J . Phys. Chem. 1991, 95, 1007-I012
1007
An Anomalous Effect in Kinetics of Polydisperse Ion Exchangers K.Bund Gesellschaft fur Strahlen-und Umweltjorschung Munchen (GSF). Institut fur Strahlenschutr. 0-8042 Neuherberg, Federal Republic of Germany (Received: April 12, 1990; In Final Form: August 15, 1990)
The kinetics of ion exchange involving simultaneously ion-exchanger particles of different diameters was investigated in well-stirred, dilute solutions, where film diffusion is the rate-determining step. A special situation arises, if sufficient ion exchanger is added to a solution to remove a given counterion species almost completely. In this case the concentration of the ions in solution still attains its equilibrium in a time as about expected from the weighted mean of the rate coefficients of the individual size fractions of the ion exchanger. I n the same system, however, the ion-exchanger particles approach their equilibrium more slowly by several orders of magnitude. The kinetic behavior of such mixtures can, therefore, not be obtained by measuring only the concentration changes in the solution phase. The above effect is demonstrated by investigating the isotopic exchange Cs+/Cs+ and the ion exchange Cs+/H+ in 10-4-10-3 M solutions (50mL each) in the presence of about 100 mg of cation exchanger, sieved to contain only particles with 0.4 and 0.8" diameter. Rate equations that describe quantitatively the experimental observations are given. If the particles exhibit different ion-exchange properties (ion-exchange capacity, separation factor), the above effect should be observable also for monodisperse mixtures. The presence of anomalous kinetics has to be considered also when the sorption of trace ions (metal ions, radionuclides) by soils or soil components is investigated.
Introduction A detailed knowledge of the kinetics of ion-exchange processes is important 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. in the soil or in biological membranes. In dilute, moderately stirred solutions and for small particles thc rate of ion exchange is controlled mainly by diffusion of the ions across a hydrostatic boundary layer (Nernst film) surrounding the particles (film diffusion). Even though the rate of film diffusion controlled ion-exchange processes has been investigated thoroughly for monodisperse systems,'-I6 much less is known for polydisperse mixtures, i.e., for systems where the size of the ion-exchanger particles is not uniform. Yet, even most commercially available ion-exchange resins (unless carefully sieved) are invariably polydisperse. This is even more true for natural inorganic and organic ion-exchanger particles in the soil. Polydisperse mixtures, not necessarily ion exchangers only, have received recently considerable attention, because of their interesting properties (see e.g. ref 17). For polydisperse systems of ion exchangers, for example, we have shown that the kinetics of film diffusion controlled ion-exchange processes in a finite bath shows several remarkable characteristic^.^^,'^ In contact with a solution ( I ) Boyd, G. E.; Adamson, A. W.; Myers, L. S. J . Am. Chem. Soc. 1947,
69, 2836.
(2) Dickel, G.; Meyer, A. Z . Elektrochem. 1953, 57, 901. (3) Dickel, G.; Nieciecki, L. Z . Elektrochem. 1955, 59, 913. (4) Helfferich, F. Ion Exchange; McGraw-Hill Book Co: New York, 1962. (5) Helfferich, F. I n Ion Exchange; Marinsky, J. A,, Ed.;Marcel Dekker: New York, 1966; Chapter 2. (6) Helfferich, F. I n Mass Transfer and Kinefics of /on Exchange; Liberti, L., Helfferich. F., Eds.: Martinus Nijhoff Publ.: The Hague, 1983; Chapter 5. (7) Liberti, L. In ref 6; Chapter 6. (8) Petruzzelli, D.: Liberti, L.; Passino, R.; Helfferich, F.; Hwang, Y. L. React. Polym. 1987, 5, 219. (9) Liberti, L.; Petruzzelli, D.; Helfferich, F.; Passino, R. React. Polym. 1987, 5, 31. (10) Jackman, A. P.; Ng, K. T. Wafer Resour. Res. 1986, 22, 1664. ( I 1) Dickel, G. In ref 6; Chapter 12. (12) Bunzl, K . 2.Phys. Chem. (Munich) 1971. 75, 118. ( I 3) Bunzl, K . J . Chromatogr. 1974, 102, 169. (14) Ogwada, R.A.; Sparks, D. L. SoilSci. Soc. Am. J . 1986,50, 1162. ( I 5) Sparks, D. L. Kinefics ofsoil Chemical Processes; Academic Press: San Diego, 1988. (16) Tsai, F. N. J . Phys. Chem. 1982, 86,2339. (17) Kofke, D. A.; Glandt, E. D. J . Chem. Phys. 1989, 90, 439.
