Subject diffusion coefficient

It will be noted that because of the low self-diffusion coefficients the numerical values for representations of self-diffusion in silicon and germanium by Anhenius expressions are subject to considerable uncertainty. It does appear, however, that if this representation is used to average most of the experimental data the equations are for silicon... 

The self-diffusion coefficient is determined by measuring the diffusion rate of the labeled molecules in systems of uniform chemical composition. This is a true measure of the diffusional mobility of the subject species and is not complicated by bulk flow. It should be pointed out that this quantity differs from the intrinsic diffusion coefficient in that a chemical potential gradient exists in systems where diffusion takes place. It can be shown that the self-diffusion coefficient, Di, is related to the intrinsic diffusion coefficient, Df, by... 

Modeling relaxation-influenced processes has been the subject of much theoretical work, which provides valuable insight into the physical process of solvent sorption [119], But these models are too complex to be useful in correlating data. However, in cases where the transport exponent is 0.5, it is simple to apply a diffusion analysis to the data. Such an analysis can usually fit such data well with a single parameter and provides dimensional scaling directly, plus the rate constant—the diffusion coefficient—has more intuitive significance than an empirical parameter like k. 

We consider again the redox reaction Ox + ze = Red with a solution initially containing only the oxidized form Ox. The electrode is initially subjected to an electrode potential Et where no reaction takes place. For the sake of simplicity, it is assumed that the diffusion coefficients of species Ox and Red are equal, i.e., D = D()s = DRcd. Now, the potential E is linearly increased or decreased with E(t) = Ei vt (v is a potential scan rate, and signs + and represent anodic scan and cathodic scan, respectively.) Under the assumption that the redox couple is reversible, the surface concentrations of Ox and Red, i.e., c()s... 

The results of experimental studies aimed at assessing the validity of Eq. (32) have, for the most part, been inconclusive. Although the exact reasons for this are not entirely clear, certain differences between the experimental conditions employed and those upon which the model is based can be readily identified. In most studies the diffusion coefficient for the particular polymeric solution used was not known and was necessarily treated as an adjustable parameter in the theory. A subjective judgment was thus required as to whether the value of the diffusivity which described the data best was a reasonable one. Generally speaking, the resulting values were unrealistically high. 

Diffusion coefficients can also be obtained from sedimentation velocity experiments, but the precision is quite low and subject to some ques-don. Quasi-elastic light scattering is a much more useful technique to obtain diffusion coefficients (see below). 

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 Gaussian plume foimulations, however, use closed-form solutions of the turbulent version of Equation 5-1 subject to simplifying assumptions. Although these are not treated further here, their description is included for comparative purposes. The assumptions are reflection of species off the ground (that is, zero flux at the ground), constant value of vertical diffusion coefficient, and large distance from the source compared with lateral dimensions. This Gaussian solution to Equation 5-1 is obtained under the assumption that chemical transformation source and sink terms are all zero. In some cases, an exponential decay factor is applied for reactions that obey first-order kinetics. A typical solution (with the time-decay factor) is ... 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Marrero and Mason [45] have reviewed the subject of diffusion in gases and give a comprehensive list of data. Diffusion coefficients of gases are inversely proportional to the pressure and vary with temperature according to a power of T between 1.5 and 2. If experimental values are not available, the Wilke—Lee method [46] predicts the diffusion coefficients of non-polar mixtures to within about 4% of their true value. 

The subject of diffusion of gases dissolved in liquids has been reviewed by Himmelblau [47] who gives many values of diffusion coefficients. 

When more satisfactory forms of diffusion coefficient for the hydro-dynamic repulsion effect become available, these should be incorporated into the diffusion equation analysis. The effect of competitive reaction processes on the overall rate of reaction only becomes important when the concentration of both reactants is so large that it would require exceptional means to generate such concentrations of reactants and a solvent of extremely low diffusion coefficient to observe such effects. This effect has been the subject of much rather repetitive effort recently (see Chap. 9, Sect. 5.5). By contrast, the recent numerical studies of reactions between uncharged species is a most welcome study of the effect of this competition in various small clusters of reactants (see Chap. 7, Sect. 4.4). It is to be hoped that this work can be extended to reactions between ions in order to model spur decay processes in solvents less polar than water. One other area where research on the diffusion equation analysis of reaction rates would be very welcome is in the application of the variational principle (see Chap. 10). 

The evaluation of transport properties, including diffusion coefficients, is the subject of Chapter 12. The objective in this section is only to provide a brief discussion to assist understanding of the following derivation of the species continuity equations. 

The region of flow where collisions of molecules with the container walls are more frequent than intermolecular gaseous collisions was the subject of detailed study by Knudsen(8) early in the twentieth century. From geometrical considerations it may be shown(9) that, for the case of a capillary of circular cross-section and radius r, the proportionality factor is Snr3/3. This results in a Knudsen diffusion coefficient ... 

Correlation of the permeation properties of a wide variety of polymers with their free volume is not possible [32], But, within a single class of materials, there is a correlation between the free volume of polymers and gas diffusion coefficients an example is shown in Figure 2.24 [33], The relationship between the free volume and the sorption and diffusion coefficients of gases in polymers, particularly glassy polymers, has been an area of a great deal of experimental and theoretical work. The subject has recently been reviewed in detail by Petropoulos [34] and by Paul and co-workers [35,36],... 

Diffusion controlled reactions have been the subject of considerable study and several models have been proposed to relate the rate constant to the appropriate diffusion coefficient. Allen and Patrick (8 ) summarize these models and show that they may be written as... 

This result says that the concentration profile is linear, as implied by Fig. 1. It says that the flux will double if the diffusion coefficient is doubled, if the concentration difference across the film is doubled, or if the thickness of the film is cut in half. This important result is often undervalued because of its mathematical simplicity. However, anyone wishing to understand this subject should make sure that each step of this argument is understood. 

Effect of Radiation Dose on Micellar Properties. Figure 1 shows the concentration dependence of the micellar diffusion coefficient at 40° as determined by quasi-elastic light scattering (QELS) for solutions subjected to radiation doses of up to 4.56 Mrad. Limiting diffusion coefficients, D0>were obtained by extrapolation of data for dilute solutions (<0.05%) to zero concentration, the critical micelle concentration (CMC) being negligibly low for this poloxamer ( 1 ). 

Effect of Temperature on Micellar Properties. Figure 5 compares the influence of temperature on the diffusion properties of the micelles in solutions previously irradiated with a dose of 4.56 Mrad with those not subjected to radiation treatment. Hydrated radii calculated from the limiting diffusion coefficients for micelles not treated with radiation remain independent of temperature over the range 25° to 40° (Table III). 

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