Multisolute solution

Equation 10.3.3 relates the chemical potential of electrolyte B in a binary solution to the single-ion chemical potentials of its constituent ions  

This relation is valid for each individual solute substance in a multisolute solution, even when two or more of the electrolyte solutes have an ion species in common. 

As an illustration of this principle, consider a solution prepared by dissolving amounts B of Bal2 and nc of Csl in an amount Ha of H2O. Assume the dissolved salts are completely dissociated into ions, with the 1 ion common to both. The additivity rule for the Gibbs energy of this solution can be written in the form 

Comparing Eqs. 10.3.13 and 10.3.14, we find the following relations must exist between the chemical potentials of the solute substances and the ion species  

These relations agree with Eq. 10.3.12. Note that /r(I ), the chemical potential of the ion common to both salts, appears in both relations. 

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Kralj, Z. I., Simeon, V. Estimation of spectra of individual species in a multisolute solution. Anal. Chim. Acta 1982, 129,191-198. 

Malcolm et al. (14) and Thurman et al. (15) noticed that the adsorption of solutes onto XAD-8 macroreticular resin could be predicted by means of a linear correlation between the logarithm of the capacity factor and the inverse of the logarithm of the water solubility of each compound. Their investigation, however, was limited to approximately 20 selected organic compounds in individual aqueous solutions. By comparing the results shown in Table II and the water solubility properties of each model compound used in this study (see Table I), it appears that the predictive model could serve for a first estimate of the recovery of multisolute solutions at trace levels. However, low recoveries and the erratic behavior of several compounds included in this study suggest that additional factors need to be considered. 

For instance, suppose we apply Eq. 10.3.16 to the solution of Bab and Csl used above as an illustration of a multisolute solution, letting Um,B be the activity of solute substance Bal2. The quantities m+ and m in the equation are then the molalities of the Ba " " and 1 ions, and y is the mean ionic activity coefficient of the dissolved Bab. Note that in this solution the Ba " and I ions are not present in the 1 2 ratio found in Bab, because 1 is a constituent of both solutes. 

This method is applicable to the complete multisolute aqueous solution described before. It is estimated that total solute concentrations up to 10 or 20 molal may be handled. The limitation on temperature results mainly from the limited temperature range for which experimental results for equilibrium constants and Henry s constants are available (cf. Appendix II and tables I and II). Although for some constants this range only extends up to 60 °C, it is expected that by an appropriate extrapolation the method may be used also at temperatures up to 170 oc. 

Some process models have more than one feasible solution. Most numerical methods have local convergence, so the solution obtained is dependent upon the initial guess for the solution before the first iteration. There is an ongoing effort to develop techniques that have global convergence or to find all solutions to multisolution problems. 

In modern crystallography virtually all structure solutions are obtained by direct methods. These procedures are based on the fact that each set of hkl planes in a crystal extends over all atomic sites. The phases of all diffraction maxima must therefore be related in a unique, but not obvious, way. Limited success towards establishing this pattern has been achieved by the use of mathematical inequalities and statistical methods to identify groups of reflections in fixed phase relationship. On incorporating these into multisolution numerical trial-and-error procedures tree structures of sufficient size to solve the complete phase problem can be constructed computationally. Software to solve even macromolecular crystal structures are now available. 

Derylo-Marczewska and Jaroniec [28] have reviewed the adsorption of organic solutes from dilute solutions and have provided a useful compilation of published experimental data for both single- and multisolute adsorption isotherms on carbonaceous adsorbents. They also presented a survey of theoretical approaches used to describe the solute adsorption equilibria, including the Polanyi adsorption model, the solvophobic interaction model, the Langmuir adsorption theory, the vacancy solution model, as well as considerations based on the energetic heterogeneity of the adsorbent. In particular, these authors emphasize the... 