containing the counterions A, the smaller, but faster reacting, particles in the mixture can initially overshoot their eventual equilibrium uptake considerably. Subsequently, however, they have to release these already sorbed ions A in favor of the larger, but more slowly reacting, particles. As a result, we found that during the second period of the ion-exchange process the concentration of the ions A in solution approached its equilibrium slightly faster than the particles approached their equilibrium. As we will show now, this latter effect is enhanced extraordinarily if there is an excess of exchanger material (in equivalents) in the system as compared to the total amount of ions A (in equivalents) initially in the solution. In this case, the concentrations of the ions A in solution become very small after attainment of equilibrium (exhaustion of the solution). As a result, we will be confronted with the anomalous situation that the concentration of the counterions in the solution attains its equilibrium rather fast, while in the same system the ion-exchanger particles approach their equilibrium extremely slowly. Situations where an excess of ion exchanger is used occur if, for example, a given ionic species has to be removed from the solution by a batch operation. In addition, an excess of particles with ion-exchanger properties is usually also present if the sorption of trace ions by soils or soil con.ponents (clay minerals, oxides, humic substances) is investigated. Especially in this latter case the ion-exchanger particles will be invariably polydisperse. In the present investigation, the anomalous kinetic effect mentioned above will be demonstrated experimentally by measuring the rates of isotopic exchange (Cs+/Cs+) and ion exchange (Cs+/H+) in a finite bath containing simultaneously two particle fractions (0.4 and 0.8 mm in diameter) of a cation exchanger. The corresponding rate equations for a polydisperse system in the presence of an excess of ion exchanger are also given and compared with the experimental results.
Theory Rare Equations. The differential equations for the rate of film diffusion controlled ion-exchange processes between two monovalent counterions A and B are given for uniform particles
where XA = CA/C is the equivalent fraction of the ions A in the (18) Bunzl, K. J . Inorg. Nucl. Chem. 1977, 39, 1049. (19) Bunzl, K. Anal. Chem. 1978, 50, 258.
0022-3654/91/2095-l007%02.50/0 .~ , 0 1991 American Chemical Society I
,
1008 The Journal of Physical Chemistry, Vol. 95, No. 2, 1991
ion exchanger, CAis the concentration of the counterion A in the ion exchanger (equiv of ion/L ion exchanger), and C the total ion-exchange capacity in equiv/L ion exchanger; cA(t) and cB(t) are the concentrations (in mol/L) of the two counterions in solution at time t . DAand DBare diffusion coefficients which are functions of the four diffusion coefficients Di> describing isothermal diffusion in the corresponding electrolyte solution involving counterions A and B and the common co-ion. They can be determined by independent measurements in this solution, but as yet very few data are available. The rate coefficient R is given as R = FDB/(tT6)
(2)
where d is the film thickness (Nernst diffusion layer) as used first in ion-exchange kinetics by Boyd.' F/ Pis the ratio of the surface to the volume of the ion-exchanger particles. For spherical particles R = & , / ( 3 r C b ) ,where r is the radius of the particles. The equivalent scparation factor c$ of the ion exchanger is given as
XA,-.~B,=
(3)
~
XB%..XA?
where x,, and xB are the equivalent fractions of the counterions A and B in solution and the subscript denotes the equilibrium a). values ( t The derivation of eq 1 is based on an unstirred Nernst film around the particles, across which the counterions and co-ions are transported according to their transport coefficients and electrochemical potentials (theory of irreversible thermodynamics). Further assumptions involve a vanishing electric current density, the appearance of a diffusion potential as a result of the different mobilities of the ions in the film, electroneutrality in the ion exchanger, and equilibrium at the interface ion exchanger/film. For ions of arbitrary valence eq 1 is only slightly more complicated. 2 ~*1 Because an ion exchanger will exchange counterions in equivalent amounts, the total equivalent solution concentration c (in equiv/L) will remain constant in a finite bath of a given volume. For the exchange of univalent ions
-
cA(?)
