Diffusion experimental

For particles of any shape at an absolute temperature T, Einstein showed that f is related to the experimental diffusion coefficient D by the expression... 

Fuller-Schettler-Giddings The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of X Vj used in this equation are in Table 5-16. 

The chemical diffusion coefficient at any concentration C in the experimental diffusion profile is dren given by... 

The silanization of the surface of a glass pipette may be necessary for different reasons [19]. If the pipette is to be filled with an aqueous solution, its outer wall should be made hydrophobic to prevent the formation of a thin aqueous film that may cause large deviations of the experimental diffusion-limiting current from the theory (see Section II.B). Experimental voltammograms were found to quantitatively agree with the theory after the aqueous layer was eliminated by silanizing the outer pipette wall. 

Diffusion of cations in a Nation membrane can formally be treated as in other polymers swollen with an electrolyte solution (Eq. (2.6.21). Particularly illustrative here is the percolation theory, since the conductive sites can easily be identified with the electrolyte clusters, dispersed in the non-conductive environment of hydrophobic fluorocarbon chains (cf. Eq. (2.6.20)). The experimental diffusion coefficients of cations in a Nation membrane are typically 2-4 orders of magnitude lower than in aqueous solution. 

The experimental diffusion battery penetration, g-s, is related to the theoretically calculated penetration probability, Pj C<1) by... 

The count median diameter, d, and the geometric standard deviation, Og, were calculated by using the experimental diffusion battery penetration values, gj, as input for a computer program based on the optimization procedure described earlier (Busigin et al., 1980). 

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

Figure 3-8 Concentration plots in Boltzmann-Matano analysis of an experimental diffusion-couple profile, (a) Plot of x versus C for the calculation of the integral Jx dC. (b) C versus x. The slope can be evaluated using this plot. As x approaches -co (that is, for large negative x<-500 /im), C is roughly 0.003. As x approaches co (that is, for large x> 2000 /im), C approaches 0.108. The data and the fit (using Dqh = 0 and Du om = oexp(aXH20t) see Section 3.3.1) are for exp Rhy-DC9 from Zhang and Behrens (2000).

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

Figure 3-35 Comparison of calailated diffusivity and experimental diffusivity of noble gas elements in water. Noble gas radius from Zhang and Xu (1995). Molecular diffusivity data are from Jahne et al. (1987) except for Ar (Cussler, 1997). A different symbol for Ar is used because different sources for diffusion data may not be consistent. The solid curve is calculated from the Einstein equation, and the dashed curve is calculated from the Sutherland equation. The curve from Glasstone et al. (1941) is outside the scale.

The following diffusion data are adapted from experimental diffusion data for water diffusion in a basaltic melt (Zhang and Stolper, 1991). The experiment was carried out at 1 300°C and the duration of the experiment is 10 minutes. Using Boltzmann analysis to obtain diffusion coefficients or water as a function of water concentration. Hint You will probably need to use a spreadsheet program to do simple integration and differentiation. You may also try to write a simple program. You may fix the concentration at one end to be 0.410 and the other end to be 0.100.)... 

In principle it is possible to verify these findings experimentally. Diffusion measurements characterizing single-crystal samples, free from defects and grain boundaries, should be able to demonstrate that smaller guests do not always diffuse faster. However, there are a number of problems in trying to find suitable experimental results with which to corroborate the claim of the theoretical simulations. One of these problems is the rather ideal distribution of cations that is assumed in the simulations of zeolites A and... 

The experimental diffusion parameters, D /r., at 30°C. are presented in Table II for all the coals. Clearly, no correlation exists between diffusion parameter and rank. If r<> is taken as the average particle radius for the 200 X 325 mesh samples, an upper limit to the values of diffusion coefficient, D, is obtained. The diffusion coefficient ranges from 1.92 X 10 9 sq. cm./sec. for Kelley coal to 1.41 X 10"8 sk. cm./sec. for the Dorrance anthracite. Our previous studies on the change of D /n with particle size suggested that n is not necessarily the particle radius (7) but is a smaller distance related to the average length of the micropores in the particles. That is, the calculated... 

The results of experimental studies of the sorption and diffusion of light hydrocarbons and some other simple nonpolar molecules in type-A zeolites are summarized and compared with reported data for similar molecules in H-chabazite. Henry s law constants and equilibrium isotherms for both zeolites are interpreted in terms of a simple theoretical model. Zeolitic diffusivitiesy measured over small differential concentration steps, show a pronounced increase with sorbate concentration. This effect can be accounted for by the nonlinearity of the isotherms and the intrinsic mobilities are essentially independent of concentration. Activation energies for diffusion, calculated from the temperature dependence of the intrinsic mobilitieSy show a clear correlation with critical diameter. For the simpler moleculeSy transition state theory gives a quantitative prediction of the experimental diffusivity. 

Table IV. Comparison of Theoretical and Experimental Diffusivities for Simple Molecules in 5A Zeolite...

This equation, with a constant value for D, has been shown to provide a satisfactory correlation of experimental diffusivity data for several zeolitic systems (6-8). For a system which obeys the Langmuir isotherm, Equation 5 becomes... 

Eerkens, 1. W. and Grossman, L. M. Tech. Report HE/150/150 (Univ. Calif. Inst. Eng. Res., Dec. 5th 1957). Evaluation of the diffusion equation and tabulation of experimental diffusion coefficients. 

Gibson, L.T., Cooksey, B.G., Littlejohn, D. andTennent, N.H. (1997a) Determination of experimental diffusion coefficients of acetic acid and formic add vapours in air using a passive sampler. Analytica Chimica Acta, 341, 1-10. 

For a system containing spherical particles, D = RT/6rrr]aNA - i.e. D oc 1/m1, where m is the particle mass. For systems containing asymmetric particles, D is correspondingly smaller (see Table 2.3). Since D = k77/, the ratio D/D0 (where D is the experimental diffusion coefficient and D0 is the diffusion coefficient of a system containing the equivalent unsolvated spheres) is equal to the... 

Table A16 Experimental diffusion coefficient of water in organic liquids at 20-25 °C at infinite dilution...

Figure 4. Comparison of experimental diffusion coefficients (o) for toluene-dg at 100 bar with predictions of toluene diffusion (----), method of Slattery and...

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

An attempt to correlate experimental diffusion data with free-volume, for the system of organic vapors with polyvinyl acetate, has been made in (57). The experiments showed that in this system, for T > Tg, the diffusion is Fickian and that the measured average diffusion coefficient steeply increases with the concentration, cs, of penetrant in the polymer. To quantify such a finding, an empirical relation has been proposed earlier (58) ... 

Where no experimental diffusion coefficient data in the polymer is available an estimation for the upper limit diffusion coefficient Dp can be made using the empirical correlation given in Eq. (15-1) (Chapter 15) ... 

Table 14-3 Experimental diffusion coefficients for styrene in polystyrene materials.

Using the experimental diffusion coefficient data for 1 1 GPPS HIPS from Linssen et al. 1992 and Eq. (14-4) ... 

Interpretation of result The calculated migration values here are realistic since results calculated using Eq. (14-4) cannot be larger than the mass balance result (Example 14-1). The calculated amount of styrene is still above the assumed sensory threshold limit of 0.1 mg/kg in the product for the worst case in step 4 but is equal to the estimation using the experimental diffusion coefficient in step 5. 

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