Complex equilibrium calculations

This chapter has shown many examples of the use of CALPHAD methods, ranging from an unusual application in a binary system, through complex equilibrium calculations to calculations for 10-component alloy systems. In all cases the use of CALPHAD methods has enhanced the understanding of processes, clearly defined alloy behaviour and provided vital information for other models, etc. It is also clear that equilibrium calculations can be used in many different areas and under a surprising number of different conditions. For numerous reasons, modelling will never completely replace experimental measurement. However, die quantitative verification of the accuracy of CALPHAD calculations now means that they can be seriously considered as an information source which can be used as an alternative to experimental measurement in a number of areas and can also enhance interpretation of experimental results. 

J. N. Butler, Ionic Equilibrium Solubility and pH Calculations (New York Wiley, 1998) and M. Meloun, Computation of Solution Equilibria (New York Wiley. 1988). For commercial software that performs complex equilibrium calculations, see http //www.rnicromath.com/ and http //www.acadsoft.co.uk/. 

The competition of metals for oxygen to form their oxides and for carbon to form their carbides is also a common problem for complex equilibrium calculations. In principle, multiple oxide and formation reactions can be considered simultaneously. This is the case for the reactions [6]... 

As indicated above, the ChemApp programmers library (in the form of a Dynamic Link Library, DLL) has been used to form the thermodynamic backbone of SimuSage. The general concept is represented in Figure 3. The ChemApp library permits the easy use of full complex equilibrium calculations within a software by way of a set of interface routines. These interface routines are used to... 

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

The methods of experimental determination and the calculation of different values associated with chemical reactions (enthalpies, entropies, calorific capacities, free and constant equilibrium enthalpies) lead to complex equilibrium calculations and their graphical representations in various forms pole figures, generalized Ellingham diagrams, binary, tertiary and quaternary diagrams. 

Chaston, S. Calculating Complex Equilibrium Concentrations by a Next Guess Factor Method, /. Chem. Educ. 1993, 70, 622-624. 

At the equivalence point, the moles of Fe + initially present and the moles of Ce + added are equal. Because the equilibrium constant for reaction 9.16 is large, the concentrations of Fe and Ce + are exceedingly small and difficult to calculate without resorting to a complex equilibrium problem. Consequently, we cannot calculate the potential at the equivalence point, E q, using just the Nernst equation for the analyte s half-reaction or the titrant s half-reaction. We can, however, calculate... 

S. Gordon and B. J. McBride, Computer Program for Calculation of Complex Equilibrium Compositions. Rocket Peformance Incident and Reflected Shocks and Chapman-]ouquetDetonations, NASA-Lewis Research Center, NASA, Airport, Md., Mar. 1976. 

The complexation of Pu(IV) with carbonate ions is investigated by solubility measurements of 238Pu02 in neutral to alkaline solutions containing sodium carbonate and bicarbonate. The total concentration of carbonate ions and pH are varied at the constant ionic strength (I = 1.0), in which the initial pH values are adjusted by altering the ratio of carbonate to bicarbonate ions. The oxidation state of dissolved species in equilibrium solutions are determined by absorption spectrophotometry and differential pulse polarography. The most stable oxidation state of Pu in carbonate solutions is found to be Pu(IV), which is present as hydroxocarbonate or carbonate species. The formation constants of these complexes are calculated on the basis of solubility data which are determined to be a function of two variable parameters the carbonate concentration and pH. The hydrolysis reactions of Pu(IV) in the present experimental system assessed by using the literature data are taken into account for calculation of the carbonate complexation. 

While these calculations provide information about the ultimate equilibrium conditions, redox reactions are often slow on human time scales, and sometimes even on geological time scales. Furthermore, the reactions in natural systems are complex and may be catalyzed or inhibited by the solids or trace constituents present. There is a dearth of information on the kinetics of redox reactions in such systems, but it is clear that many chemical species commonly found in environmental samples would not be present if equilibrium were attained. Furthermore, the conditions at equilibrium depend on the concentration of other species in the system, many of which are difficult or impossible to determine analytically. Morgan and Stone (1985) reviewed the kinetics of many environmentally important reactions and pointed out that determination of whether an equilibrium model is appropriate in a given situation depends on the relative time constants of the chemical reactions of interest and the physical processes governing the movement of material through the system. This point is discussed in some detail in Section 15.3.8. In the absence of detailed information with which to evaluate these time constants, chemical analysis for metals in each of their oxidation states, rather than equilibrium calculations, must be conducted to evaluate the current state of a system and the biological or geochemical importance of the metals it contains. 

