Steady-state approximation with reaction mechanisms

Elementary unimolecular reactions have first-order rate laws elementary bimolecular reactions have second-order rate laws. A rate law is often derived from a proposed mechanism by imposing the steady-state approximation. A proposed mechanism must be consistent with the experimental rate law. 

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). 

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. 

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... 

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. 

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. 

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. 

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... 

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. 

The Lindemann mechanism for unimolecular reactions, discussed in Section B.2.2, provides a convenient vehicle for illustrating partial-equilibrium approximations and for comparing them with steady-state approximations, even though this mechanism is not a chain reaction. To use the partial-equilibrium approximation for the two-body production of SRJ, select for example, as the species whose concentration is to be determined by partial equilibrium and use... 

The basis of the mechanism of hydrocarbon oxidation has been discussed in Chapters 1 and 2 and clearly forms the kernel of the construction of a comprehensive chemical mechanism. Traditionally, detailed mechanisms have been constructed manually , with chemical experts examining the species likely to be present in the system, and assessing which reactions they are likely to undergo under the appropriate conditions. In the precomputer era, when the steady-state approximation formed a primary tool and analytic solutions were necessary, there was a need to limit the size of the mechanism, and this was achieved in an a priori manner with the expert selecting the principal reactions on the basis of experience. The... 

In Section 18.7 we wrote a mechanism in which silver ions catalyze the reaction of TD with Ce. Determine the rate law for this mechanism by making a steady-state approximation for the concentration of the reactive intermediate Ag"". 

The specificity of the reaction mechanism to the chemistry of the initiator, co-initiator and monomer as well as to the termination mechanism means that a totally general kinetic scheme as has been possible for free-radical addition polymerization is inappropriate. However, the general principles of the steady-state approximation to the reactive intermediate may still be applied (with some limitations) to obtain the rate of polymerization and the kinetic chain length for this living polymerization. Using a simplified set of reactions (Allcock and Lampe, 1981) for a system consisting of the initiator, I, and co-initiator, RX, added to the monomer, M, the following elementary reactions and their rates may be... 

The fundamental reaction mechanism for the free-radical oxidation of hydrocarbons has been used to relate the consumption of oxygen to the formation of oxidation products in polypropylene. A kinetic interpretation is based on the steady-state approximation equating the rates of the initiation and termination reactions. With this approach it is possible to derive mathematical equations describing the consumption of oxygen or the formation of specific oxidation products. To solve the equations it is necessary to determine the most likely route for initiation of oxidation. The initiation mechanism chosen is the bimolecular reaction of hydroperoxides, reaction (1 ) of Scheme 1.55, with a rate coefficient k. ... 

The most important difference between a living ionic polymerization which has no termination or transfer mechanism and free-radical or ionic processes that do have termination or chain transfer steps is that the distributions of the degrees of polymerization are quite different. The distribution function can be derived by a kinetic approach due to Flory [6], which is analogous to that used earlier for free-radical reactions (see Problem 6.44). However, in the present case with no chain termination the simplifying steady-state approximation cannot be used. 

In fact, quenching effects can be evaluated and linearized through classic Stem-Volmer plots. Rate constants responsible for dechlorination, decay of triplets, and quenching can be estimated according to a proposed mechanism. A Stern-Volmer analysis of photochemical kinetics postulates that a reaction mechanism involves a competition between unimolecular decay of pollutant in the excited state, D, and a bimolecular quenching reaction involving D and the quencher, Q (Turro N.J.. 1978). The kinetics are modeled with the steady-state approximation, where the excited intermediate is assumed to exist at a steady-state concentration ... 

If the number of catalytic species is three, then the reaction rate becomes more complicated, but still managable to derive using a steady-state approximation. However, this laborous excercise is not needed, as we can directly use the derivation presented for the three-step catalytic sequence with linear steps, i.e. equation (4.114). Examples of direct and indirect hydrogenation mechanisms when the reaction mechanism can be expressed by a cycle with three intermediates are presented in Figure 5.10 and Figure 5.11 respectively. 

Many chemical reactions involve very reactive intermediate species such as free radicals, which, as a result of their high reactivity, are consumed virtually as rapidly as they are formed and consequently exist at very low concentrations. The pseudo-steady-state approximation (PSSA) is a fundamental way of dealing with such reactive intermediates when deriving the overall rate of a chemical reaction mechanism. 

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