Quinones et al. [17] measured by frontal analysis multisolute adsorption equilibrium data for the system benzyl alcohol, 2-phenylethanol and 2-methyl benzyl alcohol in a reversed-phase system. Data were acquired for the pure compoimds, for nine binary mixtures (1 3,1 1, and 3 1) and four ternary mixtures (1 1 3,1 3 1, 3 1 1, and 1 1 1). These data exhibited very good thermod5mamic consistency. The thermodynamic functions of adsorption were derived from the single-solute ad-... 

IAS model for dilute liquid solution The IAS method was first proposed to accoimt for the adsorption of gas mixtures. It was later extended to multisolute adsorption from dilute liquid solutions [54]. Assuming that both the solution and the adsorbed phase are ideal, the following equation can be derived to calculate multi-solute equilibriirm composition [54]. 

Muller, G., Radke, C.J., Prausnitz, J.M. (1985). Adsorption of weak electrolytes from dilute aqueous solution onto activated carbon. Part II. Multisolute systems. J. Colloid Interface Sci., 103, 483—92. 

In order to obtain a single equilibrium curve, we have to specify enough variables that only one degree of freedom remains. For binary distillation this can be done by specifying constant pressure. For absorption, stripping, and extraction we specified that pressure and tenperature were constant, and if there were several solutes we assumed that they were independent. In general, we will specify that pressure and/or temperature are constant, and for multisolute systems we will assume that the solutes are independent. 

Multisolute Adsorption from Dilute Aqueous Solutions.—Earlier work on adsorption from multicomponent systems has been reviewed in references 2 (p. 103) and 3 (p. 75). The work of Radke and Prausnitz, Fritz and Schliinder, and of Myers and Minka has been developed in a paper on the thermodynamics of multisolute adsorption from dilute aqueous solutions by Jossens, Prausnitz, Fritz, Schliinder, and Myers. The experimental work... 

An alternative approach to multisolute adsorption is by application of the Polanyi potential theory. Its use for binary solute systems was outlined in reference 3 (pp. 111-114) Rosene and Manes have now extended consideration to ternary solute systems. The same principles apply as in the binary solute case the driving force for adsorption per unit volume is... 

Dean JA (1999) Lange s handbook of chemistry, 15th edn. McGraw-Hill, Inc., New York Elliott JA, Prickett RC, Elmoazzen HY, Porter KR, McGann LE (2007) A multisolute osmotic virial equation for solutions of interest in biology. J Phys Chem Bill 1775-1785 Franke J, Hardle WK, Hafner CM (2011) Statistics of financial markets an introduction, 3rd edn. 

Prickett RC, Elliott JA, McGann Ui (2011) Application of the multisolute osmotic virial equation to solutions containing electrolytes. J Phys Chem B 115 14531-14543 Priestley MB (1981) Spectral analysis and time series Vol. 1 univariate series and vol 2 multivariate series, prediction, and control. Academic, New York Ross-Rodriguez LU (2009) Cellular osmotic properties and cellular resptmses to cooling. University of Alberta, Edmonton... 

Most aspects have not been worked out for the thermodynamic properties and equilibria of systems of macromolecules in the crystalline, glassy, or solution form. There is much uncertainty about the use of dilute solution reference states for supercritical components, particularly in multisolute, multisolvent solutions. 

Prasad and Sirkar (1985) may be consulted for details of such approximations for multisolute systems as well as for earlier references providing a quantitative basis for these approximations in a single-solute system. 

An extension to multisolute aqueous solutions with a polyelectrolyte, nonelectrolyte solutes, and low molecular weight salts might start from (48) using (49) together with (50) and (51) for the cmitributions of the polyelectrolyte, but replacing (52) by the osmotic pressure of an aqueous solution of the polyelectrolyte-free solutions, i.e., an aqueous solution of the low molecular weight salts and the other nonelectrolyte solutes. However, such an extension always suffers from neglecting the interactions between the other solutes and the polyelectrolyte. 

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