Diffusion coefficients, list

The choice of vx is a matter of convenience for the system of interest. Table 1 summarizes the various definitions of vx and corresponding, /Y, commonly in use [3], The various diffusion coefficients listed in Table 1 are interconvertible, and formulas have been derived. For polymer-solvent systems, the volume average velocity, vv, is generally used, resulting in the simplest form of Jx,i- Assuming that this vv = 0, implying that the volume of the system does not change, the equation of continuity reduces to the common form of Fick s second law. In one dimension, this is... 

The magnitude of the diffusion coefficients given in Table I can be compared with a value of 3.3 X 10 5 cm.2/sec. determined experimentally by Stokes (26) for HCl in bulk solution at infinite dilution. The pore diffusion coefficients listed in Table I for HCl vary by a factor of (2 - 4) X 10"2 from that given by Stokes. McNeill and Weiss (15) have indicated that active carbon can be considered as a weak-base anion-exchange sorbent. According to Helfferich (13), diffusion coefficients in such resins can be several orders of magnitude less than the corresponding bulk solution coefficients. The Cl" ion probably limits the rate of diffusion, since its mobility in aqueous solution is much less than that of the H30+ ion. Further evidence to support this conclusion has been obtained in the present work from determinations of pore diffusion... 

Up to this point, quite a number of diffusion coefficients, listed below, have been introduced. For the reader not well versed in the field, this could lead, understandably enough, to some confusion. The purpose of this section is to shed some light on the subject. 

In (Barone et al. 1992) / = 1.8 is given for acetone, / = 1.7 for 1,4-dioxan and R = 1 for aniline. R. L Johnson and collaborators examined drill cores for pollutant contents of a clay deposit under a hazardous waste landfill which was impacted by leachate (Johnson et al. 1989). The apparent diffusion coefficients were derived from the depth profile of the concentration of the organic pollutants. A tortuosity factor of 0.20-0.33 was found for chloride in this investigation. The following retardation factors can, therefore, be calculated from the measured apparent diffusion coefficients with the effective diffusion coefficients listed in Table 7.5 (see (Muller et al. 1997a), note 16) i = 15 (benzene), R = 56 (trichloroethylene), i = 54 (toluene), R = 21 (1,2-dichloropropane), R= 115 (ethylbenzene), i =115 (naphthalene). The following values were derived from sorption experiments in the laboratory on unaffected drill core samples i = 44 (benzene), R = 65 (trichloroethylene), i = 82 (toluene), R = 269 (ethylbenzene). 

The values of the diffusion coefficients listed in Table 17.1 are similar to the gas kinetic diffusion coefficients measured by other techniques. However, the disorientation cross-sections are quite remarkable, being many orders of magnitude smaller than gas kinetic cross-sections. In the case of Rb-He collisions, for instance,... 

The diffusion coefficients listed above are easy to accept as experimentally valuable parameters, but they are harder to understand as a consequence of molecular motion. These coefficients are most often experimental values. In some cases, they are estimated from theories which imply models for the system involved. For gases, this is the model of gas molecules colliding in space. For liquids, they most often imply a solute sphere in a solvent soup. For solids, these estimates are based on a crystal lattice. In every case, the diffusion coefficients are not very directly related to random molecular motions. 

Part AM This part lists permitted individual constnic tion materials, apphcable specifications, special requirements, design stress-intensity vafues, and other property information. Of particular importance are the ultrasonic-test and tou ness requirements. Among the properties for which data are included are thermal conduc tivity and diffusivity, coefficient of theiTnal expansion, modulus of elasticity, and yield strength. The design stress-intensity values include a safety factor of 3 on ultimate strength at temperature or 1.5 on yield strength at temperature. 

Transport numbers for several non-haloaluminate ionic liquids generated from ionic liquid self-diffusion coefficients are listed in Table 3.6-7. The interesting, and still open, question is whether the NMR-generated transport numbers provide the same measure of the fraction of current carried by an ion as the electrochemically... 

There are several correlations for estimating the film mass transfer coefficient, kf, in a batch system. In this work, we estimated kf from the initial concentration decay curve when the diffusion resistance does not prevail [3]. The value of kf obtained firom the initial concentration decay curve is given in Table 2. In this study, the pore diffusion coefficient. Dp, and surface diffusion coefficient, are estimated by pore diffusion model (PDM) and surface diffusion model (SDM) [4], The estimated values of kf. Dp, and A for the phenoxyacetic acids are listed in Table 2. 

Experimental methods for determining diffusion coefficients are described in the following section. The diffusion coefficients of the individual ions at infinite dilution can be calculated from the ionic conductivities by using Eqs (2.3.22), (2.4.2) and (2.4.3). The individual diffusion coefficients of the ions in the presence of an excess of indifferent electrolyte are usually found by electrochemical methods such as polarography or chronopotentiometry (see Section 5.4). Examples of diffusion coefficients determined in this way are listed in Table 2.4. Table 2.5 gives examples of the diffusion coefficients of various salts in aqueous solutions in dependence on the concentration. 

