J . Phys. Chem. 1987, 91, 4287-4297 work suggests that at energies 5 1 0 meV electron attachment to C7FI4occurs through two channels, one leading to the formation of long-lived C7F,4-ions, the other C7F14-ions that undergo rapid autodetachment, with the formation of long-lived C7FI4-ions accounting for approximately two-thirds of the total electron capture. If both attachment channels are operative at higher electron energies, the rate constants for electron attachment measured in swarm experiments need not correspond directly to those for the production of long-lived C7FI4-ions. This results because any short-lived C7FI4-ions formed could be rapidly stabilized by collisions with the buffer gas, thereby allowing this reaction channel to contribute to the total measured electron attachment rate. (The lower pressures employed in Rydberg-atom and TPSA studies preclude such stabilization.) If there is significant collisional stabilization of short-lived C7FI; ions occurring in the swarm experiments, then the TPSA data should be re-
4287
normalized to that part of the total thermal attachment rate constant associated with the formation of long-lived C7Fi4-ions. This would reduce the TPSA cross sections and could lead to even better agreement with the Rydberg atom data. The present measurements again demonstrate the value of Rydberg atom collision studies in the investigation of electron attachment at subthermal energies and point to the future application of Rydberg atoms to investigate a wide range of verylow-energy electron interactions.
Acknowledgment. The authors thank A. Chutjian for valuable discussions and for communicating his results to them prior to publication. This research is supported by the National Science Foundation under Grant PHY84-05945 and the Robert A. Welch Foundation. C.W.W. is the recipient of a Texaco Philanthropic Foundation Fellowship.
Micellar Effects on Reaction Rates and Acid-Base Equilibria Denver G . Hallt Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4 WT, UK (Received: May 14, 1986)
The effects of ionic surfactant micelles on reaction rates in solution and on acid-base equilibria often conform quite well to a theory known as the pseudophase ion-exchange model (PPIEM). Despite its success the theory suffers from several limitations. These are outlined and an alternative approach which overcomes them is forwarded. This new theory is based on the equilibrium formulation of transition-state theory. The philosophy underlying it is similar to that used in the treatment of salt effects on the kinetics of reactions involving ions. Attention is focused on the partition coefficients of reactants and activated complexes between micellar and aqueous environments. The author’s previous work on the equilibrium thermodynamics of micellar solutions is used to describe the dependence of these coefficientson the concentrationsof surfactants and supporting electrolyte. However, the approach is not restricted to any particular equilibrium theory. The conceptual basis of the new theory is more sound than that of the PPIEM. It successfully explains experimental data both when the PPIEM succeeds and when it fails. Hence it explains why the PPIEM often works in practice. It is equally convenient to use and is applicable to all types of micelle. Its only limitations are those of the theories on which it is based.
1. Introduction In recent years many papers concerned with the catalysis or inhibition of reaction rates by surfactant micelles have been published and numerous reviews have appeared; see, for example, ref 1-4. Prominent among theories used to interpret experimental data is one known as the pseudophase ion-exchange model (PPIEM).3,4 This model has evolved over a period of years. It incorporates many of the successful features of earlier treatments and is closely related to other approaches which recognize the same basic issues but handle some of them differently. The theory has been used to interpret kinetic measurements and studies of acid-base eq~ilibria.~ Although there are instances where it fails, the agreement with experiment is often quite good. This indicates that the theory is essentially correct and suggests that further progress is likely to arise from a refinement of the various ideas which form its basic framework. It is argued below that this may not be so. In section 2 the main assumptions of the theory are analyzed and some of its inherent weaknesses are exposed. In sections 3 and 4 an alternative treatment is forwarded. This treatment is based on the equilibrium formulation of transition-state theory6 and as such emphasizes the role of the transition state and the effects of the environment thereon. It also utilizes recent developments in equilibrium treatments of micellar systems, especially as applied to solutions of ionic surfactant^.^-'^ The philosophy it is based on is essentially the same as that underlying the application of transition-state +Alsoat Unilever Research Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside, L63 3JW, UK.
