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Kinetic Study of the Oxidation of [Ru(NH3)5pz]2+ by [Co(C2O4)3]3- in AOT-Oil-Water Microemulsions and in CTACl Micellar Solutions R. de la Vega, P. Pe´rez-Tejeda, P. Lo´pez-Cornejo, and F. Sa´nchez* Departamento de Quı´mica Fı´sica, Facultad de Quı´mica, Universidad de Sevilla, C/ Profesor Garcı´a Gonza´ lez, s/n. 41012 Sevilla, Spain Received May 19, 2003. In Final Form: October 17, 2003
The kinetics of the electron-transfer reaction between pentaaminepyrazine ruthenium(II), [Ru(NH3)5pz]2+, and trisoxalatecobaltate(III), [Co(C2O4)3]3-, was studied in sodium bis(2-ethylhexyl)sulfosuccinateoil-water microemulsions and in micellar solutions of hexadecyltrimethylammonium chloride. The results can be interpreted by taking as a starting point the pseudophase model. However, the model must be modified according to the characteristics of the reaction media. In this way, two different equations result for this model in the two reaction media studied. The influence of the surface potential on the parameters (equilibrium and rate constants) of the model is shown.
Introduction Recently, there has been a growing interest in the study of processes under conditions globally referred to as restricted geometry conditions, that is, under conditions in which a given ligand is forced (or different ligands are forced) to remain, totally or partially, bound at the surface of some substrate. In a broad sense, the restricted geometry conditions encompass phenomena such as heterogeneous catalysis,1 enzymatic catalysis,2 reactivity in micellar systems3 and microemulsions,4 molecular machines,5 molecular electronics,6 trapping of substrates by polyelectrolytes,7 conformational changes of DNA induced by the binding of solutes,8 and so forth. The effects of the restricted geometry conditions on chemical reactivity can be considered by using different approaches. One of these approaches is to use the transition state theory,9,10 that is, to employ as a starting point the Bro¨nsted equation:11
γAγB k ) k0 γ*
(1)
In this equation, k is the actual rate constant, γA, γB, and γ* represent the activity coefficients of the reactants (A * Author to whom all correspondence should be sent. Tel.: +34954557175. Fax: +34-954557174. E-mail:
[email protected]. (1) Laidler, K. J. Chemical Kinetics; McGraw-Hill: New York, 1965; p 256 and following. (2) Reference 1, p 474 and following. (3) Prado-Gotor, R.; Jime´nez, R.; Pe´rez-Tejeda, P.; Lo´pez-Cornejo, P.; Lo´pez-Lo´pez, M.; Sa´nchez, A.; Muriel-Delgado, F.; Sa´nchez, F. Prog. React. Kinet. 2000, 25, 371. (4) Luisi, P. L., Straub, B. E., Eds.; Reverse Micelles; Plenum Press: New York, 1984. (5) Frey, E.; Vilfan, A. Chem. Phys. 2002, 284, 287 and references therein. (6) (a) Amendola, V.; Fabbrizzi, L.; Mangano, C.; Pallavinici, P. Acc. Chem. Res. 2001, 34, 488. (b) Colin, J. P.; Dietrich-Buchecker, C.; Gavin˜a, P.; Jime´nez-Molero, M. C.; Sauvage, J. P. Acc. Chem. Res. 2001, 34, 477. (c) Harada, A. Acc. Chem. Res. 2001, 34, 456. (7) (a) Gavryushov, S.; Zielenkiewicz, P.; J. Phys. Chem. 1999, 103, 5860 and references therein. (b) Gharibi, H.; Safarpour, M. A.; Rafati, A. A. J. Colloid Interface Sci. 1999, 219, 217 and references therein. (8) Hamelberg, D.; Williams, L. D.; Wilson, D. J. Am. Chem. Soc. 2001, 123, 7745.
