J . Phys. Chem. 1988, 92, 4670-4676
4670
Generalized Two-Pseudophase Model for Ionic Reaction Rates and Equilibria in Micellar Systems. Hexachlorolridate(1V)-Iron( II ) Electron-Transfer Kinetics in Cationic Micelles Claudio Minero,* Edmondo Pramauro, Dipartimento di Chimica Analitica, Universitri di Torino, via P . Giuria 5, 101 25 Torino, Italy
and Ezio Pelizzetti* Istituto di Chimica Fisica Applicata, Universits di Parma, via Spezia 73, 431 00 Parma, Italy (Received: March 9, 1987; In Final Form: February 1 , 1988)
A general phenomenologicalapproach based on (i) the two-pseudophaseapproximation, (ii) a proper definition of the transfer constants between the pseudophases in term of molar fractions, (iii) an explicit hypothesis that relates the number of bound molecules or ions to the surfactant concentration, and (iv) a semiempirical calculation of the activity coefficients of bound charged species at the micellar surface permits prediction of the reactivity as a function of micelle and added salt concentrations. The application of this approach to limiting cases leads to the equations already known in the literature. Kinetic experiments were designed to test the model through simplification of the proper formulas. Good prediction of the observed rates for the reaction of hexachloroiridate(1V)-iron(II), in the presence of hexadecyltrimethylammonium (CTA) chloride and sulfate, is obtained, and the dissociation degree of the micelle is found comparable with literature values.
Introduction In recent years, reactions that take place at charged surfaces of self-assembling colloids in solution (e.g., micelles, microemulsions, vesicles) have been investigated with increasing interest.l-3 Micellar effects upon reaction rates and equilibria have generally been discussed in terms of a pseudophase Whereas the distribution of uncharged reagents can be measured directly or estimated from the binding of model compounds, thus allowing straightforward analysis of reaction rate effects, for charged reagents the competition of reactive and inert ions was soon recognized to be crucial in the analysis of reactions involving ions.' Since counterions also bind specifically to ionic micelles, it is evident that ionic binding could conceptually be separated into electrostatic and nonelectrostatic terms. The non-Coulombic contribution to binding is larger with polarizable and lowcharge-density ions.8 The binding of some ions to micelles can be estimated experimentally (electrochemical, photochemical, self-diffusion, and other techniques), but direct methods are not a~ailable.~ The extent of ion binding can be calculated from the surface potential of a micelle or evaluated through proper standard reactions (e.g., indicator equilibriumg). This approach has been proved to be useful for some specific reactions involving aquo
(1) Fendler, J. H. Membrane Mimetic Chemistry; Wiley: New York, 1982. (2) Bunton, C. A.; Savelli, G. Adu. Phys. Org. Chem. 1986, 22, 224. (3) Meisel, D.; Pelizzetti, E. Prog. Reacr. Kinet., in press. (4) Mittal, K. L. Micellization, Solubilization and Microemulsions; Plenum: New York, 1977. (5) Martinek, K.; Yatsimirski, A. K.; Levashov, A. V.; Berezin, I. V. In ref 4, Vol. 11, p 489. ( 6 ) Berezin, I . V.; Martinek, K.; Yatsimirski, A. K. Russ. Chem. Rev. (Engl. Transl.) 1973, 42, 187. (7) Romsted, L. In ref 4, Vol. 11, p 509. (8) Gamboa, C.; Sepulveda, L.; Soto, R. J . Phys. Chem. 1981,85, 1429. (9) Fernandez, M. S.; Fromherz, P. J. Phys. Chem. 1977, 81, 1755. (10) Diekmann, S.; Frahm, J. J. Chem. SOC.,Faraday Trans. I 1979, 75, 2199
(1 1) Pramauro, E.; Pelizzetti, E.; Diekmann, S.; Frahm, J. Inorg. Chem. 1982, 21, 2432.
