Thermodynamic and Kinetic Approach of the ... - ACS Publications

12844

J. Phys. Chem. B 2001, 105, 12844-12856

Thermodynamic and Kinetic Approach of the Reactivity in Micellar Media. Reaction of 1,3,5-Trinitrobenzene upon Hydroxide Ion in Aqueous Solutions of Cationic Surfactants Joe1 l Lelie` vre,* Murielle Le Gall, Anne Loppinet-Serani, Franc¸ ois Millot,† and Pierre Letellier Laboratoire d’Electrochimie et de Chimie Analytique - UMR 7575, Energe´ tique et Re´ actiVite´ aux Interfaces Boite 39 - UniVersity P. and M. Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France ReceiVed: June 26, 2001; In Final Form: October 2, 2001

The kinetics of the reaction of OH- with 1,3,5-trinitrobenzene in the presence of dodecyl and hexadecyltrimethylammonium bromide have been studied. The micellar effects on this reaction have been modelized with the help of a new thermodynamic and kinetic analysis. This modelization has been based on a profitable approach proposed by De Donder and Defay which is well suited to the kinetics in dispersed medium. The chemical exchange rates σV0 and bV0, which are, respectively, the micellar and the bulk exchange rates, have been determined. This report also reveals that the acceleration of chemical reactions is mainly due not only to a large increase of the chemical exchange rate on the micellar aggregates but also to the high stabilization of the activated complex.

I. Introduction The modification of the equilibrium constants and rate of reaction which occurs when a dispersed system is introduced into a reaction medium is a well-known phenomenon which has been much investigated and reported in the literature over the past few years. The work involves, essentially, micellar systems using both neutral and charged tensioactives,1-8 and also complex mixtures which give microemulsions.9-11 The change in reactivity can be very large (and variations of several orders of magnitude in equilibrium and rate constants may be seen) and is often explained in terms of micellar catalysis on the basis of models involving extraction, adsorption, or association between the reaction components and the amphiphiles or other constituents of the aggregates. The charge, the size, the type, the concentration of the surfactants, and the counterions are the parameters which are invoked to explain these phenomena. However, these parameters intervene simultaneously, and it is often difficult to discriminate between them. As a consequence, the interpretation of the experimental data often differs according to the picture each worker has of the nature of micelles in solution, despite the fact that the results are obtained from the same analytical techniques and involve the same variables. This suggests that quite apart from the different models of these dispersed systems, the variables involved are unique and that a purely thermodynamic analysis of the micellar catalysis should lead to a unified treatment of the experimental data. We present here such a fundamental thermodynamic and kinetic analysis, which we will illustrate by a study of the exchange of hydroxide ions, OH-, between 1,3,5-trinitrobenzene (TNB) and the Meisenheimer complex, TNBOH-, in different aqueous micellar solutions of two tensioactives reagents, dodecyltrimethylammonium bromide (DTABr) and hexadecyltrimethylammonium bromide (CTABr) in the presence of NaBr supporting electrolyte. This reaction is well-known in water and * Corresponding author. E-mail: [email protected]. Fax: 01 44 27 30 35. † We regret the passing of our colleague, Franc ¸ ois Millot.

aqueous organic media12-19 as well as in some micellar solutions.20,21 In what follows, for reasons of simplicity, we present the reaction studied by

A + OH h AOH

(1)

II. Experimental Section A. Materials. 1,3,5-Trinitrobenzene (TNB) was prepared and recrystallized from methanol. The following compounds were from Acros Organics: DTABr, CTABr, sodium hydroxide, and sodium bromide. All the solutions were prepared with distilled water. B. Kinetic Measurements were made on an Applied Photophysics SX 18 MV stopped-flow spectrophotometer. The temperature of the spectrophotometer cell was maintained at 298.2 ( 0.1 K by circulating water from a thermostated bath around the injection syringes. The solutions were prepared with the same concentration of surfactant and NaBr in the injection syringes. To reduce the mixing time, we carried out the experiments by mixing the products in the same solvent. The base concentration in all experiments reported in this paper was in large excess over that of TNB, thus ensuring pseudo-first-order conditions throughout. The reaction of OH- with TNB gives first the mono adduct TNBOH- and then the di adduct TNB(OH)22-. We were interested solely in the formation of the mono adduct TNBOH-, which has its absorption maximum at 430 nm. At high concentrations of sodium hydroxide, the mono adduct and di adduct are formed simultaneously, and the kinetic parameters were obtained from a mathematical treatment of the data using the relation:

absorbance ) a(1 - exp(-kt))+ b(1 - exp(-k′t)) + A0 in which a is the infinite absorbance of the mono adduct, b the infinite absorbance of the di adduct, k the formation rate of the mono adduct, k′ the disappearance rate of the di adduct, and A0 the absorbance at t ) 0.

10.1021/jp012452j CCC: $20.00 © 2001 American Chemical Society Published on Web 12/05/2001

Thermodynamic and Kinetic Approach in Micellar Media

J. Phys. Chem. B, Vol. 105, No. 51, 2001 12845

The measurements of solubility of TNB in water, in the salt solutions, and in the surfactant solutions were carried out by introducing an excess of TNB solid into 10 mL of solvent and stirring for 36 h in a thermostated bath. Then the solutions were centrifuged and left at rest at the chosen temperature. The supernatant was analyzed by a UV-visible spectrometer at 320 nm. The micellar concentrations and more generally the concentrations of free surfactant in solution were determined by means of a specific surfactant electrode made in our laboratory;22 the voltmeter was a LPH430T from Tacussel. The critical micellar concentration (cmc) values of DTABr in 0.1 and 0.3 mol dm-3 NaBr solutions are, respectively, 4.5 × 10-3and 2.3 × 10-3 mol dm-3. For CTABr in 0.3 mol dm-3 NaBr, the cmc is 2.8 10-5 mol dm-3. III. Reactivity in Micellar Medium; Thermodynamic and Kinetic Points of View A. Thermodynamic. A.1. Phenomenon of Retention. Let us consider the general case where ni moles of a solute i are introduced into a solution made up of n1 mol of water and n2 mol of surfactant. The Gibbs energy of this chemical system is written as

dG ) VdP - SdT + γdA + µ1 dn1 + µ2 dn2 + µi dni

(2)

γdA corresponds to possible the surface tension work, A is the area of the system, and γ is the surface tension. The partial derivatives allow us to write

( )

( )

∂µ2 ∂µi T,P,A,n2,n1 ) T,P,A,ni,n1 ∂ni ∂n2

(3)

This relation is always valid, whatever the concentration of surfactant is, above or below the cmc. Above the cmc, this equation is of particular interest in view of a decrease of the chemical potential of i with the addition of 2 in solution, the values of the variables ni, n1, T, P, and A being kept constant. This behavior is interpreted as a “retention” of the solute by the micelle, modelized by several authors as an extraction in the micellar volume23,24 using the pseudo phase model or as an adsorption supposing that i is located on the surface of the micelle25 or as a fixation in the superficial layer of the micelle.26,27 Thermodynamics does not allow any conclusion about the real nature of the phenomena or the physical reality of the retention. It simply establishes the interdependence of the variables µi and n2. For dilute solutions, the chemical potential of the solute i can be written as