and B) and the transition state (*), and k0 represents the rate constant at the reference state. The main problem with this approach is that the activity coefficients of ions cannot be measured;12 thus, it is not possible to have experimental data on γA and γB in the cases, such as the one in this work, in which the reactants are ionic species. On the other hand, the calculation of these activity coefficients, through statistical mechanical approaches, is an impossibly difficult task in systems in which restricted geometry conditions prevail. Probably, as a consequence of these difficulties, other approaches have become more used: generally speaking, reactivity under restricted geometry conditions has been rationalized using partition models, that is, by considering a ligand (reactant) distribution between different phases (or pseudophases). Assuming that this distribution is a rapid, diffusion-controlled process in relation to the characteristic reaction time, it can be considered to be at equilibrium, in such a way that the concentration in the different phases can be calculated. Then, assuming that the reactivity in the different phases is governed by a phase-specific rate constant, the measured rate constant can be expressed as a function of the rate constants characteristic of each phase and of the binding constant(s) of the reactant(s). Thus, in the case of the so-called pseudophase model,13 the following equation is found for a unimolecular process:
k)
kf + kbK[T] 1 + K[T]
(2)
In this equation, k is the actual rate constant, kf is the rate constant corresponding to the free, unbound reactant, kb is the same parameter for the bound reactant, that is, for the reaction under restricted geometry conditions, and K is the equilibrium constant for the binding of the (9) Bunton, C. A.; Robinson, L. J. Am. Chem. Soc. 1968, 90, 5872. (10) Lo´pez-Cornejo, P.; Jime´nez, R.; Moya´, M. L.; Sa´nchez, F. Langmuir 1996, 12, 4981. (11) Bro¨nsted, J. N. Z. Phys. Chem. 1922, 115, 337. (12) Koryta, J.; Dvorak, J.; Bohackova, V. Electrochemistry; Methuen: London, 1970. (13) Menger, F. M.; Portnoy, C. E. J. Am. Chem. Soc. 1967, 89, 4698.
10.1021/la0302089 CCC: $27.50 © 2004 American Chemical Society Published on Web 01/28/2004
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reactant, S, to the substrate:
Sf + T a Sb K)
[Sb] [Sf][T]
(3a) (3b)
Finally, [T] represents the concentration of the dispersed pseudophase (the substrate) to which the reactant is bound. In eqs 2 and 3b, the concentrations refer to the total volume of the system. The parameters kf, kb, and K appearing in equations of these kinds of models are generally thought to be true constants, that is, independent of the relative amounts of the phases or the reactant concentrations. However, this assumption implies an additional hypothesis on the equilibrium distribution mentioned above: the concentration of the partitioned reactant must be low enough to avoid the saturation of the dispersed (e.g., micellar) pseudophase.14 Indeed, even in this case, it is implicit that the presence of the reactant in the dispersed pseudophase neither encourages nor discourages the union of another reactant: in other words, binding of the reactant to the dispersed pseudophase is noncooperative in character.15 Moreover, to consider the binding constant and the rate constant characterizing reactivity in the dispersed pseudophase as true constants, one must assume that some characteristics (e.g., the shape, size, and charge of the micelles) are concentration-independent parameters. Thus, the assumptions implicit in the partition models do not always hold, particularly if a broad range of concentrations of the dispersed pseudophase are explored. Consequently, deviations from the predictions of these models (in their simple forms) are frequently found. This work was undertaken to consider in detail the questions previously mentioned. In this regard, we have studied the title reaction in micellar solutions of hexadecyltrimethylammonium chloride (CTACl) and in sodium bis(2-ethylhexyl)sulfosuccinate (AOT)-oil-water microemulsions. In both cases, we considered a broad range of concentrations of the dispersed pseudophase. Our main conclusion is that, although the pseudophase model, in its simple version as given in eq 2, is not adequate for the interpretation of the results, it can be adequately modified, taking into account the changes of the dispersed phase, in order to be applied to our cases. It should be mentioned that the reaction studied here has been previously studied in micellar solutions of sodium dodecyl sulfate (SDS).16 The results obtained in this work will also be considered here for comparative purposes. Experimental Section Materials. The reactants, [Ru(NH3)5pz]2+ (pz ) pyrazine), as perchlorate salt, and [Co(C2O4)3]3- (C2O42- ) oxalate anion), as sodium salt, were prepared and purified according to the procedures described in the literature.17,18 AOT and CTACl were obtained from Fluka. AOT was stored in a vacuum desiccator over P2O5 for several days before use. The solutions of CTACl were titrated by a standard procedure.19 (14) (a) Bunton, C. A.; Gan, L. H.; Hoffat, J. R.; Romsted, L. S.; Savelli, S. G. J. Chem. Phys. 1985, 85, 4118. (b) Bacaloglu, R.; Bunton, C. A.; Ortega, F. J. Phys. Chem. 1989, 93, 149. (c) Staedler, E.; Zanette, D.; Rezende, M.; Nome, F. J. Phys. Chem. 1984, 88, 1892. (d) Rodenas, E.; Vera, S. J. J. Phys. Chem. 1985, 89, 513; 1996, 90, 3414. (15) McGhee, J. D.; von Hippel, P. H. J. Mol. Biol. 1974, 86, 469. (16) de la Vega, R.; Lo´pez-Cornejo, P.; Pe´rez-Tejeda, P.; Sa´nchez, A.; Prado, R.; Lo´pez, M.; Sa´nchez, F. Langmuir 2000, 16, 7986. (17) Creutz, C.; Taube, H. J. Am. Chem. Soc. 1973, 95, 1086. (18) Cannon, R. D.; Stillmann, J. Inorg. Chem. 1975, 14, 2207. (19) Stolzberg, R. J. J. Chem. Educ. 1988, 65, 621.
