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Anal. Chem, lg82, 54, 272-277
Kinetics of Differentially Small Conversions in Contact Isotopic and Contact Ion Exchange Kurt Bunzl" and Wolfgang Schultz Gesellschaft fur Strahlen und Umweltforschung mbH Munchen, Instltut fur Strahlenschutz, P8042 Neuherberg, Federal Republic of Germany
The klnetlcs of dlfferentlaiiy small contact Ion exchange processes of two adjacent resin ion exchanger beads in pure water was investigated, uslng radloactive tracers. Interruption tests and determination of the rates, as a function of the force with which the beads were pressed together, lndlcate that diffusion of the ions across the aqueous fllm between the beads was the rate-determlning step. The experlments show that the half-times of the exchange Cs+-Na+ and Cs+-H+ decrease, If the Initial Cs+ concentration In the partlcies Is increased. With the help of the theory developed, the rate coefflclents of these processes can be evaluated from the experiments. It Is found that they depend strongly on the lonlc composnion In the case of the exchange Cs+-H+ but are almost constant when Cs+ is exchanged vs. Ne+. The rate coefflclents of differentially small contact Ion exchange processes were closely related to those of isotopic contact exchange processes. With the help of the rate coefflclents of dlfferentiai exchange processes the rates observed for nondiff erentlal exchange processes are discussed.
If ion exchanger particles of different ionic compositions are placed in pure water and allowed to touch each other, their electric double layers will intermingle and counterions are exchanged between them. In analytical chemistry this phenomenon of contact ion exchange does not seem to have been utilized often as yet, even though it should be possible to desorb in this way very selectively the ions of interest directly from other materials with ion exchange properties, as for example from inorganic and organic synthetic or natural polymers. Besides that, contact ion exchange can possibly also contribute to the exchange of nutrient and toxic ions between soil particles exhibiting ion exchange properties (clay minerals, humic substances, sesquioxides) or between these soil components and plant roots. Because, as yet, very little is known about contact ion exchange when compared to ordinary ion exchange in electrolyte solutions, it was the purpose of the present paper to obtain first a better understanding of these exchange equilibria and especially of the rates of their attainment. In the previous paper (I) the kinetics of contact ion exchange between two adjacent cation exchanger particles in pure water was studied. In this investigation each resin bead was initially loaded completely with different counterions. This had the advantage that a maximum amount of ions was exchanged after contacting the beads. Much more detailed information, however, can be obtained if the initial ionic compositions of the two beads are just slightly different. In this way, even after attainment of equilibrium, the ionic composition of both beads has changed by only a very (and in the limit, a differentially) small fraction. Measurement of the rates of differential contact ion exchange processes as a function of the initial ionic composition of the beads then offers the following advantages: (a) all quantities which may affect the rate, such as, the separation factor, the bead diameter, or the rate coefficient, and which, in general, depend 0003-2700/82/0354-0272$01.25/0
on the ionic composition of the ion exchanger can be considered constant within the small range of a differential exchange process; (b) information on the effect of the ionic composition of the ion exchanger on the rate of contact exchange; (c) a deeper understanding of all nondifferential contact ion exchange processes for which the ionic composition changes considerably during the exchange process. For this reason we investigated, in this paper, the kinetics of differential ion exchange processes for the exchange of Cs+ vs. H+and Cs+ vs. Na+ between two resin beads, laying on top of each other in pure water. In the theoretical treatment of the rates of such processes we assumed diffusion of the counterionsthrough the aqueous film between the beads to be the rate-determiningstep. Also, to verify this hypothesis some suitable experiments will be described. In addition, because the theory predicts a close relationship between the rates of contact ion and contact isotopic exchange processes, the exchange Na+/Na+ and Cs+/Cs+ between two beads was investigated.