µi ) µθi + RT ln{Ci}

(4)

where µθi is the chemical standard potential of i in the solution referring to the behavior of the diluted solutions for the molar scale, and {Ci} is the value of the molar concentration of i. The value of µi is modified by the presence of the micellized surfactant, which can be explained in several ways. The observed effect may concern the standard chemical potential only, µθi and mµθi in the presence of micellized surfactant. The observed phenomenon is described as a “medium effect“, comparable to that obtained by adding a cosolvent to the water. If one reasons in this way, the micellar solution is considered as a homogeneous medium different from water whose properties depend on the amount of added surfactant (the

“cosolvent“). The concentration of the solute i, mCi, is the same ({mCi} ) {Ci}), and the chemical potential of i in the micellar solution is thus m

µi ) mµθi + RT ln{mCi}

(5)

Traditionally, one attacks the problem in an other way by considering that one is in the presence of a heterogeneous medium on a microscopic level, in which the micellar aggregates are supposed to be scattered in the continuous aqueous medium containing monomers of surfactant. In this approach, one takes the simplifying assumption that the value of the chemical standard potential of i remains constant and equal to the value of the chemical standard potential of i in the premicellar solution, bµθ. i The decrease of the chemical potential of i is then assigned to the retention of a certain amount of i upon the micellar aggregates, varying the i concentration so that bCi < mCi

µi ) bµθi + RT ln{bCi}

m

(6)

This approach is confirmed by different studies, essentially fluorescence experiments, which show that the molecular solutes are able to interact strongly with the micellized surfactant and be placed, according to their nature, in different local environments.28,29 This does not prevent the assumption, particularly for high concentrations of surfactant, that the nature of the intermicellar phase evolves and the standard chemical potential, bµθi is modified. Even if their nature is very different, these two approaches are linked and lead to an identical interpretation of the phenomena. A.2. Effect of SolVent. Equilibrium. The chemical potential of i being independent of the way it is formalized, one can write

µi ) mµθi + RT ln{mCi} ) bµθi + RT ln{bCi}

m

To account for the variation of the standard chemical potential, a transfer activity coefficient, γtmfbi, can be introduced:

ln γtmfbi )

b θ µi

- mµθi ∆tmfbGi ) RT RT

(7)

The transfer free energy (∆tmfbGi)/(RT) is positive when the solute i is stable in the micellar medium. The value of γtmfbi depends on the amount of micellized surfactant x introduced into solution:

γtmfbi ) γi(x)

(8)

where x is the total concentration of surfactant aggregated in solution. The transfer activity coefficient γi(x) tends toward 1 as x tends toward zero. The i concentrations then are linked by

Ci ) bCi exp

(

)

b θ µi - mµθi

) bCiγtmfbi

(9)

µi ) bµθi - RT ln γtmfbi + RT ln{Ci}

(10)

m

RT

The chemical potential is written as m

For the equilibrium state of the reaction studied, the equilibrium constants in the premicellar aqueous medium, bK and micellar mK, are linked by

12846 J. Phys. Chem. B, Vol. 105, No. 51, 2001

γtmfbAOH γAOH(x) K ) bK ) γtmfbAγtmfbOH γA(x)γOH(x)

m

Lelie`vre et al.

V ) σn2σV′2 + σniσV′i

σ

(11)

(13)

where σV′2 and σV′i are the partial molar volumes of 2 and i, respectively, in the micellar aggregates:

The equilibrium constant in a micellar medium is then a function of x. A.3. Modelization of the System: Dispersed Medium. Equilibrium. Another way to approach the problem is to assume that the system is dispersed and its properties can be fitted to a model. The delicate stage in choosing a model is to find an equivalent system which has the same behavior as the real system and which corresponds to a satisfactory description of experimental reality without it necessarily being a strict representation. We will adopt the idea that the micellar system can be split into two subsystems: a homogeneous bulk of constant composition b and a “ phase“ σ of surfactant made of σn2 mol of micellized amphiphiles. The number of moles of surfactant and solute i in the micellar solution are, respectively:

( ) ( )

V′2 )

σ

V'i )

σ

∂σV T,P,σni σ ∂ n2

(14)

∂σV T,P,σn2 σ ∂ ni

(14')

Each partial molar volume is a homogeneous function of degree zero of the mass of micellized surfactant. In the case where the solute i is very dilute in the micelle, one can assume that σV′i tends toward a limit σVθi (partial molar volume at infinite dilution of i) and σV′2 toward σV/2, the molar volume of the surfactant in the micelle. The product of the terms niσVθi can be neglected in comparison with σn2σV/2. The volume σV is simply written as

V ) σn2σV/2

σ

n2 ) bn2 + σn2

(12)

ni ) bni + σni

(12′)

It can be assumed that the exchange rate between the species in the two subsystems is fast with regard to the rate of the reactions studied. This assumption is not enough to predict the properties of the system. An idea of how the solutes are bound to the micellar phase is also required. Indeed, according to the way of considering how the solutes are bound, it appears that the number of moles of i linked to the micellar aggregates is not necessarily proportional to the amount of micellized surfactant. If σni moles of a solute i are solubilized (phenomenon of volume) at equilibrium by micelles containing σn2 mol of surfactant, then λ σni moles will be solubilized by λ σn2 mol (where λ is a number) because the micellar volume is a homogeneous function of degree one of the mass. The retained amount, σni is then a homogeneous function of degree one of the mass of micellized surfactant. σni and x are proportional. On the other hand, if it is a surface phenomenon, this property will apply only if the area is itself a homogeneous function of degree unity of the mass. For this to be the case, it is necessary to assume that above the micellar threshold, an increase of the mass of surfactant has as consequence the production of identical copies of the micelles. In contrast, if this operation implies a modification of the size or of the shape of the micelles, then the area of micellar aggregates, always able to vary with the mass of micellized surfactant, is not necessarily a homogeneous function of degree unity of σn2. As a result, if one multiplies by λ the mass of the micellized surfactant, the amount of i taken up is not necessary multiplied by λ. σni and x are not proportional. The problem with a chemical reaction in this kind of medium is that each reagent or product can have different behavior according to its mode of retention by the micellar aggregate.30 So it is necessary to express the composition of the micellar aggregates by unique variables indifferent to the various forms of retention possible. Let us first assume retention by “dissolution” that affects the whole volume of the micellar aggregate. The micellar volume is written as

(15)

Then one can characterize the presence of i in the micelle by its concentration σ σ

Ci ) σ

ni 1 n2 σV/2

(16)

In the case where the concentration of i is low, the molar ratio of i can be written in the form σ

Xi ) σ

σ

σ ni ≈ σ σ n2 + ni n2

ni

(17)

or

1 C i ) σ Xi σ / V2

σ

(18)

Let us now assume that the fixation of i occurs only on the surface of the aggregate. Usually, one expresses the composition of the surface by the excess of surface area: σ