+ cB(1) = c
(4)
The system considered shall now be polydisperse; i.e., it shall consist of a solution in which n size fractions of ion-exchanger particles with different rate coefficients Ri ( i = I , 2, ...,n ) , e.g., as a result of different particle diameters, are stirred at a given speed. I n the solution volume V (L), each fraction of ion exchanger is present in amounts Q, (in equiv); if the mass of each fraction is mi (kg). Qi = miC', where C'is the ion-exchange capacity in equiv/kg. For simplicity we assume the following initial conditions at t = 0; XA,o= 0 (no ions A in the ion exchanger) ~ 6 . 0=
0;
CA.O
= c (only ions A in the solution)
Bund of time, the concentration cB(t) and c A ( t ) of the ions in solution can be calculated with help of eqs 5 and 4 . Because eqs 6a are nonlinear and inhomogeneous they have to be solved numerically, e.g., by the Runge-Kutta method.20 As shown previo~sly,'~*'~~*' eqs 6a describe quite accurately that in a polydisperse mixture of ion exchangers the smaller particles can overshoot their eventual equilibrium values XAsm and have to release subsequently (second period of the ion-exchange process) ions A in favor of the larger particles. As we will show now, a special situation arises, if, after attainment of equilibrium, the concentration of the ions A in solution, c~,..,becomes very small (in the order of M or less). Such an exhaustion of the solution will occur if either the separation factor a t is very high or if a liberal excess of ion exchanger is used as compared with the amount of ions A present initially in the solution. This latter condition reads in our notation n
C Qi >> CA,OV i= I
Thus, as long as condition 7 holds, or a is very high, it is not necessary that the initial concentration of the ion A in the solution, c ~ , has ~ , to be low. The surplus of ion exchanger, necessary to achieve a small value for cy, depends of course on the value of the separation factor. If aBis very small, the surplus has to be larger as compared to a system where a i is large. Under the above conditions, eq 6 can be simplified to yield uA,i/dt
= IR,*/ VlIa9(1 - XA,i) x n
n
Ceix~,iI- x~.i[CQix~,iII/Ia8(1 - XA,i)I I=I i= I
[CV-
(6b)
where Ri* = R~DA/DB.The advantage of eq 6b is that in this case one has to know for each size fraction i of the ion exchanger only Ri* rather than Ri and DA/DB separately is in eq 6a. As shown later, the values for Ri* can be obtained easily from kinetic measurements on the individual size fractions. Similar to eq 6a, eq 6b can be solved only numerically. An approximate solution for the first period of the reaction is given in the Appendix. Fractional Attainment of Equilibrium. The resulting equilibrium uptake XA,- of the ions A by the ion exchanger can be obtained by putting in eq 6a or 6b dXA,i/dt = 0, because in equilibrium all rates have to vanish. Because the different fractions of the ion exchanger shall be homogeneous except for its particle size, the values for XA,- have to be identical for each fraction. This quantity is obtained then for the case a t # 1 as xA.-
- [(b2 + 4eh)'/2- b ] / ( 2 e )
(8a)
where n
n
e = (1/V)(I - a$)CQi; b = at[c+ I=I
In case that
ai
(l/V)CQi];h = cap i= I
= 1 (isotopic exchange)
The material balance yields that all ions B leaving the ion exchanger must appear in solution, i.e.
Elimination of cA and cB from eq 1 with help of eqs 4 and 5 yields the system of n differential equations, describing the rate of ion exchange for each ion-exchanger fraction i in the polydisperse system as
(7)
If, as a result of condition 7, cA,proximation for eqs 8a and 8b
-
0 we can write as an ap-
The concentration of the ions in solution is obtained for the equilibrium state with help of eq 5 as n
-
+
x A , i [ ~ Q ~ A 3 i l l / ( a $ ( 1X A , ~ ) ( D B / D A )x A , ~ I ; I=
I
i = 1, 2, ..., n
(6a) These n differential equations have now to be solved simultaneously in order to obtain the rates of ion exchange for each fraction in the polydisperse mixture. After all Xi are known as a function
= ( 1 / V)XA3mCQi; CA.- = C - C6.m
CB.~
i= I
(9)
(20) Collatz, L. The Numerical Treatment of Dijjereential Equations; Springer: Berlin, 1960. (21) Wolfrum, C.; Bunzl, K.; Dickel, G . ; Ertl, G.2.Phys. Chem. (Munich) 1983, 135, 185.
