Simple parallel reactions

To this point we have focused on reactions with rates that depend upon one concentration only. They may or may not be elementary reactions indeed, we have seen reactions that have a simple rate law but a complex mechanism. The form of the rate law, not the complexity of the mechanism, is the key issue for the analysis of the concentration-time curves. We turn now to the consideration of rate laws with additional complications. Most of them describe more complicated reactions and we can anticipate the finding that most real chemical reactions are composites, composed of two or more elementary reactions. Three classifications of composite reactions can be recognized (1) reversible or opposing reactions that attain an equilibrium (2) parallel reactions that produce either the same or different products from one or several reactants and (3) consecutive, multistep processes that involve intermediates. In this chapter we shall consider the first two. Chapter 4 treats the third. 

In the simple case, B grows at the expense of A. However, in the second case involving parallel reactions, two embryos form from A. In the third case, at least two steps are involved in the two consecutive reactions where "A" trcinsforms to "B" which transforms to "C". 

Note that with Cao = Cco = Cqo = 0, Cbo = 1, ki = k3 = 0 the program models a simple first-order parallel reaction sequence. 

The term parallel reactions describes situations in which reactants can undergo two or more reactions independently and concurrently. These reactions may be reversible or irreversible. They include cases where one or more species may react through alternative paths to give two or more different product species (simple parallel reactions),... 

Simple Parallel Reactions. The simplest types of parallel reactions involve the irreversible transformation of a single reactant into two or more product species through reaction paths that have the same dependence on reactant concentrations. The introduction of more than a single reactant species, of reversibility, and of parallel paths that differ in their reaction orders can complicate the analysis considerably. However, under certain conditions, it is still possible to derive useful mathematical relations to characterize the behavior of these systems. A variety of interesting cases are described in the following subsections. 

H.3 Higher-Order Irreversible, Simple Parallel Reactions. Many simple parallel reactions... 

In this subsection we have treated a variety of higher-order simple parallel reactions. Only by the proper choice of initial conditions is it possible to obtain closed form solutions for some of the types of reaction rate expressions one is likely to encounter in engineering practice. Consequently, in efforts to determine the kinetic parameters characteristic of such systems, one should carefully choose the experimental conditions so as to ensure that potential simplifications will actually occur. These simplifications may arise from the use of stoichiometric ratios of reactants or from the degeneration of reaction orders arising from the use of a vast excess of one reactant. Such planning is particularly important in the early stages of the research when one has minimum knowledge of the system under study. 

The first point to be established in any experimental study is that one is dealing with parallel reactions and not with reactions between the products and the original reactants or with one another. One then uses data on the product distribution to determine relative values of the rate constants, employing the relations developed in Section 5.2.1. For simple parallel reactions one then uses either the differential or integral methods developed in Section 3.3 in analysis of the data. 

There are few short-cut methods for analyzing simple parallel systems. One useful technique, however, is to use stoichiometric ratios of reactants so that the ratio of the time derivatives of the extents of reaction simplifies where possible. For higher-order irreversible simple parallel reactions represented by equations 5.2.41 and 5.2.42, the degenerate form of the ratio of reaction rates becomes... 

A recent study,209 in which previous results on the complexation of a series of non-centrosymmetrical guests with CDs were re-evaluated, suggested that the two observed relaxation processes could possibly be interpreted as a mechanism involving two parallel reactions inclusion of the guest through either the wide or narrow rim of the cyclodextrin. This mechanism was shown to lead to the same dependence of observed rate constants on concentration of cyclodextrin as the consecutive mechanism. This study showed that even for seemingly simple host systems the mechanistic details for complexation can be quite complex and still controversial. 

Acid-promoted aquation of the binuclear complex Cu2L of the hexaaza macrocycle L = 2,5,8,17,20,23 -hexaaza[9.9]paracyclophane, whose half-life is of the order of a second, exhibits simple one-stage first-order kinetics. This is attributed to parallel reactions at each Cu(II) center having identical rate constants (305). The kinetics of dissociation of mono- and... 