0022-3654/88/2092-4670$01 S O / O
The other approach treats the micelle as an ion-exchange resin in which reactive ions compete with inert ones at the micellar s ~ r f a c e . ~This . ~ ~model has been adopted by a number of research groups for a variety of reactiom2 Among the hydrophilic ions, OH- was the widest investigated. It is worth mentioning that the reported ion selectivity coefficients exhibit significant differen~m.2~ Despite the large applicability of the ion-exchange model, some failures have been r e p ~ r t e d ' ~and J ~ some problems arise from the assumption of the constancy of the degree of dissociation of the micelles, irrespective of the type of counterions and of the ideal behavior of ions (this may not be the case for some species at the micellar surface even when their analytical concentration is small). Another problem arises in treating reactions where ions of different charge are present.* In this paper the effect of the micellar medium on the kinetics of the outer-sphere electron-transfer reaction IrC162- Fez+ IrCls3- Fe3+ (1)
+
-
+
was studied in the presence of hexadecyltrimethylammonium (CTA) micelles, and of added inert salt, namely, NaCl and Na2S04. The nomenclature for the chemical species and for the symbols used in the present work are reported under List of Symbols. Considering the electrostatic interactions of the reagents with the micelle, one expects the reactive counterion to be distributed between the pseudomicellar phase and the bulk and the reactive co-ions to be mainly confined in the bulk. Inhibition of the reaction rate is thus expected. Experimental Section Reagents and Procedures. A stock solution of iron(I1) chloride was prepared by dissolving a pure iron wire in hydrochloric acid (RPE Carlo Erba). Iron(I1) sulfate (RPE Carlo Erba) and iron(I1) chloride solutions were standardized by oxidimetric titration. Sodium hexachloroiridate(1V) (Merck) was used as (12) Minero, C.; Pramauro, E.; Pelizzetti, E.; Meisel, D. J . Phys. Chem. 1983, 87, 399. ( 1 3 ) Vincenti, M.; Pramauro, E.; Pelizzetti, E.; Diekmann, S . ; Frahm, J . Inorg. Chem. 1985, 24, 4533. (14) Quina, F. H.; Chaimovich, H. J . Phys. Chem. 1979, 83, 1844. (15) Nome, F.; Rubira, A. F.; Franco, C.; Ionescu, L. G. J . Phys. Chem. 1982,86, 1881. (1 6 ) Rodenas, E.; Vera, S . J . Phys. Chem. 1985, 89, 5 13.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 16, 1988 4671
Hexachloroiridate( 1V)-Iron(I1) Electron Transfer
TABLE I: Experimental cmc Values as a Function of Ionic Strength for the Two Added Salts Considered in This Work
1 0.01
cI-
0.02 2 x 10-4
5 x 10-4 3 x 10-4
502-
0.04 1.5 x 10-4 1.3 X lo4
1.8 X lo4
0.10 8X 5 x 10-5
0.07 1 x 10-4 9 x 10-5
0.15 6X 3 x 10-5
0.20 5 x 10-5 2 x 10-5
0 7 0 1
4-
2-
0
L
0
1
5
3
7
[SOgl.lOz M
Figure 3. Same legend as Figure 2 but with added Na2S04: (D) 3 X lo-) M CTA-S04; (0)5 X M CTA-S04.
0
5
IO
15 cd'
20
lo3 M
f
Figure 1. Second-order rate constant (L mol-' s-I) observed as function of the surfactant concentration at fixed added salt: ( 0 ) Na2S04;(W) NaC1. Concentrations of the added salt are indicated. Dashed lines refer to the behavior calculated according to eq 18.
0
.05
.I
.15
.2
[I1
Figure 4. Second-order rate constant (L mol-l s-I) without added surfactant observed for reaction 1 as a function of ionic strength with NaCl (W) and Na2S04(A). The fit with eq 2 is reported as a dashed line.
