Ind. Eng. Chem. Res. 1996,34, 2584-2587
2584
Calculation of the Helfferich Number To Identify the Rate-Controlling Step of Ion Exchange for a Batch Process K.Bunzl GSF-Forschungszentrumfur Umwelt und Gesundheit, Neuherberg, Institut fur Strahlenschutz, 85758 Oberschleissheim, Federal Republic of Germany
The Helfferich number He is used frequently as a valuable criterion to decide whether for a n ion exchange process film diffusion or particle diffusion of the ions is the rate-determining step. The corresponding equation given by Helfferich is restricted, however, for the boundary condition of a n infinite solution volume. In the present paper, the Helfferich number is calculated also for a finite solution volume, i.e., for a typical batch process. Because the resulting equation can be solved only numerically, the results are presented in graphical form. It is also examined for which batch processes the conventional Helfferich number already yields a conservative and thus a very simple and useful estimate of the rate-determining step.
Introduction Information on the kinetics of ion exchange reactions is required not only for the economic employment of synthetic ion exchangers in the industry and the laboratory but also for a better understanding of these processes in natural systems, as, e.g., the sorption of nutrient and toxic ions by the soil. For ordinary ion exchange processes in solution two mechanisms can be rate determining (Helfferich, 1962,1966,1983; Helfferich and Hwang, 1991): (i) interdiffusion of the counterions within the ion exchanger particle itself (particle diffusion); (ii)interdiffusion of the counterions across a hydrostatic boundary layer (Nernst film) which surrounds the particles even in well-stirred solutions (film diffusion). The observed overall rate of ion exchange will be determined by the slower of these two processes. Important quantities which will affect these rates are, at a given temperature, the concentration of the counterions in solution, the diffusion coefficients of the ions in the particles and in the aqueous phase of the film, the rate of agitation or flow, the particle size, the capacity of the ion exchanger, the solution volume, and the selectivity of the ion exchanger. Various rate laws which describe the kinetics of ion exchange for film and particle diffusion controlled processes are available (see, e.g., Helfferich, 1962; Brooke and Rees, 1968, 1969; Sparks, 1988; Bunzl and Dickel, 1969; Bunzl, 1974, 1991, 1993; Helfferich and Hwang, 1991). Before they can be applied, however, it is necessary to know which of these two processes is for a given ion exchanger material and for given experimental conditions actually the slower and thus the ratedetermining process. For this purpose, Helfferich (1962) derived first a very valuable criterion by calculating the quantity H e , the Helfferich number, which is defined as the ratio of the half-times of a film diffusion controlled ion exchange process t o the corresponding one of a particle diffusion controlled process. Provided the two counterions A and B exhibited equal mobility, no reactions were involved, the solution volume was infinite, and the ion exchanger was completely converted from the A-form to the B-form, he obtained
where 6 is the thickness of the Nernst film (cm),-c the concentration of the solution (mequiv mL-l), C the
capacity of the ion exchanger (i.e., the concentration of fixed ionic groups (mequiv mL-l)), D the interdiffusion coefficient of the ions in the ion exchanger particle (cm2 s-l),D the interdiffusion coefficient of the ions in solution (cm2 s-l), r the radius of the spherical particles (cm), and a$ the separation factor for the two counterions exchanged, defined as
where XA = CPJC and XB = C$C are the equivalent fractions of the ions A and B in the ion exchanger, XA and XB the corresponding values in the solution, and the subscript 00 denotes the equilibrium state (t -1. From the above definition it then follows that for
-
He > 1
film diffusion is rate controlling
If He 1, both film and particle diffusion will affect the rate of ion exchange. The above-mentioned boundary condition of an infinite solution volume, (Le., where the composition of the solution remains constant and free of the ions A released by the ion exchanger) can be realized experimentally by the shallow-bedtechnique. There, a thin layer of ion exchanger particles in the A-form is placed on a fine screen and the solution containing the ions B is forced through this layer a t a high flow rate. As emphasized by Helfferich, eq 1 should strictly be applied only under the boundary conditions mentioned. For the frequently used batch technique, the infinite solution volume condition is, of course, not fullfilled. There, a given amount of ion exchanger in the A-form is mixed with a given volume of solution containing the ions B at a given initial concentration. Agitation is usually provided by stirring. To obtain some information on the effects of the various kinetic parameters of a batch system (e.g., solution concentration, diffusion coefficients, etc.), it is, of course, again important to know whether film or particle diffusion is rate determining. The purpose of the present investigation was, therefore, to calculate the Helfferich number for batch systems, and to check to which extent the values given
0888-5885/95/2634-2584~~9.00/0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2585 for He by eq 1 can be used as an approximation. Because the rate equations for finite solution systems are rather complicated and cannot be solved analytically for the half-time, an analytical solution for the corresponding Helfferich number analogous to eq 1 cannot be derived. The equations can be solved only numerically, and the results are consequently presented in graphical form.