THEORY In the previous paper (I) we considered contact ion exchange between two ion exchanger particles A and B, which were saturated initially with monovalent counterions 2 and 1,respectively. If we now want to describe exchange processes between two beads, which differ in their initial ionic composition by only a small fraction, the initial equivalent fractions rl,A,O and rl,B,O of the ions 1 in particles A and B, respectively, may have any value between 0 and 1. The equivalent fractions of the ions 1in the two particles at any time shall be given as r i , = ~
CI,A/CA
r i , ~CI,B/CB
(1) where C1,A and C1,B are the concentrations of the ions 1 in particles A and B, respectively, and C A and CB their ion exchange capacities in mequiv/mL. Because during the contact ion exchange process all counterions 1leaving particle B must appear in particle A, I'l,B depends on r 1 , A according to (2) = rl,B,O - w ( r l , A - rl,A,O) where w = C , V A / C B V B . VAand VB are the volumes of particles A and B, respectively. The flow J1of the ions 1between the two adjacent beads A and B is given ( I ) as rl,B
where y1is the equivalentfraction of ion 1in the aqueous film between the two beads, y l , A and yl,Bare the corresponding equivalent fractions in the aqueous phase a t the surface of the two beads A and B, and M is a quantity which is assumed to depend, for a given ion exchanger and a given pair of counterions, on the concentration of the ions in the film and on the thickness of the film between the two particles. D 1 and D2are the diffusion coefficients of the two counterions in the film, which can be calculated from the limiting equivalent conductivities of the corresponding ions with help of the 0 1982 American Chemlcal Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982
Nernst expression. In the same way as earlier the unknown y1 at the surface are related to the measurable equivalent fractions r 1 , A and r 1 , B of the counterions inside the two ion exchangers according to rl,A(l ff12,A
- 71,A)
rl,B(l
- 71,B)
Q - (Q2 - 4 W ( a 1 2 , ~-~ - a12,B)(ff12,AWrl,A,0 2w(ff12,A
(ff'2,A
for ion exchanger B
(4)
where FAis the area of contact between the two beads. Integration of eq 3 and using eq 2, 5, and 4 we obtain for the time dependence of the measurable equivalent fraction r 1 , A in bead A
(9) = (Wrl,A,O + r l , B , O ) / ( l + wl If the two particles have also the same diameter (w = 11, we obtain (101 rl,A,-= (rl,A,O + rl,B,0)/2 rl,A,-
In the special case of I'lb,o= 0 and I'l,B,o= 1eq 8-10 reduce to the corresponding equations given earlier (I) for these equilibria. Rates of Differentially Small Conversions. Finally, we want to calculate the rates of contact ion exchange processes between the two beads for differentially small conversions of each bead in terms of the rate of attainment of the fractional equilibrium U as a function of the time. In accordance with the following experiments we will consider here only the case of contact ion exchange between two beads of the same material and diameter. However the general case ( a 1 2 b # alzg; w # 1) can be also readily derived. Defining U as usual (rl,A
- rl,A,O)/(rl,A,-
rl,A
rl,A,O
-
(12)
and substituting r 1 , A from eq 14 in eq 7, while observing that dI'l,A= edU/2, results in
where E* = E l / E 2and El = Dz (Dl - D ~ ) ( ~ I , A ,+O € ( I - 0 . 5 U ) ) / ( t ( 0 . 5 U ( f f 1 2 1) 1 - d 2 ) + r l , A , O ( l -ff12)
=D
2
(rl,A,O
and p = rl,B,O + rl,A,O. The integral in eq 7 has to be solved numerically with the help of a computer. Distribution of the Ions in Equilibrium. The equivalent fractions rib,.. and of the ions 1 in particle A and B, after the attainment of equilibrium, can be obtained either with help of eq 2 and 4 and by observing that in equilibrium 71= ~ ylg or from eq 6 by considering that at t the rate of contact ion exchange has to vanish. This latter condition is fulfilled if the argument of the logarithm becomes unity. In this way one obtains
+ U(IIl,B,O - rl,A,0)/2
where e 1, see Figure 4)) a comtightly than the Cs+ ions paratively high tendency of the H+ ions to diffuse into the film can be expected. Also, we can now understand readily, why in Figure 5 the two straight lines, obtained by regression analysis and describing R / C A as a function of res, converge at rcS = l: In this form both beads are saturated with Cs+ ions and the total concentrationof ions in the film (and hence R/CA) is the same, irrespective of whether a differentially small exchange of Cs+-H+ or Cs+-Na+ occurs. One might, therefore, predict that at rcs= 1 a differentially small exchange process between Cs+ and any other monovalent counterion should proceed with the same rate coefficient. Especially, the isotopic contact exchange Cs+-Cs+ should proceed with the same R/CA, because in this case also both beads are in the Cs+ form. For the same reason we should expect for the differential ion exchange processes Cs+-Na+ at l'Ca = 0 to observe the same value of R / C Aas for the isotopic exchange Na+-Na+, because in both cases the two beads are in the Na+ form. These isotopic contact exchange processes Cs+-Cs+ and Na+-Na+ between the two beads, under the same experimental conditions as for the other experiments, were carried out by using radioactive tracers and the resulting rate curves are shown in Figure 6. As can be seen, the rate of contact isotopic exchange Cs+-Cs+ is considerably faster than that of the exchange Na+-Na+. With help of the rate equation for isotopic
ref 1.
