PIE model

As an alternative to these PIE models, we now consider the simple inflow model of Section 8.5.2. In this model we assume... 

The effects of micelles of cetyltrimethylammonium bromide (CTABr), tetradecyl-trimethylammonium bromide (TTABr) and sodium dodecyl sulfate (SDS) on the rates of alkaline hydrolysis of securinine (223) were studied at a constant [HO ] (0.05 m). An increase in the total concentrations of CTABr, TTABr and SDS from 0.0 to 0.2 M causes a decrease in the observed pseudo-first-order rate constants (kobs) by factors of ca 2.5, 3, and 7, respectively. The observed data are explained in terms of pseudophase and pseudophase ion-exchange (PIE) models of micelles. Cationic micelles of CTABr speed attack of hydroxide ion upon coumarin (224) twofold owing to a concentration effect. ... 

For the first case, one can use the so-called pseudophase ion exchange (PIE) model.The PIE model is based on the Menger-Portnoy model but additionally allows for ion exchange to occur in the micellar Stern region where a reactive counterion competes with nonreactive counterions (Scheme 5). 

The kinetic profile of reactions involving the micelle s counterions is frequently analyzed in terms of the PIE model. Despite the known shortcomings of this model, it nevertheless typically reproduces kinetic data rather well - though one should remain conscious of the potential problems related to parameter covariance (vide supra). 

The emphasis placed on the last assumption is responsible for the name of the model. It is now well known that these assumptions, especially the first two, are reliable with impunity only over very narrow and dilute micellar concentration ranges. Nevertheless, the PIE model has provided invaluable insight over the past 25 years in elucidating micellar catalysis. Its failures [27-31] are usually attributable to clear-cut violations of its simple assumptions. Refinements or alternatives to these basic premises such as solving the nonlinear Poisson Boltzmann equation for the cell model have not proved to be particularly enlightening nor more helpful [32]. The extension of the PIE model to complicated micellar systems where anomalous rate behavior is more often than not the rule rather than the exception is probably unwarranted [33]. Sudhdlter et al. [34] have critically reviewed the Berezin model and its Romsted variation, the PIE model, as matters stood 20 years ago. In... 

The combination of the pseudophase assumption with mass action binding constants of substrates and ion exchange of reactive and nonreactive counterions is called the pseudophase ion-exchange (PIE) model [10,48,66]. It successfully fits the kinetics of many bimolecular reactions and also shifts in apparent indicator equilibria in a variety of association colloids, especially reactions between organic substrates and inorganic ions in normal micelles over a range of surfactant and salt concentrations and types (up to about 0.2 M). It has also been successfully applied to cosurfactant-modified micelles [77,78], O/W microemulsions [79-81], and vesicles [82]. 

Theoretical models similar to those applied to reactions in normal aqueous micelles have been used to simulate microemulsion effects on reactivity when the transfer equilibria of the nucleophiles could not be measured directly. The ion-exchange (PIE) model [Eq. (6)] was applied to the reaction of /j-nitrophenyldiphenyl phosphate (6) with OH" or F" in microemulsions of CTABr with 1-butanol as cosurfactant [79]. The rate data could be fitted to this model, even though in solutions of OH" the 1-butoxide ion formed by deprotonation of the alcohol is a competing nucleophile. 

Similar observations were made on reactions of OH with 2,4-dinitrochloro- and 2,4-dinitrofluorobenzene in microemulsions and alcohol-modified micelles derived from CTABr. With primary alcohols, 1-butanol, or benzyl alcohol, extensive amounts of ethers were generated by attack of alkoxide ions, but this reaction became unimportant in micelles of CTABr and tert-amy alcohol where OH was the only nucleophilic reagent and rate data were fitted by the PIE model, as for normal aqueous micelles [80]. 

The PIE model was also applied successfully to the chromic acid oxidation of alcohols in micelles of sodium dodecyl sulfate (SDS) [81]. In these systems the medium chain length alcohols act as cosurfactants. However, as noted above, key assumptions of the PIE treatment are the constancy of a and the one-to-one ion exchange, which are reasonably satisfactory for aqueous micelles but are less reliable for cosurfactant-modified micelles and O/W microemulsions, where a is relatively large and sensitive to both ionic and nonionic solutes. 

