Aggregates mole fractions

For sufficiently low values of xi, such that xie < 1, xj /N < xi for any N> 1. For such low-mole fractions, the solution will consist principally of monomer units, with mole fraction xi, dissolved in the solvent. However, this inequality no longer applies at higher concentrations for, by definition, xj /N for use in Eq. (5.4.31) cannot exceed unity, which means that the quantity in braces is not allowed to exceed unity. In other words, exp - d(l - 1 /N ) = exp (/ij — /i /RT) is to be an upper limit on xi. This cutoff value represents the critical aggregation mole fraction (CAME). Thus, whenxT is specified by Eq. (5.4.31), we find that xi Icamf for all > 1 at these values of x, any further addition of monomer molecules to the solution results solely in the formation of more aggregates, such that x remains at its upper limit... 

The pioneering work on amphiphilic polyelectrolytes goes back to 1951, when Strauss et al. [25] first synthesized amphiphilic polycations by quaternization of poly(2-vinylpyridine) with n-dodecyl bromide. They revealed that the long alkyl side chains attached to partially quaternized poly(vinylpyridine)s tended to aggregate in aqueous solution so that the polymers assumed a compact conformation when the mole fraction of the hydrophobic side chains exceeded a certain critical value. Thus, Strauss et al. became the first to show experimentally the intramolecular micellation of amphiphilic polymers and the existence of a critical content of hydrophobic residues which may be compared to the critical micelle concentration of ordinary surfactants. They called such amphiphilic polyelectrolytes polysoaps [25],... 

As has been described in Chapter 4, random copolymers of styrene (St) and 2-(acrylamido)-2-methylpropanesulfonic acid (AMPS) form a micelle-like microphase structure in aqueous solution [29]. The intramolecular hydrophobic aggregation of the St residues occurs when the St content in the copolymer is higher than ca. 50 mol%. When a small mole fraction of the phenanthrene (Phen) residues is covalently incorporated into such an amphiphilic polyelectrolyte, the Phen residues are hydrophobically encapsulated in the aggregate of the St residues. This kind of polymer system (poly(A/St/Phen), 29) can be prepared by free radical ter-polymerization of AMPS, St, and a small mole fraction of 9-vinylphenanthrene [119]. 

Species concentrations are shown in Figure 12. At 34 GPa (2.0g/cc), H2O is the predominant species, with H30+ and OH having mole fractions of ca. 5%. In addition, some aggregation has occurred in which neutral and ionic clusters containing up to six oxygens have formed. The concentrations of OH and H30+ are low for all densities investigated and nonexistent at 95 and 115 GPa (2.8 and 3.0g/cc, respectively). The calculated lifetimes for these species are well below 10 fs for the same thermodynamic conditions (less than 8 fs at 34 GPa). At pressures of 95 and 115 GPa, the increase in the O-H bond distance leads to the formation of extensive bond networks (Figure 13). These networks consist entirely of O-H bonds, whereas 0-0 and H-H bonds were not found to be present at any point. 

A mass action model (MAM) with monodisperse aggregation number N which depends on the micelle mole fraction x and the counterion binding parameter /3(x) has been developed for binary surfactants either ionic/ionic or nonionic/ionic. 

Finally, DOSY [24,25] has been used to study ion aggregation, since the ionic radii are directly correlated to the respective diffusion coefficients. In this fashion, it could be demonstrated for [C4CiIm]Br in D2O that this IL is only completely dissolved for very dilute solutions [36]. Above a mole fraction of 0.015, ion pairing is dominant. A similar effect has been reported by Pregosin et al. who used a combination of H, f F] HOESY, and E DOSY techniques [37]. 

The mole fraction of micelles is given by xmic/n for particles of aggregation n. By the same logic as used above, (/ ) describes the surfactant in a standard state of pure micelle. We can write Equation (13) per surfactant molecule by dividing it by n ... 

