The Quasi-equilibrium Approximation

In this approximation we assume that one elementary step determines the rate while all other steps are sufficiently fast that they can be considered as being in quasi-equilibrium. If we take the surface reaction to AB (step 3, Eq. 134) as the rate-determining step (RDS), we may write the rate equations for steps (1), (2) and (4) as  

In essence, we have used the Langmuir isotherms for the adsorbing and desorbing species. By substituting the coverages into the rate expression for the ratedetermining step we obtain 

If we introduce the equilibrium constant for the overall reaction in the gas phase. 

The term in parenthesis, as we saw in Eq. (8), expresses the affinity of the reaction towards equilibrium. In marked contrast to Eq. (8), however, is the term 6i, which describes the fraction of free sites available for reaction. Thus even if the rate constants and the affinity towards equilibrium are high, the rate of the process may still be low if there are insufficient free sites, if the surface is blocked such that 6 0. 

The fraction of free sites in Eq. (155) is found from the principle of conservation of sites  

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In addition to [A ] being qiiasi-stationary the quasi-equilibrium, approximation assumes a virtually unperturbed equilibrium between activation and deactivation (equation (A3.4.125)) ... 

This is a further simplification of the quasi-equilibrium approximation, in which we simply neglect the reverse reaction of one or several steps. For instance, we may envisage a situation where the product concentration AB is kept so low that the reverse reaction in step (4) may be neglected. This greatly simplifies Eq. (161) since... 

The Most Abundant Reaction Intermediate (MARI) approximation is a further development of the quasi-equilibrium approximation. Often one of the intermediates adsorbs so strongly in comparison to the other participants that it completely dominates the surface. This intermediate is called the MARI. In this case Eq. (156) reduces to... 

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

CO oxidation, an important step in automotive exhaust catalysis, is relatively simple and has been the subject of numerous fundamental studies. The reaction is catalyzed by noble metals such as platinum, palladium, rhodium, iridium, and even by gold, provided the gold particles are very small. We will assume that the oxidation on such catalysts proceeds through a mechanism in which adsorbed CO, O and CO2 are equilibrated with the gas phase, i.e. that we can use the quasi-equilibrium approximation. 

In solving the kinetics of a catalytic reaction, what is the difference between the complete solution, the steady-state approximation, and the quasi-equilibrium approximation What is the MARI (most abundant reaction intermediate species) approximation ... 

Why does the quasi-equilibrium approximation fail in the low-pressure... 

If all steps except one are fast, we can use the quasi-equilibrium approximation For the fast steps we use the corresponding equilibrium equations instead of the kinetic equations. 

This approximation will in most cases provide a very significant simplification in particular for large reaction mechanisms. In the quasi-equilibrium approximation the transient behavior is eliminated. Further, the description of changes in rate-limiting step has been lost. 

In the irreversible step approximation, we neglect the forward or backward rate for one of the steps. For small mechanisms the irreversible step approximation may be used alone, for larger mechanisms it is usually combined with the quasi equilibrium approximation... 

One of the reaction steps is assumed rate limiting, this step has a forward rate constant. This model maps onto the quasi equilibrium approximation. 

The quasi-equilibrium approximation can be applied to more than one step. As the preceding section has shown, the postulate of a rate-controlling step implies that all others are in quasi-equilibrium. 

In a number of reactions of practical interest, the offending non-simple step is a reversible dissociation of a reactant or intermediate, as in Case V in Table 6.1. Often, such a step is fast compared with the others and thus is at quasi-equilibrium. If so, the quasi-equilibrium approximation (see Section 4.2) can greatly simplify mathematics, in some instances even lead to an explicit rate equation. This has been discussed in Section 5.6. 

This problem prompted a closer examination and ultimately a rederivation of the theory describing the link between mechanistic features in generalized sequential reaction schemes and the values of experimentally accessible transfer coefficients upon which the conclusions on mechanism were based. We endeavor here to develop this hnk, which is built upon the quasi-equilibrium approximation for dealing with the kinetics of multistep reactions, clearly and concisely, giving attention to the limits of its application. We hence justify its significance in relation to determination of the reaction mechanism. 

The quasi-equilibrium approximation relies on the assumption that there is a single rate-determining step, the forward and reverse rate constants of which are at least 100 times smaller than those of all other reaction steps in the kinetic scheme. It is then assumed that all steps other than the rds are always at equilibrium and hence the forward and reverse reaction rates of each non-rds step may be equated. This gives simple potential relations describing the varying activity of reaction intermediates in terms of the stable solution species (of known and potential-independent activity) that are the initial reactants or final products of the reaction. The variation of the activities of reaction intermediates is, however, restricted by either the hypothetical solubility limit of these species or, in the case of surface-confined reactions and adsorbed intermediates, the availability of surface sites. In both these cases, saturation or complete coverage conditions would result in a loss of the expected... 

An important point concerning the set of reactions 5.2 is worth noting. As already mentioned, it is generally assumed in such reactions that the net rate of generation of an intermediate can be set to zero. In another approach, if an equilibrium step is involved, we use the quasi-equilibrium approximation, in which the ratio of the forward to backward rates of a rapid elementary step is set equal to unity. In either case, it must be remembered that the fundamental... 

The quasi-equilibrium approximation is based on assumption (3) in Section 4.2.2. One or more steps of the overall reaction are considered to be at equilibrium if their kinetic parameters, both forward and reverse, are much larger than the kinetic parameters of other steps. An example is the reaction sequence... 

The Four Kinetic Regimes of Adsorption from Micellar Solutions In the theoretical model proposed in Refs. [149,150], the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers) is avoided. The theoretical problem is reduced to a system of four nonlinear differential equations. The model has been applied to the case of surfactant adsorption at a quiescent interface [150], that is, to the relaxation of surface tension and adsorption after a small initial perturbation. The perturbations in the basic parameters of the micellar solution are defined in the following way ... 

The Quasi-Equilibrium Approximation Enzymatic Reaction Kinetics... 

What condition would make this rate law coincide with that obtained with the quasi-equilibrium approximation ... 

For the casein which [A] is held fixed, we have onlyEq. (13.3-13b) andEq. (13.3-13c) to solve simultaneously. Since we expect an oscillatory solution, we cannot apply the quasi-equilibrium approximation or the quasi-steady-state approximation. The equations must be solved numerically, using standard methods of numerical analysis to obtain the time dependence of [X] and [Y]. The concentrations of X and Y must satisfy the expression... 

Hints The quasi-equilibrium approximation can be assumed for reactants adsorption giving... 

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