σ ni Γi ) σ A

(19)

The expression of the surface area involves every constituent has a contribution in the form of partial areas σA′2 and σA′i such that

dσA ) σA′2 dσn2 + σA′i dσni

(20)

if one assumes that the area has a total exact differential, σA′2 and σA′i are, respectively, the partial areas of the surfactant and the solute i σ

A'2 ) A'i )

σ

( ) ( )

∂σA T,P,σni ∂σn2

(21)

∂σA T,P,σn2 ∂ σ ni

(21')

In a previous paper,31 we used homogeneous functions to show

Thermodynamic and Kinetic Approach in Micellar Media the evolution of the shape of the micellar aggregates as the concentration of surfactant increases in the solution. Let us, in this case, write the functions σA ) σA (σn2,σni) as homogeneous functions of degree p different from or equal to unity. Euler’s relations allow us to write the two following relations: σ σ

A ) σn2

A'2 σ σA'i + ni p p

(p - 1)dσA ) σn2 dσA′2 + σni dσA′i

(22) (22′)

The condition p ) 1 describes, among others, the particular case of a multiplication of identical micelles when reagent is added to the solution. In this case, σA′2 and σA′i are homogeneous functions of degree 0 of the mass. When p * 1, σA′2 and σA′i are always homogeneous functions but of degree p - 1. Thus, they are functions whose value changes with the amount of surfactant in the medium. For dilute solutions of i, the product σniσA′i becomes small with respect to σn2 σA′2, and we find σ

A ) n2

σ

σ

A′2 p

(23)

For the dilute solutions of i, we have

ni p p Γi ) σ σ ) σXiσ n2 A′2 A′2

the counterions at the micelle has as a consequence an increase in the area σA′2. We will consider the general case of a charged solute i partially bound by the micellar aggregates in the solution of surfactant. The condition of the equilibrium of the system depends on the transfer affinity,31,32 which must be zero, which is to say that the magnitude of tension of i are the same throughout the medium at given T and P. In the homogeneous phase b, the variable of tension of i at given T and P is its electrochemical potential:

µ˜ i ) bµθi + RT ln{bCi} + ziFbφ

zi is the charge of the ion i, and bφ is the electrical potential of the phase b. On the surface, one assumes that σni moles of i are shared out on the σn2 mol of surfactant M and on the associated β X. We take for the variable of tension of i at the surface an expression close to that proposed by Sanfeld32 that we have discussed in a previous paper31

( )

∂G T,P,n1,n2 ) σµθi + RT ln{σΓi} + zi Fσφ + σγ σA′i ∂ni

(24)

( )

If p ) 1, σA′2 is identical with the area of the polar headgroup invariant with the mass of surfactant

1 Γi ) σXiσ / A2

(28)

∂G T,P,n1,n2 ) σµθi + RT ln σXi ∂ni

(25)

Thus, in the case where the micelles copy themselves identically with an increase of surfactant concentration, the amount of i retained by a given amount of micellized surfactant can be written as a variable of molar concentration or of an excess of surface, the pertinent variable being the molar fraction of i. In the case where the micelles change their shape, the variables of volume and of surface do not lead to identical conclusions, the term σA′2 being itself variable with σn2. Thus, the range of application of relation 23 is very large. It includes the different modes of retention of volume and of surface mentioned above with as a particular case when the micelles copy themselves identically. Therefore, we explain the model of the micellar systems according to an approach considering the “micellar phase” as a surface phase of area σA containing σn2 mol of micellized surfactant. In the case where the surfactant is a salt of an amphiphile M with X as counterion, the value of σA′2 takes into account the partial areas of each ion and the association coefficient β3b. In this case, relation 20 can be written as

dσA ) (σA′M + β σA′X)dσn2 + σA′i dσni

{ } σ

σA/, 2

σ

(27)

The introduction of relation 24 gives

σ

σ


RT ln

A′2 + ziFσφ + σγσA′i (28′) p

σµθ i

is the standard chemical potential of i on the surface, σφ is the electrical potential of the micellar surface, σγ is the surface tension between the micellar surface and the phase b, and σA′i is the partial area of i at the micellar surface. The condition of equilibrium implies the equality of relations 27 and 28′, which leads to σ

ni

σ

n2

{ } ( (

) ) (

b θ A'2 µi - σµθi exp p RT

σ

) σXi ) {bCi}

exp

)

σ σ ziF(bφ - σφ) γ A'i exp (29) RT RT

Thus, the amount of i bound to the surface, σni, depends on the amount of micellized surfactant, σn2, on the concentration of i in the bulk b, on the way which the shape of the micellar aggregates develops with the concentration of surfactant,( σA′2), on the nature of the both phases, (bµθi - σµθi ), on their electrical potentials (bφ - σφ), and on the surface energy, (σγσA′i). In the general case, the factors of {σCi} are able to vary with x. To simplify this approach, we will introduce a positive and dimensionless “retention coefficient“ φi such that σ

which is

ni

A′2 ) A′M + β A′X

σ

σ

σ

σ

(26)

This relation shows that an increase in the association ratio of

with

n2

) σXi ) {bCi}φi

(30)


{ } ( σ

φi )

) (

Lelie`vre et al.

)

b θ A'2 µi - σµθi ziF(bφ - σφ) exp exp p RT RT

(

exp -

σ σ

)

γ A'i (31) RT

The chosen formalism favors the use of the variable σXi to account for the composition of the aggregates, whatever the mode of retention of i by the micellar aggregates is. One can write the total concentration of i in the micellar solution as a function of the concentration of i in the bulk. Relation 30 leads to

ni ) bCiV + σn2bCiφi where V is the volume of the phase b, which one will assume to be little different from the total volume of the solution. The total concentration of i in the micellar solution, σCi, is thus

Ci )

m

(

)

σ ni b n2 ) Ci 1+ φi V V

the term σn2/V is the number of moles of surfactant forming the surface for a volume V of solution. Thus σ

n2 m ) C2 - bC2 ) x V

σC

(32)

is the total concentration of surfactant in the system and the concentration of monomer in the phase b. Thus, one 2 can obtain 2

bC

Ci ) bCi(1 + xφi)

m

(33)

Relation 33 allows us to relate the equilibrium constants in the micellar solution and in the premicellar medium by m

K)

{CAOH}

1 + xφAOH ) bK {CA}‚{COH} (1 + xφA)(1 + xφOH)

(34)

mK

depends solely on x, but also implicitly on the terms φi, which themselves are likely to vary with the amount of surfactant in solution. To define the equilibrium condition, account should be taken of the equilibrium on the surface defined by the molar ratios, and an equilibrium constant of the surface, σK, should be defined such that σ σ

K)σ

XAOH

XA‚ XOH σ

φAOH ) bK φA‚φOH

(35)

This parameter has a fixed value if the terms φi do not vary with x. A.4. Relation between the Two Approaches. The two approaches described at the beginning of this study lead to similar formalisms. The model of the micellar solution interestingly gives an expression of the variation of the transfer activity coefficient of i with x. Indeed, if we compare relations 9 and 33, we can write m

