Isothermal diffusion theory

This formula follows directly from isothermal diffusion theory (Schottky, 1938) in which the parameter denotes the transmission coefficient for majority carriers across the barrier interface (j9 = 1 implies a transparent barrier). 

Structure and Properties of Metal-Ammonia Systems (Das) Studies in the Kinematics of Isothermal Diffusion. A Macrodynamical Theory of Multicomponent Fluid Diffusion (Lamm). ... 

Homogeneous charge transfer can take place between chemically similar redox species of one redox couple, e.g., Fe3+ and Fe2+ ions in solution or ferrice-nium and ferrocene moieties in poly(vinylferrocene) films, the electron transfer (- electron hopping or electron exchange reaction) can be described in terms of second-order kinetics and according to the - Dahms-Ruff theory [viii-x] it may be coupled to the isothermal diffusion ... 

Dahms-Ruff theory — For fast electron exchange processes coupled to isothermal diffusion in solution, the theoretical description and its experimental verification were given by Dahms [i] and by Ruff and co-workers [ii—v]. Ruff and co-workers studied the displacement of the centers of mass particles, which is brought about by both common migrational motion and chemical exchange reaction of the type... 

Adsorption kinetics and isotherms. The rate of protein adsorption onto solids is usually much slower than that predicted from the diffusion theory [85-87]. For various protein-adsorbent systems, the period of time required to obtain maximum adsorption ranges, as a rule, from several tens of minutes [10,12,14,88] to several hours [11,12,14,63,65,66,79,81,84,89,90]. More rarely, the adsorption terminates after several minutes [67,91] or continues for 24 h and longer [92,93], It cannot be excluded, however, that the initial adsorption rates should be transport limited, as has been shown by Norde et al. [94] for adsorption of lysozyme, RNase, and myoglobin on glass. The importance of diffusion is also obvious at the first step of adsorption from protein mixtures [95]. In this case the interface accommodates initially the protein molecules with the largest diffusion coefficients, and afterwards these molecules may be displaced by other molecules with higher affinity to the surface. 

In this chapter the formalism of nonequilibrium thermodynamics, is reviewed. This formalism is then applied to the theory of isothermal diffusion and electrophoresis. It is shown that this theory is important in determining the relations between the transport coefficients measured by light scattering and those measured by classical macroscopic techniques. Since much of this material is covered in other chapters, this chapter is very brief. Our presentation closely follows that of Katchalsky and Curran (1965). Other books that can be consulted are those of DeGroot and Mazur (1962) and Prigogine (1955). 

STUDIES IN THE KINEMATICS OF ISOTHERMAL DIFFUSION. A MACRO-DYNAMICAL THEORY OF MULTICOMPONENT FLUID DIFFUSION... 

Studies in the Kinematics of Isothermal Diffusion. A Macrodynamical Theory of Multicomponent Fluid Diffusion... 

The selection of a proper sorbent for a given separation is a complex problem. The predominant scientific basis for sorbent selection is the equilibrium isotherm. Diffusion rate is generally secondary in importance. The equilibrium isotherms of all constituents in the gas mixture, in the pressure and temperature range of operation, must be considered. As a first and oversimplified approximation, the pure-gas isotherms may be considered additive to yield the adsorption from a mixture. Models and theories for calculating mixed gas adsorption (Yang, 1987) should be used to provide better estimates for equilibrium adsorption. Based on the isotherms, the following factors that are important to the design of the separation process can be estimated ... 

As mentioned in Sect. 5.5, in the classical diffusion theory for a porous medium, adsorption is described by a distribution coefficient Kd resulting from the transfer of the species from the fluid phase to the solid phase through the linearized equation of equilibrium adsorption isotherm (5.113). 

Similar to the case of a binary system, the theory for finding the incubation time of the intermediate phase formation was proposed for ternary systems. This time strongly depends not only on diffusion parameters of the system (diffusivities), but also on the initial composition of the samples which undergo isothermal diffusion. Unlike the binary system, the ternary initial composition influences not only the incubation time, but the result of the whole diffusion process. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

In this section, the basic theory required for the analysis and interpretation of adsorption and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the transient adsorption of a single solute from a dilute solution in a constant volume, well-mixed batch system, or equivalently, adsorption of a pure gas. Moreover, uniform spherical particles and isothermal conditions are assumed. Finally, diffusion coefficients are considered to be constant. Heat transfer has not been taken into account in the following analysis, since adsorption and ion exchange are not chemical reactions and occur principally with little evolution or uptake of heat. Furthermore, in environmental applications,... 

Hatfield, B. and Aris, R. (1969). Communications on the theory of diffusion and reaction, part (iv), combined effects of internal and external diffusion in the non-isothermal case. Chem. Eng. Sci., 24, 1213-22. 

The results of experimental studies of the sorption and diffusion of light hydrocarbons and some other simple nonpolar molecules in type-A zeolites are summarized and compared with reported data for similar molecules in H-chabazite. Henry s law constants and equilibrium isotherms for both zeolites are interpreted in terms of a simple theoretical model. Zeolitic diffusivitiesy measured over small differential concentration steps, show a pronounced increase with sorbate concentration. This effect can be accounted for by the nonlinearity of the isotherms and the intrinsic mobilities are essentially independent of concentration. Activation energies for diffusion, calculated from the temperature dependence of the intrinsic mobilitieSy show a clear correlation with critical diameter. For the simpler moleculeSy transition state theory gives a quantitative prediction of the experimental diffusivity. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

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