Bulk diffusion step process

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. 

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. 

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... 

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. 

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] ... 

On the other hand, if the rate constant for the quenching step exceeds that expected for a diffusion-controlled process, a modification of the parameters in the Debye equation is indicated. Either the diffusion coefficient D as given by the Stokes-Einstein equation is not applicable because the bulk viscosity is different from the microviscosity experienced, by the quencher (e.g. quenching of aromatic hydrocarbons by O, in paraffin solvents) or the encounter radius RAb is much greater than the gas-kinetic collision radius. In the latter case a long-range quenching... 

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. 

An attempt was made by Doblhofer et al. [210] to separate surface from bulk charging processes for thermally prepared Ru02 using the potential step technique. These authors [210] concluded that some bulk diffusion was involved, presumably involving protons, and estimated a diffusion coefficient of 10 19 cm2 s1. Weston and Steele [213] deduced a diffusion coefficient value for protons in porous powder electrodes of Ru02 which is approximately similar to the value of Doblhofer et al. [210]. Iwakura and co-workers [214], on the other hand, employed cyclic voltammetry in deduc-... 

Before a protein molecule can adsorb and exert its influence at a phase boundary or take part in an interfacial reaction, it must arrive at the interface by a diffusion process. If we assume there is no barrier to adsorption other than diffusion, simple diffusion theory may be applied to predict the rate of adsorption. Under these conditions, after formation of a clean interface, all the molecules in the immediate vicinity will be rapidly adsorbed. The protein concentration in a sublayer, adjacent to the interface.and of several molecular diameters in thickness, will thus be depleted to zero. A diffusion process then proceeds from the bulk solution to the sublayer. The rate of adsorption, dn/dt, will be simply equal to the rate of this diffusion step given by classical diffusion theory (Crank, 1956) as... 

ONP (Fig. 3) in organic molecular crystals can be described as a three-step process (1) optical creation of electronic spin polarization in (transient) excited triplet states (2) polarization transfer to nearby nuclear spins and (3) accumulation of bulk nuclear spin magnetization in the (diamagnetic) ground state via spin-diffusion processes. Highly detailed theoretical descriptions and modeling of ONP processes observed in many systems can be found in the reviews mentioned above thus, only a brief overview is provided here. 

With the machinery of transition state theory in place, it is now possible to examine the predicted diffusion rates associated with a host of different important situations ranging from the bulk diffusion of impurities to the motion of adatoms on surfaces to the short-circuit diffusion of atoms along the cores of dislocations. It is evident that besides being of academic interest (which they definitely are), diffusive processes such as those mentioned above are a key part of the processing steps that take place in both the growth and subsequent microstructural evolution of the materials around which modern technology is built. 

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