Bulk diffusion step

For the purposes of this description, the reaction mechanism is considered to consist of two steps. The first of these takes place entirely in the liquid phase and is a bulk diffusion step. It is characterized by a mass transfer coefficient k and is a function of the hydrodynamics of the system, i.e. in this case, the agitation conditions. If the process of interest is a simple physical dissolution, then indeed only this bulk diffusion step is important and the reaction rate is the rate of dissolution. In general, however, there is also a reaction step which takes place at the surface of the solid. 

Three different cases of rate-determining step are encountered (Fig. 18.21) (a) the surface step is rate determining (the concentration gradient across the membrane is zero but the rate is infinite because the diffusion coefficient is infinite in the case considered here, surface resistances on both membrane sides are equal) (b) the bulk diffusion step is rate determining (interfaces are at equilibrium with the gas phase, whatever the gas permeation flow) and (c) this is the most common intermediate case, where both surface and bulk rate contributions are playing a role. 

In Section 14.3 we concentrate our attention on the formulation of equations for modeling gas semipermeation within perovskite membranes for high-tem-perature CO2 capture applications. In such derivations, we assume that external diffusion at both the feed/membrane and permeate/membrane sides of the membrane (steps 1 and 2, respectively) can be neglected. In such a situation, bulk diffusion (step 3) and surface exchange kinetics (steps 2 and 4) are expected to contribute only to the permeation process. 

Sinee the kinetie proeesses oeeur eonseeutively, the solution eoneentration adjusts itself so that the rates of the two steps are equal at steady state. In most eases, more than one meehanism influenees a erystal s growth rate. If the different meehanisms take plaee in parallel, then the meehanism resulting in the faster growth eontrols the overall rate. If the proeesses take plaee in series, as in the ease of bulk diffusion followed by surfaee reaetion, then the slower meehanism will eontrol the overall rate. 

The characteristic feature of solid—solid reactions which controls, to some extent, the methods which can be applied to the investigation of their kinetics, is that the continuation of product formation requires the transportation of one or both reactants to a zone of interaction, perhaps through a coherent barrier layer of the product phase or as a monomolec-ular layer across surfaces. Since diffusion at phase boundaries may occur at temperatures appreciably below those required for bulk diffusion, the initial step in product formation may be rapidly completed on the attainment of reaction temperature. In such systems, there is no initial delay during nucleation and the initial processes, perhaps involving monomolec-ular films, are not readily identified. The subsequent growth of the product phase, the main reaction, is thereafter controlled by the diffusion of one or more species through the barrier layer. Microscopic observation is of little value where the phases present cannot be unambiguously identified and X-ray diffraction techniques are more fruitful. More recently, the considerable potential of electron microprobe analyses has been developed and exploited. 

The role of bulk diffusion in controlling reaction rates is expected to be significant during surface (catalytic-type) processes for which transportation of the bulk participant is slow (see reactions of sulphides below) or for which the boundary and desorption steps are fast. Diffusion may, for example, control the rate of Ni3C hydrogenation which is much more rapid than the vacuum decomposition of this solid. 

While it is inherently probable that product formation will be most readily initiated at sites of effective contact between reactants (A IB), it is improbable that this process alone is capable of permitting continued product formation at low temperature for two related reasons. Firstly (as discussed in detail in Sect. 2.1.1) the area available for chemical contact in a mixture of particles is a very small fraction of the total surface (and, indeed, this total surface constitutes only a small proportion of the reactant present). Secondly, bulk diffusion across a barrier layer is usually an activated process, so that interposition of product between the points of initial contact reduces the ease, and therefore the rate, of interaction. On completion of the first step in the reaction, the restricted zones of direct contact have undergone chemical modification and the continuation of reaction necessitates a transport process to maintain the migration of material from one solid to a reactive surface of the other. On increasing the temperature, surface migration usually becomes appreciable at temperatures significantly below those required for the onset of bulk diffusion within a product phase. It is to be expected that components of the less refractory constituent will migrate onto the surfaces of the other solid present. These ions are chemisorbed as the first step in product formation and, in a subsequent process, penetrate the outer layers of the... 

In addition, assuming that the rate-determining step is the bulk diffusion (i.e.,ka/(DJt) lholds), we can derive the minimum dissolution current observed after the fluctuation-diffusion current, that is,... 