0022-3654/87/2091-4287$01.50/0
theory to the interpretation of ionic strength effects on reaction rates in electrolyte solutions.6 The expressions which result are in some cases very similar to those given by the PPIEM but are both more general and more rigorous. In section 5 four applications of the theory are described. The first two are concerned with fairly dilute solutions of interacting micelles. The two cases considered are that where the principal counterion is a reactant and that where one of the reactants is an indifferent coion. The third application concerns the effects of small nonmicellar aggregates. The fourth is to reactions in solutions where the concentration of supporting electrolyte is of order 1 M or more. In section 6 acid-base equilibria are discussed. The assumptions made in the theory are so few that any failure can be attributed to a breakdown of the equilibrium treatment or to a failure of transition-state theory itself. Consequently the (1) Bunton, C. A. Prog. Solid State Chem. 1973, 8, 239. (2) Berezin, I. V.; Martinek, K.; Yatsimirski, A. K. Rum. Chem. Rev. 1973, 42, 487. (3) Chaimovich, H.; Aleixo, R. M. V.; Cuccovia, I. M.; Zanette, D.; Quina, F. H. In Solution Behavior of Surfactants; Theoretical and Applied Aspects; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; Vol. 2, p 949. (4) Romsted, L. S. in Surfacrants in Solution; Mittal, K. C., Lindman, B. Eds.; Plenum: New York, 1984; Vol. 2, p 1015. (5) Romsted, L. S.J. Phys. Chem. 1985,89, 5107, 5113. (6) Glasstone, S.;Laidler, K. J.; Eying, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (7) Hall, D.G. J . Chem. SOC.,Faraday Trans. 1 1981, 77, 1121. (8) Hall, D. G. Colloids Surf. 1982, 4 , 367. (9) Hall, D. G. In Aggregation Processes in Solution; Wyn-Jones, E., Gormally, J., Eds.; Elsevier: Amsterdam, 1983; Chapter 2. (10) Hall, D. G. J . Colloid Interface Sci. 1987, 115, 110.
0 1987 American Chemical Societv
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theory has considerable power both as a predictive and as an interpretive tool. 2. Short Critical Analysis of the Pseudophase Ion-Exchange Model Consider a biomolecular reaction as A +B products
-
occurring in a micellar solution. According to the pseudophase ion-exchange model (PPIEM) the reactants A and B partition between micellar and aqueous environments and reaction occurs in both. Qualitatively, rate enhancements can arise from two effects. Firstly the rate coefficient in a micellar environment may be substantially greater than that in free solution. Secondly the local product CACB of the reactant concentrations in the micelle may be much greater than that in solution so that JCACB dVover the entire volume of the system is greatly enhanced. For unimolecular reactions only the first effect operates. Quantitatively the overall rate in solution per unit amount of solvent (usually water) is written as rate = kamAmB aqueous rate
+ kmCsmnAnB micellar rate
(1)
where mA and mB are the aqueous concentrations of A and B expressed as moles per unit amount of solvent, n A and nB are the micellar concentrations of A and B expressed as moles per mole of micellar surfactant, Csmis the concentration of micellar surfactant in moles per unit amount of solvent, and k , is the rate constant in the absence of micelles and is assumed to depend only on T and p . k , is the rate constant for the reaction in the micellar pseudophase and, for a given surfactant, is also assumed to depend only on T and p . To what extent, if any, k , is expected to depend on the nature of the surfactant is not always made clear. For uncharged species which are incorporated into the micelles it is proposed that ~A,"A
= KA
(2)
where K A is a partition coefficient sometimes assumed to depend only on T and p . Since the total concentration of A, CA,is given by CA = mA + nACsm (3) it follows that mA =
1
+ KAC,"
and that
An important feature of the PPIEM is the assumption that we may talk about micellar concentrations of hydrophilic counterion species in much the same way as for true micellar constituents such as surfactant ions and solubilized additives. Let 2 denote the counterion of the surfactant which we may often take as the most abundant counterion species in the system. For other counterion species with the same valency as 2 it is argued that
where at constant T and p , KAz and P are both constant. For 1:l surfactants /Iis the fraction of the micelle charge neutralized by bound counterions and KA2 is the so-called ion-exchange constant for the replacement of micellar 2 by A. The total concentration of 2, C,, is given by
C2 = m2 + n2Csm
(7)
Hall Equations 5-7 summarize the ion-exchange aspects of the model for reactive species which are counterions. Such species may include H+ ions for anionic micelles and nucleophiles such as CNand OH- for cationic micelles. Given eq 1-7 it is quite straightforward to obtain expressions for the apparent second-order rate constant kappdefined by rate = ka,,CACB
(8)
