Equilibrium constant-based approach

In Equation (5.13), the temperature T must be provided in K, whereas the coefficients a to e regarding reaction j are given in Table 5.2. The range of validity is 500 to 1800°C. The equilibrium constant-based calculations have the main advantages in terms of simplicity of the calculation. Deviations from equilibrium can be easily implemented by calculating equilibrium constants at other than the physical process temperature (see Section 5.6.1). However, the main drawback is that trace components may be not accurately predicted, and the implementation of non-ideal gas behavior is difficult. In more complex calculation frameworks, the equilibrium constant-based approach may also show disadvantages regarding mathematical behavior. 

Several features of equation 6.50 deserve mention. First, as the ionic strength approaches zero, the activity coefficient approaches a value of one. Thus, in a solution where the ionic strength is zero, an ion s activity and concentration are identical. We can take advantage of this fact to determine a reaction s thermodynamic equilibrium constant. The equilibrium constant based on concentrations is measured for several increasingly smaller ionic strengths and the results extrapolated... 

Kent and Eisenberg (5) also correlated solubility data in the system S+CC +alkanoleimines+ O using pseudo-equilibrium constants based on molarity. Instead of using ionic characterization factors, they accepted published values of all but two pseudoequilibrium constants and found these by fitting data for MEA and DEA solutions. They were able to obtain excellent fits by this approach and also discovered that the fitted pseudo-equilibrium constants showed an Arrhenius dependence on temperature. 

CVT rate constant can also be expressed in terms of equilibrium constant based on quasithermodynamic formulation of TST. " In this approach, the quasiequilibrium between the reactant and the GT is given by the equilibrium constant ... 

Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A , must equal the weak acid s initial concentration, Cha- ... 

In developing this treatment for determining equilibrium constants, we have considered a relatively simple system in which the absorbance of HIn and Im were easily measured, and for which it is easy to determine the concentration of H3O+. In addition to acid-base reactions, the same approach can be applied to any reaction of the general form... 

The product is equal to the equilibrium constant X for the reaction shown in equation 30. It is generally considered that a salt is soluble if > 1. Thus sequestration or solubilization of moderate amounts of metal ion usually becomes practical as X. approaches or exceeds one. For smaller values of X the cost of the requited amount of chelating agent may be prohibitive. However, the dilution effect may allow economical sequestration, or solubilization of small amounts of deposits, at X values considerably less than one. In practical appHcations, calculations based on concentration equihbrium constants can be used as a guide for experimental studies that are usually necessary to determine the actual behavior of particular systems. 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

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. 

Both these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

The most effective spectrophotometric procedures for pKa determination are based on the processing of whole absorption curves over a broad range of wavelengths, with data collected over a suitable range of pH. Most of the approaches are based on mass balance equations incorporating absorbance data (of solutions adjusted to various pH values) as dependent variables and equilibrium constants as parameters, refined by nonlinear least-squares refinement, using Gauss-Newton, Marquardt, or Simplex procedures [120-126,226],... 

In its simplest form a partitioning model evaluates the distribution of a chemical between environmental compartments based on the thermodynamics of the system. The chemical will interact with its environment and tend to reach an equilibrium state among compartments. Hamaker(l) first used such an approach in attempting to calculate the percent of a chemical in the soil air in an air, water, solids soil system. The relationships between compartments were chemical equilibrium constants between the water and soil (soil partition coefficient) and between the water and air (Henry s Law constant). This model, as is true with all models of this type, assumes that all compartments are well mixed, at equilibrium, and are homogeneous. At this level the rates of movement between compartments and degradation rates within compartments are not considered. 

The problem of predicting a rate constant thus reduces to one of evaluating Kx. There are two basic approaches that are used in attacking this problem one is based on statistical mechanics and the other on thermodynamics. From statistical mechanics it is known that for a reaction of the type X + YZ XYZ1 the equilibrium constant is given by... 

The equilibrium state is generated by minimizing the Gibbs free energy of the system at a given temperature and pressure. In [57], the method is described as the modified equilibrium constant approach. The reaction products are obtained from a data base that contains information on the enthalpy of formation, the heat capacity, the specific enthalpy, the specific entropy, and the specific volume of substances. The desired gaseous equation of state can be chosen. The conditions of the decomposition reaction are chosen by defining the value of a pair of variables (e.g., p and T, V and T). The requirements for input are ... 

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. 

