Reactivity of Anions with Organic Substrates Bound to Sodium

Langmuir 1997, 13, 687-692

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Reactivity of Anions with Organic Substrates Bound to Sodium Dodecyl Sulfate Micelles: A Poisson-Boltzmann/ Pseudophase Approach Severino Amado, Luis Garcı´a-Rı´o, J. Ramo´n Leis,* and Ana Rı´os Departamento de Quı´mica-Fı´sica, Facultad de Quı´mica, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain Received July 29, 1996. In Final Form: November 11, 1996X Kinetic studies of the reactions of the nucleophiles HO-, SO32-, and N3- with N-methyl-N-nitroso-ptoluenesulfonamide (MNTS) bound to sodium dodecyl sulfate micelles, and of HO- with 1,2-chloroethyl3-cyclohexyl-1-nitrosourea (CCNU) in the same medium, showed that for the MNTS/N3- and CCNU/HOreactions (but not the others) the distribution of anion between the aqueous phase and the aquo-micellar interface is nontrivial and must be taken into account in applying the pseudophase model of micellar media. A satisfactory distribution was calculated by numerical solution of the Poisson-Boltzmann equation for the cell model of micellar media.

Introduction

Scheme 1

A potential use of micellar media is for control of chemical reactivity. The chemical reaction rates and equilibria observed in micellar media can differ from those observed in conventional media due to solubilization of the reagents, due to the reduction of their effective concentrations through their segregation in different “compartments” of the bulk medium, due to their concentration within micelles, and in other ways. Predictions of reaction rates in micellar media are usually based on the pseudophase model,1 which treats aqueous, organic and/or surfactant components of the solvent medium as constituting distinct phases in which reactions occur, and between which reagents and products are distributed, in accordance with conventional laws of kinetics and mass transfer. This model has allowed qualitative and quantitative correlation of a large number of experimental results, often with no more than crude assumptions being made as to the distribution of reagents between pseudophases. For example, for reactions between anions and organic substrates distributed between the aqueous phase and anionic surfactant micelles, successful quantitative results are often obtained by supposing that electrostatic repulsion between the anionic reagent and the micelles leads to the reaction taking place exclusively in the aqueous phase. Sometimes, however, the persistence of non-negligible reaction rates at surfactant concentrations at which virtually all substrate must be micelle-bound shows that this assumption is too drastic and that the distribution of anion between the aqueous and micellar phases must be estimated more accurately. One approach to the problematical reactions described above has considered the consequences of autoprotolysis of water,2 but in general this effect appears to be too small to explain the observed reaction rates. Another has been to calculate the distribution of co-ions in accordance with the Poisson-Boltzmann model.3-5 This model allows a

minority of reactive ions to penetrate the micellar zone and react with the micelle-bound substrate. In this work we investigated the differences between reactions of the kind described above that can and cannot be satisfactorily treated as occurring exclusively in the aqueous phase. To this end, we studied the kinetics of the decomposition of N-methyl-N-nitroso-p-toluenesulfonamide (MNTS) under attack by the nucleophiles HO-, SO32-, and N3- in the presence of sodium dodecyl sulfate (SDS) micelles and of the decomposition of 1,2-chloroethyl3-cyclohexyl-1-nitrosourea (CCNU) by HO- in the same medium. MNTS is a bifunctional electrophile that can react with nucleophiles via either its nitroso or its sulfonyl groups6,7 (see Scheme 1), and studies carried out in our laboratory have recently investigated the mechanism of the basic hydrolysis of CCNU in aqueous media;8 both MNTS and CCNU are very poorly soluble in water. In this work we found that the reactions of MNTS with HOand SO32- can be satisfactorily assumed to take place exclusively in the aqueous phase, but not those of MNTS with N3- or of CCNU with HO-. For adequate treatment of the latter reactions we used a mixed Poisson-Boltzmann/pseudophase approach in which the distribution of substrate between the aqueous and micellar pseudophases is governed by a conventional partition constant, but not that of the reactive anion, whose concentration in the proximity of the interphase was calculated by numerical solution of the Poisson-Boltzmann equation for the cell model of micellar media.

