Individual Reaction Paths

In the next 4 sections we shall give a more detailed account of the individual reaction paths connected with hole injection when an excited dye molecule is formed at the surface of an organic crystal. 

As an example of the practical use of the above criterion, let us discuss again the electrocyclic transformation of butadiene to cyclobutene. The individual alternative reaction mechanisms are described, as in the previous chapter, by the scheme II. For evaluating relative ease of individual reaction paths, it is necessary to first calculate the density matrices PR, P, and FP which are then, in the next step converted into the similarity indices rRP, rRI and rIP. Such a calculation requires, however, the density matrices to be transformed into the common basis of atomic orbitals [33,43]. These transformations are described by the matrices TRP, TRI and TIP which has to be determined for each elementary step. 

Concurrent Reactions. If the reactants may combine with each other in two or more different ways to produce either the same or different products, the over- ill rate of disappearance of reactants will be a composite of the individual reaction paths that are accessible. Such systems are termed systems of concurrent or competing reactions, and their kinetic behavior may 26... 

The simple three-step mechanism of dissociation and recombination, given in sections 1.2.2 and 1.2.3, must be generalized to obtain a quantitative description of dissociation and recombination rates. Collisional activation and deactivation steps, (1.4) and (1.5), and the final dissociation of highly excited molecules (1.6), never proceed in a unique way. Instead, many different individual reaction paths exist which contribute to the overall reaction. This will be illustrated briefly in the following by looking at the fate of one particular molecule. 

Our goal is the representation of reaction mechanisms. An individual reaction path has many incidental features, some not compatible with quantum mechanics. Instead of a reaction path, a reaction mechanism can be better represented by a formal reaction itinerary, where the main, invariant features of the journey are relevant. Such a reaction itinerary can be represented by a whole family of similar paths, and it is natural to model a reaction itinerary by a homotopy equivalence class of paths. 

The use of homotopy equivalence classes at some energy bound A is the key step in the simplification of dealing with infinitely many possible individual reaction paths, and reducing the problem to dealing with reaction mechanisms. 

Using the CFTI protocol, we have calculated directly both the derivative of the free energy with respect to the reaction path dA/dX and the 14 individual derivatives dA/d k, k = 1,...,14 with respect to all fixed coordinates along the path ... 

One of the possibilities is to study experimentally the coupled system as a whole, at a time when all the reactions concerned are taking place. On the basis of the data obtained it is possible to solve the system of differential equations (1) simultaneously and to determine numerical values of all the parameters unknown (constants). This approach can be refined in that the equations for the stoichiometrically simple reactions can be specified in view of the presumed mechanism and the elementary steps so that one obtains a very complex set of different reaction paths with many unidentifiable intermediates. A number of procedures have been suggested to solve such complicated systems. Some of them start from the assumption of steady-state rates of the individual steps and they were worked out also for stoichiometrically not simple reactions [see, e.g. (8, 9, 5a)]. A concise treatment of the properties of the systems of consecutive processes has been written by Noyes (10). The simplification of the treatment of some complex systems can be achieved by using isotopically labeled compounds (8, 11, 12, 12a, 12b). Even very complicated systems which involve non-... 

Geochemists have long recognized the need for computational models to trace the progress of reaction processes, both natural and artificial. Given a process involving many individual reactions (possibly thousands), some of which yield products that provide reactants for others, how can we know which reactions are important, how far each will progress, what overall reaction path will be followed, and what the path s endpoint will be ... 

The reaction path of thiamine-dependent catalysis is essentially unchanged in the presence of an apoenzyme, except that the enzyme active site residues increase reaction rates and yields and influence the substrate and product specificity. The X-ray crystal structures of TDP-dependent enzymes have clarified this view and permitted an understanding of the roles of the individual amino acids of the active site in activating and controlling the thiazolium reactivity [36-40]. 

As these types of calculations are extended further, for larger hydrocarbons, not only does the complexity of individual calculations increase but so does the number of possible reaction paths. At present, calculations for... 

Thus put, details of the individual reactions—which are, in any event, certain to be complex—remain as undetermined and debatable as before. What becomes clear (and consistent with experiment) is that (a) product gases such as carbon dioxide can form via two fundamentally unrelated paths (b) humic acids can be abstracted by secondary degradation or by stripping reactions such as decarboxylation (i.e. by reactions respectively characterized by kn> fe, etc. and by k) (c) in a sequential reaction series such as Reaction 2, a zero rate of humic acid formation denotes establishment of a steady state condition rather than formation of a simple equilibrium of the type coal humic acids. 

The curved-line reaction paths are generated from individual analyses of product distributions which can be obtained either by changing catalyst volume or xylene flow rate or merely by sampling the product as the catalyst deactivates with time. Selectivity remains essentially constant during aging. 

The basic parameters which determine the kinetics of internal oxidation processes are 1) alloy composition (in terms of the mole fraction = (1 NA)), 2) the number and type of compounds or solid solutions (structure, phase field width) which exist in the ternary A-B-0 system, 3) the Gibbs energies of formation and the component chemical potentials of the phases involved, and last but not least, 4) the individual mobilities of the components in both the metal alloy and the product determine the (quasi-steady state) reaction path and thus the kinetics. A complete set of the parameters necessary for the quantitative treatment of internal oxidation kinetics is normally not at hand. Nevertheless, a predictive phenomenological theory will be outlined. 

In fluids, structural equilibration is typically rapid and the reaction paths are competitive. In this case the overall decay is exponential with an effective rate constant which is the weighted average of the individual rate constants (Figure 2b). The weighting factors are the steady-state populations of the equilibrating reactant structures. 

The spatial distribution of deposited Ni and V in the reactor bed is determined by the activity of the catalyst and phenomenologically parallels that for profiles in individual pellets. Metals will tend to deposit near the reactor inlet with a highly active catalyst. A more even distribution or one skewed toward the reactor outlet is obtained for catalyst with less activity, as shown by Pazos et al. (1983). Generally with a typical small-pore (60-A), high-surface-area desulfurization catalyst, metals will concentrate near the inlet (Sato et al., 1971 Tamm et al., 1981). Fleisch et al. (1984) observed concentration maximums a short distance into the catalyst bed, as a probable consequence of the consecutive reaction path. 

In this paper we compare behavior of catalysts in extinction recycle hydrocracking. In such a processing scheme, all of the feed is ultimately converted to product boiling below a certain predefined temperature. The reaction paths of individual feed components may differ from catalyst to catalyst. Product distributions and properties are examined to determine the general effects of changes in catalytic properties. 

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