Bodenstein intermediate

Branching may also occur from an intermediate and Scheme 9.8b shows the simplest scheme where P and Q are formed irreversibly from a Bodenstein intermediate (I). Ki-netically, this mechanism is not distinguishable from that where branching occurs at the reactant. The product ratio [P]/[Q] is the ratio of the rate constants of the forward processes at the branch point. The reaction again shows first-order behaviour with respect to the reactant R, with the overall experimental pseudo-first-order rate constant dependent on all the microscopic rate constants. 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

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... 

Third, it is often useful to assume that the concentration of one or more of the intermediate species is not changing very rapidly with time (i.e., that one has a quasistationary state situation). This approximation is also known as the Bodenstein steady-state approximation for intermediates. It implies that the rates of production and consumption of intermediate species are nearly equal. This approximation is particularly good when the intermediates are highly reactive. 

This expression contains the concentrations of two intermediate free radicals, C102 and C103. These terms may be eliminated by using the Bodenstein steady-state approximation. 

It can be solved by the so-called Bodenstein or steady-state approximation. This approximation supposes that the concentration of the reactive intermediate, in this case MS, is always small and constant. For a catalyst, of which the concentration is always small compared to the substrate concentration, it means that the concentration of MS is small compared to the total M concentration. The rate of production of products for the scheme in Figure 3.1 is given by equation (3). Equation (4) expresses the steady state approximation the amounts of MS being formed and reacting are the same. Equation (5) gives [M] and [MS] in measurable quantities, namely the total amount of M (Mt) that we have added. If we don t add this term, the nominator of equation (6) will not contain the term of k and the approximations that follow cannot be carried out. 

Trautz424 argued that there could be no true three-body reactions because of the improbability of a three-body collision, and he considered both (NO)2 and N03 as possible intermediates. Bodenstein at first rejected the idea of intermediates as being artificial, particularly because they required postulating unknown compounds. He argued that if such an intermediate formed it must be so unstable that there would be little difference between it and an NO molecule in a collision of finite duration with oxygen. Later,45 he accepted the idea of (NO)2 as the likely intermediate. In the case of either mechanism... 

The original more formal proof (31) of (44) consisted in the demonstration of its equivalence to the steady-state condition in the Bodenstein s form i.e., the condition that the rates of formation of intermediates are equal to zero. 

The conditions of steady state in the form of Bodenstein are obtained having r(Xj) = 0 in (46). In contradistinction to (42), these conditions may be called the intermediates steady-state conditions. They define only a part of the unknowns, viz., the concentration of intermediates, Xj, but there is usually no difficulty in the determination of the reaction rate from these. 

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. 

Intermediate B in Scheme 11.15 can be observed in principle but, if it undergoes rapid decay, it is usually expressed in concentrations very small compared with those of reactants and products, and it may be too dilute to be observed by instrumental techniques it is then usually called a transient intermediate. Under these conditions, the Bodenstein steady-state hypothesis applies and the rate equation for Scheme 11.15 can be solved to give Equation 11.10 (see Chapter 4) ... 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

As usual, this hypothesis is associated with the names of Bodenstein and Semenov. The latter introduced a concept of partial quasi-stationarity realized for some intermediates. Christiansen described the history of the problem as follows [36] "... the first who applied this theory was S. Chapman and half the year later Bodenstein referred to it in his paper devoted to hydrogen reaction with chlorine. His efforts to confirm his viewpoint were so energetic that this theory is quite naturally associated with his name . 

The concentration of an intermediate in a multistep reaction is always very low when it reacts faster than it is produced. If this concentration is set equal to zero in the derivation of the rate law, unreasonable results may be obtained. In such a case, one resorts to a different approximation. One sets the change of the concentration of this intermediate as a function of time equal to zero. This is equivalent to saying that the concentration of the intermediate during the reaction takes a value slightly different from zero. This value can be considered to be invariant with time, i.e., steady. Consequently this approximation is called Bodenstein s steady state approximation. 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

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) ... 

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. 

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... 

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. 

Another case in point is chain reactions (see Chapter 9). Asa rule, these are non-simple because their termination step involves two intermediates as reactants. Nevertheless, explicit rate equations can often be derived with the Bodenstein and long-chain approximations. The classical example is that of the hydrogen-bromide reaction (see Section 9.3). 

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

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 ... 

Rigorous rate equations for multistep catalytic reactions in terms of total amount of catalyst material are enormously cumbersome. Just the reduction to the level complexity of the Christiansen formula calls for the Bodenstein approximation of quasi-stationary behavior of the intermediates, requiring these to remain at trace concentrations, and that formula still entails a lot more algebra than does the general rate equation for noncatalytic simple pathways For a reaction with three intermediates, the Christiansen denominator contains sixteen terms instead of four for a reaction with six intermediates, forty-nine instead of seven Although the mathematics is simple and easy to program for modeling purposes and, usually, some... 

The concentrations of the members of the cycles can be found with a relationship expressing the concentration of an intermediate in a pathway as a function of those of the end members. Use of the Bodenstein approximation for any intermediate X in a pathway Xj — ... — Xk (reduced to two pseudo-single steps) yields... 

Software for direct, "brute-force" solution of the rate equations is available [4-9] and can be used if the network consists of only a few elementary steps. In practice, however, effective fundamental modeling usually calls for a reduction in the number of simultaneous rate equations and their coefficients. As Chapter 6 has shown, a systematic application of the Bodenstein approximation to all trace-level intermediates can achieve this, at least unless the network is largely non-simple. 

The reduced models in Table 11.1 rely on the validity of the Bodenstein approximation for all intermediates except the aldehyde in hydroformylation, but are otherwise free of assumptions. In every case, equations that are as simple or even simpler have long been derived, but only with much more restrictive assumptions, most commonly that of a single rate-controlling step and quasi-equilibrium everywhere else. Of course, such equations should be used in preference if their assumptions can be substantiated. 

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