Solving Complex Equilibria

For all but the simplest cases, there is no explicit formula and so the calculations need to be performed numerically, i.e. in an iterative process starting from initial guesses. 

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The log-log diagram can also be used to solve complex equilibrium problems. 

The next step in the procedure for calculating the stability constants is to represent the concentrations of the several complexes in terms of the equilibrium constants for their formation. We do this by solving each equilibrium constant expression for the concentration of the complex. For example, the first step has an equilibrium constant... 

One useful trick in solving complex kinetic models is called the steady-state approximation. The differential equations for the chemical reaction networks have to be solved in time to understand the variation of the concentrations of the species with time, which is particularly important if the molecular cloud that you are investigating is beginning to collapse. Multiple, coupled differentials can be solved numerically in a fairly straightforward way limited really only by computer power. However, it is useful to consider a time after the reactions have started at which the concentrations of all of the species have settled down and are no longer changing rapidly. This happy equilibrium state of affairs may never happen during the collapse of the cloud but it is a simple approximation to implement and a place to start the analysis. 

The relative ease of solving the system of non-linear equations for rather complex equilibrium problems, as indicated by the shortness of the function NewtonRaphson. m and by the inconsequentiality of poor initial guesses, is misleading. As we will see shortly, this statement is particularly pertinent to cases of general systems of m equations with m parameters. Solving systems of equations is a common task and we give a short introduction. To start with, we investigate the simple case of one equation with one parameter. 

NewtonRaphson.m, a 30 line code function that solves chemical equilibrium problems of any degree of complexity. 

This approach works well for small n. However, it requires solving n equations simultaneously and is computationally complex. Solving the equilibrium becomes unwieldy and slow as n gets large (our simulations handled up to about 400 species using this approach). 

The calculations involved in complex equilibria are the major subject of this chapter. The systematic approach to solving multiple-equilibrium problems is described. The calculation of solubility when the equilibrium is influenced by pH and the formation of complexes is also discussed. 

Chapter 6 The Systematic Approach to Equilibria Solving Many Equations Chapter 11 Complex Equilibrium Calculations Chapter 10 Complex Equilibrium Calculations... 

Several programs have been written specifically for a very restricted class of equilibrium only problems. The Pit Method of Sillen and Warnquist has been widely used to solve for equilibrium constants in inorganic systems that have one or more simultaneous reversible reactions. DeLand uses goal-seeking routines to facilitate the matching of data, but free energy data for all reactants is required. Bos and Meershoek 24) have written a PL/1 program which uses the Newton-Raphson iteration to compute equilibrium constants in complex systems. 

We can now write some general rules for solving chemical equilibrium problems, using the approximation approach. These rules should be applicable to acid-base dissociation, complex formation, oxidation-reduction reactions, and others. That is, all equilibria can be treated similarly. 

If you add the solubility equilibrium and the complex-ion equilibrium (the two equations just before this example), you obtain the overall equilibrium for the dissolving of AgCl in NH3. The equilibrium constant for this reaction equals the product of the equilibrium constants for solubility and complex-ion formation. (We will show that in the solution to this example however, we discussed the general principle at the end of Section 15.2.) Once you have the equilibrium constant, you can solve the equilibrium problem. 

Ladder diagrams are a useful tool for evaluating chemical reactivity, usually providing a reasonable approximation of a chemical system s composition at equilibrium. When we need a more exact quantitative description of the equilibrium condition, a ladder diagram may not be sufficient. In this case we can find an algebraic solution. Perhaps you recall solving equilibrium problems in your earlier coursework in chemistry. In this section we will learn how to set up and solve equilibrium problems. We will start with a simple problem and work toward more complex ones. 

The solubility of a precipitate can be improved by adding a ligand capable of forming a soluble complex with one of the precipitate s ions. For example, the solubility of Agl increases in the presence of NH3 due to the formation of the soluble Ag(NH3)2°" complex. As a final illustration of the systematic approach to solving equilibrium problems, let us find the solubility of Agl in 0.10 M NH3. 