Ci

b

Ci

) γi(x) ) 1 + xφi

(36)

In the case where the φi’s do not depend on x, the transfer activity coefficients vary linearly with x, and the behavior of the system can be described from a pseudophase model. The parameter φi is equivalent to a partition constant. In the case where the φi’s depend on the amount of surfactant in solution, the pseudophase model does not fit. The reactivity in a micellar medium for reactions involving several species of very different nature will only be interpreted if the variations of φi with x are determined independently from each other. This is often a difficult operation. Generally, experimental data in the literature are interpreted using a number of additional assumptions. To keep our study general, we have avoided this kind of approach, and we have completed this work by investigating the kinetic details of the reactivity in micellar medium. We will illustrate the theoretical results by means of the data for nucleophilic addition of OH- to TNB. B. Kinetic Point of View. The addition of the OH- ion to TNB to make the mono adduct Meisenheimer complex has been investigated in water with several concentrations of background electrolyte and the mechanism discussed in several papers.14a,15a,18,33 In the case of micellar solutions, the fundamental kinetic study can be carried out following the same approach as the previous by supposing that the medium is homogeneous and different from water or it is dispersed. Considering that medium dispersed, the reaction is able to occur in the bulk or on the micellar aggregates. In the solution, this leads one to consider different rates of reactions according to the location of the reagents and the products. To overcome this problem, we used a profitable approach proposed by De Donder34 and Defay,35 which is well suited to the kinetics in a dispersed medium. The molar rate, V, of an elementary reaction leading to an equilibrium can be written as

V)

[ (

)

(

)]

B A-B Aeq A A-A Aeq 1 dξ - exp ) Vo exp V dt RT RT

(37)

ξ is the extent of the reaction, B A is the “forward“ affinity, B Aeq is “forward“ affinity at equilibrium, A A is the “reverse“ affinity, A Aeq is “reverse“ affinity at equilibrium, and Vo is the chemical exchange rate at equilibrium. This relation proposes that the thermodynamic force appears under the form of several affinities “forward“ and “reverse”, which are the same everywhere in the solution, when one assumes very fast exchange reactions between the components of the bulk and those linked to the micellar surface. As a result, we can consider that all the reagents and the products in a micellar solution are submitted to the same thermodynamic forces wherever they are located in the medium. The reaction will take place under the control of these thermodynamic forces. The reaction rate constitutes the system response to the thermodynamic constraint. Its value depends on the local conditions of reaction. This is an important property which allows a simple characterization of the “micellar catalysis” concept. It is therefore interesting to recall that relation 37 shows how the simple presence of a surface can be expressed by “catalytic effects”. The only condition is that the surface tension (surface/solution) has to vary during the reaction without supposing that the reagents interact with the surface and, in this case, without association of the surfactants in the premicellar range. As far as we are concerned, when the concentration is



over the cmc, the liquid-vapor and micelle-solution interfacial tensions are presumed to be constant. We will consider the elementary exchange reaction of the hydroxide ion between TNB and the mono adduct. By definition, the affinity of the reaction results from the difference between its forward B A and reverse A A constituents

A)B A-A A

The differential equation is typical of a first-order kinetic equation if the OH- concentration is maintained constant (mCOH ) mCOHeq). Its integration gives an exponential function whose constant is

(

k ) mVo

m

(38)

Similarly, the reaction rate is the difference between a “forward rate” b V and a “reverse rate” a V

V)b V-a V

(39)

m

)

1 1 +m CAeq CAOHeq

As the forward and the reverse processes are supposed to result from elementary reactions, the exchange rate is linked to the kinetic constants mk1 and mk-1 by relation

Vo ) mk1 mCAeqmCOH ) mk-1mCAOHeq

m

At equilibrium, the forward rate is equal to the reverse rate:

V eq ) 0 w b V eq ) a V eq ) Vo V)b V eq - a

(40)

Vo is the chemical exchange rate which appears as a factor of the exponential terms in relation 37. At equilibrium, the reaction affinity is equal to zero

Aeq ) B Aeq - A Aeq ) 0

(41)

B.2. SolVent Effect. Kinetics. For the exchange reaction 1 considered in the micellar solution, the forward and reverse affinities are written:

B A)

∑νimµi(reagents)

) mµA + mµOH ) mµθA+mµθOH + RT ln{mCA}‚{mCOH} A A)

(42)

∑

νimµi(products)

) µAOH ) mµθAOH + RT ln{mCAOH} m

(42′)

The concentrations mCi of the various species i are those relative to the micellar solution. The standard chemical potentials mµ°i correspond to the behavior at infinite dilution of the compounds in the micellar solution in the concentration scale. Replacing the affinities in relation 37 leads to a simple relation which involves the chemical exchange rate, the values of the instantaneous concentrations of the reagents, and their values at the equilibrium, which are also constant for a closed system. For the rate of disappearance of A from the micellar medium, we can write

(

)

{mCA}‚{mCOH} {mCAOH} dmCA m ) Vo m V)dt { CAeq}‚{mCOHeq} {mCAOHeq}

[ (

The substitution of mVo by this expression in relation 45 gives m

k ) mk1mCOH + mk-1

As a result, the constant rate observed in a given medium depends linearly on the hydroxide ion concentration when the assumptions made in the calculation are verified. A.2. System Modelization: Dispersed Medium. Kinetics. Now let us suppose that the reaction is carried out in a heterogeneous medium. One must accept that the chemical exchange rates cannot be the same at all the points of the system. The whole molar reaction rate can be split up in two terms:

V)-

b σ 1 dnA 1 d nA 1 d nA )V dt V dt V dt

(48)

The amount of the micellized surfactant, σn2, is introduced into relation 48. Then the participation of the bulk and of the micellar aggregates appear in the whole rate:

dbCA σn2 dσXA 1 dnA )V)V dt dt V dt

(49)

Let the volume V of the bulk phase b be the volume of the system; the molar rate is defined in the bulk phase by

dbCA V)dt

(50)

dσXA dt

(51)

b

σ

V)-

This last choice is entirely conventional. As we have shown previously, the molar fraction of the solutes in the micellar aggregates is a pertinent variable of the system. It releases us from reaching to know the location of the reaction and the evolution of the aggregates as we add surfactant into the solution. The φi coefficients take these features into account. The whole reaction rate is simply written as

V ) bV + xσV

)

(47)

and the micellar rate by

C°A ) mCA + mCAOH

Relation 43 becomes

(46)

(43)

The units of V and Vo are mol dm-3 s-1. The ratios in eq 43 are pure numbers if the variables of the numerator or of the denominator are expressed either in concentration (mol dm-3) or with the values of the concentrations, {Ci}. To carry on the kinetic calculation, we will use the values of the concentrations (no unit) instead of the concentrations themselves (mol dm-3). The whole concentration of A, mC°A, is m

(45)

m m dmCA m m COH C°A 1 ) Vo CA m + m m m dt CAeq‚ COHeq CAOHeq CAOHeq

]

(44)

(52)

Similarly, the forward and reverse rates can be written as

V + xσ b V b V ) bb

(53)

a V) a V+x a V

(53′)

b

σ


Lelie`vre et al.