In principle, the calculation of concentrations of species of a complexation equilibrium is no different from any other calculation involving equilibrium constant expressions. In practice, we have to consider multiple equilibria whenever a complex is present. This is because each ligand associates with the complex in a separate process with its own equilibrium expression. For instance, the silver-ammonia equilibrium is composed of two steps ... 

The small amounts of gold contained in low-grade ores can be extracted using a combination of oxidation and complexation. Gold is oxidized to Au, which forms a very strong complex with cyanide anions Au ((2 q) + l CN(a q) [Au (CN)2] aq) K -lx 10 Suppose that a sample of ore containing 2.5 X 10 mol of gold is extracted with 1.0 L of 4.0 X 10 M aqueous KCN solution. Calculate the concentrations of the three species involved in the complexation equilibrium. 

Determining concentrations requires a quantitative equilibrium calculation, so we apply the seven-step strategy. The complexation equilibrium is given in the problem. 

Complex formation constants could also be determined directly from UV spectrophotometric measurements. Addition of tert.-butyl hydroperoxide to a solution of nitroxide I in heptane at RT causes a shift of the characteristic absorption band of NO at 460 nm to lower wavelengths (Fig. 9). This displacement allows calculation of a complex equilibrium constant of 5 1 1/Mol. Addition of amine II to the same solution causes reverse shift of theC NO" absorption band. From this one can estimate a complex formation constant for amine II and +00H of 12 5 1/Mol (23 2 1/Mol was obtained for tert.-butyl hydroperoxide and 2,2,6,6-tetramethylpipe-ridine in ref. 64b). Further confirmation for an interaction between hindered amines and hydroperoxides is supplied by NMR measurements. Figure 10a shows part of the +00H spectrum in toluene-dg (concentration 0.2 Mol/1) with the signal for the hydroperoxy proton at 6.7 ppm. Addition of as little as 0.002 Mol/1 of tetra-methylpiperidine to the same solution results in a displacement and marked broadening of the band (Fig. 10b). 

As an example of an equilibrium calculation accounting for surface complexation, we consider the sorption of mercury, lead, and sulfate onto hydrous ferric oxide at pH 4 and 8. We use ferric hydroxide [Fe(OH)3] precipitate from the LLNL database to represent in the calculation hydrous ferric oxide (FeOOH /1H2O). Following Dzombak and Morel (1990), we assume a sorbing surface area of 600 m2 g-1 and site densities for the weakly and strongly binding sites, respectively, of 0.2 and 0.005 mol (mol FeOOH)-1. We choose a system containing 1 kg of solvent water (the default) in contact with 1 g of ferric hydroxide. 

One of the major difficulties in Forlani s proposal of the molecular complex substrate-catalyst mechanism, to explain the fourth-order kinetics, is the assumption that this complex needs an additional molecule of amine to decompose to products. The formation of molecular complexes between dinitrohalobenzenes and certain amines (especially aromatic amines) has been widely studied, and their involvement in SwAr reaction has been discussed in Section II.E. The equilibrium constants for the formation of those complexes were calculated in several cases, and they were included in the kinetic expressions when pertinent. But in all cases, the complex was assumed to be in the reaction pathway, and no need of an additional amine molecule was invoked by the several authors who studied those reactions. 

Similarly, for the system of iron/calcium/phosphate, the percentage distribution of various complexes can also be calculated using solution equilibrium calculations as shown in Fig. 6.26. It follows that depending on solution pH, the dominant complexes is CaPO at pH= 10, whereas CaHP04(aq) and CaH2P04are dominant at pH = 8. 

It is also worthwhile to make some distinctions between methods of calculating phase equilibrium. For many years, equilibrium constants have been used to express the abundance of certain species in terms of the amounts of other arbitrarily chosen species, see for example Brinkley (1946,1947), Kandliner and Brinkley (1950) and Krieger and White (1948). Such calculations suffer significant disadvantages in that some prior knowledge of potential reactions is often necessary, and it is difficult to analyse the effect of complex reactions involving many species on a particular equilibrium reaction. Furthermore, unless equilibrium constants are defined for all possible chemical reactions, a true equilibrium calculation caiuiot be made and, in the case of a reaction with 50 or 60 substances present, the number of possible reactions is massive. 

---