Irreversible step approximation

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 

It is important to keep in mind that, in general, the model cannot describe the approach towards equilibrium, since this would violate our assumption that the product concentration is negligible. We note that Eq. (166) would also describe the case in which the adsorption-desorption equilibrium lies on the desorption side, i.e. if the temperature is such that the molecule AB hardly adsorbs on the surface. 

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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... 

If we want to determine the limiting behavior of a kinetic model very far from equilibrium, the irreversible step approximation is the appropriate limit. 

If we have difficulties making sense of a complicated reaction mechanism, the irreversible step approximation may provide a simplification, which allows us to understand the mechanism well enough to choose a better approximation. 

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. 

More than one step may be irreversible. The approximation can then be applied to each of them, with further simplification of the rate equation. 

The approximations in this section can be combined in many different ways. Significant further simplification may result. For example, for a cycle 8.34 with X as the macs and irreversible step X, + B — X2, the only remaining row of the Christiansen matrix is the second, and that row has but one single element, A qAq, (see reduced matrix 8.45). In this case, the rate equation reduces to... 

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. 

Although systems containing completely irreversible steps are an idealization, reaction systems are very numerous that contain steps with a sufficiently large change in free energy so that they may be approximated quite accurately by irreversible steps. When a species A, is connected to other species by irreversible steps, its equilibrium amount Oj is zero. WTien the equilibrium amount ai of some species is equal to zero, the matrices and do not exist and are not available for transforming the rate constant matrix K into a symmetrical matrix (see Appendix I). In this situation, we have no assurance that n independent characteristic... 

Generally such one-step approximation follows if we cut the chain right after the irreversible step (i.e. at L) Then the quasi-equilibrium of the preceding steps demands the kinetics of an eflFective reaction A L. The result for the temperature dependence of the effective rate kes is... 

The discussion of the performance of gas turbine plants given in this chapter has developed through four steps reversible a/s cycle analysis irreversible a/s cycle analysis open circuit gas turbine plant analysis with approximations to real gas effects and open circuit gas turbine plant computations with real gas properties. The important conclusions are as follows ... 

The theoretical approach involved the derivation of a kinetic model based upon the chiral reaction mechanism proposed by Halpem (3), Brown (4) and Landis (3, 5). Major and minor manifolds were included in this reaction model. The minor manifold produces the desired enantiomer while the major manifold produces the undesired enantiomer. Since the EP in our synthesis was over 99%, the major manifold was neglected to reduce the complexity of the kinetic model. In addition, we made three modifications to the original Halpem-Brown-Landis mechanism. First, precatalyst is used instead of active catalyst in om synthesis. The conversion of precatalyst to the active catalyst is assumed to be irreversible, and a complete conversion of precatalyst to active catalyst is assumed in the kinetic model. Second, the coordination step is considered to be irreversible because the ratio of the forward to the reverse reaction rate constant is high (3). Third, the product release step is assumed to be significantly faster than the solvent insertion step hence, the product release step is not considered in our model. With these modifications the product formation rate was predicted by using the Bodenstein approximation. Three possible cases for reaction rate control were derived and experimental data were used for verification of the model. 

Assuming that the second step is irreversible and using the steady-state approximation, we can determine the coverage... 

Eflornithine (Vaniqa) is an irreversible inhibitor of ornithine decarboxylase, which catalyzes the rate-limiting step in the biosynthesis of polyamines. Polyamines are required for cell division and differentiation, and inhibition of ornithine decarboxylase affects the rate of hair growth. Topical eflornithine has been shown to be effective in reducing facial hair growth in approximately 30% of women when applied twice daily for 6 months of therapy. Hair growth was observed to return to pretreatment levels 8 weeks after discontinuation. Local adverse effects include stinging, burning, and folliculitis. 

The oxidation of guanine (G) and adenine (A) follows a two-step mechanism involving the total loss of four electrons and four protons showing current peaks at approximately 0.9 and 1.2 V, respectively. However, the redox properties are dependent on the pH, the ionic strength of the electrolyte, and the electrode material.2 The reader is referred to a recent review by Palecek and coworkers for a more comprehensive discussion regarding the electrochemical mechanism of the oxidation and reduction of DNA bases on carbon and mercury electrodes.3 4 Guanine oxidation is irreversible and occurs in two consecutive steps (Fig 10.1).5... 

This represents the case where the first step is rate determining. (If this situation is known beforehand, one need not work through the steady-state approximation but merely write down dP/dt = k A since the first step is rate limiting and irreversible.)... 

In order to illustrate the steps involved in the dissociation of NO, a series of experiments was performed in which the surface prepared with molecularly adsorbed NO was heated briefly at approximately 5K/s and allowed to cool. The maximum temperature was maintained for approximately one second. The surface cooled below 1A0 K during the recording of the vibrational spectrum, so that only irreversible changes in the adsorbed layer could be observed. The vibrational spectrum is shown as a function of temperature treatment in Figs. A and 5 for surfaces prepared with NO coverages (relative to saturation) of 0.3 and 0.8, respectively. 

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