0
.05
.1
.I5
.2
[Clj M
Figure 2. Second-order rate constants (L mol-' s-I) observed as function of added NaCl at different fixed surfactant concentration: (A) 1 X lo-' M CTACI; (0)5 X M CTACI; (W) 1 X M CTACI. Dashed lines are as in Figure 1.
received, and stock solution concentrations were estimated by electronic spectroscopy with a Cary 219 spectrophotometer. Hexadecyltrimethylammonium chloride and sulfate were prepared from the corresponding bromide by passing the solution through an ion-exchange resin and then were recrystallized. Complete exchange was checked by ion chromatography (Biotronik IC 5000). Water used was doubly distilled. A Dognon-Abribat tensiometer with a platinum blade was used to measure surface tension. Calibration of the apparatus was checked by measuring the interfacial tension of triply distilled water, and cmc values as a function of the added salt are reported in Table I. Kinetics runs were carried out on a Durrum-Gibson stopped-flow spectrophotometer following the disappearance of IrC12- at 485 nm. All kinetic experiments were performed at 25.0
f 0.1 "C. The experimental reactant concentrations were [Fez+]
= (0.8-1.2) X lo4 M, [IrC16"] = (0.8-1.2) X M in the presence of micelles, and [Fez+] = (1-4) X M, [IrC16*-] = (0.5-1) X lo-' M for the homogeneous solution. Hydrogen ion activity was maintained at pH 3 by adding the acid corresponding to the salt. Kinetic Results. The experimental second-order rate constant in the presence of micelles are reported in Figure 1 as a function of the surfactant concentration at fixed molar added salt and in Figures 2 and 3 as a function of the salt concentration at fixed surfactant concentration. The inhibition factors in presence of micelle of CTA are of 2-4 orders of magnitude, which are quite relevant with respect to previously investigated Figure 4 shows the experimental second-order rate constants for reaction 1 as a function of the ionic strength in the absence of surfactant for the two salts used. No significant differences are observed between the two different salts. The salt dependence of k," (in the absence of micelles) is represented very well through a conventional Brernsted-Bjerrum treatment17a,giving values of r# = 6.1 A and of the specific rate constant at zero ionic strength ko = 6 X lo7 L mol-' s-', Alternatively, using an expression based
4672
The Journal of Physical Chemistry, Vol. 92, No. 16, 1988
on the potential energy function of Debye and the Marcus theory of outer-sphere electron-transfer reaction^"^ In k,” = In k, - A
ZLZM
(
(rL + rM)
exp(-xrL) ( l + XrM)
+ ~xP(-x~M) ( l + xrL)
)
(2)
where A = 3.576 at 25 O C . Using values of rFe2+= 3.6 A and = 4.3 8, I E b and setting k , = 1.85 X lo5 L mol-’ s-] gives rIrCI62an excellent fit of the data (see the dashed line in Figure 4). The value of the specific rate constant at infinite ionic strength can be of some interest for comparison purposes with the specific rate k, on the micellar surface where typical ionic strengths are quoted in the range 5-12 M.19
Theory If one assumes, as usual, (a) a two-pseudophase model of the solution and (b) that equilibria between the two pseudophases are always maintained, then for reaction 1, the kinetic data can be analyzed according to
where J indicates any reactive counterion or co-ion in solution and [Jl’s are their molar concentrations in the whole solution, subscripts b and f mean respectively in the micelle = bound and in the bulk = free, CJ’s are the analytical total concentrations, k, and k, are the specific rate constants in the pseudophases, and c d is the concentration of the micellized surfactant. The evaluation of the terms [J]/CJ is an equilibrium problem which can be tackled successfully if a satisfactory thermodynamic theory of the system is provided. It was shown20that for large micelles the pseudophase approximation is equally valid in the interpretation of kinetic data as in the discussion of the equilibrium. The equilibrium condition for a counterion J in the two pseudophases is lboJ