for the cases where A and B are both neutral micellar constituents or where B is a micellar constituent and A is a counterion. This summarizes the principal ideas of the PPIEM. Some criticisms are as follows. (1) The second term on the right-hand side (RHS) of eq 1 is a straightforward application of mass-action kinetics to the reaction in a micellar environment. The expression suggests that nA and nB are well-defined even if the species concerned is a counterion. This in turn implies that counterions can be assigned unambiguously to one or other of the two classes bound or free. However, such a viewpoint is hard to reconcile with the notion that the distribution of counterions around a charged micelle is governed primarily by electrostatics. It is not well supported by the kind of spectroscopic evidence that should be readily obtainable if counterions do indeed exist in two distinct environments. Also poor agreement is obtained from different methods of estimating /?.'I Examples of such methods are electrophoresis, the behavior of colligative properties, and the effect of electrolyte on the cmc. (2) Mass-action kinetics implies ideal behavior of the reactants. Hence it is difficult as matters stand to incorporate nonideality into the theory. Morever, since species which behave ideally are usually dilute this presumably restricts the applicability of the theory to systems in which the reactants are dilute in both phases. This may not be the case for some micellar reactants even when their total concentration is small. A good example of this is that in which one of the reactive species is species 2 and it is noteworthy that the theory is known to fail for some such cases but not all. (3) It is apparent from eq 22 of ref 12 that assuming KA to be independent of surfactant concentration and added salt may be suspect for uncharged species. For charged species allowance must be made for counterion effects. As we will show below the PPIEM fails to do this properly in some cases. A satisfactory theory should overcome this difficulty. (4) When one of the reactants is an indifferent coion eq 1 predicts that the micellar rate should be zero if the micellar concentration of coions is zero. In practice some such micellar rates are nonzero. Hence if the theory is correct indifferent coions should strictly be counted as micellar constituents. This notion is hardly welcome. Within the context of the theory there appears to be no obvious method of defining the micellar amounts of indifferent coions or of estimating them from equilibrium measurements. Although the Poisson-Boltzmann equation can be used to estimate coion concentrations at the micelle surface,I3 this falls outside the scope of the PPIEM as it stands. It is apparent from the above arguments that despite their success the PPIEM and related treatments are not entirely satisfactory. Evidently there is considerable scope for improving the conceptual framework and for the development of a more general treatment. We now turn to this issue.
'
3. Application of Transition-State Theory Let us suppose that the above reaction proceeds via the formation of activated complexes. In the absence of micelles we may write for the rate per unit amount of solvent rate = k a h f (9) where ml is the concentration of activated complexes which have formed from reactants and kat is a term which involves a number of quantities including the transmission coefficient and the rate at which complexes cross the top of the energy barrier whose height is the activation energy. The important features of eq 9 are that (11) Lindman, B.; Wennerstrom, H. Top. Curr. Chem. 1980, 87, I . (12) Hall, D. G.; Tiddy, G. J. T. In Anionic Surfactants; LucassenReynders, E. H.; Ed.; Marcel Dekker: New York, 1981; Chapter 2. ( 1 3 ) Bell, G . M.; Dunning, A. J. Trans. Faraday SOC.1970, 66, 500.
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4289
Micellar Catalysis
mt is a concentration and not an activity and that kat is fairly insensitive to the environment in which the reaction takes place. According to this viewpoint changes in rate at constant T a r e due almost entirely to changes in m f . When the equilibrium hypothesis is valid it can be shown that m* is related to the concentration of the reactants by an expression such as wAe + R T In mAyA + pBe + R T In mByB = yet + R T In m*yt (10) where kCLB denotes a standard chemical potential which depends only on T , p , and solvent composition and y denotes an activity coefficient. On rearrangement eq 10 gives
mt = exP [MAe
+ FBe
- Fell mAYAmByB
RT which when combined with eq 9 gives rate =
yt
kamAyAmByB Yt
(11)
(12)
where k, is the second-order rate constant givenby
Equation 12 is essentially the well-known expression used by B r ~ n s t e d and ’ ~ many others since to describe the effects of ionic strength on reactions involving charged species in fairly dilute aqueous solutions. When yt, y A , and ye are all unity we have of course mass-action kinetics. Suppose now that we have micelles present and that some of the activated complexes are found in a micellar environment. Instead of eq 9 we now write rate = kafmt+ kmfC,l