X Abstract published in Advance ACS Abstracts, February 1, 1997.

(1) (a) Romsted, L. S. Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 2, p 1015. (b) Rodenas, E.; Ortega, F.; Vera, S.; Otero, C.; Maestro, S. Surfactants in Solution; Mittal, K. L., Ed.; Plenum Press: New York, 1989; Vol. 9, p 211. (c) Bunton, C. A.; Savelli, G. Adv. Phys. Org. Chem. 1986, 22, 213. (2) Chaimovich, H.; Aleixo, R. M. V.; Cuccovia, I. M.; Zanette, D.; Quina, F. H. Solution Behaviour of Surfactants; Mittal, K. L., Fendler, E. J., Eds.; Plenum Press: New York, 1982; p 949.

S0743-7463(96)00749-4 CCC: $14.00

(3) Bunton, C. A.; Mhala, M. M.; Moffatt, J. R. J. Phys. Chem. 1989, 93, 7851. (4) Ortega, F.; Rodenas, E. J. Phys. Chem. 1987, 91, 837. (5) Blasko, A.; Bunton, C. A.; Armstrong, C.; Gotham, W.; Zhen-Min, H.; Nikles, J.; Romsted, L. S. J. Phys. Chem. 1991, 95, 6746. (6) Garcı´a-Rı´o, L.; Iglesias, E.; Leis, J. R.; Pen˜a, M. E.; Rı´os, A. J. Chem. Soc., Perkin Trans. 2 1993, 29. (7) Leis, J. R.; Pen˜a, M. E.; Rı´os, A. J. Chem. Soc., Perkin Trans. 2 1995, 587. (8) Amado, S.; Garcı´a-Rı´o, L.; Leis, J. R.; Rı´os, A. M. J. Chem. Soc., Perkin Trans. 2 1996, 2235.

© 1997 American Chemical Society

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Figure 1. Influence of micellized SDS concentration [Dn] on the first-order rate constant for the reaction of MNTS with azide ion. [Dn] ) [SDS] - cmc, cmc ) 3 × 10-3 M, [N3-] ) 3.02 × 10-2 M.

Figure 2. Influence of micellized SDS concentration [Dn] on 1/ko for the reactions of MNTS with 3 × 10-2 M SO32- (b) and 3 × 10-2 M HO- (O); cmc ) 2 × 10-3 M in both cases.

Scheme 2

Experimental Section N-Methyl-N-nitroso-p-toluenesulfonamide (MNTS), 1,2-chloroethyl-3-cyclohexyl-1-nitrosourea (CCNU), and sodium dodecyl sulfate (SDS) were supplied by Merck or Aldrich with the maximum commercially available purity. Because of their poor solubility in water, solutions of MNTS and CCNU were made up in acetonitrile (and stored at low temperature in the dark to avoid decomposition), and appropriate small volumes of these solutions were added to the reaction mixtures to initiate reaction; in no case did the proportion of organic solvent in the final reaction mixture exceed 2% by volume. The kinetics of the slower reactions were recorded at 25 °C in Kontron Uvikon 930 or Milton Roy Spectronic 3000 diode array spectrometers equipped with thermostated cell holders; those of faster reactions (ko > 10-2 s-1) were recorded using an Applied Photophysics DX.17MV sequential stopped-flow spectrofluorimeter. The reactions were followed by monitoring absorbance by MNTS (at 250-270 nm) or CCNU (at 240 nm). Pseudo-firstorder reaction conditions were in all cases ensured by using a nucleophile concentration very much greater than the concentration of MNTS or CCNU. In all cases the experimental absorbance-time data were fitted well by first-order integrated equations, and the values of the first-order rate constant ko were reproducible to within 3%. The critical micellization concentration (cmc) of SDS was determined kinetically: series of reactions were carried out at increasing SDS concentrations, and the cmc was defined as the concentration at which the observed first-order rate constant began to decline.

