Mean diffusion

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

The rate of the bimolecular reaction in this limit is -d[A]/dt = kdclA][B]0 (where dc means diffusion-controlled), which is also equal to the flux of B toward A, multiplied by [A], That is,... 

Diffusion provides an effective basis for net migration of solute molecules over the short distances encountered at cellular and subcellular levels. Since the diffu-sional flux is linearly related to the solute concentration gradient across a transport barrier [Eq. (5)], a mean diffusion time constant (reciprocal first-order rate constant) can be obtained as the ratio of the mean squared migration distance (L) to the effective diffusivity in the transport region of interest. 

The surface diffusion of defects and adsorbates is of obvious importance in heterogeneous catalysis, as this process brings the reactants together. Understanding the dynamics of molecules on oxide surfaces is also a key step toward the realization of working molecular electronics. We note here that diffusion of Ob-vacs really means diffusion of Ob into the vacancy, which leaves another Ob-vac in the position vacated by the Ob- Similarly, diffusion of OHb occurs by diffusion of the H atom. 

If the transport were dominated by diffusion of neutrals, one would have a lifetime expression differing from (99) by having D0 instead of D+ and a much smaller R instead of Rc. The mean diffusion distance before capture would be (D0t)1/2, still typically of the order of a micron. However, we shall not consider this case seriously because the variation of the profiles with diode bias clearly showed transport by H+ to be usually dominant. 

According to classical equations of transla-tional Brownian motion in three dimensions, the mean diffusion distance of a particle during time t is (6 Dt)1 2. 

Fig. 22. Diffusion weighted spectral series recorded from TA (a) and SOL b). Diffusion sensitizing gradients with /i-values of 0, 250, 500, 1000 and 2000 s/mm are applied parallel (Gy) and perpendicular (G ) to the muscle axis. In contrast to SOL, TA shows clearly anisotropic signal losses. Mean diffusion coefficients of 20 subjects are given in (c).

The mean diffusion distance over the cooling period subsequent to Tc is proportional to (fD where the integration is from the time of Tc to the time of room temperature. The concept of Tc dictates that mid-diffusion distance (fDdty subsequent to Tc is significantly smaller than the half-thickness of a thin slab (one-dimensional diffusion), or the radius of a sphere or long cylinder. Denote the half-thickness and radius as a. Therefore,... 

For a given cooling history, the diffusivity depends on time, and the mean diffusion distance may be estimated by (JD df). Thus, if we know the dependence... 

Fig. 8.1.4 Schematic illustration of the evaporation apparatus with electron beam heating, DP and MBP mean diffusion pump and mechanical booster pump, respectively. (Reprinted from J Cryst Growth, 56 S Iwama et a].. Ultrafine powders of TiN and AIN produced by a reactive gas evaporation technique with electron beam heating, pp 265-269. Copyright 1982, with permission from Elsevier Science )...

The potentially greater toxicity of peroxynitrite can be readily visualized by comparing the mean diffusion distances that various nitrogen and oxygen-centered species may traverse in one lifetime. The definition of lifetime (t) is the time required for 67% of the initial concentration to decompose, and is readily calculated as the reciprocal of the pseudo-first-order rate constant for the disappearance of the species in question. Distances were calculated from the following equation, which is readily derived from the Fick s laws of diffusion (Nobel, 1983 Pryor, 1992). 

The mean diffusion distance of various nitrogen and oxygen based species within one estimated lifetime. The diameter of the dot for hydroxyl radical is still 100-fold larger than the actual diffusion distance (Hutchinson, 1957). 

This diffusion coefficient is the mean diffusion coefficient for a solid phase loading ranging from c/ to q2-... 

By the pulsed field gradient spin echo method (1) it is possible to measure mean diffusion lengths ( /(l2)) for molecules in a given time interval in systems without a concentration gradient. The application of the method is limited to... 

Perfectly mobile equilibria the mean diffusion coefficient... 

In fact, dc0/dt equals the very small value of the difference kpcOL — kpcQ that is virtually zero. It could be decided to neglect this so-cafled source term , but it is much more appropriate to combine eqns. (181a) and (181b) in such a way that (i) the source term is eliminated and (ii) an equation in terms of the total concentration c0t is obtained. To this end, the mean diffusion coefficient is introduced, in this case defined as [11,147]... 

Section Type Auxiliary concentrations Mean diffusion Equilibrium Rate constant ... 

In computer simulations a discrete lattice of sites is considered, each site is occupied by not more than one particle. Particles A are localized in their sites for r seconds and then possess hops. Thus, the mean diffusion coefficient Da = a2/(2dr) could be introduced. We assume that particles B are immobile, Dq = 0, since it permits to reduce greatly simulation time. Moreover, for Da = Dq and d = 1,2 the kinetics turn out to be quite similar. A hop of particle A into the site occupied by particle B results in their instant recombination. 

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