Formal kinetic equations

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

As was shown in Section 2.1, in some cases thermal fluctuations of reactant densities affect the reaction kinetics. However, the equations of the formal chemical kinetics are not suited well enough to describe these fluctuations in fact they are introduced ad hoc through the initial conditions to equations. The role of fluctuations and different methods for incorporating them into formal kinetics equations were discussed more than once. 

Surface component concentrations Xk are also described by a set of formal kinetic equations in the framework of the surface site formalism. In this formalism, a set of surface centers (sites) Tn is defined for each condensed phase ( n). Each site is characterized by a set of surface substances (/) that are present at this site (Figure 9.5). 

At the remaining surface, where reactive species are adsorbed, a competition takes place between adsorbed hydrogen and ethene. The formal kinetic equation in the form of a power law is then (approx.) rate = k pgr Ph2 where a is near to zero or slightly negative. Since adsorption of hydrogen is dissociative, one would expect p 2, if H atoms are added one by one. The reasons why a can be different are discussed elsewhere (see Section 5.2 on syngas reactions). The reactive form... 

Physical characteristics of a support, namely porosity and specific surface area, have long been understood to play a key role in stabilizing active components of the catalysts in dispersed state. Explicitly or implicitly, they reflect topological properties of the carbon surface, namely the nature and quantity of (1) traps (potential wells for atoms and metal particles), which behave as sites for nucleation and growth of metal crystallites and (2) hindrances (potential barriers) for migration of these atoms and particles [4,5]. An increase in the specific surface area and the micropore volume results, as a rule, in a decrease in the size of supported metal particles. Formal kinetic equations of sintering of supported catalysts always take into consideration these characteristics of a support [6]. 

A following stage of chemisorption process on the solid surface is a chemical reaction of the reactant immediately from the gas phase (Eley-Rideal mechanism) or between the intrinsic precursor and active sites (Langmuir-Hinshelwood mechajiism). Possible mechanisms of these reactions and formal kinetic equations have been discussed previously. 

The foundation for the calculations of kinetic data from a TG curve is based on the formal kinetic equation (43)... 

The procedure for constructing kinetic equations using the generalized Langevin equation is well known - one uses as variables in this description the p/ioje-space density fields. We could of course simply use the solute phase-space fields, (7.1), and follow the methods of Section V to obtain a formal kinetic equation for their time evolution. This procedure... 

In general, formal kinetic equations are empirical, valid only within a limited domain of concentrations and temperatures. 

The experimentally obtained formal kinetic equation (Equation 2.26) can be explained by a very fast second step compared to the first one (r, Tj). In this case the overall transformation rate will be controlled by the rate of the first step as the slowest one being in agreement with the experimentally observed PRL equation. This method to derive a concentration term in the rate expression is called the rate-determining step approach. 

The present-day theory of microbial growth kinetics stems from, and is still dominated by, Monod s formulation (1942, 1949) of the function fi = /x(s), given in Equ. 2.54. Also, this relation is a homologue of the Michaelis-Menten equation Monod derived it empirically, and thus this is a formal kinetic equation. The consequence is a different interpretation of the parameters and K. The microbial growth rate is... 

Other formal kinetic equations for the quantification of lag phases in microbial growth are found in the literature. A simple extension of Monod-type kinetics using the lag time as model parameter is given by Bergter and Knorre (1972) ... 

There are a large number of formal kinetic equations in the literature ... 

Catabolite repression plays a central role in many industrial fermentations such as in yeast technology (diauxic growth) and secondary metabolite productions. Even though the exact nature of the biochemical mechanisms involved in these regulations is often unknown (see Demain et al., 1979), the following type of formal kinetic equation... 

For example, using Equ. 5.88 a formal kinetic equation can be constructed wherein the ion concentration h is treated as if it were a substrate concentration. The value of the constant K is found by a method analogous to that in Fig. 5.29 using the concentration at the half-maximum value of fi from the two parts of the curve representing stimulation and inhibition K2 (Humphrey, 1977b, 1978). Thus, a mixed inhibition function is obtained... 

We saw that formal kinetic equations apart from kinetic parameters also contain surface concentrations Cj of electrically active species. It follows from the material presented in previous chapters that differs from the corresponding bulk values because a diffusion layer with certain concentration profiles forms at the electrode surface. Moreover, another reason due to which surface concentrations change is adsorption phenomena, which form a certain structure called a double electrode layer (DEL) at the boundary metal solution. It is clear that in kinetic equations, it is necessary to use local concentrations of reactants and products, that is, concentrations in that region of DEL where electrically active particles are located. The second effect produced by DEL is related to the fact that a potential in the localization of the electrically active complex (EAC) differs from the electrode potential. Therefore, activation energy of the electrochemical process does not depend on the entire jump of the potential at the boundary but on its part only, which characterizes the change in the potential in the reaction zone. In this connection, the so-called Frumkin correction appears in the electrochemical kinetic equations, which is related to the evaluation of the local potential i// [1]. 