Results Reactions of MNTS. The reactions of MNTS with HO-, SO32-, and N3- in the presence of SDS micelles were all strongly inhibited by increasing the concentration of surfactant, in keeping with the assumption that at high SDS concentrations almost all MNTS must be micellebound and thus sequestered from attack by the nucleophile; Figure 1 shows typical results for the reaction with azide ion. The pseudophase model of these reaction systems that is shown in Scheme 2 implies that the firstorder rate constant ko is given by

ko )

kw′ + km′KS[Dn] 1 + KS[Dn]

(1)

Figure 3. Influence of micellized SDS concentration [Dn] on 1/ko for the reaction of MNTS with 2.19 × 10-2 M N3- (]), 3.02 × 10-2 M N3- (O), and 6.00 × 10-2 M N3- (4).

where KS is the equilibrium constant governing association of substrate with micelle, kw′ and km′ are the first-order rate constants for the aqueous and micellar phases, respectively, and [Dn] is the concentration of surfactant in micelle form ([Dn] ) [SDS] - cmc, where cmc is the critical micellization concentration of SDS). If it is assumed that the anionic nucleophiles are totally excluded from the micellar phase by electrostatic repulsion, so that km′ is zero, then eq 1 reduces to

ko )

kw′ 1 + KS[Dn]

(2)

Figure 2 shows that for the reactions of SO32- and HOat SDS concentrations greater than the cmc there is good fit between the experimental results and the linearized form of eq 2

KS 1 1 ) + [D ] ko kw′ kw′ n

(3)

The values of KS inferred from these fits (156 M-1 for the reaction with SO32-, 138 M-1 for the reaction with HO-) are in keeping with the value of 108 M-1 obtained in studies of the acid denitrosation of MNTS in the presence of SDS.9 Figure 3 shows that for the reaction with N3-, a plot analogous to that of Figure 2 deviates significantly from linearity at [Dn] values greater than 0.15 M. This deviation may be attributed to the reaction rate at the micellar (9) Bravo, C.; Herve´s, P.; Leis, J. R.; Pen˜a, M. E. J. Phys. Chem. 1990, 94, 8816.

Reactivity of Anions with Substrates on Micelles

Figure 4. Influence of micellized SDS concentration [Dn] on the first-order rate constant for the basic hydrolysis of CCNU in 0.04 M HCO3-/CO32- buffers of pH 9.52 (b) and pH 10.12 (O); cmc ) 2 × 10-3 M in both cases.

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Figure 6. Influence of bulk [NaOH] on the first-order rate constant for the basic hydrolysis of CCNU totally associated with SDS micelles. [SDS] ) 0.15 M.

order rate constant is given by

ko )

kwKW[HO-]w + kmKMKS[HO-]m[Dn] 1 + KW[HO-]w + (1 + KM[HO-]m)KS[Dn]

(4)

where subscripts w and m refer to the water and micellar pseudophases, respectively. In this case, the further assumption that the hydroxyl ions are totally excluded from the micellar phase by electrostatic repulsion, so that [HO-]m ) 0, reduces eq 4 to

ko ) Figure 5. Influence of micellized SDS concentration [Dn] on the first-order rate constant for the basic hydrolysis of CCNU by NaOH. Bulk [HO-] ) 0.1 M, cmc ) 4 × 10-3 M. Scheme 3