+ RT In x& + zJF$i = pro, + RT In x,J + zJF$,J
(4)
where $f is the averaged intermicellar potential, $c the surface micellar potential “sensed by the counterionsW2’in the two pseudophases, and F is the Faraday constant. The two-pseudophase approximation is here considered in order to simplify the following algebra, although other ways can be followed.22 The mass balance CJ = [Jf] [Jb], and the molar fractions xJdefined in each pseudophase as xd = nd/Cnb and x i = n,J/Cnr transform (4) into
+
where Ki = Ki”KjEW
KJEis the electrostatic-transfer constant, KIDis the nonelectrostatic (17) (a) Clark, D.; Wayne, R. P. In Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Eds.; Elsevier: Amsterdam, 1969; Vol. 11, p 326. (b) Feinberg, B. A,; Ryan, M. D. J . Inorg. Biochem. 1981, 15, 187. (18) (a) Sutin, N. Annu. Rev. Nucl. Sci. 1962, 12, 285. (b) Pelizzetti, E.; Mentasti, E.; Pramauro, E. Inorg. Chem. 1978, 17, 1181. (19) (a) Israelachvili, J. N. In Intermolecular andSurface Forces; Academic: New York, 1985; Chapter 12. (b) Jonsson, B.; Wennerstrom, H.; Halle, B. J . Phys. Chem. 1980, 84, 2179. (20) Shincda, K.; Hutchinson, J. J . Phys. Chem. 1962, 66, 577. (21) Lissi, E. A.; Abuin, E. B.; Sepulveda, L.; Quina, F. H. J. Phys. Chem. 1984, 88, 81. (22) In a cell model the chemical potential of any mobile ions is wfl = wr0J
+ kBTIn (Ti /Jexp(-z @ ( r ) / k B rdV) ) where TiJ is the number of J ions in the volume V od the cell.2dproJcorresponds to the rhs of (4) if x and 6 are mean quantities as required from a two-state model.
Minero et al. one, and W = Cnf is a constant in diluted solutions (its value is 55.5 if nr is expressed in mol L-l). K,” incorporates factors such as solvent structure changes, changes in polarizability, and specific counterion-head-group interactions. In eq 5a lateral interactions between bound species are not accounted for. It is noteworthy that in eq 5 binding constants are defined in terms of molar fractions and that the standard state as defined in eq 4 and 5 is at infinite ionic strength, where KJE = 1. As a consequence of the finite dimension of the micellar pseudophase, two different limiting hypotheses can be considered in order to solve Crib in eq 5a with a closure related to the surfactant concentration. Two additional assumptions will be introduced below which concern the relationship between the surfactant concentration and the total number of bound molecules or ions. The new variables a or u are added to the ones of eq 5 in order to obviate a detailed discussion of the term xnb. This term depends on quantities, such as water of hydration, which are hardly computable without an accurate microscopic model of the micellar surface. The values of a and u, as they will be defined, depend upon the micellar surface composition and are formally a function of the type and the number of bound molecules. Only in the case of a large excess of one bound species are a and u actually constants, although their values are specific for the type of molecule or ion considered. The two assumptions are (a) The total number of bound molecules is proportional, through the parameter u, to the micellized surfactant concentration
Crib = ucd
(6a)
Using eq 5 and 3, we obtain
where K,‘ = l/KJ. This equation is similar in its functional form to the well-known equation due to the “Russian School”.6 The definition of u in eq 6a makes eq 6b suitable for noncharged substrates; this approach and its limits have been extensively discussed, as can be seen in the literat~re.~.’When u is equal to the micellar volume, u, eq 6b is identical with the Berezin equation.6 (b) The total charges due to bound ions and molecules on a micelle are related to its effective charge, and the degree of ionization or the charge density of the micelle is constant. From the balance of charges at the micellar surface, it follows
