Concentrations Derived from Mole Fractions

The MAM described here is a generalization of the model previously published (10). Hence, only a summary of the derivation will be given here. Details can be found elsewhere (17). The basic equations are the surfactant and counterion material balances and the minimization of the Gibbs free energy of the system with respect to the micelle concentration c , and mole fraction x (11). Equation 4 from Ref. (11) has been changed to... 

This equilibrium concentration c, or the corresponding mole fraction x, of EG, water and DEG in the interface can be calculated from the vapour pressure and the activity coefficient y, derived from the Flory-Huggins model [13-17], Laubriet et al. [Ill] used the following correlations (with T in K and P in mm Hg) for their modelling ... 

The surface tension of the aqueous solution of dode-cylaitunonium chloride (DAC) — decylairanonium chloride (DeAC) mixture was measured as a function of the total molality m of surfactants and the mole fraction X of DeAC in the total surfactant in the neighborhood of the critical micelle concentration (CMC). By use of the thermodynamic equations derived previously, the mole fraction in the mixed adsorbed film was evaluated from the y vs. X and m vs. X curves. Further, the mole fraction in the mixed micelle was evaluated from the CMC vs. X curve. By comparing these values at the CMC, it was concluded that the behavior of DAC and DeAC molecules in the mixed micelle is fairly similar to that in the mixed adsorbed film. 

For mixtures of the two oxidation states, the contact shifts depend directly on the relative mole fractions of the two states, indicating that the system is in the limit of rapid exchange. From this and the concentrations of the two species, one can calculate that the second-order rate constant involving the electron transfer must be greater than 106 M l seer1 at 25°C. This limit applies not only to the phenanthroline iron(II)—(III) system but also the iron(II)—(III) complexes with the following methyl derivatives of phenanthroline ... 

The viscosities of the acetone-bromosuccinic acid mixed solvents were derived from the Jones-Dole (33) equation and data acquired by Muller, who used the special viscometer described by Tuan and Fuoss (34). The values used for the viscosities (in poise) of solvents I-V were 3.02 X 10 3, 3.05 X 10-3, 3.08 X 10-3, 3.13 X 10-3, and 3.02 X 10-3, respectively. The literature value for the dielectric constant of acetone, 20.7, was used as the dielectric constant for each solvent. This is justified because at the highest concentration of bromosuccinic acid its mole fraction is less than 0.004. 

The concentrations at the condenser Xj,c are identical to the relative fluxes %j because of total condensation of the vapor escaping from the liquid surface. Schliinder [19] derived the following relationship between mole fractions of liquid in the still, Xj s, and the mole fractions of the condensed liquid in the receiver, ocjtc ... 

Let us first derive the units of the overall mass transfer coefficients when the concentration units used are in mole fractions. Let the overall mass transfer coefficient for the gas side be Kyf and that for the liquid side be K. Gd Y is mole of solute flowing per unit time. Mass transfer is a process where mass crosses an area perpendicular to the direction of motion of the solute particles. This area is the contact area for mass transfer. Let the differential area be designated as dA. Thus, in terms of mass transfer, Gd[Y] is equal to j,/([y/] - y )dA. From this expression, the dimensions of Kyf are mole per unit time per unit mole fraction per unit square area or MItImole fraction-1. In an analogous manner, Ld X is equal to ] -... 

At infinite dilution, when a solution behaves ideally, the three activity coefficients of the solute, viz.,/x,/c and fm, are all unity, but at appreciable concentrations the values diverge from this figure and they are no longer equal. It is possible, however, to derive a relationship between them in the following manner. The mole fraction x, concentration c, and molality m of a solute can be readily shown to be related thus... 

An equivalent expression can of course be derived in terms of mole fractions rather than concentrations. If f and /2 are the respective mole fractions of monomers M and M2 in the reaction feed and F and F2 are the corresponding mole fractions in the copolymer formed from this mixture, then... 

The concentrations of the components in the vicinity of any molecule are usually called local compositions (LCs). According to the LC concept, the composition in the vicinity of any molecule differs from the overall composition. If a binary mixture is composed of components 1 and 2 with overall mole fractions x and xi, respectively, four LCs can be defined local mole fractions of components 1 and 2 near a central molecule 1 (xii and X2 ) and local mole fractions of components 1 and 2 near a central molecule 2 x 2 and X22). Many attempts have been made to express LC in terms of the bulk compositions and some intermolecular interaction parameters. " Wilson was the first to suggest expressions for the local mole fractions and to derive on their basis expressions for the activity coefficients of binary mixtures. Since then, many expressions for LC were suggested, and the LC concept proved to be a very effective... 

Recently, a method [5] for the prediction of the solubility of a solute in a SC fluid in the presence of an entrainer has been proposed. The method, based on the Kirkwood-Buff (KB) formalism, was however developed for cases in which the entrainer was in dilute amounts. The present paper is focused on the solubility of a solid in a non-dilute mixture of a SC fluid and an entrainer. The theoretical treatment, which is more complex than for the dilute case, is also based on the KB formalism. In this paper the following aspects will be addressed (1) general equations for the solubility in binary and ternary mixtures will be written for the cases involving a small amount of solute (2) the KB formalism will be used to obtain expressions for the derivatives of the fugacity coefficients in a ternary mixture with respect to mole fractions (3) these expressions will be employed to derive an equation for the solubility of a solute in a SC fluid containing an entrainer at any concentration (4) a predictive method for this solubility will be proposed in terms of the solubilities of the solute in the SC fluid and in the entrainer (5) the derived equation will be compared with experimental results from literature regarding the solubility of a solute in a mixture of two SC fluids. 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

If the isotherm is supposed to be linear, the equilibrium isotherm does not intervene in the band profile and the global effect is derived from the flow properties. The characteristic method applies. It shows that, in linear gas-chromatography, although the isotherm is linear, the sorption effect causes the velocity associated with a given concentration to decrease with increasing concentration. [22]. A slice of mobile phase having a given mole fraction, X, moves with the velocity... 

By (numerical or graphical) integration, a2 can now be derived. Figure 2.3 gives as an example the activities of sucrose solutions. It is seen that the activities greatly deviate from the mole fractions at higher concentration. For example, at x2 = 0.1, the activity coefficient of water x 0.85/0.90 = 0.94, that of sucrose x 0.26/0.1 = 2.6. For mixtures of more than two components, the activities cannot be derived in this way. 

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

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