The Mass Action Model

The Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, (3. The reaction isdd 

A surfactant solution having a concentration greater than the CMC can be considered to be a mixture containing m mol-kg-1 of the 1 1 electrolyte [M+A ] and molekg-1 of the 1 n(l - /3) electrolyte micelle. The equilibrium molalities m and mb are related to the stoichiometric molality m by 

The mole fraction of surfactant combined as a micelle is defined as 

The Guggenheim extensions of the Debye-Hiickel equations (see Section 18.1b) are used to obtain expressions for the activity coefficients. The result is 

Starting with equation (18.85), equations can be derived for j , f L, t Cp, and 4 V. The osmotic coefficient is obtained from a transformation using the Gibbs-Duhem equation. The result is 

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Fig. 4 Reaction of benzoic anhydride in CTAOH , no added NaOH O, 0.01 M added NaOH , 0.02 M added NaOH. The lines are calculated using the mass-action model. (Reprinted with permission of the American Chemical Society)...

The mass action model (MAM) for binary ionic or nonionic surfactants and the pseudo-phase separation model (PSM) which were developed earlier (I EC Fundamentals 1983, 22, 230 J. Phys. Chem. 1984, 88, 1642) have been extended. The new models include a micelle aggregation number and counterion binding parameter which depend on the mixed micelle composition. Thus, the models can describe mixtures of ionic/nonionic surfactants more realistically. These models generally predict no azeotropic micellization. For the PSM, calculated mixed erne s and especially monomer concentrations can differ significantly from those of the previous models. The results are used to estimate the Redlich-Kister parameters of monomer mixing in the mixed micelles from data on mixed erne s of Lange and Beck (1973), Funasaki and Hada (1979), and others. 

In the pure liquid state or In the micellar form ( ). The parameters, derived from the mass-action model using data from the literature (16,23), are summarized In Table II. The curves A shown In... 

Table II. Parameters from the Mass-Action Model for the Binary System at 25°C...

To this point, only models based on the pseudo—phase separation model have been discussed. Mixed micelle models utilizing the mass action model may be necessary for micelles with small aggregation numbers, as demonstrated by Kamrath and Franses ( ). However, even for large micelles, the fundamental basis for the pseudophase separation model needs to be examined. In micelles, how much solvent or how many counterions (bound or in the electrical double layer) should be included in the micellar pseudo-phase is unclear. The difficulty is normally surmounted by assuming that the pseudo—phase consists of only the surfactant components i.e., solvent or counterions are ignored. The validity of treating the micelle on a surfactant—oniy basis has not been verified. Funasaki and Hada (22) have questioned the thermodynamic consistency of such an approach. 

In the mass action model the micellar system can be described by only one parameter, and despite this simplicity, a good qualitative description of the main physical properties is obtained, for example the onset of cmc (critical micelle concentration), as shown in Figure 9.7. Notice that the formation of micelles becomes appreciable only at the cmc, and after that, by increasing further the surfactant concentration, all added surfactant is transformed directly into micelles, so that the surfactant concentration in solution remains constant at the level of cmc. 

Following this, the thermodynamic arguments needed for determining CMC are discussed (Section 8.5). Here, we describe two approaches, namely, the mass action model (based on treating micellization as a chemical reaction ) and the phase equilibrium model (which treats micellization as a phase separation phenomenon). The entropy change due to micellization and the concept of hydrophobic effect are also described, along with the definition of thermodynamic standard states. 

We see in Section 8.8 that surfactants undergo aggregation in nonaqueous solvents also, but the degree of aggregation is very much less (n < 10), and the threshold for aggregation is far less sharp than in water. The mass action model for micellization seems preferable for nonaqueous systems. 

Two principal approaches have been used to describe the thermodynamics of surfactant solutions — the pseudo-phase model and the mass action model. 

The lines through the data points in Figures 18.11 and 18.13 are fits of the mass action model to the experimental results. The agreement is excellent when we consider that only three variable parameters are required to fit each of the thermodynamic properties.hh... 

Several models have been developed to interpret micellar behavior (Mukerjee, 1967 Lieberman et al., 1996). Two models, the mass-action and phase-separation models are described here in mor detail. In the mass-action model, micelles are in equilibrium with the unassociated surfactant or monomer. For nonionic surfactants with an aggregation numb itbfe mass-action model predicts thatn molecules of monomeric nonionized surfactaStajeact to form a micelleM ... 

