Independent parallel reactions

By use of the time equations, the X- and K-diagrams of the system are given in parametric form. By an elimination of time one obtains 

In these equations, the symbols represents the degree of advancement reduced with respect to the initial concentration. In consequence in the -diagrams (reduced X-diagrams) the initial o and the final slopes are given according to eq. (2.87) as 

In Section 2,4.1 further reactions were mentioned. Their combination results in further systems with two independent parallel reactions. These types of reaction characteristically yield Jacobi matrices, which are diagonal. According to eq. (2.87) a -diagram is obtained, which is shown in Fig. 2.4. 

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Independent Parallel Reactions of Different Species on the Same Catalyst. One often requires a catalyst that promotes the reactions of one component of a feedstock but does not promote the reactions of other constituents of the mixture. For example, one might desire to dehydrogenate six-membered rings, but not five-membered rings. This type of selectivity behavior may be represented by mechanistic equations of the form... 

Independent Parallel Reactions of the Same Species. Independent parallel reactions of the same species may be represented in stoichiometric form as... 

It should be remembered that these conjugated reactions are extrinsically similar to two independent parallel reactions ... 

Direct asymmetric cpl-induced photoreactions are only observed if there are two enantiomeric ground state reactants presgntthat absorb different amounts of light. Thus the asymmetry of the cpl source is transformed into a different concentration of excited-state enantiomeric species, which becomes obvious in emitting systems in the circular polarization of luminescence [7]. These in turn react in a nonchiral environment with the same rate constants for the different deactivation channels. Thus asymmetric photoreactions are dependent or independent parallel reactions of the enantiomers with different net rates. 

In Scheme 1 the respective rate constants of thermal interconversion of R and S(k), R and S (k ), and of deactivation of R and S (B) are equal. If this were not so, spontaneous or nonpolarized-light induced deracemization would occur, which never has been observed. The interconversion rates of Pr and Ps(k3 and k 3 ) may be different, if the products are diasteromers. If the ground state enantiomerization is slow compared to the other reactions then the kinetics of independent parallel reactions apply if not, those of dependent parallel reactions. 

This scheme represents two independent parallel reactions of the enantiomers. A general treatment for thermal kinetic resolution was given by Kagan and Fiaud [42]. For unimolecular photoreactions first-order equations seem to be appropriate [40]. Accordingly the rates are... 

Additional relations are available for the inflection point, which occurs at t = 2Tmax. These relationships are valid for arbitrary analysis wavelengths including, of course, the excitation wavelength. The time-axis transformation allows rigid kinetic treatment of the independent parallel reactions it is, however, rarely applied. 

For independent parallel reactions, Scheme 5 is appropriate. The rate equations should be expressed in the time coordinate t and are... 

Four independent, parallel reactions with kinetic data, K1-K4, empirically derived from measurements of the mass loss of small samples of birch wood [9] have previously been shown to describe the pyrolysis reaction [I ]. 

A least square evaluation of the DTG curves has led to an acceptable mathematical description of the thermal decomposition of untreated and CCA treated wood [ 17]. The low-temperature pyrolysis of untreated wood is described by a. scheme of three independent parallel reactions, while a combination of two independent parallel reactions and one subsequent reaction is needed to give a good presentation of the thermal decomposition of CCA treated wood [17], Comparison between the reaction schemes for untreated and CCA treated wood, each having its own kinetic constants, reveals the following observations. 

Wheeler characterized the independent parallel reactions A B and R S 2ls type I selectivity. Type II selectivity refers to parallel reactions of the form... 

Wheeler classified multiple reaction networks into three types in order to discuss the effect of diffusion on selectivity. In the present work, we use this classification to discuss selectivity effects with deactivation in general. Type I selectivity refers to two independent parallel reactions ... 

B and D form an addition reaction with the excited states A or A", respectively, in two linear independent steps of the reaction. In this case the correlation between the degrees of advancement is equivalent to that in the case of analogous thermal reactions treated in Section 2.4.2.1 as independent parallel reactions. If the reaction takes place according to this mechanism, the relationship will differ from that in the example given above. 

Thereby a certain analogy to the discussion of the isosbestic point in the reaction spectrum can be found. Condition eq. (5.67) can be easily fulfilled, if in a certain wavelength range just a few compounds absorb. It is sufficient if in this spectral range all other elements except two become zero in one row. This will happen if independent parallel reactions take place whose reactants do not absorb in the spectral range examined. However, the rank of Q can become 1 if the Qxk become equal in two columns. This can happen for the parallel reaction treated in Section 2.1.4.2, if in some spectral regions only compounds A and D absorb. Under these conditions... 

In the case of linear independent parallel reactions the initial slope of an X-diagram depends at least on the initial concentration of one of the starting compounds. In the case where one does not And any dependencies of the quantum yields of two partial reactions on the intensity of radiation, one can shorten the equation for the derivative of with respect to time according to... 

Care must be taken in interpreting mixture effects. As one example, it is possible in principle that at one given set of conditions the overall conversion of one component A in a two-component mixture may be controlled by mass transfer to the catalyst surface and the conversion of another component B by surface reaction kinetics. Of course, the controlling regime (mass transfer or reaction) can change with temperature. Thus, for two independent parallel reactions of the form ... 

Independent Parallel Reactions of Different Species on the Same Catalyst... 

A less simplified approach was followed by Darikavis et al. [387], who fitted their experimental data by means of a multiple independent parallel reaction kinetics. Their model assumed that volatilization products were produced from a large number of independent parallel first order reactions. 

The relative point selectivity for the following independent parallel reaction network in which B is the desired product ... 

It has been shown by Wei (1962) that any complex network of first-order reactions consisting of combinations of consecutive and parallel paths including reversible paths can be decomposed into a system involving only independent parallel reaction paths of irreversible first-order reactions. Let rc be a vector of rate expressions corresponding to a vector of concentrations C for all species. Then ... 

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