Systems of first order reactions

It would be beyond our present scope to try to cover the detail of the theory of systems of first order reactions, but the basic ideas are so simple and elegant that it seems a pity that they should pass completely unnoticed. A simple eicample will show the main features. We shall take the reversible sequential reaction A B C and add a further reaction A, thus turning it into the triangular system 

The species A, B, and C might be three isomers or the scheme might be a model of a more complicated situation. The third reaction is not of course stoichiomctrically independent of the other two, but it may represent a distinct chemical reaction and in that sense be kinetically independent. Here is a case where we shall not want to get the smallest number of equations but will exploit the symmetrical aspect of the problem. 

There will be an equilibrium state of the system attained when all the reactions are separately at equilibrium. Thus, if = kjk where / = 1, 2, and 3, the three equilibrium constants are 

But multiplying the terms on the left together gives 1 therefore 

This means that not all the rate constants can be independent, but must satisfy Eq. (5.7.1), an expression of the principle of microscopic reversibility which states that at equilibrium of the whole system each reaction must be separately at equilibrium. From Eqs. (5.7.1) and (5.7.2) we can express the equilibrium concentrations in the following symmetric form (see Exercise 5.7.1)  

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An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

A. First-order Reactions General Treatment. There has been a considerable amount of work done on the solution of particular and general systems of first-order reactions. All such systems are capable of exact, explicit mathematical solutions. If we consider the most general case of a system of s components Ci, C2,. . . , C in which first-order reactions of the following type may take place between any two components... 

Wei, J., Intraparticle diffusion effects in complex systems of first order reactions. D, The influence of diffusion on the performance of chemical reactors. J. Catal. 1,526 (1962b). 

Therefore, we conclude that when k2> k, there is a low and essentially constant concentration of the intermediate. Because of this, d[B]/d< is approximately 0, which can be shown as follows. For this system of first-order reactions. 

Intraparticle diffusion effects in complex systems of first order reactions. 

Solving the model of a fluidized bed implies that a simultaneous solution of N ODEs, Equation 5.251, and 2 N algebraic Equations 5.252 and 5.253 is required. Alternatively, S ODEs and 2 S algebraic equations have to be solved simultaneously for the reduced system. Equations 5.254 through 5.256. For systems of first-order reactions, analytical solutions are achievable [18]. 

For a first-order system, earliness or lateness of mixing does not affect reactor performance, for a given residence time distribution. Therefore, when the residence time distribution is known, the exact performance of a system of first-order reactions can be calculated from the macrofluid model. 

The two factors F, and FJ are very complex and not known. It is virtually certain, however, that each contains several terms. For example, where long-lived radiation-produced species influence the yield, Fj must contain terms such as A 1 — (At + l)c (l — e ) which expresses the average age of the atoms produced. The subsequent thermal effects are often describable in terms of first-order reactions so that FJ must contain one or more terms of the form (1 — Up to the present, there has not been enough information available on any system to make careful statement of Eq. (5) worthwhile. 

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

An important question concerning energy trapping is whether its kinetics are limited substantially by (a) exciton diffusion from the antenna to RCs or (b) electron transfer reactions which occur within the RC itself. The former is known as the diffusion limited model while the latter is trap limited. For many years PSII was considered to be diffusion limited, due mainly to the extensive kinetic modelling studies of Butler and coworkers [232,233] in which this hypothesis was assumed. More recently this point of view has been strongly contested by Holzwarth and coworkers [230,234,235] who have convincingly analyzed the main open RC PSII fluorescence decay components (200-300 ps, 500-600 ps for PSII with outer plus inner antenna) in terms of exciton dynamics within a system of first order rate processes. A similar analysis has also been presented to explain the two PSII photovoltage rise components (300 ps, 500 ps)... 

Thus, for known kinetics and a specified residence time distribution, we can predict the fractional conversion of reactant which the system of Fig. 9 would achieve. Recall, however, that this performance is also expected from any other system with the same E(t) no matter what detailed mixing process gave rise to that RTD. Equation (34) therefore applies to all reactor systems when first-order reactions take place therein. In the following example, we apply this equation to the design of the ideal CSTR and PFR reactors discussed in Chap. 2. The predicted conversion is, of course, identical to that which would be derived from conventional mass balance equations. 

The optimum size ratio for two mixed flow reactors in series is found in general to be dependent on the kinetics of the reaction and on the conversion level. For the special case of first-order reactions equal-size reactors are best for reaction orders n > 1 the smaller reactor should come first for n < 1 the larger should come first (see Problem 6.3). However, Szepe and Levenspiel (1964) show that the advantage of the minimum size system over the equal-size system is quite small, only a few percent at most. Hence, overall economic consideration would nearly always recommend using equal-size units. 

In a parallel reaction network of first-order reactions, the selectivity does not depend upon reaction time or residence time, since both products are formed by the same reactant and with the same concentration. The concentration of one of the two products will be higher, but their ratio will be the same during reaction in a batch reactor or at any position in a PFR. The most important parameters for a parallel reaction system are the reaction conditions, such as concentrations and temperature, as well as reactor type. An example is given in the following section. 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Vajda, S. and Rabitz, H., Identifiability and distinguishability of first-order reaction systems, J. Phys. Chem., 1988, 92, 701-707. 

In general there is no exact solution to any sequence of consecutive higher-order equations. The reason for this is that the differential equations are no longer linear equations (as they were in the case of first-order reactions), and nonlinear equations do not have exact solutions except in very particular cases. However, two exact methods are available for studying some aspects of these systems, and there is one more commonly used... 

Prater, C. D., Silvestri, A. J., and Wei, J., On the structure and analysis of complex systems of first order chemical reactions containing irreversible steps. I, General properties. Chem. Eng. Sci. 22, 1587 (1967). 

In the special case of first order reaction, Equation 16 becomes linear and after some manipulation gives for the exit concentration a result identical to that of Equation 10 thus, for first order reaction, calculations for complete and minimum segregation yield a single result in the transient case. Inputting the reactant as a slug of amount m, c0(t) — (m/V)S(t), and Equation 10 gives for the output of either system... 

Interestingly, the result obtained here is the same as that for the constant-volume system, equation (1-48), since the expansion factor does not appear. This is a peculiarity of first-order reactions only. (First-order reactions often do not follow the patterns anticipated from the behavior of other-order reactions. Another example is the first-order half-life, which is the only one independent of initial reactant concentration.) Keep in mind also that although fractional conversions are the same in the two systems, the concentrations will be different, as shown in equation (1-54). We can again give a listing, in terms of fractional conversion and expansion factors, of the equations for non-constant-volume systems. 

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