Reaction mechanisms steady-state approximation

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

This also accounts for the production of the small amount of butane. If the reaction mechanism were steps 1, 2, 3, 4, 5a, and 5b, then applying the steady state approximations would give the overall order of reaction as 1/2. 

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. 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

The rate law of an elementary reaction is written from the equation for the reaction. A rate law is often derived from a proposed mechanism by imposing the steady-state approximation or assuming that there is a pre-equilibrium. To be plausible, a mechanism must be consistent with the experimental rate law. 

Historically, the steady state approximation has played an important role in unraveling mechanisms of apparently simple reactions such as H2 + CI2 = 2HC1, which involve radicals and chain mechanisms. We discuss here the formation of NO from N2 and O2, responsible for NO formation in the engines of cars. In Chapter 10 we will describe how NO is removed catalytically from automotive exhausts. 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

ILLUSTRATION 4.3 USE OF THE BODENSTEIN STEADY-STATE APPROXIMATION TO DERIVE A RATE EXPRESSION FROM A CHAIN REACTION MECHANISM... 

A standard kinetic analysis of the mechanism 4a-4e using the steady state approximation yields a rate equation consistent with the experimental observations. Thus since equations 4a to 4e form a catalytic cycle their reaction rates must be equal for the catalytic system to be balanced. The rate of H2 production... 

The kinetic analysis of the mechanism 6a-6e,2 is more complicated than that of the mechanism 4a-4e because of the external reaction 6e but nevertheless is feasible using the steady state approximation. By a procedure similar to the derivation of equation 5 the following equation can be derived ... 

Two-Electron Catalytic Reactions In a number of circumstances, the intermediate C formed upon transformation of the transient species B is easily reduced (for a reductive process, and vice versa for an oxidative process) by the active form of the mediator, Q. This mechanism is the exact counterpart of the ECE mechanism (Section 2.2.2) changing electron transfers at the electrode into homogeneous electron transfers from Q, as depicted in Scheme 2.9. In most practical circumstances both intermediates B and C obey the steady-state approximation. It follows that the current is equal to what it would be for the corresponding EC mechanism with a... 

Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

Traditionally, reaction mechanisms of the kind above have been analysed based on the steady-state approximation. The differential equations for this mechanism cannot be integrated analytically. Numerical integration was not readily available and thus approximations were the only options available to the researcher. The concentrations of the catalyst and of the intermediate, activated complex B are always only very low and even more so their derivatives [Cat] and [B]. In the steady-state approach these two derivatives are set to 0. 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

As indicated in Scheme 3.7, the first step of an ElcB mechanism can be reversible and therefore deprotonation at the 3-carbon does not always lead to product formation. By applying a steady-state approximation to the carbanion concentration, the following rate law is obtained for an ElcB reaction ... 

This type of reaction for which the rate equation can be written according to the stoichiometry is called an elementary reaction. Rate equations for such cases can easily be derived. Many reactions, however, are non-elementary, and consist of a series of elementary reactions. In such cases, we must assume all possible combinations of elementary reactions in order to determine one mechanism that is consistent with the experimental kinetic data. Usually, we can measure only the concentrations ofthe initial reactants and final products, since measurements of the concentrations of intermediate reactions in series are difficult. Thus, rate equations can be derived under assumptions that rates of change in the concentrations of those intermediates are approximately zero (steady-state approximation). An example of such treatment applied to an enzymatic reaction is shown in Section 3.2.2. 

In this section, the different behavior of processes with coupled noncatalytic homogeneous reactions (CE and EC mechanisms) is discussed in comparison with a catalytic process. We will consider that the chemical kinetics is fast enough and in the case of CE and EC mechanisms K (- c /cf) fulfills K 1 so that the kinetic steady-state and even diffusive-kinetic steady-state approximation can be applied. 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate complexes derives reaction rate expressions for enzymatic reactions. A possible mechanism of a closed sequence reaction is ... 

In dealing with complicated reaction mechanisms, a simplification can often be introduced that when the reaction has reached some kind of steady state (akin to an equilibrium, except that further reactions are possible beyond this equilibrium hence the term steady-state approximation (SSA) is used. Mathematically, after the reaction has started, some intermediate product B has the condition d [B]/dt = 0. This is best illustrated by an example. 

It is perhaps easiest to explain the pseudo-steady-state approximation by way of an example. Consider the simple reaction A — B + C, whose elementary steps consist of the activation of A by collision with a background molecule M (in the atmosphere M is typically N2 and 02) to produce an energetic A molecule denoted by A, followed by the decomposition of A to give B and C. Thus, we write the mechanism as... 

In the Lindemann mechanism, a time lag exists between the energisation of A to A and the decomposition (or isomerisation) of A to products. During this time lag, A can be de-energised back to A. According to the steady-state approximation (s.s.a), whenever a reactive (i.e., short-lived) species is produced as an intermediate in a chemical reaction, its rate of formation is equal to its rate of decomposition. Here, the energised species A is short-lived. Its rate of formation = kJAp and its rate of decomposition k t [A][A ] + k2[A ]. Thus,... 

Appropriate expressions for the fluxes of each of the reactions in the system must be determined. Typically, biochemical reactions proceed through multiple-step catalytic mechanisms, as described in Chapter 4, and simulations are based on the quasi-steady state approximations for the fluxes through enzyme-catalyzed reactions. (See Section 3.1.3.2 and Chapter 4 for treatments on the kinetics of enzyme catalyzed reactions.)... 

The same considerations made before are valid for this case and it is very important to have an available validated reaction mechanism. It can be obtained from three main sources (Blelski et al., 1985 Buxton et al., 1988 Stefan and Bolton, 1998) and it is shown in Table 5. With the available information about the constant k2, k, k, fcg, and k-j, it could be possible to solve a system of four differential equations and extract from the experimental data, the missing constants 4> and k (that in real terms is k /Co2)-This method would provide good information about the kinetic constants, but it is not the best result for studying temperature effects if the same information is not available for the pre-exponential factors and the activation energies. Then, it is better to look for an analytical expression even if it is necessary to make some approximations. This is particularly true in this case, where the direct application of the micro steady-state approximation (MSSA) is more difficult due to the existence of a recombination step that includes the two free radicals formed in the reaction. From the available information, it is possible to know that to calculate the pseudo-steady-state... 

The rates of the overall reactions can be related to the rate law expressions of the individual steps by using the steady state approximation. However simple kinetic data alone may not distinguish a mechanism where, for example, a metal and an olefin form a small amount of complex at equilibrium that then goes on to react, from one in which the initial complex undergoes dissociation of a ligand and then reacts with the olefin. As a reaction scheme becomes more complex such steady state approximations become more complicated, but numerical methods are now available which can simulate these even for complex mixtures of reactants. 

The classical example of a complex straight-chain reaction for which the results of the steady-state approximation agree with experimental measurements is the hydrogen-bromine reaction H2 T Br2 2HBr [5]. The inferred mechanism is... 

The partial-equilibrium approximation differs from the steady-state approximation in that it refers to a particular reaction instead of to a particular species. The mechanism must include the forward and backward steps of any reaction that maintains partial equilibrium, and the approximation for a reaction k is then expressed by setting = 0 in equation (11). It is not always proper to conclude from this that when equations (6), (10), and (11) are employed in equation (14), the terms may be set equal to zero for each k that maintains partial equilibrium partial equilibria occur when the forward and backward rates are both large, and a small fractional difference of these two large quantities may contribute significantly to dcjdt. The criterion for validity of the approximation is that be small compared with the forward or backward rate. 

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