C z ~ n d= (1 - a)cd J
(7)
The condition (7) was proved to be quite useful in analyzing data from light scattering e ~ p e r i m e n t s . In ~ ~those cases, assuming a constant degree of ionization of the micelles, permits estimation of reasonable values of a and good fits of the data over a fairly large range of surfactant and salt concentration. The hypothesis (7) works well for many types of polyelectrolyte and micellar systems,25has been supported by a modified Poisson-Boltzmann treatment,26 and has been discussed many times in terms of its applicability in kinetic experiments2’ The solution of system 5 and (7) and the suitable definition(s) of the exchange constant(s) lead to an equation for estimating (23) Marcus, R. A. J . Chem. Phys. 1955, 23, 1057. (24) (a) Corti, M.; Degiorgio, V. J . Phys. Chem. 1981, 85, 711. (b)
Minero, C.; Pramauro, E.; Pelizzetti, E.; Degiorgio, V.; Corti, M. J . Phys. Chem. 1986, 90, 1620. (c) Nicoli, D. F.; Dorshow, R. B.; Bunton, C. A. In Physics of Amphiphiles, Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; p 429. (25) (a) Manning, G. S.Q. Rev. Biophys. 1978.11, 179. (b) Lindstrom, B.; Khan, A,; Soderman, 0.;Kamenka, N.; Lindman, B. J. Phys. Chem. 1985, 89, 5313. (26) Gunnarson, G.; Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1980, 84, 3114. (27) (a) Bunton, C . A,; Gan, L. H.; Moffatt, J. R.; Romsted, L. S.; Savelli, G. J . Phys. Chem. 1981, 85, 4118. (b) Bonilba, J. B. S.; Chiericato, G.; Martius-Franchetti, S.M.; Ribaldo, E. J.; Quina, F. H. J . Phys. Chem. 1982, 86, 4941. (c) Chaimovich, H.; Cuccovia, I. M.; Bunton, C. A.; Moffatt, J. R. J . Phys. Chem. 1983, 87, 3584.
Hexachloroiridate( 1V)-Iron( 11) Electron Transfer the ionic exchange of two or more competing counterions. Other conditions such as acid-base equilibrium between counterions can be easily introduced. For example, in the simplest case of two counterions L and S the classical equation of the ionic exchangeI4 is obtained:
+
where cs= (cd 4- cmc)/z, Csalt;K*Lls = (Ks)zL/(KL)ps and KL and Ksrefer to counterions L and S according to eq 5b. The last term in ( 8 ) is not present in the equations reported in the litera t ~ r e , ' ~but * ~ a* dependence on c d identical with that of the last term of (8) is obtained by defining an empirical selectivity coefficient KLIs in terms of local molar concentrations. The primary differences are due to the definitions of the equilibrium constants. Some criticisms are reported in literature about the equilibrium constants expressed in molarity units.29 Through simple measurements of counterion activities, it was showng0that two kinds of counterions can independently associate with or dissociate from micelles, or, in other words, that there is no strict 1 : l correspondence between an association of L and a dissociation of S , or vice versa, even when they have the same charge. Thus we prefer to avoid the exchange model formalism in the definition of exchange constants and to use simple ratios between transfer constants. In the general case of two competing counterions L and S, and two competing co-ions N and M, from eq 5 and 7, eq 9 is obtained:
( 1 - ff)Cd D ZLCL
+
The Journal of Physical Chemistry, Vol. 92, No. 16, 1988 4673 where Q is the net free charge on the micellar surface, Q = aNaw; K is the Debye-Huckel reciprocal length, B, = 2/4*q,kBT is the Bjerrum length, e is the electronic charge, and rm is the micellar radius. The hypotheses on which eq 10 is based are questionable, since both the micelle and the ion are treated as point charges and the approximation is satisfactory only if the condition z e 4 / k B T > KLN,KLM hold in eq 9 and UCdKL‘ >> 1, UCdKM’ cyso1 and are not very far from values obtained by light s ~ a t t e r i n g . ~ ~ (34) Bunton, C.A.; Rornsted, L. S.;Savelli, G. J . Am. Chem. SOC.1979, 101, 1253.