(14)
where Cmtis the concentration of complexes present in the micelles expressed as moles per unit amount of solvent and k,t is the micellar analogue of kat. Since we have argued that kt can be expected to be insensitive to the environment in which the reaction takes place we might expect kat and k,t to be nearly equal. However, it is clearly more general to allow these quantities to differ, and it turns out that this does not lead to any extra complications. What is important is that k> should not alter significantly when the environment of aqueous activated complexes is altered (e.g., by the presence of mcielles) and that k,t should not alter significantly when the environment of micellar activated complexes is altered (e.g., by addition of cosurfactants, etc.). Usually the reactions of interest in micellar catalysis and inhibition are sufficiently slow that reactants in aqueous and micellar environments are in equilibrium. It follows therefore that if the equilibrium hypothesis of transition-state theory holds in both environments then micellar activated complexes * aqueous activated complexes in the sense that equilibrium is used in the equilibrium hypothesis. We now define the quantity L f by
and note that L* thus defined is an equilibrium quantity somewhat analogous to a partition coefficient between micellar and aqueous environments. The reasons for including y f will become apparent later. Equations 14 and 15 now give
Hence we may replace /calm:in eq 16 by the right-hand side of eq 12 to give
This expression is the fundamental kinetic result describing the effects of micelles on the rate of a biomolecular reaction. We note that mAyA and mBYB are the aqueous activities of A and B and that yj is the aqueous activity coefficient of the activated complexes. If A is a micellar constituent we may define the quantity LA by L A = na/mAyA (18) which together with eq 3 gives in analogy with eq 4a
Like Lt, LA closely resembles a partition coefficient between micellar and aqueous environments. Obviously LA = KA when yA = 1. Equation 19 may be used to substitute for mAyA in eq 17. If A is a counterion then mAyA is best regarded simply as the activity of A, aA. Thus if counterions can be regarded as bound or free and the activity coefficient of free counterions is assumed to be unity then a, is simply the concentration of free counterions. Otherwise it is this concentration times an appropriate activity coefficient. Alternatively if all counterions are regarded as free then mA may be taken as the total concentration of A and the effect of the micelles can then be allowed for through its effect on yA. The thermodynamic treatment outlined in ref 7-9 shows that in practice the notion of counterion binding turns out to be useful. If A is an indifferent coion it makes sense to suppose that mA is the total concentration of A and that y A is an activity coefficient which may be modified by the presence of micelles. Similar considerations apply to the activated complexes with regard to the effects of micelles on yt both when the complexes are counterions and when they are coions. It should be noted that the activity coefficients YAYB and 71 always occur together in a form that refers to an electrically neutral combination of ions. It is interesting to compare eq 17 with the PPIEM. Two points which emerge immediately are that nonideality appears naturally in the present treatment through the inclusion, where appropriate, of activity coefficients. Secondly the present treatment encounters no difficulty in handling reactions which involve indifferent coions. When A and B are both micellar constituents eq 1 and 2 together lead to the result
When y A y Band yl are all unity eq 17 and 20 are strictly equivalent if
However if k,, k,, kmt, and kat are all constants the equations are practically equivalent in the sense that kinetic experiments cannot distinguish between them if KAKB/Ltdoes not depend on the concentrations of reactants, surfactant, or added salt. If A is micellar constituent and B is a counterion present in small amounts compared with species 2, then eq 1, 2, 5 , and 6 give
which is equally applicable to unimolecular reactions. We note that eq 9-13 are equally valid whether micelles are present or not. (14) Bronsted, J. N. Z . Phys. Chem. 1922, 102, 169; 1925, 115, 337.
If instead B is the only counterion species present, eq 1, 2, and 6 give
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The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
