Diffusion within

In contrast to the cell experiments of Gibilaro et al., it is now seen from equation (10.45) that measurement of the delay time gives no information about diffusion within the pellets this can be obtained only through equation (10.46) from measurements of the second moment. As in the case of the cell experiment, the results can also be Interpreted in terms of an "effective diffusion coefficient" associated with a Fick equation for the... 

For a fluid, with no underlying regular structure, the mecin squared displacement gradually increases with time (Figure 6.9). For a solid, however, the mean squared displacement typically oscillates about a mean value. Flowever, if there is diffusion within a solid then tliis can be detected from the mean squared displacement and may be restricted to fewer than three dimensions. For example. Figure 6.10 shows the mean squared displacement calculated for Li+ ions in Li3N at 400 K [Wolf et al. 1984]. This material contains layers of LiiN mobility of the Li" " ions is much greater within these planes than perpendicular to them. 

Other limitations of electrochemical fluorination ate that compounds such as ethers and esters ate decomposed by hydrogen fluoride and cannot be effectively processed. Branching and cross-linking often take place as a side reaction in the electrochemical fluorination process. The reaction is also somewhat slow because the organic reactant materials have to diffuse within 0.3 nm of the surface of the electrode and remain there long enough to have all hydrogen replaced with fluorine. The activated fluoride is only active within 0.3 nm of the surface of the electrode. 

In this context, the relative terms far, short, small, and large can be defined as follows. Fick s second law of diffusion dictates that the distance, 5, that a species having a diffusion coefficient, D, may diffuse within a period of time, t, is given by (12) ... 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

ADvFK for volume diffusion within die neck = 2DvFK for volume diffusion... 

Concentration polarization is an electrochemical process controlled by the diffusion within the electrolyte. 

Fig. 3.20. A two-dimensional nucleus which may grow either by the addition of a new unit onto the lateral surface or by sliding diffusion within the nucleus (from [146] by permission of the publishers, Butter-worth-Heinemann Ltd. )...

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... 

Consequently, the m-diffusion model does not extend to the domain where the Hubbard relation holds. Therefore, the J-diffusion model is the only realistic description of rotational diffusion within the framework of impact theory. 

Ubiquinone 10 830 kDa — — Diffuses within lipid phase of inner membrane 6-8 —... 

Few fixed-bed reactors operate in a region where the intrinsic kinetics are applicable. The particles are usually large to minimize pressure drop, and this means that diffusion within the pores. Steps 3 and 7, can limit the reaction rate. Also, the superficial fluid velocity may be low enough that the external film resistances of Steps 2 and 8 become important. A method is needed to estimate actual reaction rates given the intrinsic kinetics and operating conditions within the reactor. The usual approach is to define the effectiveness factor as... 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

W. R. Lieb and W. D. Stem, Non-stochesian nature of the transverse diffusion within hmnan red cell membranes. J. Membr. Biol. 1986, 92, 111-110. 

Ubiquinone, known also as coenzyme Q, plays a crucial role as a respiratory chain electron carrier transport in inner mitochondrial membranes. It exerts this function through its reversible reduction to semiquinone or to fully hydrogenated ubiquinol, accepting two protons and two electrons. Because it is a small lipophilic molecule, it is freely diffusable within the inner mitochondrial membrane. Ubiquinones also act as important lipophilic endogenous antioxidants and have other functions of great importance for cellular metabolism. ... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Figure 1 General pathways through which molecules can actively or passively cross a monolayer of cells. (A) Endocytosis of solutes and fusion of the membrane vesicle with the opposite plasma membrane in an active process called transcytosis. (B) Similar to A, but the solute associates with the membrane via specific (e.g., receptor) or nonspecific (e.g., charge) interactions. (C) Passive diffusion between the cells through the paracellular space. (C, C") Passive diffusion (C ) through the cell membranes and cytoplasm or (C") via partitioning into and lateral diffusion within the cell membrane. (D) Active or carrier-mediated transport of an otherwise poorly membrane permeable solute into and/or out of a cellular barrier.

Molecular Restricted Diffusion Within an Electrostatic Field of Force... 

Cation and anion flux across cultured cell monolayers by molecular restricted diffusion within an electrostatic field of force across aqueous pores has been described with a model derived by Adson et al. (1994). The ion fluxes per cross-sectional area of the cell monolayer are defined as... 

B Amsden. Solute diffusion within hydrogels. Mechanisms and models. Macromolecules 31 8382-8395, 1998. 

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