Article Outline
Glossary
Definition of the Subject
Introduction
Fractals and Some of Their Relevant Properties
Random Walks
Diffusion-limited Reactions
Irreversible Phase Transitions in Heterogeneously Catalyzed Reactions
Future Directions
Bibliography
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Abbreviations
- Fractals :
-
Fractal geometry is a mathematical tool well suited to treating complex systems that exhibit scale invariance or, equivalently, the absence of any characteristic length scale. Scale invariance implies that objects are self‐similar: if we take a part of the object and magnify it by the same magnification factor in all directions, the obtained object cannot be distinguished from the original one. Self‐similar objects are often characterized by non‐integer dimensions, a fact that led B. Mandelbrot, early in the 1980s, to coin the name “fractal dimension”. Also, all objects described by fractal dimensions are generically called fractals. Within the context of the present work, fractals are the underlying media where the reaction kinetics among atoms, molecules, or particles in general, is studied.
- Reaction kinetics :
-
The description of the time evolution of the concentration of reacting particles (ρ i , where \({i = 1,2,\dots,N}\) identifies the type of particle) is achieved, far from a stationary regime, by formulating a kinetic rate equation \({\dot{\rho_{i}}(t) = F[\rho_{i}(t)]}\), where F is a function. This description, known in physical chemistry as the law of mass action, states that the rate of a chemical reaction is proportional to the concentration of reacting species and was formulated by Waage and Guldberg in 1864. Often, especially when dealing with reactions occurring in homogeneous media, F involves integer powers (also known as the reaction orders) of the concentrations, leading to classical reaction kinetics. However, as in most cases treated in this article, if a reaction takes place in a fractal, one may also have kinetic rate equations involving non‐integer powers of the concentration that lead to fractal reaction kinetics . Furthermore, it is usual to find that the slowest step involved in a kinetic reaction determines its rate, leading to a process‐limited reaction, where e. g. the process could be diffusion or mass transport, adsorption, reaction, etc.
- Heterogeneously catalyzed reactions :
-
A reaction limited by at least one rate‐limiting step could be prohibitively slow for practical purposes when, e. g., it occurs in a homogeneous media. The role of a good solid-state catalyst in contact with the reactants – in the gas or fluid phase – is to obtain an acceptable output rate of the products. Reactions occurring in this way are known as heterogeneously catalyzed. This type of reaction involves at least the following steps: (i) the first one comprises trapping, sticking and adsorption of the reactants on the catalytic surface. Particularly important, from the catalytic point of view, is that molecules that are stable in the homogeneous phase, e. g. H 2, N2, O2, etc., frequently undergo dissociation on the catalyst surface. This process is essential in order to speed up the reaction rate. (ii) After adsorption, species may diffuse or remain immobile (chemisorbed) on the surface. The actual reaction step occurs between neighboring adsorbed species of different kinds. The result of the reaction is the formation of products that can either be intermediates of the reaction or its final output. (iii) The final step is the desorption of the products, which is essential not only for the practical purpose of collecting and storing the desired output, but also in order to regenerate the catalytically active surface sites.
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Albano, E.V. (2012). Reaction Kinetics in Fractals. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_92
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