Generalizing, one could state that the mass-action model simulates a cooperative (all or nothing) process with respect to large n values. This model is somewhat... 

Summarizing the statements of these three most commonly used models, it appears that the so-called mass action and phase-separation models simulate a third condition which must be fulfilled with respect to the formation of micelles a size limiting process. The latter is independent of the cooperativity and has to be interpreted by a molecular model. The limitation of the aggregate size in the mass action model is determined by the aggregation number. This is, essentially, the reason that this model has been preferred in the description of micelle forming systems. The multiple equilibrium model as comprised by the Eqs. (10—13) contains no such size limiting features. An improvement in this respect requires a functional relationship between the equilibrium constants and the association number n, i.e.,... 

The physical properties of the micelle, e.g. average aggregation number, shape and formation constant, can be deduced from the chemical-shift data from the mass action model.48 Using the mass action law model for micelle formation, the... 

The mass action model describes micelle formation as an equilibrium process. The micellar aggregation number becomes an important parameter. The solubilization process can be treated as a stepwise addition of solute molecules to the micelles. However, the partition coefficient based on this model requires the aggregation number, which makes it difficult to use in practice. There are several methods of simplification. One is to define the partition coefficient as ... 

DeLisi et al. ° used the opposite approach, in the sense that the pseudophase model was used for the surfactant and the mass action model for the solute. This means that the partition coefficient is defined as in Equation 6.2. 

In the treatment by the mass action model, micellization is considered as an association-dissociation equilibrium of individual molecules or ions with micelles in the concentration range above CMC. 

According to the mass action model, the enthalpy and entropy changes are respectively given by ... 

This two-state model, which assumes that surfactant exists either as monomers or micelles, is almost certainly an oversimplification. The mass action model assumes an equilibrium between monomers, n-mers and micelles, with the proviso that the bulk of the surfactant is present as monomers or micelles. In other words micelliza-tion is considered to occur over a narrow range of surfactant concentration, at least for aqueous micelles [1,2,23]. The thermodynamics of micellization have been discussed in terms of the hydrophobic interactions and the electrostatic interactions of the head groups, and, for ionic micelles, of the counterions with the ionic head groups [18,22,24],... 

There are two basic approaches to modeling the thermodynamics of micelle formation. The mass action model views the micelles as reversible complexes of the monomer that are aggregating and predicts the sharp change in tendency of incremental surfactant to form micelles instead of monomer at the CMC this sharp transition is a consequence of the relatively large number of molecules forming the aggregate. The mass action model predicts that micelles are present below the CMC but at very low concentrations. The ocher major model used to describe micelle formation is the pseudophase separation model, which views micelles as a separate thermodynamic phase in equilibrium with monomer. Because micelle formation is a second-order phase transition, micelles are not a true thermodynamic phase, and this model is an approximation. However, the assumption that there are no mi-celies present below the CMC, and that the surfactant activity becomes constant above the CMC. is close to reality. and the mathematical simplicity of the pseudophase... 

In the literature on micelle formation two primary models have gained general acceptance as useful (although not necessarily accurate) models for understanding the energetics of the process of self-association. The two approaches are the mass-action model, in which the micelles and monomeric species are considered to be in a kind of chemical equilibrium... 

The alternative approach to modeling micelle formation is to think in terms of a phase separation model in which, at the cmc, the concentration of the free surfactant molecules becomes constant (like a solubihty limit or Ksp), and all additional molecules go into the formation of micelles. Analysis of the two approaches produces the same general result in terms of the energetics of micelle formation (with some slight differences in detail), so that the choice of model is really a matter of preference and circumstances. There is evidence that the activity of free surfactant molecules does increase above the cmc, which tends to support the mass-action model however, for most purposes, that detail is of little consequence. 

While the mass action model of Equations 4.3 through 4.10 is an improvement over the phase separation model, it clearly has significant shortcomings. The aggregation number N, for instance, is a parameter that must be determined experimentally or otherwise specified, that is, it does not arise from the analysis... 

The ligand binds to the ground state of the protein forming a Michaelis complex which then isomerizes to the active form (the inductive model). This model for substrate activation of enzymes was advocated forcefully by Citri [26] as a plausible alternative to the mass action model described by Eq. (1). 

Thus in contrast to the mass action model [13], in the inductive model the ligand has the potential to increase the rate at which the active state of the enzyme is formed. In principle, this could be an important consideration when major, and presumably therefore energetically demanding, conformational rearrangements are involved in the shift from the ground state to the active form of the protein. 

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