Corrected diffusivities

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. 

Fig. 7. Variation of (a) intracrystaUine diffusivity and (b) corrected diffusivity (Dq) with sorbate concentration for / -heptane in a commercial sample of 5A 2eolite crystals Q 409 K A, v> 439 K (ads, des) x, 462 K +, 491 K. Reproduced by permission of National Research Council of Canada from ref 17.

The temperature dependence of the corrected diffusivity follows the usual Eyring expression... 

Fig. 8. Dependence of (A) corrected diffusion coefficient (D), (B) steady-state fluorescence intensity, and (C) corrected number of particles in the observation volume (N) of Alexa488-coupled IFABP with urea concentration. The diffusion coefficient and number of particles data shown here are corrected for the effect of viscosity and refractive indices of the urea solutions as described in text. For steady-state fluorescence data the protein was excited at 488 nm using a PTI Alphascan fluorometer (Photon Technology International, South Brunswick, New Jersey). Emission spectra at different urea concentrations were recorded between 500 and 600 nm. A baseline control containing only buffer was subtracted from each spectrum. The area of the corrected spectrum was then plotted against denaturant concentrations to obtain the unfolding transition of the protein. Urea data monitored by steady-state fluorescence were fitted to a simple two-state model. Other experimental conditions are the same as in Figure 6.

In this paper only phenomena occurring on level (iii) will be discussed. In general, if sorption kinetics on porous solid particles are considered, one is dealing with complex physical phenomena which are often superimposed by external, e.g. apparatus effects. In order to obtain the correct diffusion data for the intracrystalline MS void volume, one should start therefore, from... 

This initial attempt to compute the time-correlation function was followed by a study of the Gaussian memory function with no significantly new results.67 The Gaussian memory, adjusted to give the correct diffusion coefficient, is found in exactly the same way as the exponential memory. It turns out to be... 

Figure 5 shows the comparison of the KM-corrected diffuse reflectance spectra for the 0.0% and the 1.5% vinyl silanized aluminum hydroxide concentrations. Note again that on this scale and particularly for KM-corrected spectra, the hydrocarbon absorptions between 3200 cm 1 and 2600 cm-1 are not discernible. However, upon closer inspection of the C-H stretch region and particularly the vicinity of the 3060 cm 1, the absorptions are indeed found. 

In principle the mobility B and therefore the corrected diffusivity D0 are also concentration-dependent, so Eq. (12) does not necessarily predict quantitatively the concentration dependence of D even for a system where the isotherm obeys the Langmuir equation. Nevertheless, the concentration dependence of B is generally modest compared with that of the thermodynamic factor, so a monatonic increase in diffusivity with adsorbed-phase concentration is commonly observed (Fig. 5). Clearly in any attempt to relate transport properties to the physical properties of the system it is important to examine the corrected, diffusivity D0 (or the mobility B) rather than the Fickian diffusivity, which is in fact a product of kinetic and thermodynamic factors. 

Actually, experimental results indicate that the constant in Eq. (3.9) is too small by a factor of V7/3. We now realize that this V7/3 quantity is not empirical but is the appropriate contribution due to the growth of the mercury drop into the solution away from the capillary orifice. Thus, the correct diffusion current expression for a dropping-mercury electrode is... 

D0 = RTMa is the intrinsic or corrected diffusion coefficient is the thermodynamic factor... 

Corrected Diffusion Coefficients (D x 109 cm2/s) of H-ZSM-11 and H-SSZ-24 Zeolites at Different Temperatures... 

Arrhenius plot showing temperature dependence of corrected diffusivity for N2 in various different samples of 4A crystals. 

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,... 

Care should be taken to ensure that the correct diffusion coefficient is used in applying experimental data, e.g., in modeling efforts. 

From equations given by Crank [220], Fink and Petsko [207] show that for a onedimensional sheet of thickness 21, substrate will penetrate to the centre of the lattice in a time t where t = l /D and D is the corrected diffusion coefficient in the crystal. The diffusion coefficient in solution for many compounds is of the order 1 x 10 cm second" Thus, we see that for a crystal of 0.4 mm thickness, diffusion times should be of the order of 80 seconds to 33 minutes using the correction factors 0.5-0.02 obtained by Bishop and Richards [219]. These times are in reasonable agreement with some of the times given in Table 2. Note the last two entries (7 and 8) also involve a catalytic reaction. 

Figure 2. Arrhenius plot showing comparison of NMR and sorption diffusivities for benzene and o-xylene in NaX zeolite crystals. NMR data from (1) Germanus et al. (19) and (2) Karger and Ruthven (10). Uptake (corrected diffusivity) and tracer exchange data of Goddard (11-13) (50 pm and 100 pm NaX, 250 pm faujasite). ZLC data of Eic (15,16).

Corrected diffusion coefficients were found to be essentially independent of loading within the loading range investigated and in contrast to the system benzene-HNaZSM-5... 

The diffusion coefficients D and the corrected diffusion coefficients obtained as... 

---