STRATEGY First, we write the chemical equation for the equilibrium between the solid solute and the complex in solution as the sum of the equations for the solubility and complex formation equilibria. The equilibrium constant for the overall equilibrium is therefore the product of the equilibrium constants for the two processes. Then, we set up an equilibrium table and solve for the equilibrium concentrations of ions in solution. 

Solutions to complex ionic equilibrium problems may be obtained by a graphical log concentration method first used by Sillen (1959) and more recently described by Butler (1964) and Morel (1983). These types of problems are described further in Chapter 16 as they relate to natural systems. Computer-based numerical methods are also used to solve these problems (Morel, 1983). 

Once you have the correct equilibrium constant expression, there is no difference in solving the complex ion equilibrium than any other equilibrium problem. 

Those equilibrium processes that can be resolved explicitly are straightforwardly modelled in Excel. While it is possible to solve equilibrium problems of essentially any complexity in Excel, it is virtually impossible to develop a reasonable spreadsheet for the modelling of a complex titration. Iterative methods are generally difficult to implement in Excel. 

Such information has been stored within a database in the ECES system. Then, using user input in the form of equation (1), ECES writes the expression for computing the thermodynamic equilibrium constant as a function of temperature to a file where it will eventually become part of a program to solve the many equilibria that might describe a complex system. 

Each of these dissociation reactions also specifies a definite equilibrium concentration of each product at a given temperature consequently, the reactions are written as equilibrium reactions. In the calculation of the heat of reaction of low-temperature combustion experiments the products could be specified from the chemical stoichiometry but with dissociation, the specification of the product concentrations becomes much more complex and the s in the flame temperature equation [Eq. (1.11)] are as unknown as the flame temperature itself. In order to solve the equation for the n s and T2, it is apparent that one needs more than mass balance equations. The necessary equations are found in the equilibrium relationships that exist among the product composition in the equilibrium system. 

BMOV, was reported in 1972 159) and in 1987 160). Its electrochemical preparation was described in 1978 92a), and EPR monitoring of its redox behavior, in chloroform, in 1987 160). However this now-important compound seems not to have been properly characterized until 1992 161). Since then complexes of several 3-hydroxy-4-pyridinones 162—164), and of l-hydroxy-2-pyridinone 165), have been synthesized and characterized, especially by EPR 164). VO(malt)2 exists as a cis trans equilibrium mixture in aqueous solution, and generally crystallizes as a mixture of the two isomers. However the crystal structure of the trans structure was eventually solved, confirming the expected square-pyramidal stereochemistry 166). The relative stabilities of the cis and trans forms of V 0L2 complexes depend on the nature of the bidentate ligand L , with the cis configuration favored by VO(malt)2 and VO(koj)2 167), but the trans by 3-hydroxy-4-pyridinonate ligands 164). 

Before leaving this section we consider a slightly different optimization problem that may also be expensive to solve. In flowsheet optimization, the process simulator is based almost entirely on equilibrium concepts. Separation units are described by equilibrium stage models, and reactors are frequently represented by fixed conversion or equilibrium models. More complex reactor models usually need to be developed and added to the simulator by the engineer. Here the modular nature of the simulator requires the reactor model to be solved for every flowsheet pass, a potentially expensive calculation. For simulation, if the reactor is relatively insensitive to the flowsheet, a simpler model can often be substituted. For process optimization, a simpler, insensitive model will necessarily lead to suboptimal (or even infeasible) results. The reactor and flowsheet models must therefore be considered simultaneously in the optimization. 

The K, values correspond to a given conformation of the solvated ligand (L)solv which may or may not remain the same in the complex in some cases several conformations may be imagined (see Fig. 8). Thus Ks is an average stability constant for the system at thermodynamic equilibrium with respect to both conformation and complexation. 

Blooming of dinoflagellates is a complex affair, contemplated in the paradox of plankton . That is, at the equilibrium, resource competition models suggest that the number of coexisting species cannot exceed the number of limiting resources. In contrast, within nature, more species can coexist. A rationalization of these phenomena, possibly solving the paradox, may be found in species oscillations and chaos, without the need of advocating external causes (Huisman 1999). 

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