Bases activity coefficients

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

In all of the above, the weight fraction based activity coefficient can be substituted for that derived with mole-fraction units by replacing MB with... 

The existence of UCST is typically attributed to the energy differences between polymer and solvent and of LCST to the so-called free-volume effects, which are dne to the differences in size and free-volume between polymer and solvent. Free-volume effects are small for nonpolymeric solutions and are discussed in conjunction with free-volume-based activity coefficient models in Section 16.4. The LCST was first discovered by Freeman and Rowlinson only about 40 years ago, but is now considered to be a universal phenomenon of polymer-solvent systems at high temperatures. In some cases, it is not observed if the polymer degrades before reaching the LCST. 

Dehning volume fraction-based activity coefficients for the yth species Yj. where Uj = y ( ) , leads to... 

Also, aj = YjZj for each species, jj is the true mole fraction-based activity coefficient for the 7th species, and is the thermodynamic equilibrium constant. 

The ion-association concept relies on the use of Debye-Huckel based activity coefficients to calculate aqueous activities and is one that is employed most frequently in the models used today. A primary assumption of this approach is the use of the Macinnes convention such that for an aqueous solution containing equimolal concentrations of and Cl, their activities and hence activity coefficients are equivalent. This approach and convention were reexamined by Parkhurst in this volume, who explored the issue of mean salt based activity coefficients and the apparent... 

The data were included in the overall fit of hydrolysis relevant data (see Appendix D). SIT-based activity coefficient correction were applied to each data point. In this context, it turns out that solution concentrations are not at all controlled by monomeric species, but by the trimeric species Zrj(011)9. The fit results in a solubility constant logiQ K°q = - (4.23 + 0.06) (2ct). The overall fit also showed that nitrate complexes were negligible in the entire range of the experimental data. 

One should keep in mind that mole fraction-based activity coefficients /a become very small values for common polymer solutions and reach a value of zero forMs ->oo, which means a limited applicability at least to oligomer solutions. Therefore, the common literature provides only mass fraction-based activity coefficients for (high-molecular) polymer -i- (low-molecular) solvent pairs. The molar mass Mb of the polymeric liqitid is an average value (M ) according to the usual molar-mass distribution of polymers. 

This activity coefficient is so defined that it becomes unity at infinite dilution of the solute in the solvent, in contrast to the one commonly used for liquid mixtures, which becomes unity for the pure liquid solute. The pure liquid-solute-based activity coefficient can be calculated by combining the melting data with vapour-liquid equilibria data at the melting temperature of the solvent. When vapour-liquid equilibria data are known only at higher temperatures, it is necessary to know the molar excess enthalpies of the mixture over the temperature range. 

R. M. CoNEORTi, T.A. Barbari, P. Vi-MALCHAND, and M.D. Donohue, A Lattice-Based Activity Coefficient Model for Gas Sorption in Glassy Polymers. Macromolecules, 1991, 24, 3388-3394. 

With the necessary care, all thermodynamic expressions given above can be formulated with mass or volume or segment fractions as concentration variables instead of mole fractions. This is the common practice within polymer solution thermodynamics. Applying characteristic/hard-core volumes is the usual approach within most thermodynamic models for polymer solutions. Mass fraction based activity coefficients are widely used in Equations [4.4.7 and 4.4.8] which are related to activity by ... 

One should keep in mind that mole fraction-based activity coefficients become very small values for common polymer solutions and reach the value of 0 for which... 

Whereas the models given above can be used to correlate solvent activities in polymer solutions, attempts also have been made in the literature to develop concepts to predict solvent activities. Based on the success of the UNIFAC concept for low-molecular liquid mixtures,Oishi and Prausnitz developed an analogous concept by combining the UNIFAC-model with the free-volume model of Flory, Orwoll and Vrij. The mass fraction based activity coefficient of a solvent in a polymer solution is given by ... 

From this equation the mole fraction based activity coefficients of the cation, anion and solute are derived ... 

As shown by Robinson and Stokes (C6), the mole fraction based activity coefficient, f, is related to the mean molal activity coefficient, y, by the following relationship ... 

The mole fraction based activity coefficients can thus be derived from the combined equations (5.40), (5.41) and (5.42). For the cation, anion and molecular solute they are ... 

Since polymers have no vapor pressure and as a consequence the vapor phase does not contain polymer, the equilibrium conditions for low-pressure vapor-Uquid equilibria of polymer solutions as given by Eq. (20) are only applicable to the solvent s as in Eq. (24), or in a case where the weight fraction of polymer is used as a composition variable as in Eq. (25), where f2s is the weight fraction based activity coefficient of the solvent. 

The relation between a mole fraction based activity coefficient and a weight fraction based activity coefficient Sli of component i is given by Eq. (26), where w and Mi are the weight fraction and molecular weight of component i, respectively, y and Qs can be obtained from a correlation of experimental data using a suitable model or from a predictive model (see Section 2.3). 

Often r is approximated by the ratio of the molar volumes of pure liquid polymer and pure solvent. The segment fractions of solvent and polymer are then equal to the volume fractions of solvent and polymer, and p. Equation (27) is derived using many assumptions and approximations (for a discussion, see Ref. 8), but on the basis of this rather simple expression many features of the phase behavior of polymer solutions can be explained. The expressions for the mole fraction based activity coefficients of solvent and polymer are Eqs. (30) and (31), respectively. 

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