The specific ion interaction approach is simple to use and gives a fairly good estimate of activity factors. By using size/charge correlations, it seems possible to estimate unknown ion interaction coefficients. The specific ion interaction model has therefore been adopted as a standard procedure in the NEA Thermochemical Data Base review for the extrapolation and correction of equilibrium data to the infinite dilution standard state. For more details on methods for calculating activity coefficients and the ionic medium/ ionic strength dependence of equilibrium constants, the reader is referred to Ref. 40, Chapter IX. 

Yang and Schulz also formulated a treatment of coupled enzyme reaction kinetics that does not assume an irreversible first reaction. The validity of their theory is confirmed by a model system consisting of enoyl-CoA hydratase (EC 4.2.1.17) and 3-hydroxyacyl-CoA dehydrogenase (EC 1.1.1.35) with 2,4-decadienoyl coenzyme A as a substrate. Unlike the conventional theory, their approach was found to be indispensible for coupled enzyme systems characterized by a first reaction with a small equilibrium constant and/or wherein the coupling enzyme concentration is higher than that of the intermediate. Equations based on their theory can allow one to calculate steady-state velocities of coupled enzyme reactions and to predict the time course of coupled enzyme reactions during the pre-steady state. 

Comparing the different methods we see once again that the CASSCF/L-CTD method yields the most accurate description of the potential energy curve out of aU the theories. The error at equilibrium (5.99 mEh) is better than that of CCSD (11.03 mEh) and once again this error stays roughly constant across the curve, while that of the CC-based approaches exhibit a nonphysical turnover. For comparison, the MRMP error at equilibrium is 15.41 mEh. The nonparallelity errors for CASSCF/L-CTD and MRMP are 8.9 and 8.3 mEh, respectively, demonstrating again that CASSCF/L-CTD yields quantitatively accurate curves with NPEs competitive with that of MRMP theory. 

Problems in this chapter include some brainbusters designed to bring together your knowledge of electrochemistry, chemical equilibrium, solubility, complex formation, and acid-base chemistry. They require you to find the equilibrium constant for a reaction that occurs in only one half-cell. The reaction of interest is not the net cell reaction and is not a redox reaction. Here is a good approach ... 

The second and far more common approach to testing the predicted dependence of kob on AG has been based on the so-called Marcus cross-reaction equation. The cross-reaction equation interrelates the rate constant for a net reaction, D+A- D++A ( el2), with the equilibrium constant (Kl2) and self-exchange rate constants for the two-component self-exchange reactions D+ 0 (Zen) and A0/- (k22). Its derivation is based on the assumption that the contributions to vibrational and solvent trapping for the net reaction from the individual reactants are simply additive (equation 63). The factors of one-half appear because only one of the two components of the self-exchange reactions is involved in the net reaction. The expression for A0 in equation (63) is an approximation. Note from equation (23) that k is a collective property of both reactants and the approximation in equation (63) is valid only if the reactants have similar radii. 

A different experimental approach to the relative importance of one-center and two-center epimerizations in cyclopropane itself was based on the isomeric l-13C-l,2,3-d3-cyclopropanes165"169. Here each carbon has the same substituents, one hydrogen and one deuterium, and should be equally involved in stereomutation events secondary carbon-13 kinetic isotope effects or diastereotopically distinct secondary deuterium kinetic isotope effects may be safely presumed to be inconsequential. Unlike the isomeric 1,2,3-d3-cyclo-propanes (two isomers, only one phenomenological rate constant, for approach to syn, anti equilibrium), the l-13C-l,2,3-d3-cyclopropanes provide four isomers and two distinct observables since there are two chiral forms as well as two meso structures (Scheme 4). Both chiral isomers were synthesized, and the phenomenological rate constants at 407 °C were found to be k, = (4 l2 + 8, ) = (4.63 0.19)x 10 5s l and ka = (4kl2 + 4, ) = (3.10 0.07) x 10 5 s 1. The ratio of rate constants k, kl2 is thus 1.0 0.2 both one-center and two-center... 

It is now common experimental practice to react ketones with lithium diisopropyl amide (LDA) in order to generate the enolate of the ketone. This methodology has largely replaced the older approach to enolates, which employed alkoxide bases to remove a proton alpha to the carbonyl group. Comparison of the equilibrium constants for these two acid-base reactions reveals why the LDA method is preferable. The use of the amide base leads to essentially complete conversion of die ketone to its enolate (Keq 1016). At equilibrium, there is virtually no... 

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