When yA = yB= y f = 1 and k,, kb, k,f, and k,* are all constant, eq 22 and 17 are practically equivalent if KAKBfi/m,Lfis constant. Similarly eq 23 and 17 are practically equivalent if KAP/mBLf is constant. Although eq 17 is quite general it is somewhat formal and of little use as a predictive tool unless some means can be found for calculating the various y’s and the effects of solution composition on LALB and Lf. It is worth noting, however, that this is purely an equilibrium problem which it should be possible to tackle successfully given a satisfactory thermodynamic theory for the multicomponent micelles of interest. Also the results of this section can be used with any equilibrium theory as well as with those outlined in the following sections of this paper. 4. Equilibrium Theory a. Nonionic Surfactants. For dilute solutions of nonionic surfactants the multiple equilibrium/small systems thermodynamics treatmentl5-l7 provides a formal description of their thermodynamic behavior which for most practical purposes may be regarded as exact. At constant T and p this theory leads to the following micellar Gibbs-Duhem equation
d In C,,, = ENl d In m,
(24)
I
where C, denotes the concentration of micelles, the summation includes all micellar constituents, N,is the average number of i molecules per micelle, and it has been assumed that the nonmicellar forms of the micellar constituents behave ideally. Consider now a second-order reaction A B products in a micellar solution containing only one surfactant species 1. In accordance with the equilibrium hypothesis we suppose that the activated complexes can be treated thermodynamically in the same way as any other species present in very small amounts. Hence for micelles consisting of surfactant, 1, A, B, and activated complexes we divide eq 24 by N l and rewrite it as
+
-
d[ln ml + ?tA In mA + nB In mB + n f In m f ] = In mA dnA + In mB dnB + In m* dn* + ( l / N l )d In C, (25) Subtracting d( l/N,) from both sides we obtain the more convenient form
Hall
We note that the derivatives in eq 28a,b do not include the constraint of constant nf because the minute changes in nl encountered in practice have a negligible effect on K A , L’ and N,. It does not follow from this that derivatives with respect to nf are negligible. Similar arguments apply to the constraints of constant n A and nB when these are very small. Equations 28a,b relate the dependence of K A and L: on the concentration of micellar surfactant at constant micelle composition to the respective effects of A and activated complexes on the average value of Nl. For micelles with large aggregation numbers it is clear that these effects and the corresponding effect on the micellar rate coefficient will be significant only if very large changes in N,are involved. Any effects of factors such as micelle geometry on the reaction kinetics are allowed for implicitly in the treatment. In this respect it is a complete theory. It shows that for large micelles the phase approximation’* is equally valid in the interpretation of kinetic data as in the discussion of equilibrium. Suppose now that the micelles consist of two principal components i and j . We may define x l , x,, n A ; and KA by
R
x, =
+ RJ’
x
7
N,
= N,
rc; m
nA=-
+ N,’
N A
Nl
nA
+ N]’ K A = m,
If we now regard the micelles as a separate phase by ignoring the left-hand side (LHS) of eq 25 we may write 0 = x, d In tn, + xJ d In mJ + n A d In mA (29) I
This gives on rearrangement d [x, In mi + xJ In mJ nA In mA] = -In (m,/m,) dx, - In mA dn, (30)
+
Hence
This expression shows how the dependence of KA on micelle composition is related to the effect of A on the thermodynamics of mixing of i and j in the micelles. Again KB and L* can be treated similarly. b. Ionic Surfactants. ( i ) General Treatment. For multicomponent solutions of noninteracting micelles it can be shown rigorously, by application of the Kirkwood-Buff theory of solut i o n ~ , that ’ ~ ~at~constant ~ T and p we may rite^^'*^^' RT d In C, = X ( N i + ri)dpi (32) i
On cross differentiation this equation gives
However since KA = n A / M A it follows that the left-hand side of eq 27 is equal to
Equation 32 is a micellar Gibbs-Duhem equation. The summation on the right-hand side includes all species other than solvent. fl, is the average number of i molecules or ions in a micelle and ri is the adsorption of nonmicellar i per micelle defined according to the convention that the adsorption of solvent is zero. If i is charged then as matters stand pi in eq 32 is the chemical potential of a single ionic species. However, because
x ( + ~r,)vj ~= o
(33)
i
and that
where u denotes ionic valency including the sign. We may replace the wi in eq 32 by electrochemical potentials. We may also rewrite the expression in terms of electrically neutral components in the formgi21 RT d In C, = X ( N j r i )doi (34) i
+
Likewise
where Bi = (pi - (ui/ui)pC)and c is some ionic species whose choice
(15) Hall, D. G.; Pethica, B. A. Nonionic Surfactanrs; Schick, M . , Ed.; Marcel Dekker: New York, 1967; Chapter 16. (16) Corkill, J. M.; Goodman, J. F.; Walker, T.; Wyer, J. Proc. R. SOC. A 1969, 312, 243. (17) Hall, D. G. Trans. Faraday SOC.1970, 66, 1351, 1359.
(18) Shinoda, K.; Hutchinson, E. J . Phys. Chem. 1962, 66, 577. (19) Kirkwood, J. G.; Buff, F. P.J . Chem. Phys. 1951, 19, 774. (20) Hall, D. G. Trans. Faraday SOC.1971, 67, 2516. (21) Hall, D. G. J. Chem. Soc., Faraday Trans. 2, 1972,68, 1439. 1977, 73, 897.
