Self corrected diffusivity

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

Optional At 273 K, 7 23 (for He-Ar) is 0.653 cm s . From Du, Di, and D23 at room temperature, calculate c/12, c/13, and c/23. (To correct diffusion constants from one temperature to another, assume a dependence if the temperature change is small. This is only approximate, since the ds may vaiy in some degree with the temperature.) Then obtain d-[, di, and c/3 and use these to calculate D, Di, and D, from Eq. (V-35). Determine the ratios of the self-diffusion constants to their respective viscosities. How do these compare with the theoretical ratios discussed in the introductory section of Chapter V ... 

In contrast to all the other techniques considered in this paper, in sorption experiments molecular migration is observed under nonequilibrium sorption conditions. Therefore, instead of self-diffusivities, D, in this case transport diffusivities. A, are derived. It is generally assumed (see, e.g.. Refs. 366) that the corrected diffusivities. Do,... 

Abstract Neutron scattering was first used to derive the self-diffusivities of hydrocarbons in zeolites, but transport diffusivities of deuterated molecules and of molecules which do not contain hydrogen atoms can now be measured. The technique allows one to probe diffusion over space scales ranging from a few A to hundreds of A. The mechanism of diffusion can, thus, be followed from the elementary jumps between adsorption sites to Lickian diffusion. The neutron spin-echo technique pushes down the lower limit of diffusion coefficients, traditionally accessible by neutron methods, by two orders of magnitude. The neutron scattering results indicate that the corrected diffusivity is rarely constant and that it follows neither the Darken approximation nor the lattice gas model. The clear minimum and maximum in diffusivity observed by neutron spin-echo for n-alkanes in 5A zeolite is reminiscent of the controversial window effect . 

Coherent QENS measurements and MD simulations have been performed for N2 and CO2 in silicalite [30,31]. It has been found that the self-diffusivities of the two gases decrease with increasing occupancy, while the transport diffusivities increase. For a comparison with other systems, it is appropriate to remove the influence of the thermodynamic correction factor and to discuss the collective mobility in terms of the corrected diffusivity (also called Maxwell-Stephan diffusivity). Dq(c) is directly obtained from the Simula-... 

The diffusivity measured by the FR technique, D, is a transport diffu-sivity which has to be corrected, by using the Darken Equation (Eq. 26), to obtain the so-called corrected diffusion coefficient where the diffusion is measured at an equilibriiun pressure, Pe, which is outside the Henry s law range. This corrected diffusivity is generally taken to be the equivalent of the self-diffusion coefficient Dq ... 

Figure 8 shows the self-diffusion coefficients, Db, of methane, ethane and propane as a function of soibate loading at various temperatures. These corrected diffusion coefficients were calculated from Equation 12 using diffusion coefficients obtained fi m both sorption and desorption half-cycles. These diffusion coefficients were identical. The corrected, Db, diffusion coefficients in Figure 8 are <5 smaller than the intracrystalline self-diffusion coefficients measured directly by NMR which are also included in this figure and are in close agreement with the corrected diffusion coefficients obriuned by the full FR method. 

NMR techniques provide a somewhat more convenient and widely used method for the measurement of self-diffusivities. The method is, however, restricted to species such as hydrocarbons which contain a sufficiently hi concentration of unpaired nuclear spins. In comparing the results of NMR and uptake rate measurements it follows from Eqs. (5.6) and (5.9) that one should compare the NMR self-diffusivity with the corrected diffusivity from the uptake rate measurements. Exact agreement can be expected only when the cross coefficient is zero, but this is normally a good approximation at low concentrations. 

The relationship between sorbate activity and concentration for zeolitic systems is highly nonlinear so, except at very low concentrations approaching the Henry s law region, the thermodynamic correlation factor d np/d nc [Eq. (5.6)] is large, approaching infinity in the saturation region of the isotherm. In analyzing the dependence of diffusivity on concentration and temperature, it is therefore important to consider the corrected diffusivity or the self-diffusivity rather than the transport diffusivity. 

Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restiicied to binary diffusion at infinite dilution D°s of lo self-diffusivity D -. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits neghgible composition effects and deviations from thermodynamic ideahty. Conversely, liquid-phase diffusion almost always involves volumetiic and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. 

X 10 cm by measuring molecularly dispersed water in toluene and by correcting for local viscosity differences between toluene and these microemulsions [36]. Values for Dfnic were taken as the observed self-diffusion coefficient for AOT. The apparent mole fraction of water in the continuous toluene pseudophases was then calculated from Eq. (1) and the observed water proton self-diffusion data of Fig. 9. These apparent mole fractions are illustrated in Fig. 10 (top) as a function of... 

Although the reaction scheme shows a complete hydrolysis before condensation begins, this is likely not correct as stated earlier. The relative rates and extents of these two reactions will particularly depend on the amount of water added and the acidity of the system (10,11). The high functionality of the triethoxysilane endcapped PTMO oligomer should enhance the incorporation of PTMO molecules into the TEOS network. It was also assumed that the reactivities would be the same between silanol groups from silicic acid and endcapped PTMO. Therefore, no preferential condensation was expected and the deciding factors for which type of condensation (self- or co-) took place would be the diffusivities and local concentrations. 

The diffusion coefficients for eh, H30+, OHa, H, OH, and H202, in units of 10-5 cm2s-1, are taken respectively as 4.5, 9.0, 5.0, 7.0, 2.8, and 2.2. Of these, the first three are for charged species taken from experiment. D0h is taken the same as for self-diffusion of water. Dh2o2 is derived from self-diffusion of water using Stokes law to correct for the size effect. DQh is obtained from the diffusion of He. 

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

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