Step diffusion

Because this reaction must involve two steps, diffusion of selenate into the interlayer spaces of the green rust followed by electron transfer from Fe(ll) green rust, Johnson and Bullen (2003) interpreted this result using a two-step model similar to that discussed above. The diffusion step presumably has very little isotopic fractionation associated with it. Step 2 might be expected to involve a kinetic isotope effect similar to that observed in the HCl reduction experiments. As is discussed above, if the diffusion step is partially rate-limiting, the isotopic fractionation for the overall process should be less than the kinetic isotope effect occurring at the reduction step. This appears to be the case, as the ese(vi)-se(iv) value of 7.4%o is somewhat smaller than that observed for reduction by strong HCl (12%o). 

Thus, the pressure difference on the two sides of the step is proportional to the difference of the inverse cubes of the terrace widths (neglecting possible intereactions with more distant steps). Again in the overdamped limit, the step velocity f)x/<5t is proportional to the pressure from the terrace behind the step minus the pressure from the terrace ahead of the step. Since the motion is again step diffusion, the prefactor ought to contain the same transport coefficient as that for equilibrium fluctuations, fa for EC or Ds Cs , for TD, in either case divided by keT. Alternatively, this can be described as a current produced by the gradient of achemicalpotential associated witheachstep(Rettori and Villain, 1988). 

In particular, the equilibration of isolated and pairs of steps as well as ensembles of steps as realized in gratings, wires and bumps has been discussed, for different types of kinetics such as step diffusion, evaporation-condensation and surface diffusion. 

Because of the high value of pc the cesium atoms will remain in the position c for most of the time. Since the 011 planes on the tungsten wires used by Taylor and Langmuir might have had several steps, diffusion may not have occurred on the planes themselves but along the... 

A temperature-stepping diffusion-controlled fission product absorption model, where equilibrium is taken to occur at the surface of the... 

Large-amplitude potential step diffusion layer approximation... 

Figure 3.11 Thin-layer system response to potential-step, diffusion-controlled reduction of O to R with D0 = Dr. (A) Initial condition. (B) Profiles for O (dashed lines) and R (solid line) after potential step. (C) Final condition.

The chronocoulometry and chronoamperometry methods are most useful for the study of adsorption phenomena associated with electroactive species. Although less popular than cyclic voltammetry for the study of chemical reactions that are coupled with electrode reactions, these chrono- methods have merit for some situations. In all cases each step (diffusion, electron transfer, and chemical reactions) must be considered. For the simplification of the data analysis, conditions are chosen such that the electron-transfer process is controlled by the diffusion of an electroactive species. However, to obtain the kinetic parameters of chemical reactions, a reasonable mechanism must be available (often ascertained from cyclic voltammetry). A series of recent monographs provides details of useful applications for these methods.13,37,57... 

Since cation-exchange kinetics is usually very fast, occurring in millisecond time scales (Tang and Sparks 1993), the kinetics of cesium-137 sorption on bentonite is determined by diffusion steps diffusion through the adhesion layer... 

Regardless of the stochastic nature of its elementary steps, diffusion follows well-defined dependencies [1]. The wealth and beauty of these dependencies is particularly impressive when diffusion occurs in concentrated electrolyte solutions like ionic liquids. Owing to their structural variability and importance, ionic liquids represent a particularly attractive system for the study of self-diffusion and ionic transport behavior. 

This book aims to illustrate and discuss the subject of heterogeneous catalysis and to show the current capabilities of the theoretical and computational methods for studing the various steps (diffusion, adsoprtion, chemical reaction, etc.) of a heterogeneous catalytic process. 

Consider for a moment a rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, D - id, random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so fiiequent that a particle can rotate throu only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation... 

Semiclassical theory provides a framework for understanding biological electron flow what is necessary on the experimental front are systematic investigations of the response of rates to variations in ET parameters (AG°, X, r). Early efforts involving studies of bimolecular ET reactions were frustrated by the effects of diffusion. A simple bimolecular ET reaction can be broken into a sequence of three steps diffusive formation of an encounter or precursor complex (DA) ET from donor to acceptor within the precursor complex (DA D+A ) and dissociation of the successor... 

This equation is now widely used to estimate isotopic discrimination in photosynthesis, and shows the two-step diffusion/reaction nature of the process and the dependence of each step on the local CO2 concentration gradient. Typical values used in this model for C3 plants are a = 4.4%c... 

In a heterogeneous reaction sequence, mass transfer of reactants first takes place from the bulk fluid to the external surface of the pellet. The reactants then diffuse from the external surface into and through the pores within the pellet, with reaction taking place only on the catalytic surface of the pores. A schematic representation of this two-step diffusion process is shown in Figures 10-3 and 12-1. 

As shown in Fig. 2, we have anomalous diffusion in short time steps, but in long time steps diffusion coefficients converge to certain values to give normal diffusion. 

In the liquid phase diffusion to the catalyst may become the limiting step. Diffusion limitations provides an upper bound to the observed reaction rate (see Chapter 8). It appears that some enzyme catalytic reactions are so fast, e.g. carbonic anhydrase or acetyl cholesterase, that they exhibit this phenomenon. Catalysis under such condition is called "kinetic perfection". 

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