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4291
Micellar Catalysis is arbitrary but which is preferably not a micellar constituent. Equations 32 and 34 can be used to describe the behavior of LALB and L* in the same way as eq 24. In particular, when the activated complexes are charged, a quantity B* may be defined in the same way as the 0,. Further detail on this point is given in section 5d below. For noninteracting micelles eq 32 and 34 are essentially exact and can be expected to apply at the cmc and to solutions above the cmc in the presence of sufficient amounts of supporting electrolyte to suppress intermicellar interactions. However, the summation in eq 34 includes all ionic species other than c. Strictly therefore terms due to indifferent coions must be taken into account when there is more than one such species in the system. ( i i ) Approximate Treatmentfor Fairly Dilute Solutions. When it is not legitimate to ignore interactions between charged micelles the application of the rigorous thermodynamic methods cited above leads to expressions whose usefulness in the present context is somewhat limited by their complexity. However, in ref 7-10 a simpler but effective approximate treatment is developed which is not limited in this way. As it stands this treatment applies strictly only to solutions in which there is a single counterion species. For our present purposes we will assume without proof that the approach can be extended slightly to deal with solutions which contain several counterion species. This enables us to write at constant T and p d In C, =
ZiV, d In a, + ZiVr d In a,
(35)
r
I
where the summation over i includes surfactant and other solubilized species and the summation over r includes counterions. Although it poses no special problems we will not in this paper consider the case where a given counterion species is also incorporated into the micelles. As before let Ci and C, denote total concentrations. We define the mi and m, by
mi = C, - iV,C,
(36a)
In accordance with ref 9 and 10 we suppose that
where
z is the valency of the species concerned,
I = '/z[Cmlzl2 + Em~,21 I
(39)
r
and A is the appropriate constant given by Debye-Hiickel theory. We now consider the application of eq 35 to micellar catalysis. Let 1 refer to the principal surfactant component. We may rearrange eq 35 as d In al = -Enl d In a, - En, d In a, - (1 /N,) d In C, I
(40)
r
In general we expect eq 40 to contain terms for reactants, products, and activated complexes. Bringing in these terms explicitly and dropping the term in d In C , which will usually be negligible in comparison with the others we obtain d In a, = -En, d In a, - En, d In a, - n A d In uA1
r
nB d In aB- n* d In at (41) By making the appropriate Legendre transformation we may be rewrite eq 41 as
+
d [In a , Eni In ai + nA In aA + nB In aB] = E l n a, dn, In aA dnA + In aBdnB + In a* dnt I
+
Enr d In a, r
(42)
This expression leads to several Maxwell relationships. Among the most useful are the following.
where in (43a), the n, strictly include nB and n* and in (43b) the ni include nB and nA. We now recall that
LA =
~A/QA
L* = n*/a*
(44a) (44b)
Hence the LHS of eq 43a is equal to -(a In LA/d In ar)n,,nA,as and describes the effect of added salt on LA. The R H S shows how this quantity is related to the effect of incorporating A in the micelles on the binding of the counterion species. Similar remarks apply to the behavior of 151 and LB. Hence it is apparent that the information required to make eq 17 useful can be obtained via equations such as eq 43a and b provided that we have some means of calculating the R H S of these equations. This in turn is likely to involve more specific considerations such as the thermodynamic treatment of mixing in micelles of differently charged surfact a n t ~ . ~ ~We , * will ~ discuss this type of application in the next section. A similar theory may also be useful in calculating the dependence of LAand L* on micelle composition.
5. Applications and Comparison with Experiment a. Reactions between a Solubilized Additive and a Counterion in Fairly Dilute Solutions. For reactions of this kind we have shown above that the present theory and the PPIEM agree if activity corrections are negligible and if KAKB2b/m2L* does not depend on the concentrations of surfactant and supporting electrolyte. In the presence of sufficient amounts of surfactant when the micellar mole fractions of reactants and products are small we expect 0to be constant. Also it is probably reasonable to expect KB2to be constant especially when both counterions give 6 values which are close together. We note also that the effects of surfactant concentration on quantities such as KA and L* are usually attributable to changes in m2 as effects due to changes in C, are usually small. Consequently we focus attention on how the ratio K A / L xdepends on m2. For greater ease of comparison with previous work we put activity coefficients equal to unity. The equations we require are (43a) and (43b) and our strategy is to evaluate the R H S thereof. To do this we use the model outlined in ref 22 and 23 which describes the effects of micelle composition on the effective degree of ion binding (1 - a ) . According to this model a is given by 1 _ -1+-
a
U
w
(45)
where at a given T and p , o is a constant and CJ is the charge density on the micelle surface. In general a is defined by
ENrVr (1 - a ) = --CN,Vi
(46)
I
where the summation over i includes reactants and products where appropriate as well as surfactant species. For simplicity we consider the case where there is only one surfactant species and only one counterion species whose concentration is many times that of any others. We let uo denote the charge density on the micelle surface for the pure ionic micelles and let a. be the corresponding value of a. We suppose that neither quantity depends significantly on the concentration of surfactant or sup(22) Hall, D. G.; Price, T. J. J . Chem. SOC.,Faraday Trans. 1 1984, 80, 1193. (23) Hall, D. G.; Huddleston, R. W. Colloids Surf. 1985, 13, 209.