Figure 1. Plot of conductivity versus log[CTACl]. The oil (decane) was obtained from Merck and dried over a 4-Åtype molecular sieve. This molecular sieve was activated by heating it at 200 °C under a reduced pressure for several hours and then cooled in vacuo over silica gel. The water used in the preparation of the solutions has a conductivity ≈ 10-6 S m-1 and was deoxygenated before use. Kinetic Measurements. Kinetic runs were carried out in a stopped-flow spectrophotometer from Hi-Tech or in a manual mixing system from Hi-Tech coupled to a Hitachi 150-20 UVvisible spectrophotometer. The kinetics were monitored by following the disappearance of the ruthenium complex at 477 nm. At this wavelength, the difference between the absorption coefficient corresponding to both complexes is a maximum. The concentrations of the reactants were 2 × 10-5 and 5 × 10-4 mol dm-3 for the ruthenium and cobalt complexes, respectively, in the studies in CTACl solutions and 8 × 10-5 and 4 × 10-3 mol dm-3, respectively, in the case of microemulsions. The concentrations of the reactants refer to the total volume of the solutions in the case of the CTACl micellar solutions and to that of the aqueous phase in the case of microemulsions. They were selected for the best working conditions in each reaction medium. The preparation method of the microemulsion solutions was as follows: First, a solution of AOT was prepared in oil. The surfactant concentration was maintained at a fixed value of 0.2 mol kg-1 in all the experiments. Then, adequate amounts of aqueous solutions of the reactants (to obtain the desired value of W) were added in separated vessels to the AOT solution. After these mixtures were shaken to obtain perfectly clear microemulsions, they were mixed in the reaction cuvette. The temperature was maintained at 298.2 ( 0.1 K in all the experiments. Pseudo-first-order rate constants were obtained from the slopes of the plots of ln(At - A∞) versus time, where At and A∞ were the absorbances at times t and when the reaction was finished. These plots were good straight lines for at least four half-lives. All the experiments were repeated at least five times. The estimated uncertainty in the rate constant was less than 5%. Determination of the Critical Micellar Concentration (cmc). We tried to find the cmc corresponding to the CTACl solutions through conductivity measurements. To reproduce the same conditions as those in the kinetic experiments, the solutions of CTACl contain the cobalt complex at the same concentration employed in the kinetic runs. Unfortunately, solubility problems did not permit us to work in the [CTACl] range 9 × 10-5 e [CTACl] e 5 × 10-3 mol dm-3. Thus, an extrapolation of the lines corresponding to the plots of conductivity versus log[CTACl] (see Figure 1), to obtain the cmc, did not permit an accurate determination of this parameter. For this reason, the only information obtained was that the cmc must be in the previously mentioned range of concentration of the surfactant. Bearing this in mind, the cmc value was considered an adjustable parameter of the fits (see the following).
