Fractal chemical reaction

Recently, visible progress has been achieved in the theoretical and experimental investigations of non-linear phenomena. Among vital achievements in this area, the theory of solitons, strange attractors, the theory of fractals, chemical reactions of complex dynamics should be mentioned. 

Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

Chapter 16 - It is shown, that there is principal difference between the description of generally reagents diffusion and the diffusion defining chemical reaction course. The last process is described within the framework of strange (anomalous) diffusion concept and is controled by active (fractal) reaction duration. The exponent a, defining the value of active duration in comparison with real time, is dependent on reagents structure. 

Therefore, the estimation 0 im problem brings to the question of fractal dimension Df determination. At present two methods of indicated dimension determination one exist. First method consists of using of chemical reactions fractal kinetics general relationship [9] ... 

Synthesis processes in common case can be considered as a complex system of selforganization, developing during time, that results to formation of time-dependent fractal structures [3], In such reactions the important role is played by diffusive processes, which in the considered case have very specific nature. This specificity is due to the fact, that in chemical reactions not all reagents contacts occur with proper for reaction s product... 

The thermodynamic approach considers micropores as elements of the structure of the system possessing excess (free) energy, hence, micropore formation processes are described in general terms of nonequilibrium thermodynamics, if no kinetic limitations appear. The applicability of the thermodynamic approach to description of micropore formation is very large, because this one is, in most cases, the result of fast chemical reactions and related heat/mass transfer processes. The thermodynamic description does not contradict to the fractal one because of reasons which are analyzed below in Sec. II. C but the nonequilibrium thermodynamic models are, in most cases, more strict and complete than the fractal ones, and the application of the fractal approach furnishes no additional information. If no polymerization takes place (that is right for most of processes of preparation of active carbons at high temperatures by pyrolysis or oxidation of primary organic materials), traditional methods of nonequilibrium thermodynamics (especially nonequilibrium statistical thermodynamics) are applicable. 

Power-law expressions are found at all hierarchical levels of organization from the molecular level of elementary chemical reactions to the organismal level of growth and allometric morphogenesis. This recurrence of the power law at different levels of organization is reminiscent of fractal phenomena. In the case of fractal phenomena, it has been shown that this self-similar property is intimately associated with the power-law expression [28]. The reverse is also true if a power function of time describes the observed kinetic data or a reaction rate higher than 2 is revealed, the reaction takes place in fractal physical support. 

The observed empirical models should now be employed to simulate and predict kinetic behaviors obtained with administration protocols other than that used for observation. Moreover, we must develop pharmacokinetics in a multicompartment system by including the presence of a fractal organ. We have argued that the liver, where most of the enzymatic processes of drug elimination take place, has a fractal structure. Hence, we expect transport processes as well as chemical reactions taking place in the liver to carry a signature of its fractality. 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

One of the most important assumptions in MM kinetics is that the reaction in question wiU proceed in a three-dimensional vessel filled with a well-stirred fluid that obeys Pick s law for diffusion. This is rarely the case in a living cell, where many reactions are localized to membranes (two dimensions) or to small regions somewhere within the cell, creating an effectively one-dimensional environment with little or no diffusion. To circumvent this limitation, fractal kinetics have been developed which allow for the approximation of enzymatic reaction velocities in vivo [7]. Fractal kinetics can utilize MM-type kinetic constants to create a model of events in a spatially restricted environment. Briefly, as the dimensionality of a reaction is reduced from three dimensions to one, the kinetic order of a bimolec-ular reaction, for example, increases from 2 in a three-dimensional case, to 2.46 in a two-dimensional environment (e.g., membrane), to 3 in a one-dimensional channel, up to 50 for the case where fractal dimensions are less than 1. In simple terms, the kinetic order is the sum of all stoichiometric coefficients of the reactants in a balanced chemical reaction equation. Rearranging the familiar equation for MM kinetics... 

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