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The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
porting electrolyte. We suppose also that additives with the same charge as the surfactant do not alter u, that for neutral additives u = uo(1 - .x)
(47)
and that for additives with an equal but opposite charge to that of the surfactant u
= uo(l - 2x)
zn2
(1 - ao) = __
= ( 1 - 'yo)
For neutral additives
"\
transition state
where z = 1 . If the reactant has the same charge as the surfactant, the transition state is neutral whereas if the reactant is neutral, the transition state is oppositely charged to that of the surfactant. For the neutral transition state eq 56 and 57 give
and it turns out that for the neutral reactant eq 57 and 58 give exactly the same result so that eq 59 holds in both cases. Hence for both cases
This equation shows that when a. is small (typically of order 0.2) then In (KA/L*m2) should depend hardly at all on the concentration of surfactant and added salt. Indeed the effects of nonideality may be more significant than the RHS of eq 60. Hence when a. is small the treatment based on transition-state theory and the PPIEM agree closely. It is noteworthy that under these circumstances the PPIEM also agrees well with experiment. However, when cyo is large the approaches should differ substantially. The simplest case to consider is that where the reactive counterion species is the only counterion species present in significant amounts. Whereas for this case the PPIEM predicts that the pseudo-first-order rate constant kappshould be independent of surfactant concentration or added salt the approach outlined above predicts that it should increase by an amount given by
where z = - v 2 / v 1 . This gives immediately
(1 - a ) = zn2 =
-
(49)
l + n
$)m*
+ counterion
(48)
where in both cases x denotes the micellar mole fraction of the additive. Equations 45 and 47 fit experimental data on mixtures of ionic and nonionic surfactants quite We are not aware of any tests of eq 48. According to eq 43 the quantity we wish to evaluate is (an,/ where n is the average number of additive molecules in the micelles per surfactant molecule. For additives with the same charge as the surfactant we have
z(
reactant
Hall
l+nj uo
+ w(1 + n )
Hence
which in the limit that n 0 is equal to -ao( 1 - ao). For oppositely charged additives +
(57)
The PPIEM fails for this kind of system when a. is large. A good example is cationic surfactants with hydroxide counterions which are believed to have a. values of order 0.5-0.7.24 For these systems and for fluorides also kappvalues do indeed depend markedly both on the concentration of surfactant and added At present a quantitative comparison is not feasible because reliable values of a0are unavailable. Such values should be taken from the effect of salt on the cmc, with due allowance for solution nonideality, or from thermodynamic data such as that obtained from colligative properties, light scattering, or E M F studies rather than from extrathermodynamic techniques such as conductivity or electrophoresis. Some typical predictions of eq 61 are presented in Figure 1 which shows how kappfor a reaction occurring entirely in the micelles is expected to depend on the concentration of surfactant and added salt with a common ion when a0 = 0.5. The similarity to Figure 6 of ref 27 and Figures 4 and 5 of ref 28 is quite striking. Both the trends and the order of magnitude of the observed effects are predicted. Similar considerations also apply to the effects of surfactant and salt on reactions involving counterions other than the most abundant. Hence we may conclude that as matters stand the approach described above s u d s both when the PPIEM succeeds and when it fails. Moreover it does so quite naturally without making extra hypotheses or assumptions concerning the special nature of the systems involved. The theory also predicts that the
(58)
(24) Sepulveda, L.; Cortes, J. J . Phys. Chem. 1985, 89, 5322. (25) Bunton, C. A,; Gan, L. H.; Moffat, J.; Romsted, L.; Savelli, G. J . Phys. Chem. 1981.85, 4118. (26) Bunton, C. A,; Romsted, L. S . Solution Behaviour ofSurfactants,
Expressions identical with eq 56-58 can be expected to hold for the activated complexes with L* replacing K. Consider now the reaction
Theoretical and Practical Aspects; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; Vol. 2, p 975. (27) Rodenas, E.; Vera, S . J . Phys. Chem. 1985, 89, 513. (28) Bunton,C. A,; Gan, L. H.; Moffat, J. R.; Romsted. L. S.; Savelli, G. J. Phys. Chem. 1981, 85, 4118.
which gives on rearrangement uo(l - n)* zn2 =
+ n ) + uo(l - n )
w(1
(54)
Hence [u0w(3 -
z(
2)m: =
-
2n - n2)
+ uo2(1- n ) 2 ]
(55) + n ) + uo(l - n ) ] , 0 is equal to -[ 1 + 2 4 [ 1 - ao].