The forward and reverse rates are linked to the local compositions by b

b V ) bk1bCAbCOH

(54)

σ

b V ) σk1σXAσXOH

(54′)

b

a V ) bk-1bCAOH

(55)

σ

a V ) σk-1σXAOH

(55′)

IV. Experiments

and

At equilibrium, the reaction is characterized by its chemical exchange rate so that

Vo ) b V eq ) a V eq ) bVo + xσVo

(56)

Vo ) bk1bCAeqbCOHeq ) bk-1bCAOHeq

(57)

Vo ) σk1σXAeqσXOHeq ) σk-1σXAOHeq

(58)

m

with b

σ

Then for the chemical exchange rate in the micellar medium:

Vo ) bk1bCAeqbCOHeq + xσk1σXAeqσXOHeq

(59)

Vo) bk-1bCAOHeq + xσk-1σXAOHeq

(59′)

m

m

Each concentration or molar ratio can be expressed as a function of the total concentration of the species in the system: m

(bk1 + xσk1φAφOH)

Vo ) CAeq COHeq (1 + xφA)(1 + xφOH) m

m

(bk-1 + xσk-1φAOH)

) CAOHeq m

depending upon the comparative values of the constants bk and and of the retention coefficients of the species taking part in the reaction.

σk

(1 + xφAOH)

(60)

Before setting out the experimental results, we must underline the consequences of relation 3. If the right-hand term is supposed to be negative and not zero, the left-hand term must also be negative and not zero. Above the cmc, when the component i is added to the system for fixed values of T, P, A, and fixed amounts of surfactant (n2) and solvent (n1), the value of the chemical potential of 2 decreases. This means a decrease in the value of the cmc. To limit the consequences of this property, the ionic environment of the cationic micelles is maintained constant by a support electrolyte (0.1 and 0.3 mol dm-3 NaBr), and TNB is used at low concentration (10-5 mol dm-3). A surfactant selective electrode was used to verify that the chemical potential of the surfactant does not vary when the reagents are added to the medium or when the reaction occurs. TNB concentration is much smaller than that of the hydroxide ion one. The hydroxide ion concentration is assumed to be constant during the reaction and the kinetics are then first-order rate reactions. A. Kinetic Exchange of OH- in 0.1 mol dm-3 and 0.3 mol dm-3 NaBr in Water. C. A. Bunton has intensively studied the mechanism of this reaction.19,36 He shows that the nucleophilic attack takes place in several steps involving intermediate single-electron-transfer complexes. Several studies of this reaction in various salt media lead to values of the rate constants which are quite similar. We have complemented these results with the study of the reaction between OH- and TNB in aqueous salt solutions: 0.1 mol dm-3 NaBr, 0.3 mol dm-3 NaBr, and NaBr/NaOH with the ionic strength maintained at 0.3 mol dm-3 by varying the ratios of NaBr and NaOH. All the experiments were carried out either in the presence of normal air and its CO2 or in a CO2 free atmosphere. The results show good agreement with the rate law; for OH- concentrations above 0.2 mol dm-3, the values of the rate constant vary linearly with OH-concentration, and they are in accordance with Bernasconi’s results:14a b

The apparent rate constant is then written as

k1 ) 37.5 dm3 mol-1 s-1 k-1 ) 9.8 s-1

b

k1 + xσk1φAφOH

k-1 + xσk-1φAOH m k ) mCOHeq + 1 + xφAOH (1 + xφA)(1 + xφOH) (61) b

b

If we identify relations 61 and 47, the forward and reverse apparent rate constants are linked to those of the bulk and of the surface:

k1 + xσk1φAφOH

b

k1 )

m

(1 + xφA)(1 + xφOH) k-1 + xσk-1φAOH 1 + xφAOH

(62)

b

k-1 )

m

(63)

The ratio of theses two quantities leads again to the expression of the equilibrium constant mK (relation 34). Relations 62 and 63 show that the presence of micelles in a medium can have various effects on the apparent rate constant

K ) 3.8

b

B. Kinetic Exchange of OH- in 0.1 and 0.3 mol dm-3 NaBr in Water with DTABr. For all the tensioactive solutions studied, above the cmc, the rate constant mk fits relation 47 even for the lowest concentrations of hydroxide ion. Some results are reported in Figure 1. Table 1 presents the values of bk1, bk-1, and bK obtained in 0.1 and 0.3 mol dm-3 NaBr for various concentrations of DTABr. A standard catalytic effect is observed with the addition of surfactant. This behavior has previously been described for similar systems.1,37 For both salt concentrations, an increase of surfactant concentration increases strongly the value of mk1. The higher the concentration, the higher this effect. Concomitantly, mk-1 decreases as x increases. As a consequence, the value of the


J. Phys. Chem. B, Vol. 105, No. 51, 2001 12851 However, the values of the forward rate constants are higher for CTABr than for DTABr, whereas the values of the reverse rate constants in both media are quite similar. As a result, equilibrium 1 is more shifted to the mono adduct with CTABr. V. Results and Discussion

Figure 1. Variation of mk as a function of the OH- concentration in [NaBr] ) 0,3 mol dm-3, [TNB] ) 2 10-5 mol dm-3, and [DTABr] ) (9) 7 ×10-3, (O) 8 × 10-3, (2) 10-2, and (0) 1.28 × 10-2 mol dm-3.

TABLE 1:

mk

1,

[DTABr] dm-3)

(10-3 mol

mk

-1,

(10-3

(dm3

mk 1 mol-1 s-1)

mk -1 (s-1)

mK

In Water 37.5 dm-3

7.0 8.0 10.0 12. 8 17.6 20.0

In 0.1 mol 2.5 3.5 5.5 8.3 13.1 15.5

8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0

In 0.3 mol dm-3 NaBr 5.7 72 13.7 120 21.7 160 29.7 188 37.7 212 45.7 231 53.7 246 61.7 261

NaBr 114 138 178 228 310 340

9.8 1.39 1.22 1.09 0.99 0.90 0.99 1.58 1.18 1.04 0.98 0.96 0.93 0.93 0.92

3.8 82 113 163 230 344 343 46 102 154 192 221 248 265 284

The errors in the values are estimated at ( 5%.

