Analysis of complex equilibria

T Miyajima, M Mori, S Ishiguo. Analysis of complexation equilibria of poly acrylic acid by a Donnan-based concept. J Colloid Interface Sci 187 259-266, 1997. 

Leggett, D. J., Ed. (1985). Computational Methods for the Determination of Formation Constants. Plenum Press, New York. A high-level presentation of the theory of complex equilibria and computer programs for mathematical analysis. 

Whatever the aim of a particular titration, the computation of the position of a chemical equilibrium for a set of initial conditions (e.g. total concentrations) and equilibrium constants, is the crucial part. The complexity ranges from simple 1 1 interactions to the analysis of solution equilibria between several components (usually Lewis acids and bases) to form any number of species (complexes). A titration is nothing but a preparation of a series of solutions with different total concentrations. This chapter covers all the requirements for the modelling of titrations of any complexity. Model-based analysis of titration curves is discussed in the next chapter. The equilibrium computations introduced here are the innermost functions required by the fitting algorithms. 

The analysis of coupled equilibria is the most complex problem that is considered in this chapter. It may seem surprising that such apparently straightforward systems like those shown in Eqns. 9.26 and 9.44 should present such great difficulties in analysis, the more so because it is a trivial matter to calculate the equilibrium constants, if the concentrations of the various species are known. However, that situation arises very rarely for several reasons ... 

A quantitative analysis of complexation requires knowledge of the thermodynamic constants of all equilibria involved. Some data arc available in the literature [4-6], but when it comes to the complexation of condensed species, very little data are in fact available. It is possible, however, to use a simple model in order to establish simple criteria allowing a qualitative description of the aptitude of a ligand to coordinate a monomeric or oligomeric cation. 

In the above-mentioned case of two tautomers, the C=S and the CSH form are clearly seen to be different. Calculations of IR spectra is therefore of great help in the analysis of complex spectra of tautomeric equilibria. 

The first part of this chapter is a review of the literature. It was written with the intent to summarize the important contributions that in situ derivatizations using multidentate complexing agents have made to the development of HPLC methods of analysis for metal species. The authors have not presented an exhaustive review of the literature. Rather, special consideration has been extended to those publications which describe innovative methods for separating metal cations, important applications, or studies concerned with elucidating retention mechanisms. The second and final part of this chapter deals with the mathematical language of complex equilibria and its implementation in designing separations for difficult samples of metal species. 

The problems encountered in structural analysis are well illustrated with the adducts formed between R(fod)3 and adamantane-l-carbonitrile (fig. 2), a highly structured ligand. Raber et al. (1981) reported a procedure for the evaluation of the structure of the Eu(fod)3 adduct using LIS data. Best fit of the LIS data was obtained with a Eu-N bond distance of 2.1 A. However, using H and spin-lattice relaxation rates induced by Gd(fod)3, Peters et al. (1982) determined a Gd-N bond distance of 2.6 A, which is more in line with values expected for R +-N bond distances. Peters suggested that the values of the bound shifts obtained for the europium adduct were inaccurate due to the presence of complex equilibria involving the mono and bis adducts, and free substrate, all in fast exchange on the NMR time scale. 

To summarize the analysis of pH profiles, even complex ones, is not an arcane or difficult art. Systematic analysis in terms of ionic equilibria, predominant species, and the reaction orders with respect to [H+] provides the solution. Kinetically indistinguishable alternatives can never, by definition, be distinguished from the kinetic data contained in the pH profile. Other measurements, including some alluded to earlier and others given in Chapter 10, may, however, allow these distinctions. 

Complex formation, selective precipitation, and control of the pH of a solution all play important roles in the qualitative analysis of the ions present in aqueous solutions. There are many different schemes of analysis, but they follow the same general principles. Let s think through a simple procedure for the identification of five cations by following the steps that might be used in the laboratory. We shall see how each step makes use of solubility equilibria. 

The above equation allows the calculation of Galvani potentials at the interfaces of immiscible electrolyte solutions in the presence of any number of ions with any valence, also including the cases of association or complexing in one of the phases. Makrlik [26] described the cases of association and formation of complexes with participation of one of the ions but in both phases. In a later work [27] Le Hung extended his approach and also considered any mutual interaction of ions and molecules present in both phases. Buck and Vanysek performed the detailed analysis of various practical cases, including membrane equilibria, of multi-ion distribution potential equations [28,29]. 

It is necessary to consider a number of equilibrium reactions in an analysis of a hydrometallurgical process. These include complexing reactions that occur in solution as well as solubility reactions that define equilibria for the dissolution and precipitation of solid phases. As an example, in analyzing the precipitation of iron compounds from sulfuric acid leach solutions, McAndrew, et al. (11) consider up to 32 hydroxyl and sulfate complexing reactions and 13 precipitation reactions. Within a restricted pH range only a few of these equilibria are relevant and need to be considered. Nevertheless, equilibrium constants for the relevant reactions must be known. Furthermore, since most processes operate at elevated temperatures, it is essential that these parameters be known over a range of temperatures. The availability of this information is discussed below. 

Chapter 10 provides an exhaustive description of how these techniques can be applied to a large number of industrial alloys and other materials. This includes a discussion of solution and substance databases and step-by-step examples of multi-component calculations. Validation of calculated equilibria in multi-component alloys is given by a detailed comparison with experimental results for a variety of steels, titanium- and nickel-base alloys. Further selected examples include the formation of deleterious phases, complex precipitation sequences, sensitivity factor analysis, intermetallic alloys, alloy design, slag, slag-metal and other complex chemical equilibria and nuclear applications. 

Concerning more general application of mercury electrode in the studies on com-plexation equilibria, one should mention the paper by Jaworski et al. [59], who have investigated oxidation of mercury microelectrode in solutions with thiocyanates without any background electrolyte added. In the experiments, normal pulse voltammetry and staircase voltammetry were used. The authors have developed a general procedure for the determination of the stability constants, based on the data taken from the voltammograms. They have applied it to the analysis of Hg(II)-SCN complexes. 

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