w(1
which in the limit that n Hence for additives with the same charge as that of the surfactant we have by virtue of eq 44 -+
(E ! &-
(1 - 'yo)
(56)
For a neutral additive ao(1
( = A ) = - 'yo)
z for an oppositely charged additive
A (!&-
(1
+ 2'yONl - 'yo) Z
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
Micellar Catalysis 3.5
3.0
c
E 5 2.5
-m
-a 'S L
0
n m2.0 Q
Y
1.5
4293
effects on reaction rates comparable to those of micelles. Very recently a stepwise association mechanism hs been forwarded as a basis for interpreting kinetic data on this kind of system.30 However, simplifying assumptions concerning successive equilibrium constants were introduced when the approach was applied. This detracts from its generality. Also no allowance was made for aggregate/counterion interactions. The theory developed in section 4a above makes no assumptions concerning successive equilibrium constants. It is applicable to all kinds of aggregation between uncharged species in dilute solution and includes the stepwise association mechanism cited above as a special case. According to this treatment the rate coefficient in the aggregates k , of a unimolecular reaction will increase with increasing surfactant concentration if L i / K A increases. As eq 28 shows, changes in this ratio are governed by the effects of the reactant and the activated complexes on the average aggregation number, W,, of the surfactant. This ratio will increase if aggregates containing an activated complex have a larger average N , than aggregates containing one reactant molecule. If the two values of N , are the same, then k , will not depend on surfactant concentration. Similarly it can be shown by a straightforward application of the equilibrium theory developed in ref 17 that KA will be independent of surfactant concentration in the limit that nA 0 when the average number of surfactant molecules in an aggregate containing one molecule of A is equal to the weight average aggregation number of the surfactant. A derivation of this result is given in the Appendix. The presence of the term l/Rl2on the right-hand side of eq 28 shows that the effects of surfactant concentration on rate coefficients and partition coefficients into aggregates can be expected to be greatest for surfactants which form small aggregates. For fairly dilute solutions of charged nonmicellar aggregates at moderate ionic strengths, (0.1 M or less) the treatment described in section 4b(ii) may apply. However, the approximation of ignoring the term in d In C, may not be legitimate and it is also possible that a could depend appreciably on the concentration of free counterions. When this is the case the predictive capability of the theory may be reduced somewhat but it may still be useful as an interpretative tool. When interactions between aggregates are negligible the approach in section 4b(i) applies as it stands. This will usually be the case when the concentration of aggregated material is fairly small and large amounts of supporting electrolyte are present. In some cases it may be more appropriate to apply the KirkwoodBuff theory of s o l u t i ~ n sas ' ~outlined ~ ~ ~ in ref 32 rather than take account of aggregation explicitly. d. Ionic Surfactants in the Presence of Large Salt Concentrations. Recently a failure of the PPIEM has been reported for some acid-catalyzed hydrolysis reactions under conditions where the concentration of the acid (HC1) was in the range 0.5-3 M and substantially exceeded the concentration of the surfactant (SDS).33 To explain their results the authors postulated an additional catalytic pathway across the micelle/solution interfacial boundary. We will try to show that this extra hypothesis is unnecessary and that a reasonable explanation can be given in terms of the ideas outlined above. For simplicity we will consider a reaction which is exclusively micellar and will suppose that H+ ions are the only cations in the system. The latter simplification should be reasonable when the H+ concentration is several times greater than that of Na+ and enables us to treat the system as one with a reactive counterion. In this case however, the failure of the PPIEM cannot be explained in the same way as the failure for reactions involving hydroxide counterions because there is good evidence to suggest that a. for micelles of dodecyl sulfuric acid is fairly similar to the value of about 0.2 for the sodium salt. Under these circumstances the
-
n12/n
1.0 010 0.02 0.04 0.06 0.08 0:1
CT,M. vs. micellar surfactant concentration (Csm) at several concentrations of supporting electrolyte ( c 3 )as predicted by M; (C) c3 = 5 X eq 61 with a = 0.5: (A) c3 = (B)c3 = M; (D)c3 = lo-' M. krelis the value of k,,,(micellar) when m2 = IO--'
Figure 1. k,,,(micellar)/k,f
M.
PPIEM can be expected to fail for reactions involving counterions in mixed micelles of ionic and nonionic surfactants when a2is no longer of order 0.1 or less. Since the above expressions relating the dependence of a on micelle composition take no explicit account of molecular structure and any associated specific effects, eq 56-61 may not always apply. However, the recent studies of Rathman and ScamehornZ9together with the results reported in ref 22 show that eq 45 holds quite well in mixed micelles of ionic and nonionic surfactants for a variety of systems. b. Application to Reactions Involving Coionr in Fairly Dilute Solutions. The treatment described above is readily applied to reactions involving coions. The comparison with the PPIEM can be made in much the same way as with reactions involving counterions. Again the predictions of the new theory agree with those of the PPIEM when ao2