The Journal of Physical Chemistry, Vol. 95, No. 2, 1991 1009
Kinetics of Polydisperse Ion Exchangers
3 1.0
required for the slowest fraction of the ion-exchanger particles. Finally, it should be mentioned that the slowest fraction in a polydisperse mixture is not necessarily the fraction with the smallest rate coefficient (largest particle diameter). Consider, for example, a system consisting of two fractions, where the fraction of the particles with the smaller diameter is present at a very small percentage of the larger particles. In this case the large particles and the solution will attain their equilibrium almost simultaneously after a given time. The small particles, however, which overshoot their equilibrium considerably, will release subsequently the excess of ions A taken up only very slowly. Thus, in this case. the small particles are now the fraction for which the longest time is required to attain the equilibrium. Beside the above examples, numerous other model calculations were performed, using different values for the parameters and up to IO size fractions of ion-exchanger particles. In any case, as long as sufficient ion exchanger was present in the system to remove the ions A almost completely, the anomalous kinetic effect described was found.
solution 3
051LOr 2.0 '
J
size fract;on
c=l.lO-"M
1
2
3
500
0
1000
1500 2000
Time
2500 3000
(SI
Figure 1. Computer simulation of the kinetics of a system consisting of three size fractions of an ion exchanger with different rate coefficients R in a solution ( R , > R2 > R,). The fractional attainment U of the equilibrium is given separately as a function of the time for the three size fractions and the solution. Initially. the counterions A are only in the solution at a concentration of 1 X (top) or 1 X IO4 M (bottom). In equilibrium U has to approach unity for all components.
Conventionally, the rates of ion exchange are presented in terms of the fractional attainment U(t)of the equilibrium. This quantity is given in the present case for each fraction of the ion exchanger and the solution respectively as = XA,i(t)/XA.-;
Usol(t)
= cB(t)/cB,-
-
(10)
U ( t ) thus ranges from 0 at t = 0 to 1 at t m (equilibrium). Computer Simulation. We already mentioned that for low values of cA,- a polydisperse system will exhibit the peculiar effect that after a comparatively short time the concentration of the ions in solution will be in equilibrium while the ionic composition of the various ion-exchanger fractions is still far away from equilibrium. This is illustrated in Figure 1, where we used eqs 6-10 to calculate the rates of ion exchange for a system consisting of i = 3 fractions of an ion exchanger, present in equal amounts QI = Qz= Q3= 0.4 mequiv in 0.1-L solution. The values for the other constants were R , = 2.0, R2 = 1.5, and R3 = 0.5 cm3/(s mequiv), D z / D I = 2, and at = 2. The initial concentration of the counterions A in solution was either 0.001 (Figure 1, top) or 0.0001 mol/L (Figure I , bottom). The ion exchanger contained initially only counterions B. The equivalent fraction of the ions A after attainment of equilibrium, X,,,,as calculated from eq 8 is 0.0798 for c = cA,0 = 0.001 M and 0.008 30 for c = 0.0001 M. The equilibrium values c~,.., calculated for these two initial concentrations according to eqs 9 and 4, are 4.2 X 10-5 and 4.17 X IO-' moI/L. respectively. As evident from Figure I , at c = 0.001 mol/L the fractional equilibrium U for the two fractions i = 1 and 2 with the smaller particles sizes overshoot their eventual equilibrium values. Subsequently (redistribution period), these particles have to release part of the ions A already sorbed in favor of the larger particles (fraction 3). As a result, to approach 98% of the equilibrium value, 520 s is needed for the solution phase, but 20000 s for the particles with the largest diameter (fraction 3). This anomalous behavior is even more evident if c = 0.0001 M (Figure I , bottom). In this case the solution attains 98% of its equilibrium value after about 500 s, whilc thc slowest ion-exchanger fraction ( i = 3) attains 90% of thc equilibrium only after 110000 s (30 h). If we increase the initial conccntration of the ions in solution to = 0.01 M, the anomalous kinctic effect can be observed if we increase the amount of ion exchanger corrcspondingly, e.g., to 15 mequiv of each of thc abovc thrcc fractions. Only I O s is required in this case for the solution to attain 95% of its equilibrium, while 1500 s is
