Kinetic equation homogeneous

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... 

Electrochemical reaction rates are also influenced by substances which, although not involved in the reaction, are readily adsorbed on the electrode surface (reaction products, accidental contaminants, or special additives). Most often this influence comes about when the foreign species I by adsorbing on the electrode partly block the surface, depress the adsorption of reactant species j, and thus lower the reaction rate. On a homogeneous surface and with adsorption following the Langmuir isotherm, a factor 10, will appear in the kinetic equation which is the surface fraction free of foreign species 1 ... 

Reaction order. One of the most widely used (particularly for homogeneous reactions) kinetic expressions is the power law kinetic equation. ... 

Equation (13) appears to be a good approximation for describing isothermal chemiluminescence kinetics for homogeneous systems where oxidation takes place uniformly. However, as has been shown by several authors [53-58], the different sections of a polymer sample may oxidize with its autonomous kinetics determined by different rates of primary initiation. A chemiluminescence imaging technique revealed that the light emission may be spread from some sites of the polymer film and the isothermal chemiluminescence vs. time runs are then modified, particularly in the stage of an advanced oxidation reaction [59]. 

Most electrode reactions of interest to the organic electrochemist involve chemical reaction steps. These are often assumed to occur in a homogeneous solution, that is, not at the electrode surface itself. They are described by the usual chemical kinetic equations, for example, first- or second-order reactions and may be reversible (chemical reversibility) or irreversible. 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

Katz described the homogeneous nucleation of a supersaturated vapour using J(i), the net rate at which clusters of size i grow to size i + 1 [63]. In this kinetic equation, J(i) is the difference between the rate at which clusters of size i add an additional monomer, and the rate at which clusters of size i + 1... 

The kinetic equation for S for calculating system homogeneous with respect to the translational degree of freedom and subject to no external field, the result is [4,117]... 

Reactions which are unretarded by the products will first be considered. When the adsorption of all the reacting gases is small, the numbers of molecules of each which are present on the surface of the catalyst at any moment are proportional to the respective pressures. The reaction occurring on the surface will therefore follow the same kinetic equation as that which would be followed if the identical reaction took place homogeneously. 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

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... 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

The kinetic equations are useful as a fitting procedure although their basis - the homogeneous system - in general does not exist. Thus they cannot deal with segregation and island formation which is frequently observed [27]. Computer simulations incorporate fluctuation and correlation effects and thus are able to deal with segregation effects but so far the reaction systems under study are oversimplified and contain only few aspects of a real system. The use of computer simulations for the study of surface reactions is also limited because of the large amount of computer time which is needed. Especially MC simulations need so much computer time that complicated aspects (e.g., the dependence of the results on the distribution of surface defects) in practice cannot be studied. For this reason CA models have been developed which run very fast on parallel computers and enable to study more complex aspects of real reaction systems. Some examples of CA models which were studied in the past years are the NH3 formation [4] and the problem of the universality class [18]. However, CA models are limited to systems which are suited for the description by a purely parallel ansatz. 

The kinetic equations analyzed in the previous sections are valid for a spatially homogeneous hydrolysis. However, it is possible that above a certain boundary in the reactivity (K) and sample thickness (L) map, all the water molecules penetrating in the sample are consumed in superficial layers, so that aging becomes heterogeneous (Fig. 14.9). 

We know that a PDE is stable as a linear approximation (see Sect. 2). Whence from eqns. (137) and (138) we establish that, at sufficiently low um and vout and t - oo, a solution of the kinetic equations for homogeneous systems tends to a unique steady-state point localized inside the reaction polyhedron with balance relationships (138) in a small vicinity of a positive PDE. If b(c(0)) = 6(c,n) vinjvout, then at low v,n and eout the function c(t) is close to the time dependence of concentrations for a corresponding closed system. To be more precise, if vm -> 0, uout -> 0, vmjvOM, c(0), cin are constant and c(O) is not a boundary PDE, then we obtain max c(t) — cc](t) -> 0, where ccl(t) is the solution of the kinetic equations for closed systems, ccl(0) = c(0),and is the Euclidian norm in the concentration space. 

Thus if the flow velocity in a completely flowing (homogeneous) system is higher than a certain value, the balance polyhedron contains a unique steady-state point that is globally stable, i.e. every solution for the kinetic equations (139) lying in Da tends to it at t - oo. Note that a critical value for the flow velocity at which this effect is obtained can depend on the choice of balance polyhedron (gas pressure). 

Specificity of a concrete system accounts for the source of the appearance of a small parameter and for its type. For homogeneous reactions, a small parameter is usually a ratio of rate constants for various reactions some reactions are much faster than the others. For just such a small parameter Vasiliev et al. [25] distinguished a class of chemical kinetic equations for which the application of the quasi-stationarity principle is correct (they considered a closed system). 

Modern thermodynamics provides possibilities to represent kinetic equations for all elementary reactions in various chemical (both homogeneous and heterogeneous) systems in uniform form. The same is suggested for chemical diffusion (in this connection, see ref. 1). 

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] ... 

Certain crude approaches are available to predict overall results, that is, nonequilibrium compositions. More refined techniques are available for the analysis of simplified models. Solution of the reaction kinetics of homogeneous gas phase combustion is possible through numerical solution of the rate equations. With the exception of the role of an overall highly exothermic reaction, the procedures are similar to those described in the preceding section on nozzle processes. The solution of the droplet burning problem including the role of chemical reaction rates, while not particularly tractable, is feasible. 

This would be the case for homogeneous thermal or photochemical initiation in which, over the period of time coiLsidered here, the (concentration of reactants does not change appreciably. If this is not the case, the diffusion equation above must be solved simultaneously with the kinetic equations. That would be true of flames and explosions. 

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