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Table 1. Rate Constants for the Reaction [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- in CTACl Solutions 102[CTACl] (mol dm-3)
103kobs (s-1)
102[CTACl] (mol dm-3)
103kobs (s-1)
0 0.252 0.256 0.259 0.262 0.265 0.290 0.300 0.360 0.340 0.400 0.560
11.1 1.95 1.76 1.64 1.61 0.89 0.77 0.53 0.43 0.46 0.39 0.35
0.760 0.800 1.00 1.54 2.00 2.52 3.20 7.10 10.0 13.8 17.0 20.0
0.29 0.27 0.30 0.37 0.49 0.59 0.63 1.29 2.29 2.88 3.66 4.91
Table 2. Rate Constants for the Reaction [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- in AOT-Decane-Water Microemulsions at [AOT] ) 0.2 mol kg-1 W
kobs (s-1)
W
kobs (s-1)
W
kobs (s-1)
12.2 13.2 14.5 15.2 16.8
1.32 1.26 1.19 1.16 0.97
18.3 20.4 25.1 31.0
0.79 0.76 0.48 0.32
35.4 40.0 42.8 48.0
0.25 0.19 0.20 0.21
Results The results of kinetic runs are given in Tables 1 and 2 for CTACl and AOT-oil-water microemulsions, respectively, as pseudo-first-order rate constants.
Figure 2. Plot of kobs (s-1) of the process [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- versus CTACl concentration. The points are experimental data, kobs, and the line is the best fit obtained by using eq 8.
In fact, eq 4 can be obtained from eq 1. Thus, taking as the reference state for the solutes (reactants and transition state) the aqueous pseudophase in contact with the micelles (which, of course, contains the monomers of the surfactant at the cmc), as pointed out by Ise et al.,23 ko ) kf. Thus,
γAγB k ) kf γq
Discussion (a) CTACl Micellar Solutions. As can be seen in Table 1 (see also Figure 2), kobs first decreases on increasing CTACl concentration, reaching a minimum. After this minimum, kobs increases as the CTACl concentration does. This variation of kobs is qualitatively similar to that observed for this reaction in the presence of micelles of SDS (see ref 16). Obviously, this behavior cannot be explained by the pseudophase model, in its more simple version, that is, considering that the parameters of the model are true constants, because this model predicts an asymptotic behavior, with k f kb at the higher CTACl concentrations (see eq 2). Consequently, it is necessary to conclude that these parameters (or some of them) must depend on the concentration of the surfactant. Indeed, as mentioned in the introductory section, eq 2 corresponds to a true firstorder process, and here we are dealing with a second order (bimolecular) reaction. In this regard, however, it can be shown that eq 2 is still valid under the circumstances prevailing in our study, that is, when one of the reactants has the same charge sign as the micelles (here the ruthenium complex) in such a way that this ion remains mostly in the aqueous pseudophase.20 In this case, as we showed in a previous paper, eq 2 can be put as21
1 + Kq[T] k ) kf 1 + KCo[T]
(4)
In this equation, kf is the rate constant in the aqueous pseudophase, KCo is the binding constant to the micelles of the cobalt complex, and Kq is the binding constant of the transition state.22 (20) Lo´pez-Cornejo, P.; Sa´nchez, F. J. Phys. Chem. 2001, 105, 10523. (21) Muriel-Delgado, F.; Jime´nez, R.; Go´mez-Herrera, C.; Sa´nchez, F. Langmuir 1999, 15, 4344.
(5)
and using the equation for the activity coefficients of the participants,21
γi )
1 (i ) A, B, *) 1 + Ki[T]
(6)
the Bro¨nsted equation gives
1 + Kq[T] k ) kf (1 + KCo[T])(1 + KRu[T])
(7)
where KCo is the binding constant corresponding to the cobalt complex and KRu is the one corresponding to the ruthenium complex. Bearing in mind that the latter has the same charge sign as the micelles, KRu will be small, in such a way that KRu[T] , 1.24 Consequently, in the present case
1 + Kq[T] k ) kf 1 + KCo[T]
(8)
In fact, the data corresponding to the lower concentrations of CTACl (up to 3.4 × 10-3 mol dm-3) can be fitted to eq 8 (see Figure 2) with the following values of the param(22) Equilibrium for the binding of the transition state is a consequence of the assumptions of the transition state theory and pseudophase model. According to the transition state theory, the reactants in each pseudophase are in equilibrium with the transition state formed from them in each phase. On the other hand, according to the pseudophase model, the reactants in one or the other of the pseudophases are in equilibrium. This implies according to the zero law of thermodynamics that the transition states in one or the other of the pseudophases are also in equilibrium. (23) Ise, N.; Okubo, T.; Shigeren, K. Acc. Chem. Res. 1982, 15, 171. (24) This is equivalent to say that γRu ) 1. This follows from the fact that this ion, because of its charge, remains mostly in the aqueous pseudophase.