contact exchange U = 1 - exp(-2DiRt/CA),as derived earlier (I),and the known values of DNa and Des, as given above, we can again by curve fitting evaluate R / C k The resulting rate curves calculated with the help of the above equation and using a value of R / C A = 4.9 cm-2 in the case of the exchange Cs+-Cs+ and R/CA = 3.0 cm-2 for the exchange Na+-Na+ are shown in Figure 6. As can be seen a good fit to the experimental points is obtained. It is interesting to note that even though the exchange process Cs+-Cs+ has the smaller rate coefficient than that of the exchange Na+-Na+, the rate of the exchange Cs+-Cs+ is faster than that of the exchange Na+-Na+ (see Figure 6). This is the result of the different diffusion coefficients of theses ions, DCs being about twice the value of DNa(see above). These values of R/CA can be compared now with the corresponding values observed for the differential exchange Cs+-Na+ and Cs+-H+. As shown in Figure 5 , where we plotted RIGA for the isotopic exchange processes as asterisks, we find indeed that the rate coefficients are the same for the exchange Cs+-Cs+,Cs+-Na+, and Cs+-H+ a t I'cs = 1. In the case of the exchange Cs+-Na+ at rcs= 0 (i.e., pure sodium form) we find that R/CA is to within experimental error identical with R/CA for the Na+-Na+ exchange. The comparatively small difference between the R / CA values observed for the exchange Cs+-Cs+ and Na+-Na+ is thus responsible for the fact that the RIGA determined for the various differential exchange processes Cs+-Na+ is almost independent of rCs (see Figure 5). This does, however, not mean that the rates of differential Cs+-Na+ exchange processes are also almost independent of the Cs+/Na+ ratio in the exchanger phase. The considerable increase of the rate of exchange with increasing as observed in Figure 3 is the result of the different diffusion coefficients of these ions (Des > DNA, the quantitative relation given by eq 17. It would be interesting to know also whether the rate coefficient for the differential exchange Cs+-H+ a t I'cs = 0 (Le,, in the hydrogen form) is also indentical with the corresponding one of the isotopic exchange H+-H+. An experimental examination, using, e.g., tritium, is, however, not possible because in this case the tritium ions in one bead would not only exchange with H+ ions of the adjacent bead but also with the surrounding water. An extrapolation of the R/CA values obtained for the differential exchange processes Cs+-H+ to rc,= 0 in Figure 5 predicts, however, that such an isotopic exchange H+-H+ should proceed with RIGA = 20 cm-2. Once ai2and the rate coefficient as well as their dependence on rCsare known, the half-time of any differential ion exchange process can be calculated with the help of eq 18 as a function of rcs.In the case of the exchange Cs+-H+ we obtained first by regression analysis R/CA (Crn-') = (20.2 15.5)rca(see Figure 5) and aHcs= (4.0- 0.625)rcS(see Figure 4). Substituting these two equations in eq 18,one obtains t?l2 as a function of rcs. The resulting curve is shown also in Figure 3 (lower curve). In the case of the exchange Cs+-Na+ we considered R/CA = constant = 4.1 cm-2 (see Figure 5) and
ANALYTICAL CHEMISTRY, VOL. 54, NO. 2, FEBRUARY 1982
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overall value of R/CA observed at any ionic composition r C s of the two beads R/CA = f/2([R/CAIHrH,A + [~/cA~CsrCs,A~ + y2( [R / CAI HrH,B + [R / CAI CsrCs,B) (23) where [R/CA]Hand [R/CA]C~ are the values of RIG, when both beads are in the pure H+form and Cs+ form, respectively, as obtained from differential ion exchange experiments. The factor ‘Izin eq 23 is necessary, because [R/CAIHand [ R / c ~ l c ~ were obtained from measurements, where two beads of the same diameter released simultaneously their ions into the film, while R/CA is calculated now as a result of the ions released by each bead. Because rH,A + rcsh = r H , B + rCs,B = 1and because rCs,A and rCs,B are related according to eq 2 we obtain after substitution and putting w = 1
!i -
0 0
25
50
75
100
125
150
time [ m i n 1
Figure 7. Rate of the nondifferential contact ion exchange Cs+-H+. Initially one bead saturated with H+, the other one with Cs’. Upper bead weighted wlth 2 g. Curve calculated with help of eq 15, ref 1.
