Processes Involving Two Reactants

X = 0 is stable this state loses stability at Da = DaT simultaneously, the nontrivial solution gets physically meaningful values and gains stability. 

In this section we analyze processes involving the second-order A + B — 2P reaction. Such processes have been studied, among others, by Luyben andTyreus [10]. It has been noticed [11] that certain control structures lead to state multiplicity and instability. Here, we use dimensionless models to derive general feasibility and stability conditions. The reader is encouraged to check carefully the balance equations, writing them first in the dimensional form, and then deriving the dimensionless versions. To solve these equations, software such as Maple or the symbolic toolbox of Matlab can be used. 

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Frequently, an alternate way of describing a second-order process involving two reactants is employed in which the extent of reaction, x, is used as a variable. If the reaction is one in which the balancing coefficients... 

This article considers processes involving two reactants and two reactions. It is demonstrated that plantwide control relying on self-regulation results in regions of state multiplicity or unfeasibility, even if the stand-alone reactor has a unique, stable operating point. Moreover, when selectivity reasons require low per-pass conversion, instability is very likely. 

The photosensitized electron transfer process involves two successive steps (eq. 5) In the primary event an encounter cage complex of the photoproducts is formed. This can either recombine to yield the original reactants or dissociate into separated photoproducts. The separated photoproducts can then recombine by a diffusional back electron transfer reaction to form the original reactants. We have introduced two conceptional approaches as a means for assisting the separation of the encounter cage complex and for the stabilization... 

Strictly the mathematical expression to be used for the diffusion process should take account of these constraints however, this kind of counter-diffusion involving two reactants and two products in proportions determined by the stoichiometry of the process is of a complexity which has not yet been considered theoretically. In the absence of such a theoretical treatment, Equation (3) was applied using diffusion coefficients reported in the literature for each of the components for diffusion at room temperature. A small correction for the effect of the temperature gradient in the boundary layer on the diffusion coefficient was made in a manner discussed later. 

Mechanisms for most chemical processes involve two or more elementary reactions. Our goal is to determine concentrations of reactants, intermediates, and products as a function of time. In order to do this, we must know the rate constants for all pertinent elementary reactions. The principle of mass action is used to write differential equations expressing rates of change for each chemical involved in the process. These differential equations are then integrated with the help of stoichiometric relationships and an appropriate set of boundary conditions (e.g., initial concentrations). For simple cases, analytical solutions are readily obtained. Complex sets of elementary reactions may require numerical solutions. 

While almost all synthetically useful catalytic reactions involve two reactants it will be best to first illustrate this graphical approach using a unimolecular process and the way it can be affected by inhibitors. After this some different types of bisubstrate reactions will be discussed. More detailed analyses of the kinetics of multisubstrate systems have been published in texts 13 and... 

Schwartz (37), independently proposed the route that leads to presqualene pyrophosphate and to squalene. The head-to-head carbon-carbon bond-forming reaction (Scheme 13) that produces presqualene pyrophosphate formally involves the combination of the elements of two farnesyl moieties, one as the pyrophosphate and the other as the ion from which pyrophosphate has been cleaved. In contrast, the more common head-to-tail carbon-carbon bond-forming process involves two different initial species as reactants. 

Tandem domino and tandem consecutive reactions do not necessarily involve more than one reactant and can be unimolecular, as is exemplified by such processes as polyolefin cyclization [13]. Tandem sequential reactions cannot be unimolecular processes and involve two reactants at least. Both, tandem sequential and higher-order tandem domino reactions are of foremost interest in this text, because they enable the concept of multicomponent reactions (MCRs) to be introduced [4, 7-9]. Depending on the reaction conditions, MCRs can be regarded as a subclass of either tandem sequential (Scheme 17.1a) or higher order tandem domino reactions (Scheme 17.1b). In the former, formation of intermediate AB is followed by addition of a third component C to the reaction mixture to enable formation of reaction product ABC. In the latter, components A, B, and C are simply... 

Table 11.2. Bisituselectivity in Chemical Processes Involving Two Ambident Reactants S, and S 2...

If the reaction involves two reactant feed streams, two basic flowsheets are used. Consider the reaction A + B C + D. One way to design the process is to feed an excess of one of the reactants into the reactive distillation column along with the other reactant. Figure 9.4 shows a system in which an excess of reactant B is fed. In most cases this excess must be recovered. A second distillation column is used in Figure 9.4 to achieve this recovery. The fresh feed of reactant B is mixed with the recycle of B coming from the recovery column. 

Most chemical reactions require two reactant molecules to come into close proximity before they can react with each other. If the reactants are not initially in contact then the overall process involves two steps. First, the reacting molecules must move to be next to each other, and then chemical change may, or may not, occur. 

A more complex case of so-called cross-catalytic reactions may involve two reactants A and B and two products Z and P. The intermediates are X and Y and the catalytic loop is caused by multiplication of the intermediates X, see the scheme above. Figure 63 above may well illustrate the input effect of reactant concentration within the given reaction mechanism (at the threshold concentration of A the steady sub-critical region changes from the sterile to the fertile course of action capable of oscillations in supercritical region. Although first assumed hypothetically, it enabled to visualize the autocatalytic nature of many processes and gave to them the necessary practical dimension when applied to various reality situations ... 

There is one reactive intermediate, NO3, which is produced in one step and consumed in the other step. Addition of the steps of this mechanism gives the stoichiometric equation, with cancellation of NO3. Both steps in the mechanism of Eq. (12.1-2) are bimolecular. That is, they involve two reactant particles. Unimolecular steps involve a single particle. Termolecular steps involve three particles. Termolecular processes are relatively slow because of the small probability that three molecules will collide or diffuse together at once, and these processes occur less frequently in mechanics than do bimolecular processes. Elementary processes involving four or more reactant particles probably do not occur in chemical reaction mechanisms. We now make an important assertion concerning the rate law of any elementary process In an elementary process, the order of any substance is equal to the molecularity of that substance. We now justify this assertion for bimolecular elementary processes in the gas phase. 

Inferences that oxidation takes place on the photocatalyst s surface have been made (67). No such conclusions can be drawn. Similar observations have been made in homogeneous media if a bimolecular reaction between two reactants is assumed. A Langmuir-type behavior is no guarantee of a surface occurring process. A rigorous treatment (68) of the kinetics involved in the photocataly2ed oxidations of organic substrates on an irradiated semiconductor has confirmed this. 

The simplest solid—solid reactions are those involving two solid reactants and a single barrier product phase. The principles used in interpreting the results of kinetic studies on such systems, and which have been described above, can be modified for application to more complex systems. Many of these complex systems have been resolved into a series of interconnected binary reactions and some of the more fully characterized examples have already been mentioned. While certain of these rate processes are of considerable technological importance, e.g. to the cement industry [1], the difficulties of investigation are such that few quantitative kinetic studies have been attempted. Attention has more frequently been restricted to the qualitative identifications of intermediate and product phases, or, at best, empirical rate measurements for technological purposes. 

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