptions can only represent ideal limiting rate laws. (1) G. E. Boyd, A. W. Adamson, and L. S.Myers, Jr., J . A m . Chem. SOC.,69,2836 (1947). (2) F. Helfferich, “Ion Exchange,” McGraw-Hill Book Co., Inc., New York, N . Y., 1962, Chapter 6 (also gives references to earlier work). (3) R. Schlogl and F. Helfferich, J . Chem. Phys., 26,5 (1957). (4) F. Helfferich and M. S.Plesset, ibid., 28,418 (1958). (5) F. Helfferich, J . Phys. Chem., 66, 39 (1962); 67, 1157 (1963). (6) J. C. W. Kuo and M. M . David, A.1.Ch.E. J . , 9, 365 (1963). (7) B . Hering and H. Bliss, ibid., 9, 495 (1963). (8) C . Heitner-Wirguin and G. Markovits, J . Phys. Chem., 67,2263 (1963).
(9) M. Gopala Rao and M.M. David, A.I.Ch.E. J . , 10, 213 (1964). (10) A. Schwarz, J. A. Marinsky, and K. S.Spiegler, J. Phys. Chem., 68,918 (1964). (11) R. H . Doremus, ibid., 68,2212 (1964). (12) A. S’aron and W. Rieman, 111, ibid., 68,2716 (1964). (13) T . G. Smith and J. S.Dranoff, Ind. Eng. Chem. Fundamentaid, 3, 195 (1964).
ION-EXCHANGE KINETICS
1179
Table I : Typical Ion-Exchange Processes Involving Reactions" Type of process
I
I1 I11 IV
Reactants------------.
Reaction
Resin
1 2 3 4 5 6 7 8 9 10 11
-SO3H+ -N(CH3)3+ OH-SO,H+ 4(-so3-) 2Ni2+ -COONa+ -NHZ+ C12Na+ -N(CHzC00)22-COOH -NHz 2 [-N( CHzCOO)zNi] -K(CHQCOO)zNi
+ + + + + +
------p~odu~t~-~-------Resin
Solution
+
+ N a + + OH+ H + + C1+ N a + + AcO+ 4Na+ + EDTAP+ H + + C1+ Na++OH+ Ni2+ + 2C1+ Na++OH+ H + + C1+ 4Na+ + EDTA4+ 2 H + + 2'21-
Solution
+
+ HzO + Hd
+ + + +
-SOsNa+ -N(CHa)3+ C1.+ -SO,Na+ + 4(-SOS-) 4Na+ + -COOH -NHz . + -N(CHzCOO)2Ni -COONa+ -+ -NHs+ C1+ 2[-N(CHzC00)zz-] + -N( CHZCOOH)~ .+
+ AcOH + NilEDTA + N a + + Cl+ N a + + C1- + H 2 0 + 2Xa+ + 2C1-
-C
. +
+ +
+ HzO
+ 4Na+ + Ni2EIJTA + Nil+ + 2C1-
a From F. Helfferich in "Advances in Ion Exchange," Vol. I, J. A. Marinsky, Ed., Marcel Dekker, Inc., New York, published. Reproduced with permission of Marrel Dekker, Inc.