a N 2(2.4 - 0.84)rCs(see Figure 4). With these values eq 18 yields the corresponding curve for the half-times as shown in Figure 3 (upper curve). As mentioned, the rates of the differential contact ion exchange processes shown in Figure 3 increases with increasing rCseven though, as we have seen in the case of the exchange Cs+-H+, the rate coefficient decreases in this direction. However, such a result should not always be observed. Equation 18shows that in principle tl12 can, at constant R/CA, either decrease or increase with increasing rl,A,O, depending on the particular values of Dl/Dz and alp In some cases tl12 may even pass through a maximum for which the necessary condition can be derived from eq 18 as
(24/&)/(1
+ D1/D2) < alp < (1 + D1/D2)/2
(22)
Nondifferential Ion Exchange Processes. Once RIGA, as a function of res, has been evaluated experimentally from the differential contact exchange processes, it should be possible, at least, to estimate the rates of exchange reactions during which the ionic composition of the beads changes substantially (nondifferential ion exchange). We determined, experimentallyfor this purpose, the contact exchange between two beads where initially one bead is in the Cs+ form and the other bead is in the H+ form. The rate observed for the exchange process thus initiated is shown in Figure 7. After attainment of equilibrium both beads will contain Cs+ and H+ ions in equal amounts (I’C8,- = r H , m = 0.5). The curve plotted in Figure 7 through the experimental points has been calculated with the help of the rate equation given previously for such conversions (eq 1in ref l),using the above values for DH, Des, and a~~~and evaluating the rate coefficient by curve fitting to obtain R/CA = 10.0 cm-’. Because, when compared to the differential ion exchange processes, all experimentalconditions (e.g., material and radius of the beads, force between the beads) remained unchanged, we assume that the above value of R/CA resulted, essentially, from a particular concentrationof the ions attained in the film between the two beads. The linear dependence of R/CA on I?&, as shown in Figure 5 for the differential exchange Cs+-H+, indicates that the total concentration of the ions in the film results from a simple superposition of the Cs+ and H+ions released by each particle A and B into the film according to their ionic composition. We will write, therefore, for the
R/CA
1/2([R/cA]H[2 - rCs,A,O - rCs,B,OI + [R/CAICs[rCs,B,O + rCs,A,OIj (24)
For the special case of the nondifferential exchange process considered above where, initially, bead A is in the H+ form and bead B is in the Cs+ form we have rCs,A,O = 0 and rCs,B,O = 1. With these values eq 24 becomes (25) R/CA = ([R/CAIH + [R/CA]CS)/~ For the above nondifferential exchange process the rate coefficient is, therefore, independent of the ionic composition rcSof the beads, which changes of course considerably during this reaction. It is, therefore, not surprising that we were able to fit the corresponding rate curve in Figure 7 with a constant value R/Ck The values of [R/CAIHand [R/CAlCsas obtained from the differential exchange experiments (see Figure 5) by extrapolation to l?cs = 0 and 1 were 20.2 and 4.7 cm-2, respectively. Substituting these values in eq 25 yields R/CA = 12.4 cmW2.This is in satisfactory agreement with the experimental value of R/CA = 10 cm-2, considering that eq 23 is only a first approximation. Finally, we wanted to relate the differential exchange experiments Cs+-Na+ described here with a nondifferential exchange experiment, reported previously, where initially one bead was in the Na+ form and the other in the Cs+ form. In this case we observed again that the rate curve could be fitted with a constant value of R/CA = 4.9 cm-2. In this case the explanation is even simpler, because the corresponding differential exchange experiments reveal that within experimental error R/CA is always around this value, independent of r C s (see Figure 5). Equation 24 may, therefore, together with, e.g., eq 7, serve to calculate the rates of any nondifferentia1 exchange processes. LITERATURE CITED (1) Bunzl, K.: SchuCz, W. J . Inorg. Nucl. Chem. 1981, 43, 791-796. (2) Helfferich, F. “Ion Exchange”; McGraw-Hill: New York, 1962. (3) Bunzl, K.; Dickel, 0. Z . Nafurtorsch., A 1962, 24, 109-117. (4) Kressmann, T. R. E.; Kitchener, J. A. Discuss. faraday SOC. 1949, 7 , 90-94. (5) Robinson, R. A.; Stokes, R. H. “Electrolyte Solutions”; Butterworth: London, 1968.
RECEIVED for review June 29, 1981. Accepted October 14, 1981.