The Thiele modulus and the effectiveness factor, respectively, were calculated for the three CO conversions X = 5,40, and 80%. The H2, CO, and H20 gas phase concentrations as well as the respective H2 concentration at the gas-wax phase boundary were taken from Table 12.3. The value of the diffusion coefficient Dm, is listed in Table 12.1. 

There is interesting physics, not all of it yet elucidated, in the interpretation of a number of the experiments listed in Table 2. We shall give a brief discussion of some of the issues in Sections 3b through 3e, with particular attention to a few studies that have inferred especially large diffusion coefficients (as opposed to the typical lower limits, of order say 10 12 cm2/ sec at 150°C, obtained from many experiments). But in order to present the first main conclusion of this section as quickly as possible, we shall start out with a discussion of the one experiment to date that can use specific empirical evidence on the density of mobile hydrogen in order to extract a specific diffusion coefficient. 

The list below shows the last position reached, in units of the jump step a, during a random walk for 100 atoms, each of which makes 200 jumps. If the jump time is 10-3 s and the jump distance, a, is 0.3 nm, estimate the diffusion coefficient (a) in units of a2 s-1 and (b) in units of m2 s-1 ... 

The first seven iterations produced from an arbitrary trial value r = 0.0001 are listed in Table 3.6. The final result is exact to better than four decimal places. This result allows the diffusion coefficient to be calculated as... 

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. 

There are a host of physical questions that cannot be easily answered just by knowing the rates listed in Fig. 6.13. For example, once an Ag atom is deposited on the surface, how long will it be (on average) before that Ag atom visits a site adjacent to a Pd surface atom How many different Pd surface sites will an Ag atom visit per unit time on the surface What is the net diffusion coefficient of Ag atoms on this surface To answer these questions, we need a tool to describe the evolution in time of a set of Ag atoms on the surface. 

We listed many examples early in this chapter, and they can be quite different in practice, from the msting of iron to the toasting of dough to the roasting of ore. The reactant A could be a gas or a liquid, and the film could be a sohd or liquid. The migration of A through the reacted film could be diffusion of A dissolved in C or permeation of A through a porous film of C. We describe this by a diffusion coefficient D s, but the value of D/ s and the mechanism by which transport occurs will not be discussed here. 

Rotational correlation times are listed in Table 3. As with the diffusion coefficients, the effect of the ordering imposed by the clay surface is clearly evident. 

The diffusion of metal ions in vitreous silica has not been studied as extensively as that of the gaseous species. The alkali metals have received the most attention because their behavior is important in electrical applications. The diffusion coefficients for various metal ions are listed in Table 5. The general trend is for the diffusion coefficient to increase with larger ionic sizes and higher valences. 

The transition time in the galvanostatic mode is listed in Table El. The concentration of electroactive species is 0.1 M and the diffusion coefficient is 10-5 cm2/s. Find the number of electrons transferred and draw a current-time response in a potentiostatic mode. 

Isotopic exchange reaction rates and oxygen diffusion coefficients of Pt/M-CZ, S-CZ and R-CZ at 41 are listed in Table V (Dong et al., 2004a). The surface diffusion rate for R-CZ (Rg) is nearly 4 times larger than the bulk diffusion rate or the equilibrium isotopic exchange rate (R ), while it is... 

Debye—Smolucholowski rate coefficient does indicate that these reaction rates are in broad agreement (see Table 2). If it is accepted that some hydrodynamic repulsion occurs, less than implied by the Deutch and Felderhof analysis [70], but as suggested by Wolynes and Deutch [71], then the reaction radii are as listed in Table 2. However, if allowance is also made for the larger size of some reactants than the hydrated electron, then the agreement between experiment and theory becomes satisfactory. Nevertheless, the uncertainty of diffusion coefficients and the crystallographic or true reaction radii, R, let alone the rate of reaction of encounter pairs, makes a comparison of these relatively small effects difficult. 

Effective Diffusivity. The effective diffusivity for N2/He at 25° C was calculated from the slope of the straight-line portion obtained in the high velocity region of a van Deemter plot [height of an equivalent plate vs. interstitial velocity 14, 15)]. A binary diffusion coefficient for N2-He of 0.717 cm2/sec was computed from Ref. 21, and the partition coefficient was taken as the reciprocal of the particle porosity (Table III) on the assumption that the adsorption of N2 at 25° C can be neglected. The calculated diffusivities are listed in Table III. 

It is useful to list certain special results for the case of viscous flow and molecular diffusion. Results for turbulent flow profiles and diffusion coefficients can be obtained by numerical integration as has been done by Taylor (1954) and more recently by Tichacek and others (1958). For molecular diffusion ipj = 1, and for the problem of diffusion only in the absence of a second phase, R = 13 = y = 1 giving a single factor k = - 2kx2 + i3. 

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