Examples of kinetic equations without diffusion

On the basis of known mechanism of a reaction, kinetic equations describing changes in concentration of reagents taking place during the reaction can be written. When the effects related to diffusion are not taken into consideration we speak, none too precisely, about reaction equations without diffusion (a so-called point system of kinetic equations) whereas in the case of accounting in kinetic equations for diffusion we deal with reaction equations with diffusion (a so-called non-point system of equations). In the present section we shall discuss properties of equations of reactions without diffusion. The equations of reactions with diffusion and the problem of a correct limit transition from equations with diffusion to diffusionless equations will be examined in the following section. 

Consider an example describing a sequence of isomerization reactions A B C (4.19) 

The system of equations (4.20) falls into a class of systems of ordinary differential equations it is an autonomous dynamical system, see equation (1.6) in Section 1.2. Furthermore, system (4.20) is linear — all unknowns are in the first power. As will be shown later, the systems of equations of this type may be readily solved. Linearity of the system of kinetic equations is a consequence of unimolecularity of elementary reactions in the mechanism (4.19). 

By and large, reacting systems cannot be described by linear differential equations. In typical cases, quadratic terms resulting from bimolecular interactions of reagents occur in equations (molcularity of a reaction equal to two). For example, the mechanism of the Edelstein enzymatic reaction is of the form 

The system of equations (4.22) is a non-linear system of ordinary differential equations non-linearities derive from bimolecular steps of reactions (4.21a)-(4.21c). 

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