Experimental Section Apparatus. All kinetic experiments were performed in a glass-jacketed cylindrical beaker (12 X 5 cm diameter) thermostated at 21 f 0.5 OC. The outlet in the spherical bottom consisted of a quick-opening Teflon valve, covered with a nylon screen (mesh opening 25 pm). The ground glass lid of the vessel had inlets for the paddle stirrer (two swing-out blades) and the thermometer. Material. The ion exchanger employed was AG-50W-X8 (Bio-Rad Laboratories), a sulfonated polystyrene-type cation exchanger. The spherical beads were converted to the Cs+ form by 1 M CsCl solutions, washed with deionized water, wet-sieved to 0.4 f 0.02 and 0.8 f 0.03 mm in diameter, and stored in deionized water. Cracked beads were removed under a microscope. For some experiments these beads were converted subsequently to the H+ form by 1 M HCI solutions. Procedure. To investigate the rates of isotopic exchange (Cs+/Cs+) and ion exchange (Cs+/H+), the reaction vessel was filled always with V = 50 mL of a CsCl solution (c = 1 X 10-3 to 1 X M, labeled with 13'Cs), stirred at 730 rpm, and thermostated. Q,and Q2 mequiv wet ion exchanger with particle diameters 0.4 and 0.8 mm (in either the Cs or the H form), respectively, were added simultaneously at t = 0. After predetermined time periods the ion-exchange process was interrupted by opening the outlet valve and removing the solution rapidly by suction, followed by a few portions of deionized water, which were collected, however, in a separate beaker and discarded. The ion exchanger was then removed from the reaction vessel and separated in its two fractions with help of a nylon screen (mesh opening 0.7 mm). The I3'Cs activity of these two size fractions as well as that in the solution was determined subsequently separately by a germanium detector and a multichannel analyzer. From the count rates determined initially in the solution (u,,~),after a given time (a,) and in equilibrium (aS,-),the fractional attainment for the solution was obtained as U ( t ) = ( u , , -~ U , ) / ( U ,-, ~a,,-). The corresponding values for the two ion-exchanger fractions were obtained as U,(t) = u , / u , , ~where , a, and a,,- are the count rates at time t and in equilibrium, respectively ( i = I , 2 ) . For a comparison of the experimental and theoretical results, the rates of isotopic exchange and ion exchange were also determined, when the two size fractions were present individually in the solution. In this case it is sufficient, if only the fractional attainment of the solution is measured. The experimental error for the fractional attainment U of the equilibrium was about 5%. The error in the time measurement was about 3 s. Results and Discussion Isotopic Exchange. In Figure 2 the experimentally observed rates for the isotopic exchange of Cs ions are shown, if the two size fractions are added to the solution individually (monodisperse systcm). The amounts of ion cxchangcr used were Q,= 0.37 mequiv (0.4 mm in diameter) and Q2= 0.404 mequiv (0.8 mm
Bunzl
1010 The Journal of Physical Chemistry, Vol. 95, No. 2, 1991
2.0 I 1.5 0.8”
3
0.50
+
experimental -calculated
1
t
experimental -calculated
I
I
1.0
0.5 0
200
600
COO
800
1000
Time I s )
I
Figure 2. Fractional attainment of the equilibrium LI as a function of time for the isotopic exchange Cs+/Cs+for ion-exchanger particles (0.4 and 0.8 mm in diameter), when added separately to a CsCl solution (e = 1 X IO-‘ M). (+) Experimental values. Solid lines: Fit of the experimental values according to eq 1 1 to obtain the rate coefficients of the two size fractions.
