Diffusivity solution

Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. is a reference diffusion coefficient discussed hereafter is the activity of component A and iC is a constant. By assuming that could be predicted by Eq. (5-223) with P = 1, iC was found to be equal to 0.5 based on five binaiy systems and vahdated with a sixth binaiy mixture. The limitations of Eq. (5-225) using and K defined previously have not been explored, so caution is warranted. Gurkan showed that K shoiild actually be closer to 0.3 (rather than 0.5) and discussed the overall results. 

Another subsidiary field of study was the effect of high concentrations of a diffusing solute, such as interstitial carbon in iron, in slowing diffusivity (in the case of carbon in fee austenite) because of mutual repulsion of neighbouring dissolved carbon atoms. By extension, high carbon concentrations can affect the mobility of substitutional solutes (Babu and Bhadeshia 1995). These last two phenomena, quenched-in vacancies and concentration effects, show how a parepisteme can carry smaller parepistemes on its back. 

Consider a lean phase, j, which is in intimate contact with a rich phase, i, in a closed vessel in order to transfer a certain solute. The solute diffuses from the rich phase to the lean phase. Meanwhile, a fraction of the diffused solute back-transfers to the rich phase. Initially, die rate of rich-to-lean solute transfer surpasses that of lean to rich leading to a net transfer of the solute from the rich phase to the lean phase. However, as the concentration of the solute in the rich phase increases. 

FIGURE 5A.2 A dialysis experiment. The solution of macromolecules to be dialyzed is placed in a semipermeable membrane bag, and the bag is immersed in a bathing solution. A magnetic stirrer gently mixes the solution to facilitate equilibrium of diffusible solutes between the dialysate and the solution contained in the bag. 

The effect, which arises in cases where the interfacial tension is strongly dependent on the concentration of diffusing solute, will generally be dependent on the direction (sense) in which mass transfer is taking place. 

Fick s law, as modified to describe the diffusion of available (diffusible) solutes in soil and written in Cartesian coordinates is therefore... 

Therefore if the carbon substrate is present at sufficiently high concentration anywhere in the rhizosphere (i.e., p p, ax), the microbial biomass will increase exponentially. Most models have considered the microbes to be immobile and so Eq. (33) can be solved independently for each position in the rhizosphere provided the substrate concentration is known. This, in turn, is simulated by treating substrate-carbon as the diffusing solute in Eq. (32). The substrate consumption by microorganisms is considered as a sink term in the diffusion equation, Eq. (8). 

Figure 3 shows a steady diffusion across a membrane. As in the previous case, the membrane separates two well-mixed dilute solutions, and the diffusion coefficient Dm is assumed constant. However, unlike the film, the membrane has different physicochemical characteristics than the solvent. As a result, the diffusing solute molecules may preferentially partition into the membrane or the solvent. As before, applying Fick s second law to diffusion across a membrane, we... 

By integrating the flux with respect to time, we obtain the cumulative mass of diffusing solute molecules per unit area ... 

Tubes are much more sensitive to convection effects than capillaries, but capillaries contain much smaller amounts of solution for analysis. Transport by convection when tubes are used can be accounted for by experimental evaluation. A disadvantage of both methods is the amount of time required for the experiment, which may be hundreds of hours. The investigator must address the possibility of adsorption of the diffusing solute onto the glass surfaces. In addition, the dimensions of the glass capillary must be known with considerable accuracy. 

It is fortunate that theory has been extended to take into account selective interactions in multicomponent systems, and it is seen from Eq. (91) (which is the expression used for the plots in Fig. 42 b) that the intercept at infinite dilution of protein or other solute does give the reciprocal of its correct molecular weight M2. This procedure is a straightforward one whereby one specifies within the constant K [Eq. (24)] a specific refractive index increment (9n7dc2)TiM. The subscript (i (a shorter way of writing subscripts jUj and ju3) signifies that the increments are to be taken at constant chemical potential of all diffusible solutes, that is, the components other than the polymer. This constitutes the osmotic pressure condition whereby only the macromolecule (component-2) is non-diffusible through a semi-permeable membrane. The quantity... 

