Trace-level intermediates

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

Other means are available to study highly reactive intermediates that may be present at trace and ultra-trace levels. One may attempt to divert the intermediate from its normal course. We call this a trapping or scavenging experiment. If T symbolizes a species that rapidly reacts with intermediate I, one possible scheme is then... 

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... 

In treating parallel reaction, two concepts are often used (i) the concept of rate-determining path, in which the fastest path is the rate-determining path, and (ii) the concept of steady state, also called the concept of quasi-stationary states of trace-level intermediates. 

Fig. 6. Same as Fig. 5 intermediate noise level. Traces 1, 2, and 3 are for identical constraints and noise level. 

Pentachloroethane was produced commercially as a chemical intermediate, and occupational exposure may have occurred. Trace levels have been reported in ambient air and water (lARC, 1986). 

The title of the book refers to multistep reactions, defined as all kinds of reactions that involve more than a single molecular event such as rearrangement or break-up of a molecule or transformation resulting from a collision of molecules. Some standard texts speak instead of complex reactions and multiple reactions, depending on whether or not the mechanism involves trace-level intermediates. The term multistep reactions comprises both these categories. 

Homogeneous reactions occur within one phase, here taken to be fluid. Included are reactions in which a reactant is supplied from another phase by mass transfer. Multistep reactions consist of a combination of elementary steps. No distinction is made between complex reactions (with trace-level intermediates) and multiple reactions (with intermediates at higher than trace concentrations). 

If a reaction intermediate X is so unstable that it decomposes practically as soon as it is formed, its concentration necessarily remains quite small. The same must be true for its net rate of formation rx If that rate were large and positive, the concentration would rise to large values, which it is known not to do if that rate were large and negative, the concentration would have to become negative, which it cannot. Accordingly, provided the intermediate is and remains at trace level, its net formation rate rx is small compared separately with its rates of formation and decay ... 

The net rate of formation of an intermediate that is and remains at trace level is negligible compared with its contributing formation and decay rates. 

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. 

The three principal tools for reduction of mathematical complexity of rate equations are the concepts of a rate-controlling step, of quasi-equilibrium steps, and of quasi-stationary states of trace-level intermediates. 

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 notation Xj will remain reserved for trace-level intermediates, with sequential indices. In indices of concentrations, rates, rate coefficients, etc., X is suppressed for simplicity. For instance, the concentration of Xj is given as Cj the formation rate of Xj, as r, and the rate coefficient of a step Xj— Xk, as fcJk. For consistency in indices of sums and products, rate coefficients of steps A — X, are given as kQl, those of steps Xk, — P as fck l k (P = end product). Intermediates that are not, or do not remain, at trace level are designated K, L, etc. 

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 ... 

As a rule, a reduction to a single, explicit rate equation (plus algebraic equations for stoichiometric constraints and yield ratios) is not achieved. Rather, the equations for the end members of the piecewise simple network portions must be solved simultaneously. Nevertheless, The concentrations of all trace-level intermediates that do not react with one another have been eliminated by this procedure and, in many cases of practical interest, the reduction in the number of simultaneous rate equations and their coefficients is substantial. 

The concept of "simplicity" of a pathway or network is introduced. For a pathway to be "simple," all its intermediates must be and remain at trace level, and no step may involve two or more molecules of intermediates as reactants. The first condition ensures... 

Many reactions of practical interest have non-simple pathways or networks, i.e., the concentration of an intermediate rises above trace level or a (forward or reverse) step involves two or more molecules of intermediates as reactants. If the majority of the steps meet the simplicity conditions, a significant reduction in mathematical complexity can still be achieved by cutting the network into piecewise simple portions at the offending steps. In some other instances, the quasi-equilibrium or long-chain approximations can be invoked in order to obtain explicit rate equations although the network is non-simple. On the other hand, if a majority of steps in a network are non-simple, the tools described here are of little use. 

The rules deduced in this subsection are exclusively for simple pathways. A pathway or network is "simple" if all its intermediates are and remain at trace level and if no step involves two or more molecules of intermediates as reactants (see definition in Section 6.1). 

