Consecutive reactions algebra

Such was the state of the art when Amundson and Bilous s paper was published in the first volume of the newly founded A.I.CH.E. Journal (Bilous and Amundson, 1955). This for the first time treated the reactor as a dynamical system and, using Lyapounov s method of linearization, gave a pair of algebraic conditions for local stability. One of these corresponded to the slope condition of previous analyses, and there was a brief flurry of attempts to invest the other with a similarly physical explanation. For the global picture they introduced the phase plane (another feature of the theory of dynamical systems) and, with consummate skill, Bilous conjured the now classic figures from a Reeves electronic analogue computer. Even in this early paper, they had touched upon the consecutive reaction scheme A - B - C and had shown that up to five steady states might be expected under some conditions. 

DerivatioH of rate equations for consecutive reactions by matrix algebra... 

For a single equation, Eqs. (7-36) and (7-37) relate the amounts of the several participants. For multiple reactions, the procedure for finding the concentrations of all participants starts by assuming that the reactions proceed consecutively. Key components are identified. Intermediate concentrations are identified by subscripts. The resulting concentration from a particular reaction is the starting concentration for the next reaction in the series. The final value carries no subscript. After the intermediate concentrations are ehminated algebraically, the compositions of the excess components will be expressible in terms of the key components. 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

In chemical degradation kinetics and pharmacokinetics, the methods of eigenvalue and Laplace transform have been employed for complex systems, and a choice between two methods is up to the individual and dependent upon the algebraic steps required to obtain the final solution. The eigenvalue method and the Laplace transform method derive the general solution from various possible cases, and then the specific case is applied to the general solution. When the specific problem is complicated, the Laplace transform method is easy to use. The reversible and consecutive series reactions described in Section 5.6 can be easily solved by the Laplace transform method ... 

We have seen in Sec. III.9 that the kinetics of reactions which involve two or more consecutive intermediates is capable of a simplified treatment if the intermediate concentrations are small compared to both reactants and products. In such cases we can make the assumption that the rate of change of the concentrations of the intermediates with time is zero, a procedure which then permits us to solve the kinetic equations for the stationary concentrations of the intermediates and then eliminate them algebraically from the system of differential equations. ... 

A similar analysis of non-first-order reactions leads to quite cumbersome equations for [ ]n/[ o]- As the order increases, the telescoping functions involved in these equations become progressively more unwieldy, so that a simple expression for [ ]n/Mo] for a series of N reactors similar to Equation 10.15 for a first-order reaction becomes impossible. It is necessary in such cases to resort to step-by-step algebraic calculations. The equation for a single step in any such sequence is known as the recursion equation. Such equations for Mli-i/MJi (for two consecutive stages) for different types of reactions are listed in Table 10.1. 

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