surface being non-negligible and is examined quantitatively in the Discussion below. Volume reduction of the aqueous phase at high surfactant concentration could be significant. However, in Figure 2 where this effect is not being taken into account, a good fit for HO- and SO32- is observed while deviations are found for N3-. This means that deviations in the later case cannot be attributed to changes in the volume of the aqueous phase that will be neglected in the following discussion. (Easy calculations assuming volume additivity show that in the working conditions used in this paper the volume of the water phase is reduced less than 6%.) The value of KS inferred from the data for [Dn] < 0.07 M, 148 M-1, agrees well with those obtained using SO32- and HO-. Reaction of CCNU. The basic hydrolysis of CCNU in 0.04 M HCO3-/CO32- buffers of pH 9.52 and pH 10.12 and in unbuffered medium of pH 13 was inhibited between 5and 10-fold by [Dn] greater than about 0.15 M (Figures 4 and 5). The pseudophase model represented in Scheme 3 assumes that the mechanism of the reaction in the micellar phase is the same as in aqueous media8 (a fast deprotonation equilibrium followed by slow decomposition of the CCNU anion). Thus, assuming Scheme 3, the first-

kwKW[HO-]w 1 + KW[HO-]w + KS[Dn]

(5)

However, Figures 4 and 5 show that the best fits of eq 5 to the experimental data are unsatisfactory; in particular, predicted reaction rates at high surfactant concentration are smaller than experimental values, suggesting that the reaction rate in the micellar phase is not, in fact, negligible. This hypothesis is further supported by the fact that the value of ko in the high-surfactant limit (i.e., when all the substrate is micelle-bound) increases significantly with pH, as is shown both in Figure 4, in which the limiting rate increases from 1 × 10-4 s-1 at pH 9.52 to 2 × 10-4 s-1 at pH 10.12, in Figure 5 to 0.05 s-1 at pH 13, and in Figure 6, which shows the results of experiments carried out with various NaOH concentrations at [Dn] ) 0.15 M. Alternatively, distribution of CCNU- between the two pseudophases could be considered with deprotonation of CCNU by HO- wholly in water, in which case the ratio between the micellar and water rates should be constant for a given Dn and HO- concentrations. In Figures 4 and 5 it is clear that reaction in the micellar pseudophase is more important at low pH. Discussion The fact that in the presence of SDS micelles the reaction of MNTS with azide ion appears to occur in both the aqueous and micellar phases, whereas its reactions with SO32- and HO- appear to occur exclusively in the aqueous phase, is qualitatively explicable with reference to the ambident nature of MNTS, its location in the micelle, and the charge properties of the nucleophilic anions. In a recent 1H NMR study10 we concluded that MNTS associated with SDS micelles is located with its aromatic ring between the β and γ carbons of the SDS chain and its polar N-methyl-N-nitrososulfonamide moiety projecting (10) Bravo, C.; Garcı´a-Rı´o, L.; Leis, J. R.; Pen˜a, M. E.; Iglesias, E. J. Colloid Interface Sci. 1994, 166, 316.

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out through regions of progressively higher water and anions content. Since HO- attacks MNTS at its sulfonyl group, whereas N3- and SO32- attack its nitroso group (Scheme 1), the absence of micellar-phase reaction between MNTS and HO- may be attributed to the relatively deep location of the sulfonyl group, which prevents contact with hydroxyl ions. The fact that, of the two nucleophiles attacking the relatively exposed -NO group, one reacts in the micellar phase but the other does not, may be explained in terms of their charge properties: SO32- is strongly repelled from the micellar surface because of its double charge; N3-, on the other hand, is not only singly charged but also is highly polarizable due to its extensive delocalization of charge, both of which circumstances mean that contact with the NO group of micelle-bound MNTS is easier for the azide ion than for SO32-. As regards the reaction between CCNU and HO(assumed to have the same mechanism in the micellar phase as in water), it may be pointed out that the observed existence of micellar-phase reaction shows that the reactive center of CCNU bound to SDS micelles lies in a region of non-negligible water content, i.e., further from the micelle center than the sulfonyl group of micelle-bound MNTS. For a quantitative explanation of the kinetics of the MNTS/N3- and CCNU/HO- reactions, which both involve non-negligible contributions by the micellar interphase, the average concentration of nucleophile in this zone must be calculated. This may be done in three steps by (i) obtaining the electric potential in the medium (by solving the Poisson-Boltzmann equation for an appropriate model of the medium); (ii) defining the boundaries of the “micellar phase” in an appropriate way; and (iii) integrating the Boltzmann equation over the micellar phase, once defined, using the potential obtained in step (i).11-14 The model of the medium used for these calculations in this work (more exactly, the model of the neighborhood of each micelle in the medium) was the so-called cell model,15 in which each micelle is treated as an impermeable uniform sphere of radius Rm at the center of a ball (the cell) whose total volume is the volume of the whole medium divided by the number of micelles in the medium, so that