The kinetics of acid hydrolysis of the p-methoxybenzaldehyde-O-acyloximes (7) in SDS micelles modified by BuOH has also been fitted to the PIE model [86]. The substrates differ only in their hydrophobicities, and while the acetyl derivative partitions between water and micelles, the octanoyl derivative is wholly micelle-bound. The simple PIE model fits rate data in dilute HCl [Eq. (6)], but it underpredicts observed rate constants in more concentrated acid. This increased rate was analyzed in terms of a model that does not involve a constant value of a but allows concentrations of reactive and inert ions, and Na" ", in the micellar pseudophase to increase, following Langmuir isotherms [106]. This model was reasonably satisfactory except at high 1-butanol concentration. Alternatively, the rate data in more concentrated acid can be fitted in terms of Eq. (12). 

Kinetic data obtained under these conditions have been fitted by pseudophase models in terms of k by solving the PBE or in terms of k / Vm by using the PIE model [10], However, because these treatments contain reasonable but unproven assumptions [64] and because values of parameters such as //, V, and are only approximate, values of k may not be unique [123], Extensive evidence shows that k kw for many reactions of anionic nucleophiles, but this generalization does not hold for anionic electrophiles [83,124,125], Therefore, although a great many kinetic data are consistent with the assumption that counterions concentrate at surfaces of ionic association colloids, it is difficult to obtain quantitative estimates of interfacial ion concentrations from measured rate constants. 

Figure 6 Top Cartoon illustrating locations of an organic substrate (see below), surfactant, colons, and counterions across a small cross section of a cationic micelle interface. Bottom Illustration of radial counterion distributions at three salt concentrations as described by the solution of the Poisson-Boltzmann equation in spherical symmetry (solid lines) and by the PIE model assuming that = 0.75, cmc = 0, and interfacial counterion concentration = 4 M (broken lines). A = 2.4 A, a typical assumed width of the reaction region in the PBE model. (Reproduced from Ref. 71 American Chemical Society, 1991.)...

Second-order reactions and the pseudophase ion exchange (PIE) model... 

The PIE model is essentially an extension of the PP model and therefore contains all the assumptions involved in the PP model and a few more, as clearly expressed in several excellent reviews. These additional assumptions may be summarized as follows ... 

In view of the PIE model, the concentrations of a reactive anion, Y (anionic reactant), and an inert anion, X (counterion of cationic micelle), in the micellar and aqueous pseudophases are governed by an ion-exchange equilibrium process. Equation 3.18 ... 

The values of my at different [D ] values can be calculated from Equation 3.24 for the given values of Kx and p, and these my values can be subsequently used to calculate kM" and Kg from Equation 3.12 (with mR representing my), using the nonlinear least-squares technique. This is the general practice used in applying the PIE model for kinetic analysis of the rates of appropriate bimolec-ular reactions. " ... 

Although the PIE model has been extensively used in kinetic data analysis for semiionic bimolecular reactions (i.e., with one of the reactants being ionic) in the presence of normal ionic micelles,its use has been extended to such reactions in the presence of reversed micelles, - microemulsions, " cosurfac-tant-modified micelles, and vesicles. " ... 

Because of such inherent weakness in the PIE model, it has now been suggested that out of the usual five disposable parameters (CMC, Kx p, k, , and Ks) encountered in the use of PIE model, at the most only two disposable parameters, such as Jqy, and Kg or kM and Kx or any other two parameters, should be considered to be unknown, and the remaming three should be deter-mined independently under similar conditions, using nonkinetic techniques. However, in practice, it is not so easy — and almost impossible — to determine the... 

Vera and Rodenas studied the effects of [KBr] on the rate of hydroxide-ion-catalyzed hydrolysis of a few anionic moderately hydrophobic esters (S) in the presence of CTABr micelles. The k bs vs. [KBr] profiles at constant [CTABrjj were explained in terms of the PIE model considering the usual ion-exchange Br/HO coupled with an empirical equation (Equation 3.26), which gives the measure of the ion-exchange Br/S ... 

Pseudo-first-order rate constants (k bs) for the reaction of HO" with ionized V-hydroxyphthalimide (4) in the presence of varying concentrations of CTABr micelles at a constant [NaOH] and [NaBr] were explained in terms of the PIE model, considering Br/HO ion exchange coupled with Equation 3.26. The fit... 

Why the PIE Model Does Not Give a Better Data Fit than the PP Model in a Reaction System with Plausible Occurrence of an Ion-Exchange... 

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