Bodenstein approximation

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

The theoretical approach involved the derivation of a kinetic model based upon the chiral reaction mechanism proposed by Halpem (3), Brown (4) and Landis (3, 5). Major and minor manifolds were included in this reaction model. The minor manifold produces the desired enantiomer while the major manifold produces the undesired enantiomer. Since the EP in our synthesis was over 99%, the major manifold was neglected to reduce the complexity of the kinetic model. In addition, we made three modifications to the original Halpem-Brown-Landis mechanism. First, precatalyst is used instead of active catalyst in om synthesis. The conversion of precatalyst to the active catalyst is assumed to be irreversible, and a complete conversion of precatalyst to active catalyst is assumed in the kinetic model. Second, the coordination step is considered to be irreversible because the ratio of the forward to the reverse reaction rate constant is high (3). Third, the product release step is assumed to be significantly faster than the solvent insertion step hence, the product release step is not considered in our model. With these modifications the product formation rate was predicted by using the Bodenstein approximation. Three possible cases for reaction rate control were derived and experimental data were used for verification of the model. 

The rate expression for each intermediate in Figure 3.2C can be derived based on the Bodenstein approximation of quasi-stationaiy states of trace-level... 

The rate equation is presented in equation (7). It can also be derived from v = k2[MS][H2] when we substitute for [MS] the equilibrium fraction of the catalyst that occurs as MS. Note that we do not fulfil the Bodenstein approximation, but our general formula (6) is still valid. If the free eneigy of... 

The Bodenstein approximation recognises that, after a short initial period in the reaction, the rate of destruction of a low concentration intermediate approximates its rate of formation, with the approximation improving as the maximum concentration of intermediate decreases (see Chapters 3 and 4). Equating rates of formation and destruction of a non-accumulating intermediate allows its concentration to be written in terms of concentrations of observable species and rate constants for the elementary steps involved in its production and destruction. This simplifies the kinetic expressions for mechanisms involving them, and Scheme 9.3 shows the situation for sequential first-order reactions. The set of differential equations... 

Scheme 9.3 Application of the Bodenstein approximation to an intermediate in sequential first-order reactions.

For each kinetic scheme in Scheme 9.4, the rate law obtained by applying the Bodenstein approximation to the intermediate (I) is presented and, for this discussion, we consider that the reactant R is the component whose concentration can be easily monitored. The reactions are all expected to be first order in [R], but the first-order rate constants show complex dependences on [X] and, in two cases, also on [Y]. All the rate laws contain sums of terms in the denominator, and the compositions of the transition structures for formation and destruction of the intermediate are signalled by the form of the rate law when each term of the denominator is separately considered. This pattern is general and can be usefully applied in devising mechanisms compatible with experimentally determined rate laws even for much more complex situations. 

Included in Scheme 9.9 are expressions for the rates of formation of both P and Q, and for disappearance of reactant, derived by applying the Bodenstein approximation to the intermediate. Contributions to reactivity from the pathways not involving the intermediate (dashed lines) are gathered in the starred term of each of these relationships. A number of cases may be recognised. 

The condition expressed by the Bodenstein approximation rx = 0 is often misleadingly called a steady state. It is not. It is not a time-independent state, only a state in which a specific variation with time (or reactor space time) is small compared with the others. In fact, some older textbooks applied what they called the steady-state approximation to batch reactions in order to derive the time dependence of the concentrations, unwittingly leading the incorrect presumption of a steady state ad absurdum. And a continuous stirred-tank or tubular reactor may, and usually does, come to a true steady state, even if the Bodenstein approximation is and remains inapplicable. [The approximation compares process rates r, it is irrelevant for its validity whether or not the reactor comes to a steady state, that is, whether the rates of change, dC /dr, become zero.]... 

The Bodenstein approximation can be applied repeatedly to different trace-level intermediates in succession. Each application removes one rate equation and the concentration of one trace-level intermediate. This makes the Bodenstein approximation especially useful because trace-level intermediates are difficult to detect and their concentrations can rarely be measured accurately. 

The Bodenstein approximation is accurate within reason, provided the intermediate is and remains at trace level, and with the exception of a very short initial time period in which the quasi-stationary state is established [13-15], It is left to the practitioner to decree how low a concentration must be to qualify as "trace " the more generous he is, the less accurate will be his results. For the pathway 4.20 with one trace-level intermediate, the error introduced can be estimated in the same way as for rate control by a slow step (see Section 4.1.1) ... 

The following points should be observed when the Bodenstein approximation is applied ... 

To illustrate the first two points above, the following example shows how the Bodenstein approximation can be applied repeatedly to a reaction that includes reversible steps. Note, however, that the same result can be obtained much more easily with a general formula for "simple" pathways, to be developed in Section 6.3. 

Example 4.4. Nitration of aromatics of intermediate reactivity. In Example 4.1 the concept of a rate-controlling step was used to obtain simple rate equations for nitration of aromatics of either low or high reactivity. For aromatics of intermediate reactivity, no single step is rate-controlling. However, if the concentrations of H2N03+, NOz+, and ArN02+ in the pathway 4.6 remain at trace level—this is a judgment call—the Bodenstein approximation can be applied repeatedly to obtain an explicit, closed-form rate equation. 