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Table 3. Surface Potential (Ψ) Values at Different CTACl Concentrations 103[CTACl] (mol dm-3)
Ψ (mV)
103[CTACl] (mol dm-3)
Ψ (mV)
0.0 15.4 20.0 25.2 32.0
183.9 180.8 178.3 175.4 171.8
71.0 100 138 170 200
153.5 142.5 131.3 124.8 121.1
eters: kf ) 2.7 × 10-3 s-1, Kq ) 122 mol-1 dm3, KCo ) 4700 mol-1 dm3, and cmc ) 2.4 × 10-3 mol dm-3. However, as can be seen in Figure 2 at the higher concentration of CTACl, the values of kobs are above the curve corresponding to this fit. This behavior can be rationalized considering that, at these values of the surfactant concentration, the surface potential of the micelles changes as a consequence of counterion condensation. These surface potentials (from ref 25) are given in Table 3. Taking into account that the cobalt complex and the transition state are charged species, their binding constants will depend on the surface potential of the micelles. Thus, we ascribe the raising part of the experimental points in Figure 2 to a decrease of these binding constants. This decrease of the binding constants corresponds obviously to a change of the free energy of the process 3a. This free energy can be written as the sum of two contributions: (i) a potential-independent contribution, ∆Gnel (nonelectrostatic or intrinsic) and (ii) a potential-dependent (or electrostatic) contribution, ∆Gel:
∆G ) ∆Gnel + ∆Gel
(9)
The latter contribution can be expressed as
∆Gel ) zRFΨ
(10)
where z is the charge of the species, Ψ the surface potential, and R gives the fraction of the surface potential (determined with a given probe) that determines ∆Gel. This parameter is introduced because the location at the interface of the probe used in the determination of Ψ is not necessarily the same as those of the reactants and transition state. If K0, the nonelectrostatic binding constant, is defined as
K0 ) e-∆Gnel/RT
(11)
it follows from previous equations that
K ) K0e-zRFΨ/RT
(12)
For concentrations of CTACl in which Ki[CTACl] . 1, it follows from eqs 8 and 12 that
(K0)* e-z*R*FΨ/RT k ) kf (K0)Co e-zCoRCoFΨ/RT
(13)
(K0)* -FΨ(z*R*-zCoRCo)/RT e k ) kf (K0)Co
(14)
or
Or, in a more compact form
k ) Ae-BΨ
(15)
with A ) kf[(K0)*/(K0)Co] and B ) F(z*R* - zCoRCo)/RT.
Figure 3. Plot of ln(kobs) (kobs in s-1) versus the surface potential of the CTACl micellar solutions for the process [Ru(NH3)5pz]2+ + [Co(C2O4)3]3-.
To check that the raising part of the curve (experimental points) in Figure 2 is due to a change of the binding constant we have plotted ln kobs vs Ψ. This plot according to eq 15 must be a straight line, as it is (see Figure 3). At this point, it seems of interest to refer to the previous study of this reaction in SDS micellar solutions. In this case, the rate constants corresponding to the lower concentration of the surface are fitted by the equation
k ) kf
1 1 + KRu[T]
(16)
The difference with the case presented here is that, in the case of SDS micellar solutions, the ruthenium complex is the only participant bearing a charge of the opposite charge sign from the micelles, in such a way that the other two participants, the cobalt complex and the transition state, remain mainly in the aqueous pseudophase (the reference state). Consequently, their activity coefficients γi ) 1. Of course in this case, for the raising part of the curve, eq 15 also holds. Before closing this section, a few words on the value of the cmc obtained in the fitting (2.4 × 10-3 mol dm-3) seem in order. This value is in the range established from conductance experiments. However, this value is greater than those of the cmc of CTACl in the absence of added electrolytes (1.3 × 10-3 mol dm-3).26 Although this could seem, at first sight, an abnormal result because, in general, anions decrease the cmc of cationic surfactants, the situation is different for anions which form an insoluble salt with the surfactant. This situation has been carefully analyzed in ref 27. (b) AOT-Oil-Water Microemulsions. Figure 4 gives the plot of kobs versus 1/W corresponding to microemulsions. The points in the figure, including the point on the Y axis, are experimental data. The line corresponds to an interpolation because it was impossible to maintain the stability of the microemulsions containing the reactants for W > 48. The similarity of Figures 2 and 4 is striking. Thus, it seems possible to advance the idea that the situations in both reaction media must be similar. (25) Lo´pez-Cornejo, P.; Pe´rez, P.; Garcı´a, F.; de la Vega, R.; Sa´nchez, F. J. Am. Chem. Soc. 2002, 124, 5154. (26) van Os, N. M.; Waak, J. R.; Rupert, L. A. M.; Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactants; Elservier: Amsterdam, 1995. (27) Stellner, K. L.; Scamehorn, J. F. Langmuir 1989, 5, 70.