N 4 πRc3 ) 3 3 10 NA[Dn]

(6)

where Rc is the cell radius in meters (unless otherwise stated we work throughout in SI units), N is the aggregation number of the micelle, and NA is Avogadro’s number. Ions in the extramicellar part of the cell are treated as point charges, possible interactions between micelles were not considered, and all micellar charge is assumed to be uniformly distributed on the micelle surface. For the above spherically symmetric model, the relevant form of the Poisson-Boltzmann equation expresses the dependence of the electric potential ψ on the distance r from the center of the cell:

0r

1 d

( )

r2 dr

2

r

dψ dr

)-

( )

∑i zieci0 exp

-zieψ kT

(7)

where 0 is the permittivity of the vacuum, r is the relative (11) Gunnarsson, G.; Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1980, 84, 3114. (12) Bell, G. M.; Dunning, A. J. Trans. Faraday Soc. 1970, 66, 500. (13) Mille, M.; Vanderkooi, G. J. Colloid Interface Sci. 1977, 59, 211. (14) Almgren, M.; Linse, P.; Van der Auweraer, M.; De Schryver, F. C.; Gelade, E.; Croonen, Y. J. Phys. Chem. 1984, 88, 289. (15) Bell, G. M. Trans. Faraday Soc. 1964, 60, 1752.

permittivity of the extramicellar medium (assumed equal to that of water), k is Boltzmann’s constant, T is absolute temperature, e is the charge of the proton, zi is the charge number of the ith ionic species in the extramicellar medium, and ci 0 is the number concentration of species i at the cell boundary. The ci 0 are unknown, but integration of the Boltzmann equation for each species i over the extramicellar volume relates ci 0 to ni, the total number of free ions of species i in this volume

∫RR

ni ) 4πci0

c

m

e-zieψ/kT r2 dr

(8)

where ni values can be calculated from the known bulk concentrations of the components of the solution. In this work, the only cations present are the highly hydrophilic Na+, and hence, the micellar surface charge cannot be neutralized by specific association with cations in the Stern layer, as considered in similar treatments by other authors16,17 for more hydrophobic ions. Under this assumption, the SDS micelles should be considered like spheres of charge equal to the aggregation number (N). Nevertheless, experimental measurements with different micelles show that a fraction β of the micellar charge is always neutralized.1 The Poisson-Boltzmann model is a purely electrostatic treatment, and in our opinion, one cannot ignore that in average a fraction β of counterions must remain at the surface in order to stabilize the micelles and, hence, the effective concentration of free Na+ will be reduced. We have included these considerations in the electrical neutralization condition for the cell as a whole, as reflected in eq 9.

∑i nizi ) (1 - β)N

(9)

Once the ni values have been calculated, eqs 7 (which have nontrivial analytical solution) and 8 can be solved simultaneously, subject to appropriate boundary conditions, by iterative numerical procedures (see below). The ion concentrations ci 0 at the cell boundary can be used in eq 7 because, by virtue of symmetry with neighboring cells, ψ must satisfy the boundary condition

dψ/dr ) 0

at r ) Rc

(10)

Application of Gauss’s law at the micelle surface affords a second boundary condition

dψ σ )dr 0r

at r ) Rm

(11)

where the surface charge density σ is given by

σ)-

Ne(1 - β) 4πRm2

(12)

A third boundary condition can be imposed by setting the value of the arbitrary additive constant in ψ

ψ)0

at r ) Rc

(13)

Since the cell model treats all ions as point charges, it is not in fact necessary to distinguish between different ions of the same charge for the purposes of computing ψ (16) Blasko, A.; Bunton, C. A.; Wright, S. J. Phys. Chem. 1993, 97, 5435. (17) Rodenas, E.; Dolcet, C.; Valiente, M. J. Phys. Chem. 1990, 94, 1472.