Most solutes in soils are to some extent adsorbed on the soil solid only a small fraction is in the solution in the pores. However some adsorbed solutes, particularly exchangeable cations, can have considerable mobility on soil surfaces (see below), so it is important to consider the solid phase pathway as well as the solution. Because the diffusing solute passes rapidly between the solid and solution, the two pathways partly act in series. In such a heterogeneous medium as soil it is not realistic to account for the mobilities and concentration gradients of solutes in all the constituent parts. But if the soil volumes and reaction times... 

Nye, 1979, Section V.B). The values of /l for the more coarse-textured soil, Iloilo, come closest to this line, but the values are progressively far from it for the more clayey soils, and they are not parallel to it in any of the soils. There are several reasons for this. In soils electrostatic and viscosity interactions between diffusing solutes and solid surfaces are important and tend to diminish /l at a... 

This assumes that the depth of penetration of the diffusing solute molecules is at least equal to 8, i.e., that 3.6 > 8. Equation (16) is... 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

Fig. 4.2.2 A more realistic profile of the diffusive solute flux about a tabular grain. (From Ref. 6.)...

The linear driving force model has much more physical significance. It has been derived from a two-dimensional model of intra-particle diffusion, solution of which is a series development. The particle size appears explicitly. The effective diffusion coefficient is related to the particle porosity and to the size of the adsorbate molecule. Thus it makes sense to search for correlation of with these properties. However such relations are complex and it is rather difficult to predict for a given carbon and a given molecule. 

Figure 8.1 shows, in graphical terms, the concentration gradients of a diffusing solute in the close vicinity and inside of the dialyzer membrane. As discussed in Chapter 6, the sharp concentration gradients in liquids close to the surfaces of the membrane are caused by the hquid film resistances. The solute concentration within the membrane depends on the solubility of the solute in the membrane, or in the liquid in the minute pores of the membrane. The overall mass transfer flux of the solute J(kmol h m" ) is given as... 

Diffusion occurs down a concentration gradient as shown in Fig. 9. Fick demonstrated that the diffusive flux is proportional to the magnitude dc/dx of the concentration gradient of the diffusing solute... 

The left-hand side of the above equation gives the Ilkovic expression. It is thus possible to consider the right-hand side as correction terms to the Ilkovib equation. These corrections must take into account spherical diffusion, solution depletion in the neighbourhood of the drop due to previous drops, contact area with and shielding due to the capillary, and solution stirring. 

Analyses of the climbing rates of many other dislocation configurations are of interest, and Hirth and Lothe point out that these problems can often be solved by using the method of superposition (Section 4.2.3) [2]. In such cases the dislocation line source or sink is replaced by a linear array of point sources for which the diffusion solutions are known, and the final solution is then found by integrating over the array. This method can be used to find the same solution of the loopannealing problem as obtained above. 

In this paper we have used the quantity (1 — vp0) in writing equations for sedimentation equilibrium experiments. Some workers prefer to use the density increment, 1000(dp/dc)Tfn, instead when dealing with solutions containing ionizing macromolecules. This procedure was first advocated by Vrij (44), and its advantages are discussed by Casassa and Eisenberg (39). Nichol and Ogston (13) have used the density increment in their analysis of mixed associations. The subscript p. means that all of the diffusible solutes are at constant chemical potential in the buffer... 

Thus far we have examined diffusion under infinite conditions, where no phase boundaries exist. Some practical situations may be described by the above treatment. More frequently, the diffusion process will be initiated in the neighborhood of one or more phase boundaries as, for example, in chromatography and electrochemistry. The phase boundaries may be either permeable or impermeable to the diffusing solute. In electrochemical techniques, the boundary (e.g., the working electrode) is usually impermeable however, this is not always so (e.g., some ion-selective electrodes, membranes, liquid-liquid interfaces). In the... 

Liquids are simple, well defined systems and provide the starting point for modem theories of diffusion. An early and still fundamentally sound equation was derived by Einstein who applied simple macroscopic hydrodynamics to diffusion at the molecular level. He assumed the diffusing solute to be a sphere moving in a continuous fluid of solvent, in which case it can be shown that... 

Longitudinal molecular diffusion. Solute molecules are engaged in ceaseless Brownian motion, which is responsible for diffusion. The component of this erratic motion along the column axis, superimposed on the downstream displacement caused by flow, is one source of zone broadening. 

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