Two rate-determining steps

The real power of digital simulation techniques lies in their ability to predict current-potential-time relationships when the reactants or products of an electrode reaction participate in some intervening chemical reaction. These kinetic complications often result in a fairly difficult differential equation (when combined with the conditions for diffusion or convection encountered in electrochemical problems) that resists solution by ordinary means. Through simulation, however, the effect of any number of chemical steps may be predicted. In practice, it is best to limit these predictions to cases where the reactants and products participate in one or two rate-determining steps each independent step adds another dimensionless kinetics parameter that must be varied over the range of... 

This nudeation kinetic mechanism is based on the activation of reaction sites, followed by growth of the product nuclei (B4C, in this case) through chemical reaction. The global rate constant, k, describes either of these two rate determining steps for the reaction mechanism. The values of m corresponds to... 

Description of a Method Allowing the Evaluation of Two Rate-Determining Steps... 

A mode is known as mixed if two steps have their rate constants finite whereas those of the other steps are infinite. A mixed mode is thus characterized by two rate-determining steps. The resolution of such a system requires the solution of at least one difierential equation. We will see examples in the smdy of gas-solid reactions of oxidation of metals (see Chapter 15). In the majority of the cases, however, we will be satisfied with the pseudo-steady state mixed modes. 

Theorem.- In isothermal condition and with constant concentrations of the main components of the reaction, in a pseudo-steady state mixed mode for which the slowness law is apphcable, the concentrations of the intermediate species consumed and/or produced by the steps placed between the two rate-determining steps are stationary oidy if the two space functions of the two steps do not vary with time or if they are constantly equal to each other. 

Take a catalytic fraction of a mechanism such as the multiplying coefficients of the two rate-determining steps, which are identical. For exanple, the mechanism containing the following steps ... 

On the other hand, in the particular case where the two space functions of the two rate-determining steps keep equal values constantly, expression [7.58] is simplified as ... 

The first considers pseudo-steady state modes not with two rate-determining steps but with three or more. It is shown in this case that we can generalize the slowness theorem and extend the sum of slownesses to all pure modes (provided that all the multiplying coefficients of the selected steps are equal). 

However, we can obtain the separable rate only when there exists a rate determining step or in a pseudo-steady state mixed model in which the space functions of the two rate determining steps are constantly equal with one other (section 7.10). 

When we examine the results of the two preceding cases, we note that the reactivities obtained by equations [12.38] and [12.42] are identical. In addition, the two rate-determining steps selected in each of the two cases are exactly the reverse of each other. Although they are located in zones with different radii of curvatures, it is difficult to make the assumption that one of them is constantly at equilibrium, whereas the other proceeds at slow speed and is thus a rate-determining step, and vice versa. Indeed, the rate constants of these steps for the infinite radii of curvatures are same and thus, inevitably, of the same order of infinitude. This result is checked each time that two steps of a mechanism are exactly the reverse of each other. 

We saw that the pure modes with a single rate determining step, if they made it possible to explain a certain number of laws, did not cover all of them, in particular the paralinear law and the complete parabolic law (see section 15.2.1.2). From this the idea to complex the solutions utilizing mixed modes in which the kinetics is governed by two rate determining steps, others being constantly at equilibrium. 

The two rate determining steps are the internal and external interface reactions. We will consider again the reasoning of section 7.8. 

A pseudo-steady state mode is said to have two rate-determining steps or be in the mixed pseudo-steady state if it meets the following assumptions ... 

We cannot take a determining step in a loop and one outside the loop, but it is possible to have a loop between the two rate-determining steps, like in the scheme below. 

Some of the state modes involving two rate-determining steps (mixed) do not eompel psendo-steadiness on all of the intermediate species. This is especially the case with mixed state modes that are not psendo-steady on an intermediate. We will examine this case in order to develop the method and show that the pseudo-steady state mode appears to be the limit when time tends to infinity. Om goal is not to estabUsh general relations but to explain the procedure to be adopted. 

Two rate-determining steps occur in the iterative SCF procedure. The first is the formation of the Fock matrix, and the second is the solution of the pseudo-eigenvalue problem. The latter step is conventionally done as a diagonalization to solve the generalized eigenvalue problem (Eq. [7]), and thus, the computational effort of conventional SCF scales cubically with system size [0(M )j. 

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