Steps quasi-equilibrium

We will list the elementary steps and decide which is rate-limiting and which are in quasi-equilibrium. For ammonia synthesis a consensus exists that the dissociation of N2 is the rate-limiting step, and we shall make this assumption here. With quasi-equilibrium steps the differential equation, together with equilibrium condition, leads to an expression for the coverage of species involved in terms of the partial pressures of reactants, equilibrium constants and the coverage of other intermediates. 

Surface Coverage and Reaction Rate. If precursor complex formation is fast relative to electron transfer and product release, it can be treated as a quasi-equilibrium step ... 

Catalyst Step 1 forward activation Step 3 quasi-equilibrium Step 4 quasi-equilibrium Relative error (%) Sticking coefficient... 

Figure 9. Visualisation of the rate determining step and quasi-equilibrium steps.

As for pathways of irreversible steps, the more general rule allowing for reversibility remains restricted to sequential steps, and rate control may shift to a different step with temperature or concentration. Each quasi-equilibrium step introduces an error into the approximation as will be discussed in the next section. 

The concept of the rate-controlling step singles out one step as much slower than all others. The concept of a quasi-equilibrium step does the opposite It singles out one or several steps as much faster than all others. If this is so, the slowness of the other steps gives the fast steps time enough to come essentially to equilibrium ... 

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. 

The three principal tools of reduction of complexity, discussed in Chapter 4, are the approximations of a rate-controlling step, of quasi-equilibrium steps, and of quasi-stationary behavior of intermediates. The Christiansen formula has already invoked the last of these three. The other two can be used for additional simplification. A further, new and very powerful tool is the concept of relative abundance of catalyst-containing species. Moreover, much can sometimes be gained if one or several steps can be taken as irreversible. To summarize ... 

Example 8.6. Rate for a hypothetical cycle with quasi-equilibrium steps. Assume that the second and third steps in the four-membered cycle 8.34 are in quasi-equilibrium and the fourth step is irreversible (k = 0) ... 

A catalyst may exist in two or more forms with different catalytic activities. In the simplest systems of this type, two such forms interconvert in a quasi-equilibrium step. The conversion may or may not involve other species. It may, for example, be a ligand exchange. The two catalyst species may catalyze the same reaction or different ones. A network of this type, with ligand exchange and different reactions A — P and B — Q, is... 

A general formula for single catalytic cycles with arbitrary number of members and arbitrary distribution of catalyst material has been derived by Christiansen. Unfortunately, the denominator of his rate equation for a cycle with k members contains k2 additive terms. Such a profusion makes it imperative to reduce complexity. If warranted, this can be done with the concept of relative abundance of catalyst-containing species or the approximations of a rate-controlling step, quasi-equilibrium steps, or irreversible steps, or combinations of these (the Bodenstein approximation of quasi-stationary states is already implicit in Christiansen s mathematics). In some fortunate instances, the rate equation reduces to a simple power law. 

Multireaction systems often have some quasi-equilibrium steps whose forward and reverse rates greatly exceed the net rate TZj at all conditions of interest. For such a reaction, the approximation... 

Possible Rate-Determining Steps (with Preceding Quasi-Equilibrium Steps Where Present) for the Oxidation of Propane and Platinum in Phosphoric Acid, 80°C-150°C and 0.30-5.0 V (Reversible Hydrogen Scale) (Work by G. Stoner)... 

Chemical reactions may involve large numbers of steps and participants and thus many simultaneous rate equations, all with their temperature-dependent coefficients. The full set of rate equations is easily compiled as shown in Section 2.4, and to obtain solutions by numerical computation poses no serious problems. With a large number of equations, however, it may become too much of a task to verify the proposed network and obtain values for all its coefficients. Therefore, every available tool must be brought to bear to reduce the bulk of mathematics, and that without unacceptable sacrifice in accuracy. The present chapter critically reviews the principal tools for such a purpose stoichiometric constraints and the concepts of a rate-controlling step, quasi-equilibrium steps, and quasi-stationary states. Other tools useful in catalysis, chain reactions, and polymerization will be discussed in the context of those reactions (see Sections 8.5.1, 9.3, 10.3, and 11.4.1). 

The approximations of a rate-controlling step, quasi-equilibrium steps, and long chains in chain reactions and the concept of relative abundance of catalyst-containing species in catalysis or of propagating centers in ionic polymerization can often be used for additional simplification (see Sections 4.2, 4.3, 8.5, 9.3, 10.3, and 11.4.1). A procedure suited in many cases consists essentially of the following steps [11] ... 

Eq. (6.36) can also be derived from the general equation for the three-step sequence. In the treatment above, the quasi-equilibrium steps were defined as the ratio of reactants to the ratio of products following the bio logic to have an analogy with the Michaelis Menten constant. In the Lineweaver-Burk coordinates, eq. (6.36) becomes... 

The reaction rate is given by the rate of the slowest (not quasi-equilibrium) step 2... 

The analysis is straightforward in cases where adsorbate interaction effects do not play a role. This is the case either under Langmuirian adsorption conditions or in the saturation limit, 6>o 1. The former is an unrealistic assumption, since it is known that Temkin conditions prevail, illustrated in the seminal work of Sepa et al. (1981). Under the latter assumption, one can proceed to determine values of the effective transfer coefficient in the Butler-Volmer equation. Each quasi-equilibrium step, represented by a factor K in vqrr, contributes an amount of ( — to... 

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