Reactivity space function

Rate-determining steps Reactivity Space function... 

Layer thickness Reactivity Space function for plates Kinetic laws for plates... 

Table 15.5. Reactivities, space functions, and kinetic laws for plates in the various...

Figure 19. Relation between and for a bound-state system. The functions in the single space (a) can be mapped onto the double space (b) where they have opposite symmetries under R2n, and belong to different symmetry blocks of the double-space Hamiltonian matrix (c). Unhke reactive wave functions, bound-state functions cannot be unwound from around the Cl.

We haveemployed a variety of unsupervised and supervised pattern recognition methods such as principal component analysis, cluster analysis, k-nearest neighbour method, linear discriminant analysis, and logistic regression analysis, to study such reactivity spaces. We have published a more detailed description of these investigations. As a result of this, functions could be developed that use the values of the chemical effects calculated by the methods mentioned in this paper. These functions allow the calculation of the reactivity of each individual bond of a molecule. 

In this chapter, we will examine how, starting from a mechanism of a process described in elementary steps, to solve the process model and obtain the rate laws of a process according to time and the various physicochemical variables (temperature, partial pressures, or concentrations), specifying the assumptions that make it possible to obtain analytical solutions. We will introduce the concepts of separable rate, reactivity, and space function that simplify modeling. 

Theorem.- For a reaction proceeding in only a single zone of diffusion, the reactance at any time is put in the form of the pixKluct of two functions reactivity, a function of only the intensive properties at this time, and the space function, a function of only the shapes and sizes, that is, only morphology of the zone at the considered time. 

If sizes of the zone vary with time, then the space function is a function of time. In addition, the reactivity can be or caimot be a function of time. If the reactivity is independent of time (this is the case when all intensive properties are kept constant), the variation of the reactance with time is due to the space function oidy, and we say that the rate is separable. 

Theorem.- In a pure mode, the rate of the reaction is the product, balanced by the reverse of the multiplying coefficient, of the reactivity of the rate-determining step (other steps being at equihbrium) and the space function relating to the zone where the rate-determining step proceeds. 

So, at constant physicochemical conditions, the solution is put in the form of a product of reactivity, independent of time, and a space function that can... 

In this case specifically, we can then speak of the reactivity and of the space function of the general reactiom... 

Thus, we realize that the rate of a simple linear mechanism in pure mode far from equilibrium, with constant space function, and thus, the reactivity follow the law of Arrhenius with an apparent energy of activation, which is the sum of the energy of activation of the rate-determining step, and of the balanced enthalpies of the steps that precede the rate-determining one ... 

Looking at these equations, we note that even in a psendo-steady state mode with constant intensive parameters (pressures, concentrations, temperatures) during the experiment (which makes the reactivities in pure modes independent of time), the reactivities of the steps of the mixed mode are not independent of time, if, at least, one of the two space functions varies with time. Thus, although the expression rate can be written in the form

[Pg.240]

We note that in the general case of a pseudo-steady state mixed mode, whose rate has the form of relation [7.58], we cannot define any more a reactivity and one space function for the reaction that would make the reactance separable. 

Remark.- We can also check that, if the two space functions are constantly equal to each other, the reactivities of the rate-determining steps are independent of time. For example, applying relation [7.59], for the reactivity of step i for the pseudo-steady state mixed mode i, j we obtain ... 

This reactance is not separable and has the form iR = q>E only if all the space functions Ep are equal to each other at any time. In this case, the reactivity follows a generalized slowness law ... 

Now if we assume that all the steps from i to j have a pseudo-steady state behavior, then according to the slowness theorem and taking into account the common space function, we can write the reactivity of the subset as ... 

With these reactions, if the experiments are carried out at constant intensive variables, the reactivity does not depend on time and possibly only the space function can depend on time. 

In this case the rate is the product of the reactivity of the rate-determining step by its space function, divided by its multiplying coefficient the rate has the form [7.43] ... 

A linear mechanism with a general pseudo-steady state mode if the space functions are equal to each other at any time and with all equal multiplying coefficients. The total reactivity is given by equation [7,63] it is independent of time at constant temperature, pressure, and concentration and the reactance is separable. 

In the case of a separable rate, the first kind of changes of laws relates the reactivity and can produce a change of space function. 

In the reactions with separable rate, the second kind of changes of laws relate to the space function, there is no change of mechanism nor of mode, thus, the reactivity is not affected. 

The third kind of changes of laws result from a change of mechanism and thus, relate at the same time to the reactivity and the space function, if the rate is separable for both mechanisms. 

As we have just observed, in all the cases, two phenomena intervening during nucleation, that is, the creation of the precursors and their condensation, occur in zones that keep their dimensions constantly equal to each other, so we can define the reactivity (surface or volume according to whether nucleation occurs on the surface or the bulk) and the space function Consequently, for nucleation on the surface, if... 

In all the pseudo-steady state modes of nucleation, as the space function always has the same value for the whole of the steps, the theorem of the equahty of the speeds (see section 7.4.2) can be applied to the reactivities and thus... 

To find the form of the solution of a pure mode, we assume that the rate constants k and k are infinite except if p = i. As in nucleation, the various steps proceed all in the same zone or zones having same sizes (same space function), the pseudo-steady state conditions, and as the multiplying coefficients all are identical, we can thus have the equality of the reactivities... 

We expect to find the form of the solution of pure mode such that the speed constants k and k" are infinite except p = I. The quasi-steady state mode being considered because the system is closed with regard to the potential nuclei [SOU 90, p. 286], the space functions of the various steps are all identical, as well as the multiplying coefficients, we can thus have roughly the equality of the reactivities... 

The modeling of the space function of growth can develop in a generic way by considering the reactivity of growth, specific to a particular reaction under well-defined conditions, such as a parameter likely to vary with the intensive constraints (partial pressures, concentrations, temperature) and independent of time insofar as these constraints are maintained constant. 

To derive the space function, we assume isothermal and isobaric conditions so that the reactivity of growth remains constant. We will successively study both isotropic and radial anisotropic models of growth. 

We will indicate by the reactivity of growth and by the space function for a grain /, with... 

Important remark.- In all the cases (see tables of Appendix 3) of this kind of relationship between the rate and the fractional extent, we note that the space function depends only on the fractional extent and thus does not depend on the reactivity of growth, whatever is the past of the sample, and the space function is completely determined by the fractional extent. The past of the sample has no influence on speed. Consequently, the variations of the rate with an intensive variable will give (with a coefficient which will be the space function) those of the rate of growth on the condition of studying these variations with constant space function, that is, with constant fractional extent. 

We assume that the reactivity of growth and the specific frequency of nucleation are independent of time (pseudo-steady state modes at constant tenperature and partial pressures). We will thus refer to relations [10.16] and [10.18], but in this case, a nucleus corresponds to a grain we can thus reveal in these expressions the space function of growth of a grain. 

Remark.- This chapter is based on the assumptions of pseudo-steady state and separable rate. If these assumptions are not checked, in particular the second one for the growth, we no longer define the reactivity of growth and the space function, chemistry and morphology are narrowly frays, and some solutions are discussed only in some simple cases of massive plate samples, and for pseudo-steady state mixed modes. We will discuss such an approach in Chapter 15. 

Insofar as the rate is separable, we can determine, directly by the experiment, with an unknown constant multiplicative factor E, the variations of the reactivity of growth with an intensive variable. For this, it is sufficient to measure the variations of the rate with this variable at a constant space function. Two methods are... 

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