Theory To facilitate a comparison of the Helfferich number for a finite and an infinite solution volume (as given by eq 11, we will use the same initial condition as selected by Helfferich (19621, namely, all ions A are initially in the ion exchanger at a uniform concentration and in the solution only ions B but no A are present. In addition, we consider, in analogy to Helfferich, isotopic exchange reactions for the interdiffusion process within the particle, and the exchange of ions of equal mobilities for the film diffusion process. The corresponding rate equation for a particle diffusion controlled process under finite solution volume conditions is given as (Helfferich, 1962) 2
U(t)= 1 - 3w n
-
2 is related to l/a%.)
B = -aBA(c
~
E = 1- a B A S = [(B2/4)- GAll”
--
The equilibrium ionic composition of the ion exchanger a t t is obtained from eq 2 and by observing the material balance (i.e., all ions leaving the ion exchanger have t o appear in the solution) as
XB,- = { [ b 2+ MLI1/’ - b}/(2M)
(6)
where
b = aBA (c
+ QN,
M = Q(l - aBA)N
= l ~ +Sn2/9w(w
+
L = 4aB,c 6 is consistent also with eq 5 for the limit t -Equation Equation is identical to the rate equation given for 00.
5 ions of equal mobilities by Helfferich (eq 6-48 in Helfferich (1962)). The half-time t l / z , ~ iis l ~obtained by calculating first from eq 6 XB,-, and substituting then in eq 5 XB= 0.5X~,,,.Finally, from eqs 4 and 5 the Helfferich number is calculated as Hebakh = tl/2,Film/tl/2,Part.,which can be written in the form
Hebakh= [(VdD)/(VrD)Nw,aBA)
+
where a and are the roots of the equation x2 3wx 3w = 0. Because neither eq 3 nor eq 4 can be solved analytically for the half-time t1/2,part, of ion exchange, i.e., t at U = 0.5, they have to be solved numerically. The rate equation of a film diffusion controlled ion exchange process has been derived by Bund and Dickel (1969). The solution of this rate equation for a finite solution volume can be found in Bund and Schimmack (1991)for the general case of ions of different mobilities. For the case considered here, i.e., for ions with equal mobilities and for the above initial conditions this equation becomes
E Rt = -1n 2A
mB2+~XB+G
G
AaAB- EBl2
2As
In
+
[s- (Bl2) - A x B ] [ s + (B/2)] (5) [S - (B/2)1[S + (B/2) + A x B ]
where the rate coefiicient R = 3D/(rCd). Q is the total amocnJ of ion exchanger (in mequiv) in the system i.e., Q = CV, and a B A is again the separation factor. If ions B are preferred by the ion exchanger with respect to ions A, then a B A > 1;if ions A are preferred with respect to B, then a B A < 1. (Note that a% as used in eqs 1and
=
aBA
+ QN,
G=caBA
(3)
wher_e-U(t)is th_e fractional attainment of equilibrium, w = CV/cV, T = D t P , and the quantities S, are the roots of the equation S, cot Sn = 1 Sn2/_3w; t is the time (s), Vis the solution volume (mL), and Vis the total volume of ion exchanger material (mL). The other quantities have been given above. For 0 < z < 0.1, and thus for a range which extends well beyond the half-time of exchange (Helfferich, 19621, Paterson (1947)gives as an approximation which converges rather quickly:
used in eq 5 by the relation
A = (aBA - 1)QN
exp(-Sn2z)
+ 1)
aBA
(7)
where flw,aBA) cannot, however, be given as an analytical expression.