It should be noted that there are cases in which some selectivity will be lost in choosing a semi-batch mode over a simple batch reactor. If the desired product decomposes by a consecutive reaction, the yield will be higher in the batch reactor [177]. If, on the other hand, the reactants are producing by-products by a parallel reaction, the semi-batch process will give the higher yield. In any case, if the heat production rate per unit mass is very high, the reaction can then be run safely under control only in a semi-batch reactor. 

Various ways to modify ZSM-5 catalyst in order to induce para-selectivity have been described. They include an increase in crystal size (15,17,20) and treatment of the zeolite with a variety of modifying agents such as compounds of phosphorus (15,18), magnesium (15), boron (16), silicon (21), antimony (20), and with coke (14,18). Possible explanations of how these modifications may account for the observed selectivity changes have been presented (17) and a mathematical theory has been developed (22). A general description of the effect of diffusion on selectivity in simple parallel reactions has been given by Weisz (23). 

It would be an oversimplification to assume that these simple coupling and electron-transfer reactions reflect all the steps occurring in front of the electrode. In addition, parallel reactions... 

If the rate constants for parallel reactions are to be resolved, then analysis of the products is essential (Sec. 1.4.2). This is vital for understanding, for example, the various modes of deactivation of the excited state (Sec. 1.4.2), Only careful analysis of the products of the reactions of Co(NH3)jH20 + with SCN, at various times after initiation, has allowed the full characterization of the reaction (1.95) and the detection of linkage isomers. Kinetic analysis by a number of groups failed to show other than a single second-order reaction.As a third instance, the oxidation of 8-Fe ferredoxin with Fe(CN)g produces a 3Fe-cluster, thus casting some doubt on the reaction being a simple electron transfer. 

We could write species mass-balance equations (S = 6 in this example) on any such reaction sequence and solve these (/ = 4 are inseparable) to find Cj x), and in most practical examples we must do this. However, there are two simple reaction networks that provide insight into these more complex networks, and we wiU next consider them, namely, series and parallel reaction networks (Figure 4-3). 

When selectivity and yield of a given product need to be maximized, the design issues become more complicated. While rninimum T is frequently desired, it is usually more important to obtain maximum selectivity to a desired product and niinimiim selectivity to undesired products. For simple series and parallel reaction systems, we can fairly easily summarize the choices. 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

The intrinsic stability of the aromatic n system has two major consequences for the course of reactions involving it directly. First, the aromatic ring is less susceptible to electrophilic, nucleophilic, and free-radical attack compared to molecules containing acyclic conjugated n systems. Thus, reaction conditions are usually more severe than would normally be required for parallel reactions of simple olefins. Second, there is a propensity to eject a substituent from the tetrahedral center of the intermediate in such a way as to reestablish the neutral (An + 2)-electron system. Thus, the reaction is two step, an endothermic first step resulting in a four-coordinate carbon atom and an exothermic second step, mechanistically the reverse of the first, in which a group is ejected. The dominant course is therefore a substitution reaction rather than an addition. 

Diagnostic plots for heterogeneous catalytic electrode reactions at the RRDE have many features in common with those for simple parallel reactions [178]. This type of analysis is important in the investigation of the oxygen electrode reaction where non-electrochemical surface processes can occur. 

Blanding (10) first proposed the second order cracking kinetics for FCC. Krambeck (11) theoretically demonstrated that conversion in systems with a large number of parallel reactions can be approximated by simple second order kinetics. More recently, Ho and Aris (12) have developed a further mathematical treatment of this concept. An inhibition term was incorporated into the second order cracking kinetics for gas oil conversion to account for competitive adsorption. The initial cracking rate is then given by ... 

The first reaction type is when the reactants form, not just the desired products, but also other undesired products in parallel with the main reaction. We want to show here the implications of parallel reactions, so we consider a simple batch isothermal reactor at constant volume ... 

An increasing number of groups are generating libraries in which parallel reactions occur in solution. Until very recently, solution-based syntheses were primarily utilized for the preparation of simple structures. However, the introduction of resin-bound reagents, scavenger resins, and parallel purification schemes is allowing more complex chemistries to be performed in parallel solution format. 

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