Activity coefficients micelle concentration

A similar multiphase complication that should be kept in mind when discussing solutions at finite concentrations is possible micelle formation. It is well known that for many organic solutes in water, when the concentration exceeds a certain solute-dependent value, called the critical micelle concentration (cmc), the solute molecules are not distributed in a random uncorrelated way but rather aggregate into units (micelles) in which their distances of separation and orientations with respect to each other and to solvent molecules have strong correlations. Micelle formation, if it occurs, will clearly have a major effect on the apparent activity coefficient but the observation of the phenomenon requires more sophisticated analytical techniques than observation of, say, liquid-liquid phase separation. 

It is important to realize, however, that the determination of the substrate-micelle binding constant from solubility data relies entirely on data for saturated solutions and that, in the case of ionic surfactants, differences in the counterion interactions with the micelle and the micelle-substrate complex and activity coefficient effects may seriously complicate the results. In these respects, distribution studies with varying substrate and surfactant concentrations are certainly preferable. In view of the assumptions involved in the derivation and application of equations (10) and (11), the agreement between the K values obtained from kinetic data (equation 10) and those obtained from solubility measurements (equation 11) for several substrate-micelle interactions is certainly both remarkable and significant. 

If the chemical is surface active, for example an alkyl benzene sulfonate used in detergents, it will form micelles above a critical micelle concentration (CMC). This is effectively a solubility limit for such substances and it is essential that the test conditions be below the CMC, otherwise the BCF will be underestimated. Finally it should be noted that actual concentrations in the water may differ considerably from nominal concentrations deduced by adding a known mass of chemical to a known volume of water, because much of the chemical may sorb to the walls of the tank and to pumps and filters. Further, substances of relatively high air-water partition coefficients will evaporate appreciably from solution especially as a result of aeration. For these reasons actual concentration measurements are essential, and nominal values should not be trusted. 

FIGURE 2.7 Hypothetical examples of nonideality of a solute (2) in aqueous solutions. The activity coefficient is given as a2/x2 versus the mole fraction x2. (a) High concentration. (b) Adsorption or binding, (c) Self-association, especially micellization. (d) Electric shielding the broken line is for a case where other salts are present at constant concentration. 

Much less is known about micellar charge and counterion binding in the case of bile salts. Based on the result of ionic self-diffusion measurements [20,163,173], conductance studies [17,18,187], Na, and Ca activity coefficients [16,19,144,188,189] and NMR studies with Na, Rb and Cs [190], a number of generalities can be made. Below the operational CMC, all bile salts behave as fully dissociated 1 1 electrolytes, yet interionic effects between cations and bile salt anions decrease the equivalent conductance of very dilute solutions [17,18,187]. With the onset of micelle formation, counterions become bound to a small degree values at this concentration are about < 0.07-0.13 and are not greatly influenced by the species of monovalent alkali cations [163,190]. At concentrations above the CMC, values remain relatively constant to 100 mM in the case of C and this... 

It is noteworthy that the fn in Eq. (5.36) correspond to the real concentrations of ions CtotiVi(l-ai), whereas the experimental values are referred usually to the stoichiometric concentrations CtotiVi. This results in an increase of each activity coefficient by a factor of 1- aj. Then, neglecting the contribution of micelles in Eq. (5,36), the experimental mean activity coefficient of the surfactant ions in micellar solution can be written as... 

If it is assumed that the activity coefficients are one, that the concentration of micelles is constant at the CMC, and that the standard state for the micelles is chosen to make a = under these conditions. 

In Figure 2 the sodium-ion, dodecyl sulfate ion activities and the mean activity of the NaDS solution are plotted against the total surfactant concentration at a constant (0.224%) PVA concentration. The activity coefficients are relative to that of a 5.0xl0"3 moLkg" NaDS solution (a ). The shape of the experimental activity curves is similar to that obtained by the model calculations. At about 6 mmoLkg l concentration, which is still less than the critical micelle formation concentration (cjy = 8.1 mmol.kg ), complex formation occurs between PVA and NaDS, and above this point the mean activity increases only slightly. 

The next problem for nonideal multicomponent mixtures is to solve the n activity coefficients for the x, values at the total surfactant composition and concentration. To solve the n activity coefficients and the n mole fractions, we need 2n equations n equations of (10.33) and n equations of (10.31) or (10.32), with the constraint that the sum of the x, values equals unity. A numerical solution of multiple equations for multiple unknowns can be reached efficiently using the Nelder-Mead simplex technique." Once the y, values have been determined, the mole fraction in micelle x, and the monomer concentration C, for a multicomponent surfactant solution are easily determined by (10.29) and (10.30). The former values are, of course, obtained together with the y, values. Figure 10.4 shows the CMCs determined by this procedure for the ternary mixture of CioH2i(CH3)2PO/CioH2i(CH3)SO/Ci2H25S04Na. For this ternary mixture, fiq. (10.33) is written as... 

Counterion binding is not a well defined quantity, with various experimental techniques weighing the ion distribution slightly differently. Thermodynamic methods (e.g. ion activities or osmotic coefficients) monitor the free counterion concentration, transport methods (e.g. ion self diffusion or conductivity) the counterions diffusing with the micelle, and spectroscopic methods (e.g. NMR) the counterions in close contact with the micelle surface. Measurement of the effect of Na+ counterions on the symmetric S-O stretching modes would also be expected to be highly dependent on the distance of the counterion from the micelle surface (similar to the NMR method). 

Though this model was proposed first for nonionic surfactants, it has been applied frequently to dissociating surfactants. This can be justified if the counterions are mobile to a sufficient extent in order to arrange themselves almost instantaneously to the distribution of surface active ions or if they are in excess. There are attempts for a construction of a more general theory taking into account the surfactant dissociation. One approach is based on the consideration that counterions are a separate component and on the application of the kinetic theory of two-component micelles [119, 120]. The obtained relations for the relaxation times are essentially more cumbersome and contain a number of coefficients with uncertain concentration... 

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