Kinetic equation of the model

An arbitrary rank of the orientational relaxation and the orienting field of more general form than in (7.5) will be considered below. Corresponding to (7.4) interaction 

The isotropy of the liquid rigidly restricts the conditional probability density f(e, e) for the cell axis to change as a result of an elementary jump. It depends only on the angle between successive directions of the field 

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The two-variable model for birhythmicity is built on the basis of eqns (2.7) by incorporating into them a term related to the transformation of product into substrate, in a reaction catalysed by an enzyme whose cooperative kinetics is described by a Hill equation, characterized by a degree of cooperativity n. The kinetic equations of the model thus take the form of eqns (3.1) where the various parameters remain defined as for eqns (2.7) and (2.11) ... 

Incorporating the variation of the parameters into the kinetic equations of the model... 

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. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

The aim of the multivariate evaluation methods is to fit a reaction model to the measured reaction spectrum on the basis of the Beer-Lambert law and thus identify the kinetic parameters of the model. The general task can be described by the non-linear least-squares optimisation described in Equation 8.20 ... 

Out of all catalytic processes the lattice-gas model has been most often used for describing CO oxidation on platinum metals. Actually, this reaction is the one that serves to develop various theoretical approaches. The simplest model is based on the idea of a homogeneous surface of a metal face that undergoes no rearrangement during the catalytic process. The kinetic equations of the three-stage mechanism have the form [136] ... 

Using a similar approach to that used in the derivation of the Michaelis-Menten model, the kinetic equations of the peroxidase-catalyzed removal of an aromatic substrate were derived from the reaction pathways illustrated in Fig. 2 [92]. The rate of change of the aromatic compound concentration can be written as... 

The kinetic equations of the two-variable Lengyel-Epstein model become (Horsthemke and Moore, 2004)... 

The grain growth models predict a kinetic equation of the form ... 

The extended liquid-solid BET isotherm describes well the adsorption behavior corresponding to types II or III isotherms of the van der Waals classification of isotherms (see Figure 3.1). Its expression parallels that of the BET isotherm model which is often applied in gas-solid equiUbtia [3]. It assiunes the same molecular description the solute molecules can adsorb from the solution onto either the bare surface of the adsorbent or a layer of solute already adsorbed. The equation of the model is derived from kinetic adsorption-desorption relationships, assuming first order kinetics [10,85]. The expression obtained after a rather lengthy derivation is... 

The second equation of the model relates the two concentrations in Eq. 10.1. Lapidus and Amundson [5] chose a linear kinetic model... 

Lee also extended the non-equilibrium theory developed originally by Gid-dings [10] to obtain H in/ the plate height contribution due to the mass transfer resistances and to axial dispersion, the non-equilibrium contribution. He started from the kinetic equation of the lumped rate constant kinetic model ... 

Mathematical Approaches. - The complexity of mathematical modeling of catalyst deactivation is mainly due to developing kinetic equations of the deactivation phenomena and measurement or estimation of the various parameters. When two or more different deactivation processes occur at the same time, this adds another level of difficulty and complicates the interpretation of experimental results. 

An analogous equation applies for B. After linearization of the above equation (Equation 2.1-34), the adsorption constants Ka and Kb can be obtained by plotting Pa/ a versus Pa or Pg and, after gas-kinetic derivation of the model [Jakubith 1998], interpreted in the form of Equation (2.1-35). 

It is important to understand clearly what is the molecular basis of the difference between the simple pore and the simple carrier. Formally the difference is expressed in the difference between the mathematical equation describing the kinetic predictions of the model (Eqns. 13 and 30, respectively). Only the latter contains terms involving the product 8,82, such terms allowing for the mutual interaction of the substrate species at the two membrane faces. But why should such terms not appear in the equation for the simple pore If in Eqn. 29 for the simple carrier, we let the rate constants /c, and kj become very large, only terms containing these constants will remain in the equation so that, after simplifying, we will obtain ... 

The same treatment can be done for the other member of the redox couple, and, by making the difference between the two substance flows, the kinetic equation of the electrochemical reaction is obtained. The Butler -Volmer (oc, constant) or more sophisticated models (o variable, as in the Marcus theory) are easily derived from it. 

Two basic concepts are used for the interpretation of the postpolymerization process (1) diffusion-controlled reactions (DCR) and (2) the microheterogeneous model. With regard to DCR, the basis I or the analysis of experimental data is the famous kinetic equation of the initial state of the stationary process. The parameters of this equation are the function of the mobility of macroradicals. 

Depending on the photocatalytic reaction involved and on the type of membrane module used, the kinetic model changes, and consequently modeling a membrane photoreactor requires knowledge of the kinetic equations of the catalyst, the membrane and the reactor configuration. 

Figure 8. Two solutions for the ordinary kinetic equations of the Eigen mechanism [3], eq 1 (dashed curves), as compared with the exact numerical solution for the Smoluchowski equation (full curves). Both models have the same /Cj and /Cr, but different values for the complex separation rate constant in the kinetic scheme were employed (a) giving the same area or (b) the same initial transient behavior as compared with the exact solution for the R OH decay [10b].

In this section, at every fixed Cp, the Gibbs energy is minimized. Naturally, the following question remains is such quasi-equilibrium process kinetically possible The answer to this question can be given by solving the kinetic equations of the Fokker-Planck type or by modeling the corresponding kinetics in an alternative way (Section 13.7). But it must be recalled that in the present section the kinetic aspect is not considered. 

We considered the nonequilibrium size distribution function/(N, t), the number of new phase droplets consisting of N stractural units at time t. The evolution of the ensemble of clusters formed by nucleation and growth processes is described by the kinetic equation of the Fokker-Plank type [94,95]. Here we present the case of particles of equal size. The main task of such a kinetic model is to describe the volume fraction p of the new phase 1 during the temperature cycling of the isolated nanoparticle ensemble (Figure 13.18). The volume fraction of the new phase p is obtained as a function of T and No. In this section, we report the obtained kinetic result size-induced hysteresis. 

The kinetics of nucleation/growth and separation processes in an ensemble of nanoparticles under time-dependent temperature is presented. Grounding upon our numerical studies, we conclude that under temperature cycling one should observe a hysteresis behavior. Such hysteresis is conditioned by the finite size and depletion effects and is described on the basis of the kinetic equation approach. The model shows that the width of hysteresis loop depends on (i) thermodynamic... 

Thus, the kinetic equation of the quasiequilibrium adsorbate model involves a set of adsorption coefficients, co-existence parameters 02o ic aJid the gas ph tse pevrameters T, P. 

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