Diffusion and chemical reaction

The point of this terse introduction is that cellular automata represent not just a formalism for describing a certain particular class of behaviors (lattice gas simulations of fluid dynamics, models of chemical reactions and diffusion processes, etc.), but a much more general template for original and heretofore untapped ways of looking at a large class of unsolved or only poorly understood fundamental problems. 

Walker, R. E., Chemical reaction and diffusion in a catalytic tubular reactor, Phys. Fluids 4 (1961) 1211-1216. 

The differential equation governing simultaneous chemical reaction and diffusion then becomes... 

We start with the case where the initial electron transfer reaction is fast enough not to interfere kinetically in the electrochemical response.1 Under these conditions, the follow-up reaction is the only possible rate-limiting factor other than diffusion. The electrochemical response is a function of two parameters, the first-order (or pseudo-first-order) equilibrium constant, K, and a dimensionless kinetic parameter, 2, that measures the competition between chemical reaction and diffusion. In cyclic voltammetry,... 

Chemical reaction and diffusion determine the leaching rate... 

When chemical reaction is the rate controlling mechanism, then the increase in molecular weight is linear with time. This was shown to be the case at 160 °C with a pellet size <2.1 mm [29], However, under normal industrial SSP conditions, where the standard pellet diameter is between 2 and 3 mm and temperatures are >200 °C, the reaction rate decays over time. Typically, the molecular weight increase is proportional to the square root of time, as shown in Figure 4.6. This has been confirmed in other studies [15, 36-38], Such behaviour is said to be typical for a reaction involving both chemical reaction and diffusion within the material [29],... 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

A sensitive parameter in the coupling between chemical reaction and diffusion can be the temperature. In many cases the temperature coefficients are markedly different, and a shift of temperature can have a striking effect on systems coupling, compared with temperature effects on simpler molecules. 

What are the additional assumptions concerning the presented coupling of chemical reactions and diffusion ... 

We shall now illustrate these aspects through the far-from-equilibrium reaction-diffusion systems, which have been studied in great detail in recent years and which may be taken as a kind of archetype in this context. If chemical reaction and diffusion are the only relevant dissipative processes to be considered, the equations that describe the spatiotemporal variation of the concentrations X (Xt, X2,. . . , A ) are... 

Fluctuations in nonequilibrium systems have been studied mainly through two approaches the master equation approach16 and more recently the Ginsburg-Landau functional approach.17 In the master equation approach, the microscopic transition probabilities for chemical reactions and diffusion are taken to be given, and a master equation for the spatiotemporal variation of the probability distribution is obtained. Though the explicit solution of the master equation is difficult to obtain, some important general features could be deduced from it. One can show... 

Carrier facilitated transport involves a combination of chemical reaction and diffusion. One way to model the process is to calculate the equilibrium between the various species in the membrane phase and to link them by the appropriate rate expressions to the species in adjacent feed and permeate solutions. An expression for the concentration gradient of each species across the membrane is then calculated and can be solved to give the membrane flux in terms of the diffusion coefficients, the distribution coefficients, and the rate constants for all the species involved in the process [41,42], Unfortunately, the resulting expressions are too complex to be widely used. 

Quantification and Elucidation of Rate-Limiting Steps 109 Chemical Reaction and Diffusion 112 Rates of Ion Exchange on Soils and Soil Constituents 113 Mineralogical Composition 114 Ion Charge and Radius 116 Binary Cation and Anion Exchange Kinetics 117... 

Of course, for reactive flow calculations a new model would have to be constructed based on these techniques which used instead the equations governing compressible fluids and which contained the added chemical reactions and diffusive transport effects. 

Gas liquid reactions may conform to various mechanisms. Under certain conditions, the absorption and reaction may conform to a slow" reaction mechanism. By this term, we mean that a gaseous species A is absorbed, diffuses through the film,.and tben reacts ip the bulk liquid. Thus, according to film theory, the processes of chemical reaction and diffusion become two steps in series for a slow reaction. The absorption rate in this case is almost unaffected by a chemical reaction. In the limiting case, where the concentration profile of the absorbing species in the liquid film is flat, the reaction is often called a very slow reaction, and the process of absorption is said to be in the kinetically-controlled regime ... 

D.M. Auslander, G.F. Osten, A. Perelson, G. Kliford, On Systems with Coupled Chemical Reactions and Diffusions. Bond Graph Modeling for Engineering Systems, eds. D, Karnopp and R. Rosenberg. New York. USA. 

Temperature the results compiled in Tables 4.1-4.6 were obtained at different temperatures, and in some studies the temperature was not controlled. The results reported in Table 3.11 and Fig. 3.104 indicate that the PZC of oxides and related materials shifts to low pH when the temperature increases (with a few exceptions). Most surfaces carry more negative charge at elevated temperature (at given pH), and this creates favorable conditions for adsorption of cations and unfavorable conditions for adsorption of anions. Therefore elevated temperature would enhance uptake of cations, and low temperature would enhance uptake of anions at constant pH, if the electrostatic interaction was the only factor. On the other hand, the rate of chemical reactions and diffusion is enhanced at elevated temperatures. Thus, the kinetic and electrostatic effect on cation adsorption add up and the uptake increases with temperature. With anions these effects act in opposite directions the uptake increases with temperature when the kinetic factor prevails the uptake decreases with temperature when the electrostatic factor prevails, finally the both effects can completely cancel out. 

Danesi PR, Horwitz EP, Vandergrift GF, Chiarizia R, Mass transfer rate through liquid membranes Interfacial chemical reactions and diffusion as simultaneous permeability controlling factors. Sep. Sci. Technol. 1981 16 201-210. 

Also included for convenience in this chapter is a small section on miscellaneous applications of Mossbauer spectroscopy to subjects such as surface states, chemical reactions, and diffusion in liquids, which are of chemical interest but do not as yet represent major areas of study. 

Chemical reaction and diffusion inside a catalyst particle... 

Albright, Hanson, et al (1,2,3 ) have reviewed the work of previous investigators and concluded that the criteria sometimes used to determine the rate-limiting step were not adequate. The reaction kinetics measured by some workers (4,5 ) exhibit considerable differences even though they claimed to be measuring intrinsic kinetics. Such differences may result from diffusion effects. Recently, Cox and Strachan (6 ) reported the nitration of chlorobenzene to be kinetically controlled at a nitric acid concentration of 0.032 mole/liter in 70 weight percent sulfuric acid for the nitration of toluene in the same acid, the rates of chemical reaction and diffusion were of comparable magnitude. 

If the effects of the chemical reaction and diffusion are combined into a coefficient reff for the effective reactivity, the result can be illustrated by an Arrhenius diagram (Fig. 1.4) for the three above-mentioned temperature ranges. 

Both chemical reaction and diffusion are involved in concentration changes at any position X and time f. The resulting reaction-diffusion equation can be written as follows ... 

Equation 6.2 states that the local interfacial concentration Ej changes as a result of inteifacial convection, interchange with the bulk phases due to convection and diffusion, inteifacial chemical reaction, and diffusion within the interface. When the last effect is important, some relation is required between 7, and Ej. Usually Jis is taken as -ZJjsVjEj, where is an inteifacial diffusion coefficient. 

Biological membranes show anisotropy, as their molecules are preferentially ordered in a definite direction in the plane of the membrane, and the coupling between chemical reactions and diffusion flow can take place. Almost all outer and inner membranes of the cell have the abiUty to undergo active transport. Sodium and potassium pumps operate in almost all cells, especially nerve cells, while the active transport of calcium takes place in muscle cells. The proton pumps operate in mitochondrial membranes, chloroplasts, and the retina. 

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