Reaction mechanisms rate-limiting step approximation

Both the steady-state approximation and the rate-limiting step approximation require the use of the order = moleculaiity property of elementary reactions. For either of these tools to be useful, each reaction in the mechanism must be elementary and the reaction mechanism must be correct. Use of the screening criteria discussed in Section 5.1.3 oftbis chapter can avoid wasting time and energy deriving a rate equation for a proposed mechanism that contains nonelementary reactions. It is important to apply these screening criteria to each reaction in the presumed mechanism, before derivation of the rate equation is begun. 

Remarks The aim here was not the description of the mechanism of the real methanol synthesis, where CO2 may have a significant role. Here we created the simplest mechanistic scheme requiring only that it should represent the known laws of thermodynamics, kinetics in general, and mathematics in exact form without approximations. This was done for the purpose of testing our own skills in kinetic modeling and reactor design on an exact mathematical description of a reaction rate that does not even invoke the rate-limiting step assumption. 

The kinetics of the contributory rate processes could be described [995] by the contracting volume equation [eqn. (7), n = 3], sometimes preceded by an approximately linear region and values of E for isothermal reactions in air were 175, 133 and 143 kJ mole-1. It was concluded [995] that the rate-limiting step for decomposition in inert atmospheres is NH3 evolution while in oxidizing atmospheres it is the release of H20. A detailed discussion of the reaction mechanisms has been given [995]. Thermal analyses for the decomposition in air [991,996] revealed only the hexavanadate intermediate and values of E for the two steps detected were 180 and 163 kJ mole-1. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

Bond energy considerations indicate that the initiation reaction (4.2.2) should be quite slow because its activation energy must be quite high (at least equal to the bond dissociation energy). If one were dealing with an open sequence reaction mechanism, such a step would imply that the overall reaction rate would also be low because in these cases the overall reaction becomes approximately equal to that of the rate limiting step. In the case of a chain reaction, on the other hand, the overall reaction rate is usually much faster because the propagation steps occur many times for each time that an initiation step occurs. 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

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 most cases of protein HX studies, the exchange reaction is the rate-limiting step (called the EX2 mechanism as will be discussed later, see also Sections 1.3 and 3.2.2). Because the solvent deuterium concentration in an HX experiment is usually in large excess and remains approximately constant, the exchange reaction of each exchangeable hydrogen is a pseudo first-order reaction and follows... 

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. 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

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