TABLE 2:

mk

1,

[CTABr] (10-3 mol dm-3) 1.0 2.0 3.0 4.0 5.0 a

x mol dm-3) -

0

a

and mK in DTABra

mk

-1,

A. Exchange Rates. The phenomenon called “micellar catalysis” can be analyzed using relations 62 and 63. Thus, the values of the retention parameters, φi, of the reactants and of the products and the values of the local rate constants must be determined separately. The variables mk1, mk-1, bk1, bk-1, and x are determined experimentally. The complete solution of the system requires knowledge of the values of the other parameters: φA, φOH, φAOH, σk1, and σk-1. These five parameters are linked by only two equations. Therefore, three variables have to be determined separately to obtain values for all the parameters. This poses an evident analytical problem. We have thus chosen to make the analysis of the effect of micellar catalysis in an easier and more realistic way. This was possible, thanks to the thermodynamic calculation and hypothesis leading to the relation of Marcelin-De Donder.39 In a micellar solution, the affinity is a thermodynamic force identical at all points of the system. The chemical exchange rate, which depends on the location of the reaction, characterizes the catalytic effect due to the micelles in solution. The micellar effect can be easily known by comparing the exchange rates σVo and bVo. Relations 57 and 58 allow us to write σ

Vo

σ σb

Ro ) b

mk

1 (dm3 mol-1 s-1)

In 0.1 mol dm-3 NaBr 0.972 188 1.972 259 2.972 351 3.972 380 4.972 438

mK

166 229 311 336 388

k-1

(64)

(65)

Vo )

kT b { CT*eq} h

(65′)

The exchange rates ratio is linked to the retention coefficient of the activated complex by σ

The errors in the values are estimated at (5%.

equilibrium constant increases with x. This expresses a modification of the kinetic and thermodynamic parameters of the reaction. An increase in the NaBr concentration in the solution decreases the size of these effects. This phenomenon has already been noted by Gaboriaud et al.38 It is assigned to the increased amount of Br- in the vicinity of the DTA+ which decreases the difference of potential between the micelle and the solution. C. Kinetic Exchange of OH- in 0.1 and 0.3 mol. dm-3 NaBr in Water with CTABr at 303.2 K. Kinetic experiments were carried out at 303.2 K to avoid the recristallization of CTABr in the solution. The results are listed in Table 2. The system’s behavior is similar to that observed for DTABr solutions.

b

kTσ X h T*eq

Vo )

b

1.13 1.13 1.13 1.13 1.13

k-1φAOH

The value of this ratio takes account of the effect of the micelles on the probability of species collision and on their ability to react depending on the location of the reaction. According to Eyring’s rate theory, we can write that the exchange rates are linked to the surface molar ratio and to the molar concentration of the activated complex T* at the equilibrium. These exchange rates are σ

mk -1 (s-1)

σ

) ) Vo bk-1bCAOHeq

and mK in CTABra

x (10-3 mol dm-3)

k-1σXAOHeq

Ro ) b

σb

Vo

Vo

σ

σ

)

k-1φAOH b

k-1

)

XT*eq

{bCT*eq}

) φT*

(66)

which is supposed to be independent of the extent of the reaction. Relation 66 shows that the more the activated complex is held by the micellar aggregates, the higher the exchange rate σV will be. A similar result can be obtained by considering the o solution as a homogeneous medium. In this situation, the exchange rate in the micellar solution can be compared to that in the premicellar solution. Relation 67 is thus

Vo

m mb

Ro )

Vo

b

m

)

k-1 (1 + xφAOH) ) γtmfbT* ) 1 + xφT* k-1 (67)

b


Lelie`vre et al.

Relation 37 becomes

[ (

V ) bVo(1 + xφT*) exp

)

(

)]

(68)

)

(

)]

(68′)

B A-B Aeq A A-A Aeq - exp RT RT

or again

[ (

V ) bVoγtmfbT* exp

B A-B Aeq A A-A Aeq - exp RT RT

Relations 68 express a global approach to a reaction rate. This underlines the contributions both of kinetics and thermodynamics to the effect of micellar catalysis. The reaction rate in a micellar medium can vary: (i) because the exchange rate increases, which is due to stabilization of the activated complex by the micellar aggregates. When it is not so stabilized, the chemical exchange rate is equal to the premicellar exchange rate. (ii) because the affinity terms vary: the chemical potentials of the reactants and of the products of this reaction can be modified by the detergent in the solution. From this point of view, micellar catalysis is not chemical catalysis in the usual sense. The additive (the surfactant) is capable to modify the equilibrium position. It is preferable, as we have already pointed out, to talk of a medium effect on the condition under which the reaction evolves. Relation 68 is only of interest when φT* and γtmfbT* can be calculated. This is possible if we can determine separately σk-1 and φAOH when φA, the retention coefficient of TNB, is known. Its value can be obtained by solubility measurements. For these experiments, the chemical potential of the hydroxide ion was supposed to be independent of the presence of DTABr micelles (xφOH is very much smaller than unity).40 In these conditions, relation 34 simplifies to

K)

{mCAOH}

m

{ CA}‚{ COH} m

m

1 + xφAOH

) bK

(1 + xφA)

(69)

φAOH can be obtained using the value of φA. Relation 63 allows the determination of σk-1: m σ

k-1 )

k-1(1 + xφAOH) - bk-1 xφAOH

(70)

and then we can obtain φT* and γtmfbT*. B. Retention Coefficient of TNB, the Mono Adduct, and the Activated Complex. The solubility of TNB was measured in pure water, in 0.1 and 0.3 mol dm-3 NaBr solutions, in DTABr solutions in 0.1 and 0.3 mol dm-3 NaBr at 298.2 K, and in CTABr solutions in 0.1 mol dm-3 NaBr at 303.2 K, above and below the cmc. According to relation 33, the concentrations of TNB at saturations in a micellar solution, mCAsat and bCAsat, are linked by m

CAsat ) bCAsat(1 + xφA)

(71)

We verified that the solubility of TNB is not really influenced by the presence of salt or of surfactant below the cmc. Above the cmc, the solubility increases with x. The values of the ratios of the solubility, mCAsat/bCAsat as a function of x, are reported in Figure 2 for the DTABr solutions in 0.1 and 0.3 mol dm-3 NaBr at 298.2 K and for the CTABr solutions in 0.1 mol dm-3 NaBr at 303.2 K. Relation 71 is valid for both of the surfactants and in all the media studied. Constant values of the retention parameters were found. The values are φA ) 59 for DTABr at

Figure 2. Variation of the ratio of the TNB solubility values as a function of x in DTABr with [NaBr] ) (2) 0, (O) 0.05, (+) 0.1, ([) 0.2, and (9) 0.3 mol dm-3.