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Table 1. Calculation of k2m, the Bimolecular Rate Constant for the Reaction of MNTS with Azide Ion at the Surface of SDS Micelles, from Equation 1 (km′ ) k2m[N3-]m)a [SDS]/M

ko/s-1

k2m[N3-]m/s-1

-ψ(r)Rm)/(mV)

[N3-]m/M

k2m/M-1 s-1

0.145 0.165 0.186 0.207 0.227 0.248 0.269 0.285

5.25 × 10-5 4.83 × 10-5 4.25 × 10-5 3.96 × 10-5 3.78 × 10-5 3.56 × 10-5 3.37 × 10-5 3.28 × 10-5

4.50 × 10-6 4.96 × 10-6 5.37 × 10-6 5.85 × 10-6 6.37 × 10-6 6.91 × 10-6 7.49 × 10-6 8.02 × 10-6

70.24 68.82 67.63 66.30 65.23 64.00 62.81 61.97

2.93 × 10-3 3.14 × 10-3 3.37 × 10-3 3.60 × 10-3 3.80 × 10-3 3.98 × 10-3 4.22 × 10-3 4.44 × 10-3

1.40 × 10-3 1.83 × 10-3 1.36 × 10-3 1.52 × 10-3 1.77 × 10-3 1.78 × 10-3 1.75 × 10-3 1.80 × 10-3

a k ′ ) 1.01 × 10-3 s-1, K ) 148 M-1; [N -] values were calculated by the method described in the text; ψ(r ) R ) is the potential w S 3 m m in the micellar surface.

and the desired average micellar-phase concentrations; all ions with the same charge number can be treated as belonging to the same “charge species”. Once eqs 7 and 8 have been solved numerically and the zone in which “micellar phase” reactions occur has been defined as a spherical shell of thickness ∆ around the micelle, the average number concentration of charge species i in this micellar reaction zone, ci, can be computed by integration as

ci )

4πci 0 Vm

∫RR +∆ e-z eψ/kT r2 dr m

i

m

(14)

where Vm is the volume of the micellar reaction zone. [Zi j]m, the molar micellar reaction zone concentration of the jth ionic species contributing to charge species i, is simply the proportion of this species among all the species contributing to charge species i in the extramicellar volume

[Zi j]m )

10-3ci[Zi j]free k

NA

(15)

[Zi j]free ∑ j)1

k [Zij]free is the mean molar concentration, in the where ∑j)1 extramicellar space, of the kth species contributing to charge species i. In this work, the above calculations were carried out for the reactions of MNTS with N3- ([N3-] ) 3.02 × 10-2 M) and CCNU with HO- ([HO-] ) 0.1 M) in media in which the cmc’s of SDS were respectively 3.0 × 10-3 and 4.0 × 10-3 M. The aggregation number of SDS was taken as 7018 and was assumed to be independent of SDS concentration. (Although different and quite scattered values of N have been reported in the literature for SDS micelles,19,20 they seem to increase slightly with increasing surfactant concentration (from 60 to 80). Nevertheless, increasing N would imply larger micelle radius, which means a surface charge density about constant for the concentrations of SDS used.) The radius of “bare” SDS micelles was taken as 18 Å13 and the thickness of the Stern layer as 1 Å, so that the value of Rm used was 19 Å. We assumed that there was effectively only one cationic species in solution, i.e., that the reactive anion was to be added as its sodium salt; computations were accordingly performed considering just two charged species, one comprising only Na+ ions and the other comprising the reactive anion and unmicellized dodecyl sulfate. Following Bunton et al.,3,16 we also assumed that the interaction between anions and micellized dodecyl sulfate was purely electrostatic, with no specific absorption of any of the ions