I n particular, 1 he assumptions of isothermal behavior and absence of swelling changes are less justified here than in ordinary ion exchange, since the ionic reactions niay release considerable heat and may make the ion exchanger swell or shrink appreciably. Moreover, although most reactions of ions and particularly neutralization and hydrolysis are fast, reactions of certain complexes are slow and may become rate-controlling. To systems involving such slow reactions the theoretical treatment is not applicable. For simplicity, all rate laws will be derived for systems with the ion exchanger initially containing counterions of species A only and solutions initially containing counterions of species B only. Extensions to other initial conditions-provided the initial distribution within the ion exchanger is uniform-can in most cases be achieved without difficulties. Type I : Consumption of the Counterion Released by the I o n Exchanger. Processes of type I can be written in general form
A
+ B + Y .-+
B
+ AY
As in ordinary ion exchange, the rate can be controlled by diffusion within the particle or in an adherent Nernst "film. "1, 'l With particle-diflusion contl"olJ the reaction consuming A does not interfere with interdiffusion in the ion exchanger because the co-ion Y is efficiently excluded from the interior. The reaction does affect, however, the boundary condition at the particle surface. Here, A is eliminated as soon as it is released by the ion exchanger, at least as long as the solution still contains Y. The boundary condition thus is
N.Y., to be
even for exchanges with solutions of limited volume, in which the concentration of A at the interface would otherwise build up. The numerical solutions tabulated for ordinary ion exchange with the simple constant boundary condition thus apply even if the solution volume is not large compared to the amount of ion exchanger. Provided that CV > i.e., that the amount of B in the solution ie sufficient to convert the ion exchanger completely to the B form, fractional attainment F of equilibrium for any solution volume is therefore given by (l)2*7314*3b
m,
F(t)
=
G(t) (CV
>W)
where G ( t ) is the tabulated fractional attainment of equilibrium for the corresponding ordinary ion exchange with infinite solution volume. If, in a solution of limited volume, the amount of B (and Y) is smaller than the amount of A in the ion exchanger ( C V < W )then, , of course, the ion exchanger cannot be completely converted to the B form. Conversion then proceeds with a finite rate until all B and Y is used up, and equilibrium is attained in a finite time t,. Since the boundary condition (1) remains valid to the end, the function G(1) still describes fractional conversion of the ion exchanger, but equilibrium is attained when
Accordingly, one has for fractional attainment of equilibrium (14) (a) M. S. Plesset, F. Helfferich, and J. N. Franklin, J. Chem. Phys., 29, 1064 (1958); (b) F. Helfferich, ibid., 38, 1688 (1963).
Volume 60.22'umber 4
April 1066
F. HELFFEHICH
1180
0 5 t 5 t,, F(t)
=
CT’ - G(t)
cv
(CV
2 to, F ( t )
t
=
< cv)
(4)
1
The time t, for complete exhaustion of the solution is readily found as the value of t for which the tabulated function G ( t )reaches the value stated in eq. 3. With jilm-diflusion control, the reaction consuming A interferes deeply with the mechanism of ion exchange, as may be illustrated with the neutralization reaction 1 in Table I. Electroneutrality in the film requires that CH
+
CiXa = C O H
purposes, the latter concentrations thus are negligible compared to the bulk-solution concentrations. With the usual assumptions of quasi-stationary film diffusion and negligible film curvature1a2 one then obtains the following rate laws by integration of eq. 9: for constant solution concentration (CV >> cT)
0
for CV
(5)
< t,,
However, C H and COH are also linked by the dissociation equilibrium
CHCOH= KH%O
mole2/1.2
t,
(7) and for CV
It follows from the conditions (6) and (7) that, in the film CH
5 10-7~
J X =~ JOH = -D grad CN&
(9)
where D is the “Sernst” diffusion coefficient ~DN~DoH = constant15 DOH
+
(10)
The boundary condition at the particle surface is also particularly simple. Since the ratio C,,/CH at the particle surface will not greatly exceed the ratio CNJCH in the ion exchanger (unless the latter has an unusually high preference for H+), eq. 5 and 7 limit C N and ~ COH A4 until at the particle surface to below about conversion is almost complete. For all practical The Journal of Physical Chemistry
cv
roSV
= __
cv
3DP In CV -
CTI
< F(t) = 1 - exp( -
(8)
Accordingly, as long as the bulk-solution concentration of XaOH is well above lo-’ M , H + cannot make headway into the film but is consumed right at the particle surface. The physical process occurring thus is diffusion of S a + and OH- across the film to the particle surface, where Ka+ continues into the ion exchanger while OH- reacts with H + from the ion exchanger to form H20. In general terms, the ratecontrolling step is film diffusion of the counterion and coion from the solution, rather than interdiffusion of the two counterions as in ordinary ion e ~ c h a n g e . ~ Since, throughout the film, CH is negligible compared to C N a and COH,film diffusion obeys the well-known equation for diffusion of a binary electrolyte
D =
F(t) =
(6)
COH
~
> CV
0 5 t
_
