First order reversible series reactions

In Example 2.1, Maple was used to solve two simultaneous first order ODEs. The same methodology can be used to solve more than two simultaneous ODEs. Eor example, the material balance equations for the time dependent concentration of each species (A, B, and C) in an isothermal batch reactor with reversible series 

In this case, the initial conditions are Ca(0) = 1 mol/1 Cb(0) = 0 and Cc(0) = 0. One might ask What are the values of parameters (ki... ki), if any, that would produce a maximum in concentration of species B This question can be answered by using Maple to obtain a solution to the equations in (2.10) given initial conditions for the concentrations. 

This reaction scheme (equation (2.10) is simulated below by following the procedure described for the previous example (see Example 2.1). 

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The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

As we have seen before, the enzymatic reaction begins with the reversible binding of substrate (S) to the free enzyme ( ) to form the ES complex, as quantified by the dissociation constant Ks. The ES complex thus formed goes on to generate the reaction product(s) through a series of chemical steps that are collectively defined by the first-order rate constant kCM. The first mode of inhibitor interaction that can be con-... 

Bimolecular reactions of two molecules, A and B, to give two products, P and Q, are catalyzed by many enzymes. For some enzymes the substrates A and B bind into the active site in an ordered sequence while for others, bindingmay be iii a random order. The scheme shown here is described as random Bi Bi in a classification introduced by Cleland. Eighteen rate constants, some second order and some first order, describe the reversible system. Determination of these kinetic parameters is often accomplished using a series of double reciprocal plots (Lineweaver-Burk plots), such as those at the right. 

Many degradation and biotransformation reactions follow reversible and/or consecutive series pathways. Consider, for example, two reversible and consecutive first-order reactions ... 

In some cases it may be desirable to use a series of stirred-.tank reactors, with the exit stream from the first serving as the feed to the second, and so on. For constant density the exit concentration or conversion can be solved by consecutive application of Eq. (4-6) to each reactor. MacDonald and Piret have derived solutions for a number of rate expressions and for systems of reversible, consecutive, and simultaneous reactions. Graphical procedures have also been developed. The kinds of calculations involved are illustrated for the simple case of a first-order reaction in Example 4-9. 

Mechanistic Studies. - The mechanism of the reaction of tetra-zole-activated phosphoramidites with alcohols has been studied. A series of diethyl azolyl phosphoramidites (85) was prepared from diethyl phosphorochloridite and fully characterized, and the same compounds shown to be formed from the phosphoramidite (86) and azole. The degree of formation of (85) from (86) increases with the acidity of the azole, and the proposed mechanism is a fast protonation of (86), followed by a slow, reversible formation of (85) and a fast reaction of (85) with alcohols. Another study was concerned with the influence of amine hydrochlorides on the rate of methanolysis of the phosphoramidites (87) or (88), or tris(diethylamino)phosphine.The chloride content was measured to be 10-20 mM in doubly distilled samples which explains that "uncatalysed alcoholysis is possible. Intensive purification, including treatment with butyllithium and distillation from sodium, brought the chloride content down to 0.1-1 mM. The methanolysis reaction, in methanol as the solvent, was found to be first-order in catalyst concentration. An aJb initio calculation on N- and P-protonated aminophosphine (89) gave similar proton affinities for N and P this contrasts with earlier MNDO calculations which had ff-protonated species as the most stable. The M-protonated compound had an electronic structure reminiscent of a phosphenium ion-ammonia complex. 

The net rate of the first reversible reaction can be given as rmt i = kl Ca Cb — k2 Cd Ce. The second reaction is in series with the first and we find it has kinetics that are given by r = k3 Cd Ce. It is irreversible. The third reaction, A to G, is parallel to that of the first reaction and it too is reversible. This reaction is second order in the forward direction and first order in the reverse direction. 

Let us begin simply Consider a series of first-order reactions as in fig. 5.1, which shows an unbranched chain of reversible reactions. We shall not be restricted to first-order reactions but can learn a lot from this example. Let there be an influx of ko molecules of Xi and an outflow of molecules of Xs per unit time. We assume that the reaction proceeds from left to right and hence the Gibbs free energy change for each step and for the overall reaction in that direction is negative. The mass action law for the kinetic equations, say that of X2, is... 

The problem of reactions that do not go to completion is a frequently occurring one. We have shown here only the mechanics of deahng with a reversible system in which the reaction in each direction is first-order. Other cases that might arise are reversible second-order reactions, series reactions in which only one step is reversible, etc. These cases are quite complicated mathematically, and their treatment is beyond the scope of this book. However, many such systems have been elegantly described (see, for example, Schmid and Sapunov, 1982). The interested reader is directed to these worked-out exercises in applied mathematics for more details. 

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