Non-simplicity is caused by intermediates whose concentrations rise above trace level, or by steps in which two or more molecules of intermediates function as reactants. Non-simplicity caused by the first of these possibilities usually becomes apparent immediately, when the known participants in a reaction are sorted into reactants, products, intermediates, and possibly catalysts and silent partners. Where this is not so, say, because the number of participants is very large—not uncommon in hydrocarbon processing and combustion—, Delplot rank ordering can help to distinguish intermediates from end products (see Section 7.1.2). Nonsimplicity caused by reactions of trace intermediates with one another may not be apparent at the outset, only to turn up as the mechanism becomes clearer. If so, the kineticist will have to cross that bridge when he comes to it. 

In practice, many reaction systems involve non-trace intermediates, but no obvious non-simple reactions of intermediates. A good strategy in such situations is to cut the overall reaction network into portions at the non-trace intermediate or intermediates (see Section 6.5), then reduce the portions as described for simple networks in Section 6.4.1. Network reduction makes it unnecessary to keep track of trace intermediates (except those reacting in a non-simple manner) and so obviates much of the hard work Trace intermediates are the more troublesome ones in network elucidation because they are difficult or impossible to detect, identify, analyze for, or synthesize, tasks that usually do not pose problems with intermediates that rise above trace level. Often, the network portions will turn out to be "piecewise simple" (see Section 6.5). If not, further cutting at additional nonsimple steps is called for when these become apparent. 

Synthesis of intermediates. An excellent technique for confirming or refuting a postulated pathway is to synthesize intermediates and use them as starting materials. Often, a key intermediate that is reactive enough to remain at trace level under reaction conditions is stable at very low temperatures (e.g., that of liquid nitrogen) and can be synthesized. If the reaction starting with the postulated intermediate yields the same products in the same ratios, this can be taken as evidence in favor of the presumed pathway. For example, the essential features of the Heck-Breslow mechanism of hydroformylation (see Example 6.2 in Section 6.3) with cobalt hydrocarbonyl catalysts have been verified in this way by synthesis and use of the alkyl-and acyl-cobalt species [42]. 

A further complication arises if a significant fraction of the total catalyst material may be present in the form of reaction intermediates rather than as the free catalyst. If the catalyst is highly active, a minute amount suffices to produce a high reaction rate, and even a trace-level intermediate may then contain a large fraction or possibly most of the catalyst material. Such behavior is typical for enzyme catalysis, but not confined to it. In such cases, the concentration of free catalyst may vary with conversion, may not be known, and may be very difficult to measure. Rather, what is known is the total amount of catalyst material added, and rate equations in terms of the latter are therefore needed. Such systems will be discussed in the later sections of this chapter. 

The assumption that the second step is rate-controlling follows tradition, but is needlessly restrictive. Granted that the intermediates remain at trace level, as they almost certainly do, the observed kinetic behavior results even if the first step is not close to equilibrium. For the cycle 8.4 with irreversible second step, as assumed, the general formula 6.4 to 6.6 gives... 

As has already been pointed out, any rate equation containing the concentration of the free catalyst is of little practical use if that concentration is unknown, is difficult or impossible to measure, and may vary with conversion, as is the case if a significant fraction of the total catalyst material is present in the form of intermediates of the reaction. This is often true in catalysis by enzymes or other trace-level catalysts. To be sure, the equations in terms of free-catalyst concentration remain correct. However, unless practically all the catalyst material is present as free catalyst, they no longer reflect the actual reaction orders. This is because the concentrations of the participants affect the rate not only directly as expressed explicitly in those equations, but also indirectly and implicitly through their effect on the free-catalyst concentration As the reactant concentration decreases, so do those of the intermediates in turn, this produces an increase in the free-catalyst concentration that boost the rate and, thereby, decreases the apparent reaction order. To reflect this facet correctly, what is needed are rate equations in terms of the total amount of catalyst material, a quantity that is constant and known. 

Equation 8.23 is the most general rate equation for a trace-level catalyst cycle A <— P with one intermediate. It reduces to the Briggs-Haldane equation 8.21 if fcPX - 0 or CP = 0, that is, if the second step is irreversible or only the initial rate is considered. It reduces further to the Michaelis-Menten equation 8.18 if, in addition, kxr A xa, that is, if the first step is in quasi-equilibrium. 

As this chapter has shown, rate equations of multistep homogeneous catalysis are still relatively simple if the catalyst-containing intermediates are at trace level, but the free catalyst is not. In heterogeneous catalysis this corresponds to an almost entirely unoccupied catalyst surface. Since adsorption is prerequisite for reaction, low surface coverage results in low rates and therefore is of practical interest only in exceptional situations. Heterogeneous catalysis cannot avoid dealing with substantially covered... 

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