Results and Discussion According to_eq 7, the calculated Helfferich numbers fkb,t,h(VrD)/(VdD) are plotted in Figure 1as a function of w (i.e., the total amount Q = CV of ion exchanger with respect t o the total amount, Vc, of ions in the solution phase). The separation factor a B A was selected as parameter. For comparison we show in Figure 1also the resulting values of the conventional Helfferich number, He, as obtained as an approximation under the infinite volume condition according to eq 1and marked in the-following with the subscript inf. The quantity Hei,f.(VrD)l(VdD),as plotted in Figure 1, is according to eq 1simply given by w[5 2a$l or by w[5 2/a:~l). For the ion exchange capacity we used the value C = 2.8 mequiv mL-l. For the separation factor we selected a B A = 10 (Figure la), a B A = 2 (Figure lb), a B A = 1 (Figure IC),and a B A = 0.1 (Figure Id). Because the denominator of eq 6 approaches infinity for a B A 1, the curve in Figure ICwas calculated by putting a B A = 1.001. The resulting curves as shown in Figure 1reveal that for a B A > 1, i.e., if the ions B taken up by the ion exchanger are preferred with respect to A, Hebatch is always larger than Heinf.. The calculations also showed
+
+
-
2686 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
bl finite
I=
P 5
u
'r. 10 X
a2 I
/
owox.
Figure 1. Helfferich nu_mberfor batch processes, +-&h, and conventional Helfferich number under the infinite volume condition,Heinf., plotted as He(VrD)/(VGD),as a function of w = CV/cV = Q/cV. Parameter is the separation factor aBn,as given in the figure.
that if a B A < 0.55, this is not always observed (see Figure Id, where a B A = 0.1). In any case, however, for very large values of V, i.e., for very small values of w , Hebatchapproaches, as expected, Heinf.. To demonstrate to which extent the Helfferich number, Heinf.,as obtained under the condition of an infinite solution volume, can be used as an approximation for the Helfferich number Hebatch of a batch experiment with a finite solution volume, we will conside_rthe following example: The system shall consi_st of V = 10 mL ion exchanger in the A-form (capacity C = 2.8 mequiv mL-', r = 0.02 cm) stirred in V = 100 mL solution of concentration c = 0.01 mequiv mL-l. If we further assume a B A = 2 (that is, the ions B which are originally only in solution are preferred by the ion exchanger), we have t o use Figure lb- and obiain for w = 2.8-10/ (
[email protected])-= 28: Heinf,(VrD)/(VdD)= 168 and Hebatch(VrD)/(VGD)= 3030. If we further put, e.g., DID = 10, 6 = 0.001 cm, and r = 0.02 cm, this yields Heinf.= 168(1004.001)/(
[email protected]~10)= 8.4 and Hebatch = 3030(1004.001)/(104.02~10)= 152. (The value for Heinf. can also be obtained, of course, immediately from eq 1, but note that a$ = 1/21. In the present example Heinf. is larger than unity and thus indicates a film diffusion controlled ion exchange process. The even larger Hewerich number for the batch process thus confirms the approximate prediction of the Heinf,value. As evident from Figure h-C, Heinf.is always Smaller than Hebatch at any value of w as long as a B A > 1. Heinf. is thus always a conservative estimate of Hebatchas long as Hehf. =- 1 (i.e., as long as film diffusion is rate controlling). If, however, the value of Heinf.is smaller than unity (suggesting particle diffusion as rate controlling), the opposite behavior is observed. Suppose, we use V = 10 mL ion exchanger in V = 100 mL solution, but put c = 0.1 mequiv mL-'. This yields w = 2.8~1_o/c0.1~100~ = 2.8. From Figure l b we then obtain He,,f.(VrD)/(VdD)= 17