The rate equations 4.10 and 4.14, derived previously with the assumption of a single rate-controlling step, turn out to be special cases of eqn 4.31, obtained with the Bodenstein approximation If k23 is very small (third step slow), eqn 4.31 reduces to eqn 4.10 if kn and k2l are very small (second step slow), it reduces to eqn 4.14. This comparison demonstrates that the Bodenstein approximation leads to results of greater generality. 

Extensive examples of applications of the Bodenstein approximation, with or without the general formula for simple pathways, will be found in later chapters. 

Any highly reactive intermediate that is and remains at trace level attains a quasi-stationary state in which its net chemical rate is negligibly small compared separately with its formation and decay rates. This is the basis of the Bodenstein approximation, which allows the rate equation of the intermediate to be replaced by an algebraic equation for the concentration of the intermediate, an equation which can then be used to eliminate that concentration from the set of equations. The approximation can be applied in succession for each trace-level intermediate. It is the most powerful tool for reduction of complexity. It is the basis of general formulas to be introduced in Chapter 6 and widely used in subsequent chapters. 

If the intermediate K in the pathway 5.72 remains at trace level, the Bodenstein approximation can be used (see Section 4.3) and gives... 

The first of these conditions ensures that the Bodenstein approximation of quasi-stationary behavior (see Section 4.3) can be used for all intermediates, the second guarantees that the algebra is linear. If both conditions are met, explicit equations or algorithms for rates and yield ratios of all reactants and products can be given, regardless of the actual complexity of the network. 

Indices 0 and k in X coefficients are used for A and P, respectively, to avoid complications in later formulas with sums and products.) If the end members A and P additionally act as co-reactants and co-products, the respective rates rA or rP must be replaced by (l/ A)rA or (1 lnP)rP to account for the stoichiometry. After elimination of the concentrations of all the intermediates by repeated application of the Bodenstein approximation, the set of rate equations can be reduced to the single rate equation... 

In this fashion, the set of rate equations of any simple pathway (unless it is part of a network) can be reduced to a single rate equation and the algebraic equations expressing the stoichiometry. To illustrate how much work can be saved in this way, let us return to the Gillespie-Ingold mechanism of nitration of aromatics, for which a repeated application of the Bodenstein approximation provided a rate equation in Example 4.4 in Section 4.3. 

A necessary corollary of the Bodenstein approximation in a pathway is that the net rates of conversion are the same for all steps (an intermediate with higher formation than decay rate would not remain at trace level). In matrix form this condition is ... 

Derivation of equation 6.19 [8]. The Bodenstein approximation for the node intermediate Xk is... 

For reactions with non-simple pathways or networks, the formulas and procedures described so far are not valid. Any step involving two or more molecules of intermediates as reactants destroys the linearity of mathematics, and any intermediate that builds up to higher than trace concentrations makes the Bodenstein approximation inapplicable. Such non-simple reactions are quite common. Among them are some of the kinetically most interesting combustion reactions, detonations, periodic reactions, and reactions with chaotic behavior. However, a discussion of more than only the most primitive types of non-simple reactions is beyond the scope of this book. The reader interested in more than this is referred to another recent volume in this series [1], in which such problems are specifically addressed. 

Regardless of these complications, the Bodenstein approximation of quasi-stationary states of trace-level intermediates remains the principal tool for reduction... 

The Bodenstein approximation is not invalidated by this complication As long as the total amount of catalyst is extremely small compared with those of the reactants, as is largely true in enzyme kinetics, the catalytic cycle attains quasi-stationary conditions. (Exceptions are fast biological reactions of substances themselves at very low concentrations, a topic beyond the scope of this book.)... 

As first pointed out by Briggs and Haldane [30], the assumption of quasiequilibrium in the first step is inconveniently restrictive. They relaxed that postulate by replacing the quasi-equilibrium condition 8.15 with the Bodenstein approximation for the trace intermediate X ... 

As a rule, enzyme reactions are reversible. To account for the reverse reaction at significant conversion and so remove the restriction to initial rates, the reverse step P + cat — X in the cycle 8.14 must be included. Both the rate equation 8.17 and the Bodenstein approximation 8.20 then contain an additional term to account for this reverse step. The resulting rate equation is... 

Equation 8.23 derived in the present section differs by having the total instead of the free catalyst concentration in the numerator, as well as two additional terms in the denominator. The equivalence of the two equations can be shown as follows Replacement of Ccat in eqn 8.25 with use of eqn 8.16 and subsequent replacement of Cx with the Bodenstein approximation (eqn 8.20 with additional term kvxC Cr) yields eqn 8.23. Equation 8.25 is simpler and therefore preferable if one can be sure that all but an insignificant fraction of catalyst is in free form. Equation 8.23 is not subject to this restriction, an advantage one must pay for with the inconvenience of having to handle two more denominator terms. 

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