Although there are some aspects of micellization that we have not taken into account in this analysis —the fact that n actually has a distribution of values rather than a single value, for example —the above discussion shows that CMC values expressed as mole fractions provide an experimentally accessible way to determine the free energy change accompanying the aggregation of surfactant molecules in water. For computational purposes, remember Equation (3.24), which states that x2 n2/n, for dilute solutions. This means that CMC values expressed in molarity units, [CMC], can be converted to mole fractions by dividing [CMC] by the molar concentration of the solvent, [solvent] x2 [CMC]/[solvent] for water, [solvent] = 55.5 mole liter... 

Also Fryar and Kaufman8 studied the solvent effect on the stability of barium dinonylnaphthalene sulfonate in toluene, toluene/methanol, and methanol solutions by ultracentrifugation and viscometry. The aggregation number of the micelles reduced from about 10 in toluene to about 4 when the mole fraction of free methanol in the solvent mixture was approximately 0.03. In pure methanol BaDNNS micelles did not exist. 

Mole fraction of DMSO in chloroform where half of the aggregate has decomposed into components. 

In the preceding equations, the subscript g refers to a droplet containing a total of g molecules (g = gs + gA + go + gw Xgo is the mole fraction of aggregates of size g in the continuous oil (O) phase,Xsw andAAw are the mole fractions of singly dispersed surfactant and alcohol molecules, respectively, in the dispersed water phase, // w and //A y are their standard chemical potentials, and ysw and yAW are their activity coefficients. X0o denotes the mole fraction of oil in the continuous oil phase, /Iqq is its standard chemical potential, and yoo is its activity coefficient. 

In the first expression, Xww can be replaced by (1 — XgVi —.Xaw — XgW) because the sum of all mole fractions should be unity. In the second expression, the total mole fraction IJXjMi of alcohol in the oil phase accounts for the alcohol present as both monomers and aggregates in the oil phase and is calculated on thebasis of the continuous-association model16 as described in Appendix D. 

In what follows, we provide the derivation of an equation, which relates the concentration of alcohol in water and the oil phases, on the basis of a model developed earlier.28 Let us assume that No° molecules of oil and Na° molecules of alcohol are present in the oil phase. Denoting by A a° the number of aggregates containing / molecules of alcohol, the mole fraction of the aggregates of size j is given by... 

The effect of methanol on micellar solutions is slight at the low concentration used (5% v/v = 1.3 mol/L = 0.022 mole fraction). The effect of NaCl however, is more significant the CMC is greatly decreased, the degree of counterion binding and the aggregation number are increased. 

Considering the importance of micellar aggregates in separations, it is unfortunate that our knowledge of solute-micelle equilibria is quite limited, both as regards the dependence of solute activities on the intramicellar mole fractions of surfactant and organic compound, and in relation to the influence of total... 

C3fO (mO) Aggregation number Fraction of charge JHm, kcal mole- ... 

Wilsons equation and the modification proposed by Renon and Prausnitz (8) use the local mole fraction concept, produced because molecules in solution aggregate as a result of the variation in intermo-lecular forces. The local mole fraction concept results in a more useful description of the behavior of molecules in a non-ideal mixture. 

As the total mole fraction cannot exceed one, there appears automatically an upper boundary for Xj, also known as a critical aggregation concentration, or critical micelle concentration, X which is given by... 

Addition of small amounts of dipolar non-HBD solvents to solutions of the sodium 9-fluorenone ketyl radical in toluene, in which diamagnetic dimers or higher aggregates are present, gives rise to well-resolved HFS patterns. The splitting first decreases with an increase in the mole fraction of the dipolar non-HBD solvent. A limiting value is reached at mole fraction x = 0.2...0.3, due to dissociation into... 

In Chapter 7, we observed that U and H are state properties of a species that is. their values depend only on the state of the species—primarily on its temperature and state of aggregation (solid, liquid, or gas) and, to a lesser extent, on its pressure (and for mixtures of some species, on its mole fraction in the mixture). A state property does not depend on how the species reached its state. Consequently, when a species passes from one state to another, both Af/ and for the process are independent of the path taken from the first state to the second one. 

Fig. 2.—Concentration of amphiphiles in spherical micelles of aggregation number N. The curves are plotted for three assumed values of dielectric constant s. The curve for e = 0 is equivalent to ignoring curvature corrections to the electrostatic energy. The curves are normalised to the same maximum value. Xm is taken as 10 mole fraction (5 mmol dm ).

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