Equilibrium Calculations—A General Approach

From this value we would expect the dissociation of HaS into HS and H+ to be at equilibrium in a well-mixed solution only a fraction of a second after it is added. If HaS were bubbled into a solution, the rate at which HS would be produced would be controlled by the efficiency of the mixing device, even for the most efficient mixing system. 

Because acid-base reactions in solution generally are so rapid, we can concern ourselves primarily with the determination of species concentrations at equilibrium. Usually, we desire to know [H+], [OH ], and the concentration of the acid and its conjugate base that result when an acid or a base is added to water. As we shall see later in this text, acid-base equilibrium calculations are of central importance in the chemistry of natural waters and in water and wastewater treatment processes. The purpose of this section is to develop a general approach to the solution of acid-base equilibrium problems and to apply this approach to a variety of situations involving strong and weak acids and bases. 

Let us consider first the equations that describe a solution which results when an acid, HA, or a salt of its conjugate base, MA (where M is a cation) is added to water. 

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So far, we have examined specific cases to illustrate the effect of kinetics vs. thermodynamics upon reacting systems (butadiene) and how the thermodynamic property Gibbs energy allows us to calculate equilibrium compositions by quantifying the trade-off between energy and entropy (HCl). We now wish to develop a general approach so that we can analyze the chemical reaction equilibria for any system of interest. 

The general approach illustrated by Example 18.7 is widely used to determine equilibrium constants for solution reactions. The pH meter in particular can be used to determine acid or base equilibrium constants by measuring the pH of solutions containing known concentrations of weak acids or bases. Specific ion electrodes are readily adapted to the determination of solubility product constants. For example, a chloride ion electrode can be used to find [Cl-] in equilibrium with AgCl(s) and a known [Ag+]. From that information, Ksp of AgCl can be calculated. 

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

Our final task in this chapter is to demonstrate how partition constants/coefficients can be used to calculate the equilibrium distribution of a compound i in a given multiphase system. As already pointed out earlier, for simplicity, we consider only neutral species. As we will see in Chapter 8, the equilibrium partitioning of ionogenic compounds (i.e., compounds that are or may also be present as charged species, as, for example, acids or bases) is somewhat more complicated to describe. However, the general approach discussed here is the same. 

Ideal Adsorbed Solution Theory. Perhaps the most successful general approach to the prediction of multicomponent equilibria from single-component isotherm data is ideal adsorbed solution theory. In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equilibrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equilibrium pressure for the pure component at Ike same spreading pressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption. Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, are not consistent with an ideal adsorbed phase and there is no way of knowing a priori whether or not a given system will show ideal behavior. 

Some years ago, Sillen published (56) a most fundamental contribution to chemical oceanography. Assuming a simple equilibrium model he was able to obtain almost correct values for the concentrations of many major and of some minor components of sea water. In comparing Sillen s paper with our approach, the reader will find that a seven-year progress in equilibrium chemistry has generally confirmed and justified his model. There are, however, some indications that the ocean represents a steady-state system rather than a equilibrated solution. Whatever the accuracy of our calculations, the relationship between oversaturation and the rate of transport cannot be ignored. 

Nonetheless, equilibrium considerations can greatly aid attempts to understand in a general way the redox patterns observed or anticipated in natural waters. In all circumstances equilibrium calculations provide boundary conditions toward which the systems must be proceeding, however slowly. Moreover, partial equilibria (those involving some but not all redox couples) are approximated frequently, even though total equilibrium is not approached. In some instances active poising of particular redox couples allows one to predict significant oxidation-reduction levels or to estimate properties and reactions from computed redox levels. 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

For multistep complexation reactions and for ligands that are themselves weak acids, extremely involved calculations are necessary for the evaluation of the equilibrium expression from the individual species involved in the competing equilibria. These normally have to be solved by a graphical method or by computer techniques.26,27 Discussion of these calculations at this point is beyond the scope of this book. However, those who are interested will find adequate discussions in the many books on coordination chemistry, chelate chemistry, and the study and evaluation of the stability constants of complex ions.20,21,28-30 The general approach is the same as outlined here namely, that a titration curve is performed in which the concentration or activity of the substituent species is monitored by potentiometric measurement. 

A simple model of the chemical processes governing the rate of heat release during methane oxidation will be presented below. There are simple models for the induction period of methane oxidation (1,2.>.3) and the partial equilibrium hypothesis (4) is applicable as the reaction approaches thermodynamic equilibrium. However, there are apparently no previous successful models for the portion of the reaction where fuel is consumed rapidly and heat is released. There are empirical rate constants which, due to experimental limitations, are generally determined in a range of pressures or concentrations which are far removed from those of practical combustion devices. To calculate a practical device these must be recalibrated to experiments at the appropriate conditions, so they have little predictive value and give little insight into the controlling physical and chemical processes. 

For the reaction Ha + Ha- Hs + H,270 VBCI calculations were performed on all the points considered. Five attitudes of approach of H2 and Hi and three attitudes of retreat of and H were considered, while the geometries of H2, Hi, and H+ were fixed at their respective calculated equilibrium values. The steepest approach potential is found to be one in which the H+ axis bisects the Ha axis. The steepest retreat potential is one in which the H atom leaves the triangular H, ion on the same axis on which it entered. With this general reaction path defined, the calculations were repeated with the optimization of the H-H bond distance. A minimum was found at H-H and H-H + distances of 1.66 a.u. and 4.11 a.u., respectively, while the distance from the leading H atom of the H+ ion to the bisector of the Ha molecule is 1.44 a.u. The binding energy is found to be 3.6 kJ mol-1. 

An alternative approach is by the application of an approximate theory. At present, the most useful theoretical treatment for the estimation of the equilibrium properties is generally considered to be the density functional theory (DFT). This involves the derivation of the density profile, p(r), of the inhomogeneous fluid at a solid surface or within a given set of pores. Once p(r) is known, the adsorption isotherm and other thermodynamic properties, such as the energy of adsorption, can be calculated. The advantage of DFT is its speed and relative ease of calculation, but there is a risk of oversimplification through the introduction of approximate forms of the required functionals (Gubbins, 1997). 

The approach of Dainton and Ivin [1] is general, simple, and formally quite correct. Practically it really yields only a limited amount and quality of information on the polymerizing systems. The mechanistic approach of Eisenberg and Tobolsky [3] is more specialized it only applies to living systems. However, it yields information not only on monomer-polymer equilibria but also on the equilibrium distribution of molecular mass. The work of Tobolsky was extended by Wheeler et al. who further refined equilibria calculations in homopolymerizations [4, 5] a general solution of equilibrium copolymerizations in living media was developed by Szwarc [6]. These latter developments are not based on formal thermodynamics. 

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