Hexachloroiridate( 1V)-Iron(I1) Electron Transfer TABLE 111: Literature Values for Parameters of ion C1-
so:-
The Journal of Physical Chemistry, Vol. 92, No. 16, 1988 4675
Eq 18 and the Exchange Constant Calculated from Literature Data
A.
Nag@
I,,
9 5‘ 1OOb 100’ 127’
26” 29b 27.2’
a
0.27b 0.40-0.45d 0.07b
0.36c 0.22d 0.26‘
KC,lSO,
0.37c 0.50e 0.3Sb
0.32f 0.01-0.05*
0.46g
“Estimated from eq 3-6 of ref 41. bReference 35. ‘Reference 37. dReference 39. ‘Reference 38. ’Reference 40. ECalculated from Table I of ref 8. *Calculated from Table I1 of ref 8. ’Reference 36. ’Reference 42. 1.5 VI
l$
> 1
0.5
0
r
3 1
1
$. 2
, /
/e’
.//O
0
1
2
x .lo+
2
3
Figure 5. Plot of experimental kinetic data according to eq 18: (A) refers to chloride; (B) refers to sulfate. Y is the Ihs term and X the rhs bracketed term in eq 18. Values of parameters, slopes, and intercepts are reported in Table 11.
Intercepts, KLso values and a values are used to calculate the rate constants in the presence of micelles and salt. The predicted behavior is reported in Figures 1-3 as dashed lines. The agreement with experimental points is good. We note that the rate increase obtained with added sulfate (Figure 3) is fit beautifully and conclude that the hypotheses made in deriving eq 18 are strongly supported by the quality of the fit on the two set of data. The values listed in Table I1 suggest some other considerations. The “hydrophobic” exchange constants calculated from the slopes (KL1$ = 1.04 X lo4 and KLlw,O = 1.42 X IO“) indicate a stronger partitioning of IrC162-with respect to chloride or sulfate ions, as expected. By comparison of the exchange constants for chloride and sulfate, one concludes that the hydrophobic binding constant of the former is 1.4 f 0.2 greater than that of the latter. No literature values of these hydrophobic constants are reported that can be compared with this result, although it seems sensible because its magnitude is not too high and sulfate ion, through hydrogen bonds, can show a more marked hydrophilic character than the chloride ion. The exchange constant KLs takes into account the electrostatic contributions and is salt-dependent. The values of Kcllso, range from 0.25 at I = 0.01 to 1.3 at I = 0.2 and are comparable with the reported exchange constants which include implicitly the electrostatic contribution. In fact, from data we (35) Dorshow, R. B.; Bunton, C. A.; Nicoli, D. F. J. Chem. Phys. 1982, 76,175; J. Phys. Chem. 1983, 87, 1409. (36) Biresaw, G.; McKenzie, D. C.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1985,89, 5144.
calculated values of K C I I S = O ~0.32-0.46. From intercept and slope values one obtains the ratio (k,K~’o/v)~~/(kmKM’o/v)sol = 2 f 0.8. This supports the hypothesis that, if values of k, are almost the same, there will be an increase in the presence of chloride of the effective binding to micelles of iron(I1) with respect to sulfate medium. However, this effect is small and confirms that the use of eq 18 is correct, although there are some little specific interactions in homogeneous solution.44 It is noteworthy that the absolute values KM’ O could be estimated by assuming k, = k, (see eq 2). Fitting of kinetic data can be further improved by allowing NW to grow and a to diminish with increasing salt concentration. In this way a stronger linearization of the kinetic data according to eq 18 for chloride counterion is obtained. Yet, we found by numerical calculation that these two effects largely compensate, and we preferred to avoid the introduction of further hypotheses and parameters in the fit. In the framework of the pseudophase model, a promising approach for calculating the partitioning of reagent ions is the numerical solution of the Poisson-Boltzmann (PB) equation in a cell model. The limitations of this treatment in analyzing kinetic data are (a) the need for imposing as adjustable parameter the volume of the pseudomicellar phase or of the micellar radius45 and (b) the time-consuming calculation which makes it scarcely suitable for fitting many sets of experimental data. The numerical solution of the PB equation modified to account for the finite dimensions of the ions (MPB)46was applied t a kinetic data obtained for reaction 1 in the presence of SDS micelles.” The calculated ratio [Fe(II)],/ [Fe(II)lf was quite good in explaining kinetic data. The use of the MPB in the case of CTACl and CTA-SO4 is more problematic. In fact, one expects that the interaction of IrC162-ion with the CTA micelle is in part hydrophobic (or, in other words, “specific”), in contrast to that of the hydrated Fez+ ~ ~added a Volmer-type ion with SDS. Recently, B u n t ~ nhas adsorption isotherm to PB treatment. A model where such an isotherm is not put “a priori” can be obtained by using eq 9 and, instead of eq 14, by calculating with the MPB the quantities KjE.
