Rate-limiting diffusion implications

SOME IMPLICATIONS OF A RATE-LIMITING DIFFUSION 7.5.1 Bioavailability of Colloidal Bound Metal... 

The second boundary condition indicates diffusive flux is exactly balanced by rate of decomposition at the wall. Under the stipulated conditions of diffusion-limited, the implication is that k so that for the present analysis... 

As far as we know, transformations of steroids, carried out with intact microbial cells, occur within the cell and not in the medium surroundii the cell. To enter the cell the steroid being transformed must dissolve to some extent in the medium so that it can diffuse through the cell wall and into the enzyme-rich interior. The practical implication of this requirement is that solubility and rate of diffusion may become the rate-limiting factors for transformation. Most steroid substrates ordinarily employed have modest, though measurable, solubilities in water and in the aqueous media used for microbial culture. To ensure saturation of the mediiun and to minimize this rate-limiting effect, steroids are often introduced into reactions in micronized form or, more conveniently, in solution in a water-miscible solvent from which precipitation in very fine particles occurs upon dilution with the aqueous medium containing the microorganism. 

In summary, although intraparticle diffusion is a rate-limiting mechanism for sorption in porous minerals, implications in using the diffusion model include accounting for such effects as coprecipitation, adsorbate type, potential exchange reactions, sorbent and solution chemistry, and the stability of the particle size distribution. 

Very recently, Lavenda devised an interesting method of solution of the Kramers problem in the extreme low-friction limit. He was able to show that it could be reduced to a formal Schrddinger equation for the radial part of the hydrogen atom and thus be solved exactly. One particular form of the long-time behavior of the rigorous rate equation coincides with that obtained by Kramers with the quasi-stationary hypothesis and may thus clarify the implications of this hypothesis. The method of Lavenda is reminiscent of that used by van Kampen but applied to a Smoluchowski equation for the diffusion of the energy. 

This uniformity in deposition has important implications with respect to understanding the relative contribution of hydrodynamic factors that may be involved in thrombus formation. If diffusion of clotting factors or platelets to a surface limited the rate of thrombus formation, then the profile of thrombus probably would be maximum at the forward end of the rod and diminish toward the distal end, proprotional to L — lA (where L is the distance from the tip of the rod) (5). The absence of this pattern indicated that factors other than diffusion dominated the process of thrombus formation in this system. 

In this equation is the purely transport controlled limiting current, D is the diffusion coefficient assumed to be identical for all species, and k is the first-order rate constant for the chemical reaction step. A plot of the observed limiting currents versus 5 determined from the first one-electrode process is shown in Fig. 10. A theoretical line can be fitted to the experimental points and a rate constant of k = 600 100 s-i is in good agreement with the value reported in the literature, k = 740 200 s [67]. For the reduction of ortho-bromonitrobenzene, a rate constant ofk = 200 50 s was determined, which is also in close agreement with the literature value of k = 250 s [71]. The implication is that ultrasound facilitates the measurement of fast rate constants under steady state conditions at... 

Heterogeneous catalytic reactions in supercritical solvents Obviously, a solid catalyzed reaction takes place only on the active sites of the porous catalyst with the implication of some mass and heat transport steps prior to and after the reaction. The first step is the diffusion of the reactants through the film surrounding the catalyst particle to the external surface of the catalyst, followed by diffusion of the reactants into the catalyst pore to the active site in the pores. These steps are limited by the dif-fusivity and viscosity of the reactants. In the case of a supercritical fluid phase reaction, the diffusivity is higher than the liquid diffusivity, viscosity is less than the liquid viscosity and therefore, the rate of transfer to the active site will be higher. After the adsorption, reaction and desorption steps, the products have to diffuse out of the pore, and again... 

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