Solution We suppose that the mass transfer and diffusion steps are fast compared with bulk transport by convection. This is the design intent for ion-exchange columns. The reaction front moves through the bed at a speed dependent only on the supply of fluid-phase reactants. Assuming piston... 

Rate of Formation of Primary Precursors. A steady state radical balance was used to calculate the concentration of the copolymer oligomer radicals in the aqueous phase. This balance equated the radical generation rate with the sum of the rates of radical termination and of radical entry into the particles and precursors. The calculation of the entry rate coefficients was based on the hypothesis that radical entry is governed by mass transfer through a surface film in parallel with bulk diffusion/electrostatic attraction/repulsion of an oligomer with a latex particle but in series with a limiting rate determining step (Richards, J. R. et al. J. AppI. Polv. Sci.. in press). Initiator efficiency was... 

Kinetics of chemical reactions at liquid interfaces has often proven difficult to study because they include processes that occur on a variety of time scales [1]. The reactions depend on diffusion of reactants to the interface prior to reaction and diffusion of products away from the interface after the reaction. As a result, relatively little information about the interface dependent kinetic step can be gleaned because this step is usually faster than diffusion. This often leads to diffusion controlled interfacial rates. While often not the rate-determining step in interfacial chemical reactions, the dynamics at the interface still play an important and interesting role in interfacial chemical processes. Chemists interested in interfacial kinetics have devised a variety of complex reaction vessels to eliminate diffusion effects systematically and access the interfacial kinetics. However, deconvolution of two slow bulk diffusion processes to access the desired the fast interfacial kinetics, especially ultrafast processes, is generally not an effective way to measure the fast interfacial dynamics. Thus, methodology to probe the interface specifically has been developed. 

The reaction is carried out in close-loop reactor connected to a mass spectrometer for 1S02, 180160 and 1602 analyses as a function of time [38], The gases should be in equilibrium with the metallic surface (fast adsorption/desorption steps 1 and f ) If the bulk diffusion is slow (step 6) and the direct exchange (step 5) does occur at a negligible rate, coefficients of surface diffusion Ds can be calculated from the simple relationship between the number of exchanged atoms Ne and given by the model of circular sources developed by Kramer and Andre [41] ... 

When surface diffusion is the only process of exchange, ag tends to an equilibrium value a at t - oo. In most cases, after a rapid step of surface diffusion, it can be observed that a% continues slowly decreasing. This phenomenon corresponds to a slow step of bulk diffusion (coefficient l)h). A model of bulk diffusion in spherical grains was developed by Kakioka et al. which led to the following equation [43] ... 

The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form ... 

The value of[Tl ] in the solution bulk remains essentially constant since only a tiny proportion of the overall amount of Tl is oxidized, but at the surface of the electrode we can say, to a good approximation, that [Tt ] = 0. Very soon after the potential is stepped, Tl from the bulk diffuses toward the electrode, thereby attempting to even out the concentration gradient, i.e. to replenish the Tt" that was consumed at the commencement of the step. W need to recognize, however, that these thallium ions will not remain as Tl for long as they will be oxidized immediately to form Tl, i.e. as soon as they impinge on the electrode. The end result is that a concentration gradient will soon form after the potential has been stepped. 

The overall reaction rate may be affected by diffusion if the mass transfer from the bulk fluid to the coal surface and/or the pore diffusion steps are relatively slow. I have made tentative estimates of both these diffusion effects. The following equation can be written (2) for the rate of oxygen uptake of a porous solid such as coal, assuming the oxygen to be consumed in an irreversible first-order reaction. 

Figure 50 shows three examples for (different) kinetic situations in which bulk diffusion, surface reaction, and transport across a grain boundary are the sluggish steps. Nonetheless, the other parameters can also be evaluated. This becomes especially clear from the top figure, where the nonunity intercepts reveal surface effects. Similarly, the nonzero bending of the profiles in the other two figures indicates transport resistances. 

Figure 4. Diffusion coefficient vs concentration of C0CI2 (wt%) in membrane for the first rapid diffusion step (bulk region).

Having established relationships for the step density, or at least the dependence of y0 on S, it is possible to consider in more detail the mechanisms described in Sect.5. For the moment, only the diffusion of growth units across the surface to steps and their subsequent incorporation will be considered. In a later section, the process of bulk diffusion will be considered in more detail. 

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