Conclusions A two-pseudophase approach which implies the proper definition of the transfer constants, the separation of hydrophobic and electrostatic contribution, the calculation of activity coefficients of all the species at the micellar surface, and, finally, the balance of charges at the micellar surface was developed. The model gives an exceedingly good fit to the experimental kinetic data obtained in the presence of two hydrophilic counterions over a significant (37) Sepulveda, L.; Cor&, J. J . Phys. Chem. 1985, 89, 5322. (38) Bunton, C. A.; Romsted, L. S.; Sepulveda, L. J . Phys. Chem. 1980, 84, 2611. (39) Fabre, H.; Kamenka, N.; Khan, A.; Lindblom, G.; Lindman, B.; Tiddy, G. J . Phys. Chem. 1980, 84, 3428. (40) Bartet, D.; Gamboa, C.; Sepulveda, L. J. Phys. Chem. 1980, 81, 272. (41) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983, 87, 1264. (42) Wasik, S . P.; Hubbard, W. D. J. Res. Natl. Bur. Stand. A 1964, 68, 359. (43) Cordes, E. H.; Gitler, C. Prog. Bioorg. Chem. 1973, 2, 1. (44) Sillen, L. G.; Martell, A. E. In Stability Constants of Metal-Ion Complexes; Special Publications No. 17 (1964) and No. 25 (1971); The Chemical Society: London. ( 4 9 Bunton, C. A.; Moffatt, J. J. Phys. Chem. 1986, 90, 538. (46) Frahm, J.; Dieckmann, S. J. J . Colloid Interface Sci. 1979, 70, 440.
4676
J . Phys. Chem. 1988, 92, 4676-4679
range of inert salt and surfactant concentration. It is noteworthy that the only parameters, besides the micellar rate constant which cannot be measured independently, are the ratios of the hydrophobic contributions to the transfer constants. Moreover, our generalized two-pseudophase approach reduces to the well-known equations of Berezin and of ionic-exchange models under limiting conditions and provides a very reasonable estimate of the degree of dissociation of the micelle from the fit of the kinetic data. Acknowledgment. We are grateful to CNR, MPI, and the European Research Standardization Group of the U S . Army under Contract No. DAJA 45-85-C-0023 for support of this work. We acknowledge pleasant discussion with Prof. L. Romsted during the Euchem Conference in Assisi, June 1987.