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total number of moles of S in a droplet ) VFwθ V[ST] ) V[Sf] + β (23) MwW In the present case, θ , 1 (even if all of the ruthenium complex was bonded). Thus, (1 - θ) = 1 and, from eq 20, it follows that [Sf] ) θ/K. In this way, eq 23 becomes
V[ST] )
VFwθ θ V+β K MwW
(24)
being
θ) Figure 4. Plot of k (s-1) of the process [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- versus 1/W. The dots are experimental data, kobs, and the solid line is the best fit using eq 28. The dashed line is an extrapolation of the experimental results for values of W > 48. The point on the Y axis is the value of kobs (s-1) of the process studied in aqueous solution.
Obviously, the classical equation of the pseudophase model cannot be applied in the case of microemulsions because the concentration variable of the dispersed pseudophase cannot be employed in the case of microemulsions. Consequently, these equations, but not their foundations, need some modifications. We have developed this new formulation in a previous paper.25 Here, we briefly outline it: assuming, as is habitual, that all the surfactant and water molecules are incorporated into the droplets, it is really shown that
moles of surfactant in a droplet )
VFw MwW
(17)
Mw and Fw being the molar mass and the density of water and V the volume of the water in the droplet. This number is equal to, or proportional to, the number of binding sites for the ruthenium complex in the present case, in the droplet. Thus,
VFw moles of binding sites in a droplet ≡ N ) β MwW (18) where β is a proportionality constant introduced to take into account the possibility of a reactant binding to more than one polar head. Now, we consider eq 3a written in a somewhat different way: K
Sf + site 798 S/site Assuming a Langmuir-type adsorption isotherm, write
K)
θ [Sf](1 - θ)
(19) 28
we
[ST] (1/K) + (βFw/Mw)(1/W)
)
K[ST] 1 + Kβ′(1/W) (β′ ) βFw/Mw) (25)
and
[Sf] )
[ST] θ ) K 1 + Kβ′(1/W)
(26)
From eqs 25 and 26, it follows
k)
kf + kbK 1 + Kβ′(1/W)
(27)
And using eq 12 for K, we obtain finally
k)
a + be-zRFΨ/RT 1 + ce-zRFΨ/RT(1/W)
(28)
with a ) kf, b ) kbK0, and c ) K0β′. This equation fits our data, using the data of the potential in ref 25 (see Table 4), with a ) kf ) 1.32 s-1, b ) kbK0 ) 7 × 10-11 ≈ 0, c ) K0β′ ) 0.337, and R ) 1 (the solid line in Figure 4 corresponds to this fit). Notice that according to these values of the parameters the surface reaction is insignificant, as expected taking into account that the transition state and the cobalt complex have the same charge sign as the surface, in such a way that the binding constants for these species will be small. In other words, kb is very small because it contains the concentration of the anionic reactant at the negative interface. It is of interest to note that a better fit is obtained employing the equation
k)
kf0[1 + m(1/W)] 1 + ce-zRFΨ/RT(1/W)
(29)
This equation is obtained from eq 28 when the term be-zRFΨ/RT is dropped in the numerator of the latter (taking into account that b ≈ 0, this seems reasonable) and a linear dependence of kf on 1/W is allowed:
(20)
kf ) kf0[1 + m(1/W)]
moles of Sf in a droplet ) [Sf]V
(21)
VFwθ moles of S/site ) Nθ ) β MwW
(22)
where m is an adjustable parameter. This dependence is supported by the following argument: the reaction studied is a process between two ions and, thus, must be sensitive to the ionic strength. A change in W would produce a change in the ionic concentration at the water pool and, thus, in kf (Figure 5 compares the calculated values of k with the experimental ones). These calculated values were
θ being the fraction of occupied sites. On the other hand,
and, evidently,
(30)
(28) Barrow, G. M. Physical Chemistry; McGraw-Hill, New York: 1961; pp 626-629.
