Diffusion coefficients cross-term

In principle, Equations 4-14 and 4-15 could be used to determine the relationship between the cross-term diffusion coefficients, although the thermodynamic information necessary to evaluate the partial derivative of chemical potential with concentration is usually not available. 

The tracer diffusion coefficient in Equation 4-21 is less than the mutual diffusion coefficient in Equation 4-24 by a factor equal to the sum of the crossterm diffusion coefficients defined in Equation 4-15. If the cross-term diffusion coefficients are small, then the tracer and mutual diffusion coefficients are approximately equal. 

The diagonal elements of D are called the main-term diffusion coefficients and the off-diagonal elements are called the cross-term diffusion coefficients or crossdiffusion terms. The cross-diffusion term liiiks the gradient of species... 

Experimental values of ternary diffusion coefficients characteristic of liquids are shown in Table 7.4-2. In cases like KCl NaCl water, KCl sucrose-water, and tolu-ene-chlorobenzene-bromobenzene, the cross-term diffusion coefficients are small, less than ten percent of the main diffusion coefficients. In these cases, we can safely treat the diffusion as a binary process. 

The cross-term diffusion coefficients are much more significant for interacting solutes. In cases like HBr-KBr-water and H2S04-Na2S04-water, this interaction is ionic in other cases, it may involve hydrogen-bond formation. Cross-term diffusion coefficients and the resulting ternary effects should be especially large in partially miscible systems, where few measurements have been made. 

In a typical tracer diffusion experiment, the total concentration of labeled and unlabeled solute is uniform throughout the system, so that the fluxes and the local concentration gradients are equal and opposite, and the tracer diffusion coefficient, )a > equal to the main-term diffusion coefficients minus a cross-... 

This gives the mass of solute diffused per unit time through a cross-section S, in terms of the concentration gradient dcfdx in the direction x perpendicular to (he cross section. D is a constant for (he given solute and solvent at a given temperature, and is called the diffusion coefficient. For any one pair of substances. D is found to be proportional to the absolute temperature. It should be stated that these statements apply only to nonelectrolytic solutions. 

Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneous action of molecular diffusion and variation of the velocity of the solvent. A new basis for his analysis is presented here which removes the restrictions imposed on some of the parameters at the expense of describing the distribution of solute in terms of its moments in the direction of flow. It is shown that the rate of growth of the variance is proportional to the sum of the molecular diffusion coefficient, D, and the Taylor diffusion coefficient ko2U2/D, where U is the mean velocity and a is a dimension characteristic of the cross-section of the tube. An expression for k is given in the most general case, and it is shown that a finite distribution of solute tends to become normally distributed. 

Each of these fluxes can be expressed according to Fields law in terms of the corresponding concentrations and of a corresponding diffusion coefficient which takes into account also the truly available cross section and tortuosity of any passages through the membrane. 

To obtain a useful expression for EM in terms of measurable parameters, it is convenient to introduce the permeability coefficient for species j, Pj. In Chapter 1, such a permeability coefficient was defined as DjKjlAx, where Dj is the diffusion coefficient of species jy Kj is its partition coefficient, and Ax is the membrane thickness (Eq. 1.9). Upon comparing Equation 3.8 with Equation 1.1 (J) = -Djdcjldx), we see that UjRT takes the place of the diffusion coefficient of species jy D y as already indicated (Section 3.2A). The quantity KjUjRTlAx can thus be replaced by the permeability coefficient, Pj. In this way, the unknown mobility of species / in a particular membrane, the thickness of the membrane, and the unknown partition coefficient for the solute can all be replaced by one parameter describing the permeability of the solute crossing that membrane. 

Thermal diffusion coefficients should not be confused with the thermal diffusivity, a quantity defined in terms of the thermal conductivity and referring to conduction of heat (see Section E.5). Thermal diffusion one of the cross-transport effects, is a physical process entirely separate from heat conduction. It tends to draw light molecules to hot regions and to drive heavy molecules to cold regions of the gas. Hydrogen is a species that is... 

In the above equations n and N are the atom fractions of nitrogen-15 species in the gas phase and the liquid phase respectively c (moles/cc.), d (cm.2/sec.), b (cm.) for the gas phase are respectively the concentration of oxides of nitrogen, the diffusion coefficient, and the thickness of the boundary layer, while C, D, and B are the same quantities for the liquid phase k (cc./moles-sec.) is a rate constant for the exchange of oxides of nitrogen between the gas and liquid phase. The specific transfer rate kr (moles/sec.-cm. ) when multiplied by the interfacial area a (cm.2/cc.) in a 1 cm. length of column per cm. of cross-sectional area gives an interphase transfer rate fc a (moles/sec.-cc.). If chemical reaction is rate limiting, fc a will be determined by the first term of Equation 25, otherwise it will be determined by the diffusion terms. 

Sturz and DoUe measured the temperature dependent dipolar spin-lattice relaxation rates and cross-correlation rates between the dipolar and the chemical-shift anisotropy relaxation mechanisms for different nuclei in toluene. They found that the reorientation about the axis in the molecular plane is approximately 2 to 3 times slower than the one perpendicular to the C-2 axis. Suchanski et al measured spin-lattice relaxation times Ti and NOE factors of chemically non-equivalent carbons in meta-fluoroanihne. The analysis showed that the correlation function describing molecular dynamics could be well described in terms of an asymmetric distribution of correlation times predicted by the Cole-Davidson model. In a comprehensive simulation study of neat formic acid Minary et al found good agreement with NMR relaxation time experiments in the liquid phase. Iwahashi et al measured self-diffusion coefficients and spin-lattice relaxation times to study the dynamical conformation of n-saturated and unsaturated fatty acids. 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

From the experimental data we were able to determine both the intramolecular and intermolecular relaxation rates as a function of pressure and temperature. The availability of shear viscosities and self-diffusion coefficients of EHB, which were measured earlier in our laboratory, provided the opportunity to test the dependence of the experimental cross-relaxation rates on viscosity and/or diffusion of EHB. The reorientational correlation time Tc describing overall molecular motion is coupled to the rj/T term through the Debye equation, which in a modified form is ... 

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