analytical concentration of the micellized surfactant (mol L-1)
analytical concentration of the J species (mol L-]) analytical concentration of the added salt; Csrefers to the analytical concentration of the inert ion as in its definition in eq 8 distribution coefficient defined in eq 9 ionic strength generic species J total transfer constant of the j species hydrophobic-transferconstant electrostatic-transfer constant for all kinds of K is the reciprocal of K, K' = 1 / K KLM = KL/KM KLN
=
KL/KN
K , = KLIKS
List of Symbols Boltzmann constant second-order rate constant on the micellar surface second-orderrate constant observed in the presence of surfactant value of k," in the aqueous pseudophase at the ionic strength calculated by using eq 15 normalized k,; k,' = k,/( 1 - cdu') observed second-order rate constant in homogeneous solution specific second-order rate constant at zero ionic strength specific second-order rate constant at infinite ionic strength number of molecules or moles radius specific micellar reaction volume (L mol-I) specific micellar volume (L mol-') mole fraction ionic charge constant in eq 2 constant in eq 14 Bjerrum length, BI = e2/4rtt,,kBT
"exchange constant" of U and V defined as the ratio Kv/Ku (also subscript) reactive counterion (Le., IrCI:-) (also subscript) reactive co-ion (i.e., Fez+) (also subscript) inert co-ion (Le., Na+) aggregation number of the micelle correlation coefficient (also subscript) inert counterion (Le., CI- or SO:-) =
aNaggr
bracketed quantity in the rhs term of eq 18 Ihs term in-eq 18 ionization degree of the micelle activity coefficient of the j species Debye-Hiickel reciprocal length proportionality constant in eq 6a potential subscript: bound or in the micellar pseudophase subscript: free or in the aqueous pseudophase standard state of nonelectrostatic (or hydrophobic) contributions, defined at infinite ionic strength subscript: refers to micellar phase or to micelle subscript: refers to aqueous phase
Surface Recombination Velocity Measurements of CdS Single Crystals Immersed in Electrolytes. A Picosecond Photoluminescence Study D. Benjamin and D. Huppert* Sackler Faculty of Exact Sciences, School of Chemistry, Ramat- Aviv, Tel- Auiu 69978, Israel (Received: June 8, 1987; In Final Form: January 19, 1988) The effect of solution composition and concentration on the luminescence decay profile is measured for CdS single crystals immersed in various aqueous solutions. The surface recombination velocity is strongly dependent on the ionic solution composition and concentration. The experimental data are explained in terms of the chemisorption of ions on the CdS surface.
Introduction In previous studies, the photocurrent and photopotential transients of CdSe and CdS photoelectrochemical cells were measured with time resolution of a few nanoseconds.'V2 It has been noticed that significant photocharge recombination occurred within shorter time periods, probably of the order of the charge separation time. To increase the time resolution and observe ultrafast recombination processes, we measured on the picosecond time scale the timeresolved photoluminescence of CdS crystals immersed in aqueous electrolyte solution^.^^^ The luminescence time dependence is determined by bulk and surface recombination rates, the carriers ( 1 ) Harzion, Z.; Croitoru, N.; Gottesfeld, S. J . Electrochem. SOC.1981, 128. 551. (2) Harzion, Z.; Croitoru, N.; Huppert, D.; Gottesfeld, S . J . Efectroanal. Chem. 1983, 150, 511. (3) Evenor, M.; Gottesfeld, S.; Harzion, Z.; Huppert, D. J . Phys. Chem. 1984, 88, 6213. (4) Huppert, D.; Gottesfeld, S.; Harzion, Z.; Evenor, M. Ultrafast Phenomena; Auston, D., Eisenthal, K. B., Eds.; Springer-Verlag: Berlin, 1984; Vol. IV, p 181.
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diffusion constant, and the laser absorption cross section. The mathematical analysis assumes a semifinite uniform semiconductor that is irradiated by a picosecond laser pulse. Electron-hole pairs are generated at a rate g(x,t) and immediately thermalized to give free carriers. The concentration of the excess carriers is very high and therefore eliminates the space charge region by complete band flattening. Also, due to this high excess carrier concentration, the bulk recombination obeys pseudo-first-order kinetics with a characteristic lifetime Q,. The time-dependent carrier concentration, An(x,t) = Ap(x,t) can be described by the continuity equation5
where D*, the ambipolar diffusion coefficient: is 1.2 cm2/s (ref 7-9) for CdS. The instantaneous luminescence intensity, taking (5) Vaitkus, J. J . Phys. Status Solidi 1976, 34A, 169. (6) Van Roosbroeck, W. Phys. Rev. 1953, 282, 91.
0 1988 American Chemical Society
