Analysis of Solution Equilibria

In this chapter you ll learn how to calculate species distributions of polyprotic weak acid species, how to apply Gran s method for the estimation of end-points in titrations and a general method for the calculation of titration curves. 

---

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. 

Parallel events in the field of in situ IR spectroscopy (for a review of sulfate IR studies, see Ref. 26) resulted in a coupled shift to molecular level bi-sulfate anion was recognized in sulfate adlayer even in solutions with predominating sulfate. It was com-pletey new situation, when chemical equilibrium is affected by adsoibate-surface interactioa In usual terms of solution equilibria, the effect corresponds to increase of pKa from its bulk value (ca. 2) to 3.3-4.7 (pKa is potential-dependent). To agree this situation with bulk thermodynamics, one should simply use electrochemical potential instead of chemical. The phenomena of adsorption-induced protonation is relative to UPD, when adsoibate-surface interaction shifts redox equihbria. In more molecular terms, the species determined as bi-sidfate ions are probably interfacial ion pairs, i.e., the phenomenon can be considered as coadsorptioa This situation is screened in purely thermodynamic analysis, as excess surface protonation is hidden in Gibbs adsorptions of sulfate and H. However it becomes important for any further model consideration, as it can affect lateral interactions and the order in the adlayer. The excess adsorption-induced protonation of various anions is a very attractive field. In particular it is the only chance to explain why multicharged oxoanions can form complete mono-layers on platinum. 

Inducing equilibria displacements such as those mentioned above is the basis of the most fundamental kind of reasoning followed in the analysis of solutions. 

The protonation equilibria for nine hydroxamic acids in solutions have been studied pH-potentiometrically via a modified Irving and Rossotti technique. The dissociation constants (p/fa values) of hydroxamic acids and the thermodynamic functions (AG°, AH°, AS°, and 5) for the successive and overall protonation processes of hydroxamic acids have been derived at different temperatures in water and in three different mixtures of water and dioxane (the mole fractions of dioxane were 0.083, 0.174, and 0.33). Titrations were also carried out in water ionic strengths of (0.15, 0.20, and 0.25) mol dm NaNOg, and the resulting dissociation constants are reported. A detailed thermodynamic analysis of the effects of organic solvent (dioxane), temperature, and ionic strength on the protonation processes of hydroxamic acids is presented and discussed to determine the factors which control these processes. 

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. 

After completing our analysis of the effects of the dominant equilibrium, we may need to consider the effects of other equilibria. The calculation of [H3 O ] in a solution of weak base illustrates circumstances where this secondary consideration is necessary. Here, the dominant equilibrium does not include the species, H3 O, whose concentration we wish to know. In such cases, we must turn to an equilibrium expression that has the species of interest as a product. The reactants should be species that are involved in the dominant equilibrium, because the concentrations of these species are determined by the dominant equilibrium. We can use these concentrations as the initial concentrations for our calculations based on secondary equilibria. Look again at Example for another application of this idea. In that example, the dominant equilibrium is the reaction between hypochlorite anions and water molecules H2 0 l) + OCr(c2 q) HOCl((2 q) + OH ((2 q) Working with this equilibrium, we can determine the concentrations of OCl, HOCl, and OH. To find the concentration of hydronium ions, however, we must invoke a second equilibrium, the water equilibrium 2 H2 0(/) H3 O (a q) + OH (a q)... 

Licht et al. [17] developed a method of numerical analysis to describe the above-quoted equilibria of the 11 participating species (including alkali metal cations) in aqueous polysulfide solution, upon simple input to the algorithm of the temperature and initial concentration of sulfur, alkali metal hydroxide, and alkali metal hydrosulfide in solution. The equilibria constants were evaluated by compensation of the polysulfide absorption spectrum for the effects of H8 absorption and by computer analysis of the resultant spectra. Results from these calculations were used to demonstrate that the electrolyte is unstable, and that gradual degradation of polysulfide-based PECs (in the long term) can be attributed to this factor (Chap. 5). 

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]. 

If one wants to use only a metal halide as the initiator for an alkene polymerisation, an analysis similar to ours can be made for the metal halide alone. In addition, there is a 1 2 equilibrium for the organic halide [35, 36] and a molecular aggregate <-> single molecule equilibrium associated with the metal halide. Thus, a solution of a carbocation salt is more exactly described by the series of linked equilibria summarised in Scheme 5. 

There are many types of data in chemistry that are not specifically covered in this book. For example, we do not discuss NMR data. NMR spectra of solutions that do not include fast equilibria (fast on the NMR time scale) can be treated essentially in the same way as absorption spectra. If fast equilibria are involved, e.g. protonation equilibria, other methods need to be applied. We do not discuss the highly specialised data analysis problems arising from single crystal X-ray diffraction measurements. Further, we do not investigate any kind of molecular modelling or molecular dynamics methods. While these methods use a lot of computing time and power, they are more concerned with data generation than with data analysis. 

Many of the undesirable substances present in gaseous or liquid streams form volatile weak electrolytes in aqueous solution. These compounds include ammonia, hydrogen sulfide, carbon dioxide and sulfur dioxide. The design and analysis of separation processes involving aqueous solutions of these materials require accurate representation of the phase equilibria between the solution and the vapor phase. Relatively few studies of these types of systems have been published concerning solutions of weak electrolytes. This paper will review the methods that have been used for such solutions and, as an example, consider the alkanolamine solutions used for the removal of the acid gases (H2S and C02) from gas streams. 

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. 

---