Diffusional independence

The following results are applicable to any electrode modified with a sparse distribution of microparticles. The mass transport to a single, diffusionally independent microparticle can (in theory) be treated on an equal basis as a microparticle within an independent diffusional zone in the experimental time scale with respect to its neighbors. Therefore, many theoretical results produced for microparticle arrays of diffusional categories 1 and 2 (see Section 6.3.2.2.2) are also valid for single particles. 

Case 2 behaviour, depicted in Figure 10.4(b), is observed when the size of the discs is small compared with y/lX. Under this category, each disc behaves as a microelectrode and diffusion to it is convergent rather than linear. In both Cases 1 and 2, neighbouring discs are sufficiently far apart that they are effectively diffusionally independent of the experimental time scale. The voltammetric response in Case 2 is therefore that of a single... 

Fig. 9.17 Diffusion at a nanoparticle array. Case 1 almost planar diffusion at an isolated nanoparticle where the diffusion layer thickness is small compared to the nanoparticle radius. Case 2 convergent diffusion to diffusionally independent nanoparticles. Case 3 partially overlapping diffusion layers between adjacent nanoparticles. Case 4 heavily overlapping diffusion layers leading to effectively linear diffusion to the array as a whole. Reproduced from Y.G. Zhou etal, Chem. Phys. Lett. 497 (2010) 200, with permission from Elsevier.

The importance of timescale on diffusional independence was also highlighted in an earlier paper by these authors, in which the transient potential step behaviour of electrode arrays were compared by theory and experiment. It is noteworthy that the electrode separation required for true diffusional independence under conventional experimental timescales (vol-tammetric scan rates in the tens of mV s ) is typically of the order of hundreds of microns, a fact that can quite easily be verified by considering Einstein s relation for diffusion lengths, 6 = -yJnDt. Hence, it is clear that electrode spacing and experimental timescale are critical considerations when exploiting the mass transport advantages inherent to nanoelectrodes, a fact that will be addressed further in Section 4.2. 

In the arrays and ensembles presented in the previous paragraphs, aU the micro- and nanoelectrodes are polarized at the same potential, and their geometry were designed to amplify the current signal (if the micro- and nanoelectrodes are diffusionally independent) or to minimize the signal-to-noise ratio (if the diffusion... 

Volt mmetiy. Diffusional effects, as embodied in equation 1, can be avoided by simply stirring the solution or rotating the electrode, eg, using the rotating disk electrode (RDE) at high rpm (3,7). The resultant concentration profiles then appear as shown in Figure 5. A time-independent Nernst diffusion layer having a thickness dictated by the laws of hydrodynamics is estabUshed. For the RDE,... 

Figure 6.3.2 shows the feed-forward design, in which acrolein and water were included, since previous studies had indicated some inhibition of the catalytic rates by these two substances. Inert gas pressure was kept as a variable to check for pore diffusion limitations. Since no large diffusional limitation was shown, the inert gas pressure was dropped as an independent variable in the second study of feed-back design, and replaced by total pressure. For smaller difftisional effects later tests were recommended, due to the extreme urgency of this project. 

The section discusses diffusion across a number of diffusion barriers in parallel. Diffusion across the skin represents one of the best examples to illustrate steady diffusion involving two or more independent diffusional pathways in parallel [6],... 

The packing itself may consist of spherical, cylindrical, or randomly shaped pellets, wire screens or gauzes, crushed particles, or a variety of other physical configurations. The particles usually are 0.25 to 1.0 cm in diameter. The structure of the catalyst pellets is such that the internal surface area far exceeds the superficial (external) surface area, so that the contact area is, in principle, independent of pellet size. To make effective use of the internal surface area, one must use a pellet size that minimizes diffusional resistance within the catalyst pellet but that also gives rise to an appropriate pressure drop across the catalyst bed. Some considerations which are important in the handling and use of catalysts for fixed bed operation in industrial situations are discussed in the Catalyst Handbook (1). 

The rate is independent of particle size. This is an indication of neghgible pore-diffusion resistance, as might be expected for either very porous particles or sufficiently small particles such that the diffusional path-length is very small. In either case, i -> 1, and ( rA)obs = ( rA)inl for the surface reaction. 

Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the reactor will be determined solely by its initial condition and its residence time, independently of what comes before or after. Practically, of course, some interaction will occur at the boundary between successive inputs of different compositions or temperatures. This is governed by diffusional behaviors which are beyond the scope of the present work. 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

Because formation ofexcimer E is a diffusion-controlled process, Eqs (4.11)-(4.13) apply to the diffusional rate constant ki for excimer formation. Under the approximation that ki is time-independent, the d-pulse responses, under the initial conditions (at t = 0), [M ] = [M ]o and [E ]o = 0, are... 

Let us recall that, if the bimolecular process is diffusion-controlled, kq is identical to the diffusional rate constant Iq. If Iq is assumed to be independent of time (see Chapter 4), it can be expressed by the following simplified form (Smoluchowski relation) ... 

Thus, th in a kinetically controlled regime is described by a dl law. Furthermore, th is found to be inversely proportional to pressure (for a first-order reaction) under kinetically controlled combustion, and in contrast, independent of pressure under diffusionally controlled combustion (since D P-1). In the kinetically controlled regime, the burning rate depends exponentially upon temperature. 

The issue of activity coefficient measurement for binary droplets was addressed by Allen et al. (1990). For low-vapor-pressure species, their diffusional fluxes in the gas phase are independent because of their low gas phase concentrations, and for a quasi-steady process each flux may be written... 

It can be proposed that superimposed upon the intrinsic random walk of molecules in the NFI channel network are processes of non-diffusional molecular re-orientation leading to an optimal sorbate arrangement. These processes are slow for the relatively "stiff" 2-butyne molecule (due to its triple bond) but fast for the "flexible" n-butane, i.e. the additional regime of sorption kinetics becomes observable if the time constant of diffusion (k /D) is small compared to the time constant of re-orientation. Since the latter process should be Independent of crystal size, size variation will give further evidence for appropriate systems. 

Activities of tri-n-butylammonium and tri-n-butylphosphonium ions with two different spacer chain lengths are compared in Table 8 1I8). The greater activity of the phosphonium ions is opposite to what has been reported for analogous soluble phase transfer catalysts119). Activities of the catalysts bound to silica gel were as high as activities of soluble catalysts adsorbed to silica gel118). Without some independent determination of the role of intraparticle diffusion it is not possible to determine whether the reduced activity of the adsorbed catalysts is due to lower intrinsic activity at the silica gel surface or to diffusional limitations. The size selectivity for alkyl bromides suggests that intraparticle diffusion was not a problem. 

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