0.5 O.8mt-n 0
500
I
1000 1000
I
1
5000
Time
10,000
30,000
50,000
9000
(SI
Figure 3. Fractional attainment L/ of the equilibrium for the isotopic when ion-exchanger particles of 0.4 and 0.8 mm in exchange Cs+/Cs+, diameter are added simultaneously to a CsCl solution (e = 1 X IO4 M). Solid lines: calculated by using eq 6a and the rate coefficients obtained from the individual particles (Figure 2). (+) Experimental values. in diameter). The initial concentration of the Cs ions in the solution was c = 1 X 10aM, and the solution volume was 50 mL. In this case the exchange process is fairly rapid for both size fractions. After about 1000 s the equilibrium is attained to more than 90%, the size fraction with the smaller particles reacting of course somewhat faster. The rate curves for the ion-exchanger particles and the solution are in this case identical. To obtain form these rate curves the rate coefficients R* of the two size fractions, we can use eqs 6b and IO. In the present case (initial conditions as described, only one size fraction present (i = I), isotopic exchange ( D A / D s = I ; a$ = 1; R* = R)) eq 6a or 6b can be solved analytically to yield
U ( f )= 1 - exp[R*(c + Q / v ) f ]
(11)
Because with the exception of R* all quantities are known, we can use these equation to evaluate R* by curve fitting. If we put R,* = 0.70 and R2* = 0.32 cm3/(s mequiv) (for the 0.4- and 0.8-mm fraction, respectively), a good fit between the experimental points and the calculated ones (solid lines) was obtained (Figure 2). I f both size fractions of the ion exchanger were added simultaneously to the solution, the rate curves shown in Figure 3 are obtained. The amounts of ion exchanger Q, and Q2 used were each 0.41 mequiv for the 0.4- and 0.8-mm fraction. The initial Cs concentrations in the solution was again c = =1X M. Because in this case an excess of ion exchanger is in the system ( C Q i = 0.82 mequiv) as compared to the total amount of Cs ions in solution ( VcAj0= 0.005 mequiv), condition 7 holds and the anomalous kinetic behavior is observed. While the solution was in equilibrium after about 600 s, the ion-exchanger particles had sorbed at that point of time either too much l3’Cs (0.4-mm fraction) or too little (0.8-mm fraction). They approached sub-
0
I 1000
2000
3000
LOO0
5000
6000
Time ( S I Figure 4. Fractional attainment U of the equilibrium for the isotopic when ion-exchanger particles of 0.4 and 0.8 mm in exchange Cs+/Cs+, diameter are added simultaneously to a CsCl solution (e = 1 X IO-) M). Solid lines: calculated by using eq 6a and the rate coefficients obtained from the individual particles (Figure 2). (+) Experimental values. sequently their equilibrium ( U = 1) so slowly that 95% of the equilibrium value are attained only after about 50000 s. Equation 6b can now be applied for a comparison of the experimental results with the theoretical predictions. For this purpose we first use eqs 8c and 9 to calculate the equilibrium values XA,and cB,-. Next, the two differential equations (6b) have to be solved simultaneously numerically to obtain XAqiand x A , 2 as a function of f . This is possible, because all quantities in eq 6b are known. (The values for R I * and R2* were obtained from the kinetic measurements on the individual size fractions; see above.) Usol(t) is then obtained with help of eqs 5 and IO, and U l ( t )and U 2 ( t )with help of eq IO. The rate curves thus calculated are also shown in Figure 3 as solid lines. The agreement between the experimental and calculated results is quite good, even though no parameter was adjusted. If the concentration of the Cs ions in the solution is increased M (Figure 4), the rate curves observed are in to c = 1 X principle similar, but the two fractions of the ion exchanger approach their equilibrium faster, even though still considerably slower than the solution. Again we can calculate the rate curves in the same way as above. The amounts of ion exchanger were slightly different than in the preceding experiment (Q, = Q2 = 0.366 mequiv), but the same rate coefficients R* as above can be used. A good agreement between the experimental and theoretical rate curves is again obtained (see Figure 4). The faster rates observed for the ion-exchanger particles during the redistribution period ( t > 1000 s) is not surprising because in this experiment the condition requiring an excess of ion exchanger (see condition 7 ) does not hold as well as for the case c = 1 X IO4 M. If we had increased the amount of ion exchanger in the system also by a factor of IO, very similar rate curves as shown in Figure 3 would have been obtained again. Ion Exchange. To investigate the rates of ion exchange Cs+/H+, the resin beads in the H+ form were added to the CsCl solution, labeled with I3’Cs. If the two size fractions were put individually in the solution ( V = 50 mL), the rate curves shown in Figure 5 were observed. The quantities Q of ion exchanger used were 0.34 and 0.30 mequiv for the 0.4- and 0.8-mm fraction, respectively. The initial Cs concentration of the solution was 1 X M. From the observed equilibrium uptake of Cs by the ion exchanger, the separation factor (see eq 3) was obtained as a? = 3.5. To obtain from these rate curves the values of R* for the individual size fractions, we used a rate law that can be derived from eqs 6b and 8c for the limiting case XI,A
