Diffusion in mixtures

He, Q Johnson, CS, Two-Dimensional Electrophoretic NMR for the Measurement of Mobilities and Diffusion in Mixtures, Journal of Magnetic Resonance 81, 435, 1989. 

The Onsager irreversible thermodynamics approach in terms of the Fick s law methodology is, between the most frequently applied procedures to describe diffusion in mixtures. For a binary mixture (i.e., two types of diffusing molecules A and B) it is given by [124,125] ... 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

Chapman, Enskog, and Cowling have developed a general method for dealing with nonequilibrium states in not-too-dense gases which allows the approximate evaluation of the proper distribution functions to be used in place of the equilibrium distribution functions. While we shall not go into the details of the method, one of the results is of direct interest in the problem of diffusion in mixtures. 

In the first place, diffusion in mixtures with more than two components may give rise to complications when the mole fractions of the transported components are not small. In the simple examples given above there are already four components i4, B, P and the solvent. Generally a high diffiisional flow of one component in a given direction may hinder the diffusional flow of another component in the opposite direction. A rapid production of the product P will cause an accumulation of P that will diffuse away from the reaction zone, thus limiting the diffusion of both reactants to the reaction zone, and limiting the reaction rate. For rapid reactions, especially if there is no diluent (e.g., in gas mixtures), this effect may be considerable. 

Easteal, A.J. Woolf, L.A. (1984). Diffusion in mixtures of hard spheres at liquid densities A comparison of molecular dynamics and experimental data in equimolar systems. Chem. Phys.,88,101-111. 

The type of treatment described here was originally introduced by Scott and Dullien [4], who confined attention to isothermal isobaric diffusion in binary mixtures. Similar equations were independently published shortly after by Rothfeld [5], and the method was later extended to multi-component mixtures by Silveston [6], Perhaps the most complete exposition is given by Mason and Evans [7],... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Che pore size distribution and Che pore geometry. Condition (iil). For isobaric diffusion in a binary mixture Che flux vectors of Che two species must satisfy Graham s relation... 

Condition (iv). For equimolar counter diffusion in a binary mixture, the... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

They then compared measured and predicted fluxes for diffusion experiments in the mixture He-N. The tests covered a range of pressures and a variety of compositions at the pellet faces but, like the model itself, they were confined to binary mixtures and isobaric conditions. Feng and Stewart [49] compared their models with isobaric flux measurements in binary mixtures and with some non-isobaric measurements in mixtures of helium and nitrogen, using data from a variety of sources. Unfortunately the information on experimental conditions provided in their paper is very sparse, so it is difficult to assess how broadly based are the conclusions they reached about the relative merits oi their different models. 

Chromizing. The other principal method of obtaining a chromium-rich surface on steel is by chromizing (29). The material to be treated is embedded in a mixture of ferrochromium powder, a chromium halide, alumina, and sometimes NH Cl. The chromium is diffused in by a furnace... 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

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. 

Figure 3.5 The apparent vapour pressure of gold in gas transportation measurements as a function of the gas flow rate. Low flow rates, which were used earlier to assure equilibrium, are now known to be too high as a result of thermal diffusion in the gas mixture which is saturated with gold vapour...

In view of Swalin s treatment of diffusion in liquid metals. Are latter seems to be a better description. In binaty mixmres such as NaCl-KCl the equivalent conducAvities are a linear manner, but the KCl-CdCla mixture shows a marked negative departure from linear behaviour, probably because of the formation of the complex ion CdCh. ... 

The tubular (plug flow) reactor in which piston flow of the reacting mixture is assumed, and there is neither mixing nor diffusion in the flow direction. 

Note Equation (4.241) characterizes diffusion when the mixture element is in steady state with no turbulence. Diffusion in a pipe can be represented by Eq. (4.241) in convective mass transfer the flow and turbulence are important. 

Figure 4.4-1 Self-diffusion and mutual diffusion in a binary mixture. The self-diffusion coeffi-...

All the expressions for % — evidently represent the dissipation of energy which occurs when the gases are allowed to mix by diffusion in the specified manner. It follows from the principle of dissipation of energy that work will have to be spent in separating the mixture into its constituents, and, conversely, work should be obtained if the gases are allowed to mix in a suitable manner. The first quantity of work will be a minimum, the latter a maximum, and both equal and opposite, when the processes are conducted reversibly. 

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