298.2 K and φA ) 228 for CTABr at 303.2 K. We checked with the surfactant electrode that the solubilization of TNB, which is at saturation in the micelles, does not modify the concentration of the amphiphile monomer in the premicellar solution. The constant value of φA can be interpreted in different ways. That TNB solubility is independent of NaBr concentration in the premicellar solutions means that the values of Setchenov coefficients are quite small in these systems. The standard chemical potentials of TNB in the solutions of 0.1 and 0.3 mol dm-3 NaBr in the premicellar solutions and in pure water are quite similar. The behavior of the standard chemical potential must be similar in the other aggregated solutions for the different NaBr concentrations. The retention coefficient is independent of the amount of micellized surfactant. This can be explained in two different ways: TNB retention by the micellar aggregates is due either to solubilization (volume phenomenon) or to surface fixation associated with the identical duplication of the micellar aggregates. In this last case, the value of the degree of counterion binding is unchanged. However, it is not possible to choose between these hypotheses at this point of the discussion. Nevertheless, it should be underlined that TNB solubility in a CTABr solution is 4 times greater than that in a DTABr solution, though admittedly the temperature is lower by 5 degrees. But assuming that the nature of micellar aggregation of these two surfactants is similar, the difference in solubility could result from a difference in the interfacial energy between the two spherical aggregates. As noted before by N. Nishikido,41 the smaller an aggregate, the less its extraction power. Finally, whatever assumption is used the retention coefficient can be fitted from a pseudophase model. By knowing the values of φA, the values of φAOH, σk-1, φT*, and γtmfbT* can be determined. These values are reported in Table 3a,b. We have plotted in Figure 3 the variations of the transfer activity coefficients of the activated complex versus x in order to reveal the differences in the behavior of the systems studied. On one hand, for a fixed x value, the activated complex T* is more stabilized in a CTABr solution than in a DTABr one at a salt concentration of 0.1 mol dm-3. On the other hand, in a solution of DTABr, the activated complex T* is more stabilized at a salt concentration of 0.1 mol dm-3 than in one of 0.3 mol dm-3. Similar behavior is obviously observed for the exchange rates.



TABLE 3a: φA, φAOH, σk-1, φT*, and γtmfbT* in DTABr x (10-3 mol dm-3)

φAOH

σ k-1 (dm3 mol-1 s-1)

2.5 3.5 5.5 8.3 13.1 15.5

In 0.1 mol dm-3 NaBr 9505 1.04 9965 0.97 10148 0.93 10743 0.89 12175 0.84 11084 0.94

5.7 13.7 21.7 29.7 37.7 45.7 53.7 61.7

In 0.3 mol dm-3 NaBr 2662 1.04 3470 1.00 4212 0.94 4649 0.92 4947 0.91 5257 0.89 5394 0.90 5605 0.89

φT* 1005 990 967 977 1049 1062 282.1 353.6 405.9 434.6 460.7 479.0 495.1 511.5

γtmfbT* 3.5 4.5 6.3 9.1 14.7 17.5 2.6 5.8 9.8 13.9 18.4 22.9 27.6 32.6

TABLE 3b: φA, φAOH, σk-1, φT*, and γtmfbT* in CTABr x (10-3 mol dm-3) 0.963 1.963 2.963 3.963 4.963

φAOH

σ k-1 (dm3 mol-1 s-1)

In 0.1 mol dm-3 NaBr 54284 0.964 43930 1.029 45943 1.066 42219 1.078 43652 1.090

φT*

γtmfbT*

5341 4615 4999 4645 4855

6.2 10.1 15.8 19.4 25.1

Relations 68 can be used to examine these values and to determine the reasons for the surfactant effect on the reaction rate. In our system, both the kinetic and thermodynamic contributions are in the same direction. The activated complex is stabilized by the micellar aggregates. Simultaneously, the chemical equilibrium is shifted to the mono adduct formation. The exchange rate increases, and the exponential terms are greater than 1. Relations 68 also explain the inhibitor effects observed for some reactions30 when a surfactant is added to the solution. In such cases, one can assume that a part of the reagents but not of the products is stabilized by the micellar aggregates. As a result, the chemical equilibrium is shifted to the favor reagents. Even if the activated complex is supposed to be stabilized by the micellar aggregates, the exponential terms would be less than 1, and the reaction rate in the micellar solution could be lower than in the premicellar one. The effect of “negative micellar catalysis” is mainly a thermodynamic but not a kinetic effect. In Table 3a,b, one can observe that the system behaves differently between the two salt solutions. In both surfactant solutions in 0.1 mol dm-3 NaBr, the mono adduct and the activated complex are strongly extracted by the micellar aggregates. Kinetically, this means a chemical exchange rate on the surface very much higher than that in the premicellar solution and a low value of the reverse rate constant σk-1. This last rate is independent of x. The values of the retention coefficients of AOH and T* depend very little on the amount of surfactant micellized. As a result, a pseudophase model can explain the kinetic and thermodynamic behavior of the exchange equilibrium of OH- between TNB and the mono adduct in the presence of surfactant. The retention coefficient of the activated complex, the mono adduct, and TNB are about 4 times higher in a CTABr solution than in a DTABr one. The retention of the ionic or molecular species increases then with the length of the surfactant chain. This suggests that the interfacial solution/ micelle energy act directly by influencing the value of the retention coefficients of the ionic and molecular species (relation 31). The smaller the radius of the micelle, the lower the retention

Figure 3. Variation of γT* in function of x as: (2) CTABr, [NaBr] ) 0.1 mol dm-3; (9) DTABr, [NaBr] ) 0.1 mol dm-3; (b) DTABr, [NaBr] ) 0.3 mol dm-3.

Figure 4. (a) Variation of ln φAOH as a function of x in ([) DTABr, [NaBr] ) 0.3 mol dm-3. The curve was calculated according to the relation 74. (b) Variation of ln φT* as a function of x in ([) DTABr, [NaBr] ) 0.3 mol dm-3. The curve was calculated according to the relation 74.

capacity of the micelle, whatever the nature of the species which is a consequence of Laplace’s Law. Such a result allows us to assume that in a solution of 0.1 mol dm-3 NaBr, the various reaction species are located on the surface of the micellar aggregates. It means too that the micelles duplicate identically when the surfactant concentration is increased. This kind of behavior is generally admitted in these systems.42-45 In the case of DTABr solutions in a salt


Lelie`vre et al.

TABLE 4: Fitting Parameters to the Experimental Data compound

ln A

p-1

AOH T*

9.95 7.80

0.38 0.38

γ Ri/RT

σ σ

0.71 1.48

concentration of 0.3 mol dm-3, the values of the retention coefficients of AOH and T* are lower than those in 0.1 mol dm-3, and they increase with the amount of micellized surfactant. The pseudophase model cannot be used for this system. The decrease of the values of the extraction coefficients with an increase of salt concentration results from a decrease in the difference of micelle/solution chemical potential: the counterions partly screen the repulsive interactions between the “charged heads” of the micelle (relation 31). The increase of φAOH and φT* with x can be explained by a change in the nature of the micellar aggregates as they become more concentrated in the solution. In fact, if the difference of the electric potential, the difference of the standard chemical potential of i between the micelle and the solution, and σγ are supposed to be constant and independent of x in relation 31, then the retention coefficient of i can be written in a simplified form:

{ } ( σ

φi ) constant

)

σ σ A′2 γ A′i exp p RT

(72)

This relation clearly shows the partial areas of the surfactant and the solute. Their values depend on one hand on the aggregate shape and on the other hand on their mass. Each partial area will be supposed to be a homogeneous function of same degree p of the amount of micellized surfactant in accordance with relation 22

A′2 ) R2xp-1

(73)

A′i ) Rixp-1

(73′)

σ

σ

VI. Conclusion

R2 and Ri are respectively two specific coefficients of the micellized surfactant 2 and the solute i. The variation of the retention coefficient and x (Figure 4a,b) are then linked:

γ Ri p-1 x RT

σ σ

ln φi ) ln A + (p - 1) ln x -

The system’s behavior can then be described with the assumption that the area of the micelle/solution interface is a homogeneous function of x of degree 1.38. This means that the area of the micellar aggregates increases faster than their mass in this medium. The values of both terms linked to the partial areas (functions of x) are listed in Table 5 in order to judge the size of the contribution of each factor to the variation of the retention coefficient. When x increases, the absolute value of the term linked to the partial area of 2 decreases, which has the effect of increasing the value of φi. In the same way, if the absolute value of the term linked to the interfacial energy increases, this decreases the value of φi. The variations of the retention coefficient with the amount of surfactant are the result of two opposed contributions which partially cancel. For both AOH and T*, the main influence on the variations of φi is the change of the partial area of the surfactant. In the case of TNB, whose retention coefficient seems to be independent of x, it is possible to find 3 parameters (ln A ) 6.6; p - 1 ) 0.38; σγσRi/RT ) 4) where the contributions balance themselves totally (ln φA ) 4.15 ( 0.07). This implies that it is not possible to conclude whether the micelles copy themselves identically when the mass increases, for a constant value of the retention coefficient. An increase in the value of the partial areas with the amount of surfactant (homogeneous degree equal to 0.38) means that the area attributable to each aggregated amphiphile increases with the concentration of aggregates in the solution. We must then suppose that the micelle/solution interface is continuously shifted to the bulk when the surfactant concentration increases and when strong interactions are established between the micellar aggregates. Whatever the cause of this phenomenon, relation 74 accounts accurately for the variations of the retention coefficients of T* and AOH with x.

(74)

A least-squares method was used to solve this equation. The parameters in Table 4 allow an accurate description of the experimental results.

The present study is designed to show the different aspects of reactivity in micellar media and, in particular, the phenomenon of micellar catalysis. It seems important to us to show that the reactivity in a micellar medium can be treated either kinetically or thermodynamically in exactly the same way as that used for mixed solvents, such as aqueous organic solvents for example, by the introduction of transfer activity coefficients. The action of the micelles is then related to a simple medium effect, which explains both the modification of the reaction rate and changes in equilibrium.

TABLE 5: Computed Area Terms to Relation 74 compound AOH

T*

x (10-3 mol dm-3)

(p - 1) ln x

-(σγσRi/RT)xp-1

ln φi (calculated)

ln φi (experimental)

5.7 13.7 21.7 29.7 37.7 45.7 53.7 61.7 5.7 13.7 21.7 29.7 37.7 45.7 53.7 61.7

-1.693 -1.630 -1.455 -1.336 -1.246 -1.172 -1.111 -1.058 -1.963 -1.630 -1.455 -1.336 -1.246 -1.172 -1.111 -1.058

- 0.208 - 0.290 - 0.345 - 0.389 - 0.426 - 0.458 - 0.487 - 0.513 - 0.099 - 0.139 - 0.166 - 0.186 - 0.204 - 0.220 - 0.234 - 0.246

5.629 5.880 5.999 6.075 6.128 6.169 6.201 6.228 7.887 8.181 8.329 8.427 8.500 8.558 8.605 8.645

5.642 5.868 6.006 6.074 6.133 6.172 6.205 6.237 7.724 8.152 8.346 8.444 8.506 8.567 8.593 8.631

Thermodynamic and Kinetic Approach in Micellar Media Modeling the micellar medium as a dispersed medium is of interest principally because it gives a physicochemical significance to the magnitude of these transfer parameters through the introduction of retention coefficients whose form depends on the composition variables used (x) and the thermodynamic forces involved. The model we propose takes into account the values of tension likely to exist in a dispersed, charged system, including those relating to the surface energy, and uses a system describing composition which is independent of the localization of the solutes which are attached to the micellar aggregates. By expressing the areas micelle/solution by means of homogeneous function, of a degree which differs from unity, of the amount of tensioactive component aggregated, we described systems whose behavior cannot be explained by a pseudophase model. This approach allows us to propose a general expression for the variation of the retention coefficients with the concentration of micellized tensioactive. For instance, we show that the area of the micellar aggregates increases more rapidly than the mass of tensioactive micellized, following a homogeneous function of degree 1.38, for DTABr in a solution 0.3 mol dm-3 in NaBr. The use of the kinetic approach of Marcelin-Dedonder has allowed us to show that the phenomenon of the acceleration of chemical reactions in certain systems when a tensioactive is added has two sources: (i) A kinetic source. We have shown that the essential effect of the tensioactive is to increase the rate of chemical exchange and, thus, to transform, by analogy with electrochemistry, a system which is low in water into a fast system, no doubt because of a phenomenon of a reduction in space,46 since the reactions take place in confined areas whether on the surface of the aggregates or in the Stern layer. The ratio of the exchange rates indicates the ratio of the probabilities of collisions between reactants, whose ratios are similar to the retention coefficient for the complex activated by the micelle. We have thus been able to calculate the values of these coefficients for the different systems studied and have deduced from them the values of the transfer activity coefficients. (ii) A thermodynamic source. Marcelin-Dedonder’s relation shows explicitly how the chemical affinities appear directly in the expression for rate reaction. In the exchange case studied, the chemical equilibrium is strongly displaced in favor of the formation of the final product, which gives a large thermodynamic contribution to the increase in rate of attack of the hydroxide ion on TNB. This underlines the whole ambiguity of the name “micellar catalysis” when the presence of micelles modifies at the same time the kinetic and thermodynamic characteristics of the reaction. For reactivity in a micellar medium, it seems better to use the term “medium effect”. The approach which we propose is general and can be applied to other dispersed systems with pure or mixed tensioactive components. We are thus currently examining in our laboratory the effect of mixtures of cationic and neutral micelles upon the same reaction for the exchange of OH- between TNB and TNBOH for a range of micellar compositions. VII. Glossary Glossary bX mX σX

variable relative to the premicellar medium variable relative to the homogeneous micellar solution (medium different from water) variable relative to the “phase” constituted of the aggregated surfactants in aqueous solution

J. Phys. Chem. B, Vol. 105, No. 51, 2001 12855 G µi µϑi

x {Ci} γtmfbi K k1 k-1 σA′i bφ σ φ σγ ξ B A B Aeq A A A Aeq eq b V b Veq a V a Veq Vo Ro φi

Gibbs Energy or Free Enthalpy chemical potential of the solute i standard chemical potential of the solute i in solution referring to the behavior of the diluted solutions for the molar scale total concentration of surfactant aggregated in solution value of the molar concentration of i transfer activity coefficient of i from the micellar medium to the phase b equilibrium constant forward rate reverse rate partial area of the solute i at the micellar surface electrical potential of the phase b electrical potential of the micellar surface surface tension between the micellar surface and the phase b extent of reaction “forward” affinity “forward” affinity at equilibrium “reverse” affinity “reverse” affinity at equilibrium at equilibrium “forward” rate “forward” rate at equilibrium “reverse” rate “reverse” rate at equilibrium chemical exchange rate at equilibrium exchange rates ratio retention coefficient of i by the micelles

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