Diffusion coefficients, determination with

Figure 15. Relationship between the Stokes radius and the number of amino groups succinylated in bovine serum albumin. O, Values obtained from a calibrated Sephadex G-200 column x, values calculated from diffusion coefficient determined with the ultracentrifuge. From Habeeb ( 1967a ).

Figure 9 shows the water self-diffusion coefficients determined with pulsed gradient spin echo NMR by Zhao and Benziger, as functions of water activity and temperature. The diffusivity increases by almost two orders of magnitude as water activity increases from near a = 0.05 to = 0.9. The effective long-time... 

SNA Snabl, M., Ondrejeek, M., Chab, V., Chvoj, Z., Stenzel, W., Conrad, H., Bradshaw, A. M. Surface diffusion ofK on Pd(III) coverage dependence of the diffusion coefficient determined with the Boltzmann-Matano method. J. Chem. Phys. 108 (1998) 4212. 

The diffusion coefficient varies with temperature and pressure. The temperature dependence is expressed by the following expression first determined by Arrhenius... 

The question about the difference between the macroscopic and microscopic values of the quantities characterizing the translational mobility (viscosity tj, diffusion coefficient D, etc.) has often been discussed in the literature. Numerous data on the kinetics of spin exchange testify to the fact that, with the comparable sizes of various molecules of which the liquid is composed, the microscopic translational mobility of these molecules is satisfactorily described by the simple Einstein-Stokes diffusion model with the diffusion coefficient determined by the formula... 

In Section IB we presented experimental evidence that diffusion coefficients correlate with PVC main-chain polymer motions. This relationship has also been justified theoretically (12). In the previous section we demonstrated that the presence of CO2 effects the cooperative main-chain motions of the polymer. The increase in with increasing gas concentration means that the real diffusion coefficient [D in eq. (11)] must also increase with concentration. The nmr results reflect the real diffusion coefficients, since the gas concentration is uniform throughout the polymer sample under the static gas pressures and equilibrium conditions of the nmr measurements. Unfortunately, the real diffusion coefficient, the diffusion coefficient in the absence of a concentration gradient, cannot be determined from classical sorption and transport data without the aid of a transport model. Without prejustice to any particular model, we can only use the relative change in the real diffusion coefficient to indicate the relative change in the apparent diffusion coefficient. 

The F uptake of flint takes a much longer time than that for bone. Fluorine diffusion into the depth of flint material is controlled by defect clusters. The diffusion coefficient determined by implanting a model compound (amorphous silica bombarded with heavy ions and hydrated at 100°C) is 9.10—21 cm2/s at room temperature. The corresponding penetration depth of F under ambient conditions in a 1000-year-old artefact can be estimated via x — (Dt)1/2 = 0.17. im [50], Thus, F accumulates only in the first micrometre of the surface. The surface of ancient flint artefacts can be altered by dissolution. The occurrence of this phenomenon is especially important in basic media. However, in some cases, the thickness of the dissolved layer can be neglected compared to the F penetration depth at low temperatures. Therefore, Walter et al. [35] proposed relative dating of chipped flint by measuring the full width at half maximum (FWHM) of F diffusion profiles in theses cases. 

The ApBq and A Bn layers are seen to grow parabolically, whereas the thickness of the ArBs layer will gradually decrease with passing time. Eventually, this layer will disappear. It is easy to notice that in this case the values of the diffusional constants k[A2 and kim can readily be determined from the experimental dependences x2- t and z2- t, respectively, using an artificially prepared specimen A-ApBq-ArBs-A iB,-B or A-A,B-B. It is essential to mention that both the ApBq and AtBn layers must be the first to occur at the A-B interface. The diffusional constant k[A2 thus obtained is the reaction-diffusion coefficient of the A atoms in the ApBq lattice, while the diffusional constant klB3 is the reaction-diffusion coefficient of the B atoms in the Afin lattice, to be compared with respective self-diffusion coefficients determined using radioactive tracers. 

Figure 6 shows the influence of the bulk diffusion coefficient, Dhi on the metal deposition profiles. Obviously, by decreasing the diffusivity the metal deposition process becomes more diffusion rate-determined. With decreasing diffusivity the transport of reactant and intermediates is decreased resulting in a less deep penetration into the catalyst pellet. Therefore, the metal deposition maximum is shifted further to the exterior of the catalyst pellet. 

Table 4 contains a collection of diffusion coefficients determined experimentally for a variety of adsorbate systems. It shows that the values may vary considerably, which is of course due to the specific bonding of the adsorbate to the surface under consideration. Surface diffusion plays a vital role in surface chemical reactions because it is one factor that determines the rates of the reactions. Those reactions with diffusion as the rate-determining step are called diffusion-limited reactions. The above-mentioned photoelectron emission microscope is an interesting tool to effectively study diffusion processes under reaction conditions [158], In the world of real catalysts, diffusion may be vital because the porous structure of the catalyst particle may impose stringent conditions on molecular diffusivities, which in turn leads to massive consequences for reaction yields. 

Fig. S. Comparison of the true Henry s law mode diffusion co licfent, Di> with the diffusion coefficient determined from time lag measurements at low nessure...

For an unbiased symmetric random walk, P(S) = P(—S), the second term on the right vanishes and taking the time-continuous limit of small r one obtains a diffusion equation with diffusion coefficient determined by the variance of the jumps D = 5%)/(2r). In d dimensions the result is D = (<52)/(2g t). In the random walk context, the dispersion in Eq. (2.13) is giving the second moment of the position of a random walker which started at r = 0 ... 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

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