and Hebatch(v?'D)/(v6D)= 54. If we further put DID = 100, the resulting values for the Helfferich numbers are Heinf, = 17(1004.001)/(104.02*100)= 0.085 and Hebatch = 54(100.0.001)/(10.0.02~100) = 0.27. Heinf.thus suggests particle diffusion as rate controlling. This prediction is now not confirmed by the corresponding (larger) Hebatchvalue, which indicates that both film and particle diffusion affect the rate of ion exchange. The situation becomes somewhat more complicated if the ions B taken up by the ion exchanger are no longer preferred with respect t o A, i.e., a B A < 1. Here we observe that for a B A 5 0.55 the curves for Heinf.and Hebabhbegin to intersect each other. This is illustrated in Figure Id, where we put a B A = 0.1. The above examples then suggest the following procedure for the prediciton of the rate-determining step for an ion exchange batch experiment from known values of the conventional Helfferich number Hei,f. as defined by eq 1: 1. Heid, >> 1(suggesting film diffusionis rate controlling): 1.1. Value of w is in a region where according to Figure 1Hebakh > Hex.. Result: Film diffusion will be rate controlling also for the batch process. This is, for example, the case if the ions B taken up by the ion exchanger from the solution are preferred by the ion exchanger with respect t o ions A originally in the ion exchanger ( a B A > 1). 1.2. Value of w is in a region where according to Figure 1Hebatch < Heinf.. Result: Film diffusion will not necessarily be rate controlling also for the batch process. The exact value for Hebatch has to be determined for the system from Figure 1. 2. Heinf. Heid.. %Sulk Particle diffusion will not
Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 2687 necessarily be rate controlling also for the batch process. The exact value for Hebatch has t o be determined for.the system from Figure 1. This is, for example, the case if the ions B taken up by the ion exchanger from the solution are preferred by the ion exchanger with respect to ions A originally in the ion exchanger (aB*> 1). 2.2. Value of w is in a region where according to Figure 1 Hebatch < Heinf..Result: Particle diffusion will be rate controlling also for the batch process.
Literature Cited Brooke, N. M.; Rees, L. V. C . Kinetics of ion exchange, Part 1. Trans. Faraday Soc. 1968, 64,3383. Brooke, N. M.; Rees, L. V. C . Kinetics of ion exchange, Part 2. Trans. Faraday Soc. 1969, 65, 2728. Bunzl, K.Kinetics of differential ion exchange processes in a finite solution volume. J . Chromatogr. 1974, 102,169. Bunzl, K. An anomalous effect in kinetics of polydisperse ion exchangers. J. Phys. Chem. 1991,95, 1007. Bunzl, K.Kinetics of ion exchange in a stirred-flow cell. J. Chem. Soc., Faraday Trans. 1993, 89, 107. Bunzl, K.; Dickel, G. Zur Kinetik der Ionenaustauscher 11. Die Filmdiffision bei differentiellen Umbeladungen. (Kinetics of ion exchange 11. Film diffusion during small conversions.) 2. Naturforsch. 1969, 24a, 109.
Bunzl, K Schimmack, W. Kinetics of ion sorption. In Rates of Soil Chemical Processes; Sparks, D. L., Suarez, D. L., Eds.; SSSA Special Publication 27; Soil Science Society of America, Inc.: Madison, WI, 1991. Helfferich, F. Zon exchange; McGraw-Hill: New York, 1962. Helfferich, F. Ion exchange kinetics. In Zon exchange; Marinsky, J., Ed.; Marcel Dekker, Inc.: New York, 1966. Helfferich, F. Ion exchange kinetics-Evolution of a theory. In Mass Transfer and Kinetics of Zon Exchange; Liberti, L., Helfferich, F., Eds.; Martinus Nijhoff Publ.: The Hague, 1983. Helfferich, F.; Hwang, Y.-L. Ion exchange kinetics. In Zon Exchange; Dorfner, K., Ed.; Walter de Gruyter: Berlin, 1991. Paterson, S. The heating or cooling of a solid sphere in a wellstirred fluid. Proc. Phys. SOC. (London) 1947, 59, 50. Sparks, D. L. Kinetics of Soil Chemical Processes; Academic Press: San Diego, 1988. Received for review September 14, 1994 Revised manuscript received December 8, 1994 Accepted December 30, 1994@
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*Abstract published in Advance ACS Abstracts, J u n e 15, 1995.