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Table 4. Surface Potential (Ψ) Data in AOT-Decane-Water Microemulsions at [AOT] ) 0.2 mol kg-1 and Different W Values W
-Ψ (mV)
W
-Ψ (mV)
W
-Ψ (mV)
12.2 13.2 14.5 15.2 16.8
11.0 19.0 28.0 32.5 41.1
18.3 20.4 25.1 31.0
47.7 54.8 64.8 70.9
35.4 40.0 42.8 48.0
73.0 74.2 74.7 75.1
obtained from eq 29 with the following values of the parameters in the equation: kf0 ) 0.017 s-1, m ) 1038, c ) 0.084, and R ) 1. This is a point that deserves special comment. The positive value of m would imply a positive salt effect on the reaction at the water pool. Taking into account that the reaction is a process between ions of opposite charge sign, this is rather unexpected according to the classical theory of salt effects. However, it can be shown, taking into account the Marcus theory29 of electron-transfer reactions, that a positive salt effect is possible for electrontransfer reactions between ions of opposite charge when the oxidant is an anion, as it is here, and the reductant a cation. This follows from the fact that the anions become more oxidant when the ionic strength increases, and the opposite is true for cations. This argument receives support from the data in Figure 6 where kobs, in water, for this reaction is plotted versus the concentration of a salt (Na2SO4): a linear variation with concentration, the positive salt effect, exists.30
Figure 5. Plot of kcalc (s-1; calculated from eq 29) versus kobs (s-1) for the process [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- in microemulsions.
(29) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155 and references therein. According to the Marcus theory, the activation free energy, ∆Gq, for an electron process depends on two free energies: the reorganization free energy, λ, which always increases when the salt concentration in the reaction medium increases, and the reaction free energy, ∆G°′: ∆Gq ) (λ + ∆G°′)2/4λ The free energy ∆G°′ increases or decreases for anion/cation reactions when the salt concentration is changed. For a reaction, as here, in which the oxidant is an anion and the reductant a cation, this reaction free energy decreases (thus, the reaction becomes more favorable) as the concentration of salt increases. This is a consequence of the fact that ∆G°′ ) -nF(Eox°′ - Ered°′) Ei°′ being the standard formal redox potential of the redox couple i. This redox potential is given by Ei°′ ) Ei° + (RT/F) ln(γox,i/γred,i) For the Ru(NH3)5pz3+/2+ couple, both γox and γred decrease with increasing salt concentration. But the decrease in γox is more marked because of the higher charge of the oxidized form of this couple. Thus, the Ru(NH3)5pz2+ becomes more reductant when the salt concentration increases. The opposite is true for the anionic couple because in this case the (absolute) value of the charge is higher for the reduced form of the couple. Consequently, the cobalt complex, Co(C2O4)33-, becomes more oxidant when the concentration of salt increases. Thus, ∆G°′ becomes more favorable. The positive salt effect is, thus, a consequence of the more favorable driving force, ∆G°′, of the reaction when the salt concentration increases (see the equation for the activation free energy). (30) de la Vega, R.; Pe´rez-Tejeda, P.; Sa´nchez, F. Unpublished results.
Figure 6. Plot of kobs (s-1) versus the salt concentration for the process [Ru(NH3)5pz]2+ + [Co(C2O4)3]3- studied in the presence of Na2SO4 as background electrolyte.
Thus, we have shown that the pseudophase model is flexible enough to be adapted to a variety of situations. This adaptation must be done considering that when a broad range of concentrations of the dispersed pseudophase are explored, the parameters of the pseudophase model are not true constants, and this fact needs to be properly taken into account. Acknowledgment. This work has been financed by the DGICYT (BQU 2002-01063) and the Consejerı´a de Educacio´n y Ciencia de la Junta de Andalucı´a. LA0302089