Thus the kinetic equation may be derived for operator (7.21), though it does not exist for an average dipole moment. Formally, the equation is quite identical to the homogeneous differential equation of the impact theory with the collisional operator (7.27). It is of importance that this equation holds for collisions of arbitrary strength, i.e. at any angle of the field reorientation. From Eq. (7.10) and Eq. (7.20) it is clear that the shape of the IR spectrum... 

The present appendix represents a detailed derivation of the kinetic equations of the fluctuating liquid cage model in the classical formalism. A natural generalization is done for the case of partially ordered media, e.g. nematic liquid crystals. One of the simplest ways to take into account the back reaction is demonstrated, namely to introduce friction. 

To close the list of formal properties of the kinetic equations for stereoisomerizations with particular ligand partitions, let us simply recall that the solution of process P ... 

It should be also noted that the kinetic equation derived from the notions of formal kinetics is applicable over a wide range of concentrations of active particles at temperatures above 200 C in case of adsorbtion of O- and N-atoms on ZnO films. It may be expressed as follows ... 

As noted above, often the kinetic equations are written as a function of i0 rather than k°. One of the advantages of using i0 is that the faradaic current can be described as a function of the difference between the potential applied to the electrode, E, and the equilibrium potential, Eeq, rather than with respect to the formal electrode potential, E01, (which, as previously mentioned, is a particular case of equilibrium potential [COx(0,f) = CRed(0,t)], and at times may be unknown). In fact, dividing the fundamental expression of i by that of i0 one obtains ... 

We now turn attention to a mechanism where the sequence of substrate binding is not rigorously preserved, random binding. The formal kinetic scheme corresponding to Equation 11.37 is ... 

The above equation can be used as a formal kinetic approach without assuming a mechanism. This equation should be incorporated into the mass balance equations for oxygen in which the enhanced oxygen transfer rate due to the dispersed phase should also be considered. 

Formally, we call the set of reaction solvable, if there exists a linear transformation of coordinates a- a such that kinetic equation in new coordinates for all values of reaction constants has the triangle form ... 

The formal apparatus for defining the kinetic equations (294) and (295) with the reaction described by eqn. (296) is now in place. It remains to solve these equations by means of suitable approximations. 

The complex nature of heterogeneous catalytic reactions, which consist of a sequence of at least three steps (adsorption, surface reaction and desorption), the possible intervention of transport processes and the uncertainty about the actual state of the surface makes every attempt to obtain a complete formal kinetic description without simplifying assumptions futile. In this situation, some authors prefer fully empirical equations of the type... 

Experimental kinetic results (a) Formal rate equations... 

For a formal kinetic description of vapour phase esterifications on inorganic catalysts (Table 21), Langmuir—Hinshelwood-type rate equations were applied in the majority of cases [405—408,410—412,414,415]. In some work, purely empirical equations [413] or second-order power law-type equations [401,409] were used. In the latter cases, the authors found that transport phenomena were important either pore diffusion [401] or diffusion of reactants through the gaseous film, as well as through the condensed liquid on the surface [409], were rate-controlling. 

As appears from the examination of the equations (giving the best fit to the rate data) in Table 21, no relation between the form of the kinetic equation and the type of catalyst can be found. It seems likely that the equations are really semi-empirical expressions and it is risky to draw any conclusion about the actual reaction mechanism from the kinetic model. In spite of the formalism of the reported studies, two observations should be mentioned. Maatman et al. [410] calculated from the rate coefficients for the esterification of acetic acid with 1-propanol on silica gel, the site density of the catalyst using a method reported previously [418]. They found a relatively high site density, which justifies the identification of active sites of silica gel with the surface silanol groups made by Fricke and Alpeter [411]. The same authors [411] also estimated the values of the standard enthalpy and entropy changes on adsorption of propanol from kinetic data from the relatively low values they presume that propanol is weakly adsorbed on the surface, retaining much of the character of the liquid alcohol. 

These mechanisms are in formal agreement with kinetic equations assuming surface reaction between molecularly adsorbed reactants besides the group of catalysts used by Mochida et al. [406], such equations were also found to fit the kinetic data for silica—alumina [405] and bauxite [414] (see Table 21). 

The discussion up to this point has been concerned essentially with metal alloys in which the atoms are necessarily electrically neutral. In ionic systems, an electric diffusion potential builds up during the spinodal decomposition process. The local gradient of this potential provides an additional driving force, which acts upon the diffusing species and this has to be taken into account in the derivation of the equivalents of Eqns. (12.28) and (12.30). The formal treatment of this situation has not yet been carried out satisfactorily [A.V. Virkar, M. R. Plichta (1983)]. We can expect that the spinodal process is governed by the slower cation, for example, in a ternary AX-BX crystal. The electrical part of the driving force is generally nonlinear so that linearized kinetic equations cannot immediately be applied. 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

Following Zeldovich and Ovchinnikov [35], let us consider the role of reactant diffusion in establishing equilibrium in a reversible A B + B reaction. In terms of formal kinetics, it is described by the equations... 

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