(18) Bravo, C.; Leis, J. R.; Pen˜a, M. E. J. Phys. Chem. 1992, 96, 1957. (19) Grieser, F.; Drummond, C. J. J. Phys. Chem. 1988, 92, 5580. (20) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153.

present. The fraction of micellar charge neutralized by Na+ in the Stern layer, β, was taken, in keeping with the results of several mutually independent experimental methods,18 as 0.7. The numerical solution of eqs 7 and 8 was based on the Newton-Raphson solution of eq 8 for the ci 0, starting from appropriate initial values. In each iteration, the values of ψ required for calculation of the right-hand sides of eq 8, and of the Jacobian matrix d(r,ni)/dcj 0, were obtained by numerical solution of eq 7 using the fourthorder Runge-Kutta-Fehlberg method with a constant step size small enough to ensure that ψ was always within 25 × 10-5 mV of the results given by the fifth-order method. The integrals on the right-hand sides of eq 8 were evaluated by Simpson’s method (hence the use of a constant step size in the Runge-Kutta-Fehlberg calculations). The Jacobian d(r,ni)/dcj 0 was computed using a five-point formula. The halt condition for the whole procedure was that the right-hand sides of eq 8 should be within 0.01% of the known values of the ni. When this model is applied to a reactivity problem, it is necessary to define a volume element of reaction. In fact, the Poisson-Boltzmann model allows the calculation of concentrations of reactive anions in a volume element of thickness ∆. This volume must be a region where the reactive center of the bound substrate and the reactive ions are located. At least for MNTS, taking into account the location of this substrate in the micelle,10 a reasonable ∆ value would be 1 Å, and therefore, this value will be used in our calculations. So, the micellar reaction zone is thus defined as the zone between r ) 19 Å and r ) 20 Å. Once c-1, the mean number concentration of the negative charge species in this zone had been calculated as in eq 14, the mean molar concentration of reactive anion in the zone, [Z-1, j]m (i.e. [HO-]m or [N3-]m) calculated from eq 15, in which, because of the assumption that the anions had no specific interaction with the micelle, each [Z-1, j]free was set equal to the bulk concentration of Z-1, j multiplied by Rc3/(Rc3 - Rm3). For the reaction of MNTS with N3-, the pseudophase model of Scheme 2 implies that the first-order rate constant is related to [Dn] by eq 1, in which kw′ is given by experiments in the absence of surfactant as 1.01 × 10-3 s-1, KS has been obtained from the linear part of Figure 3 as 148 M-1, and km′ ) k2m[N3-]m, where k2m is the bimolecular rate constant in the micellar phase. Table 1 lists, for each value of [SDS], the experimental value of ko, the value of [N3-]m calculated for the given reaction conditions by the numerical methods described above, and the resulting estimate of k2m. The mean estimate of k2m, 1.65 × 10-3 M-1 s-1, is about 20 times less than the value observed in water. For the reaction of CCNU with HO- under conditions in which all substrate is micelle-bound, eq 4 implies that the first-order rate constant is related to [Dn] by

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Figure 7. Influence of [HO-]m on the first-order rate constant for the basic hydrolysis of CCNU totally associated with SDS micelles. [SDS] ) 0.15 M. [HO-]m values were calculated, by the method described in the text, for the bulk [NaOH] values shown in Figure 6.

ko )

a[HO-]m 1 + b[HO-]m

(16)

where a ) kmKM and b ) KM. Figure 7 shows the plot of the observed values of ko against the values of [HO-]m calculated for the corresponding values of bulk HOconcentration. For low [HO-]m values (