The Diffusion Process

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the net change in mass per unit time in the slice (dx) thick will be 

this is a standard differential equation and one solution to this equation, which can be proved by appropriate differentiation, takes the Gaussian form as follows  

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The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

Before pursuing the diffusion process any further, let us examine the diffusion coefficient itself in greater detail. Specifically, we seek a relationship between D and the friction factor of the solute. In general, an increment of energy is associated with a force and an increment of distance. In the present context the driving force behind diffusion (subscript diff) is associated with an increment in the chemical potential of the solute and an increment in distance dx ... 

The diffusion process has not been designed to ensure sterility, although temperatures above 65°C significantly retard microbial activity. Sulfur dioxide, thiocarbamates, glutaraldehyde, sodium bisulfite, and chlorine dioxide are all used, occasionally disregarding their redox incompatibilities, to knock down or control infections. The most common addition point is to the water from the pulp presses as it is returned to the diffuser. Surfactants ate almost... 

The procedure in use here involves the deposition of a radioactive isotope of the diffusing species on the surface of a rod or bar, the length of which is much longer than tire length of the metal involved in the diffusion process, the so-called semi-infinite sample solution. 

According to the transition state theoty, the diffusion process can be described by the equation... 

The heart of the energy-dispersive spectrometer is a diode made from a silicon crystal with lithium atoms diffiised, or drifted, from one end into the matrix. The lithium atoms are used to compensate the relatively low concentration of grown-in impurity atoms by neutralizing them. In the diffusion process, the central core of the silicon will become intrinsic, but the end away from the lithium will remain p-type and the lithium end will be n-type. The result is a p-i-n diode. (Both lithium-... 

Driven by the concentration gradient, solutes naturally diffuse when contained in a fluid. Thus, a discrete solute band will diffuse in a gas or liquid and, because the diffusion process is random in nature, will produce a concentration curve that is Gaussian in form. This diffusion effect occurs in the mobile phase of both packed GC and LC columns. The diffusion process is depicted in Figure 6. 

Thus, treating the diffusion process in a similar way to that shown in Figure 4 the total variance due to longitudinal diffusion in a column of length (1) is given by equation (7), viz.,... 

The diffusion process in natural and polychloroprene rubber adhesives can be explained by Campion s approach [1] which considers the concept of molecular free volume. This free volume is mainly affected by the solvent mixture of the adhesive (which will determine the degree of uncoiling of rubber chains) and by the ingredients in the formulation (mainly the amount and type of tackifier). 

The rate of reaction is controlled by the diffusion process, as the sulfide ion must first diffuse to the surface of the zinc oxide to react. High temperature (>250°F) increases the diffusion rate and is normally used to promote the reaction rate. 

In addition to the adsorption and desorption explained in Sec. II A, we can also include the diffusion process within our master-equation formahsm (10) [47]. For this purpose, we must only include the supplementary channel of the diffusive transition into the right-hand side of Eq. (10). A diffusion jump... 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

For such a condition of equilibrium to be reached, the atoms must acquire sufficient energy to permit their displacement at an appreciable rate. In the case of metal lattices, this energy can be provided by a suitable rise in temperature. In the application of coatings the diffusion process is arrested at a suitable stage when there is a considerable solute concentration gradient between the surface and the required depth of penetration. 

Interstitial diffusion is rarely possible when two metals interdiffuse, since their atomic radii are usually of the same order. Several mechanisms have been proposed, but it is now generally accepted that interdiffusion is due to the motion of vacant sites within the lattice, solvent and solute atoms moving as the vacant sites migrate. The diffusion process is thus dependent upon the state of imperfection of the solvent metal and the alloy being formed. 

The kinetics of the diffusion process (whether interstitial or substitutional) can be expressed by Pick s equations ... 

Although zinc has an appreciable vapour pressure at the temperatures of treatment, it is unlikely that zinc vapour plays any significant part in the diffusion process and it is generally accepted that the mechanism relies almost exclusively on intimate contact of hnely divided zinc dust with the steel surface. In spite of this requirement, coatings of even thickness and composition are obtained on the most intricate shapes, on fine threads, inside blind holes, and in the bore of small-diameter tubes. Large articles of uniform section, e.g. rods, tubes, etc. can be coated by this process. 

The practical importance of vacancies is that they are mobile and, at elevated temperatures, can move relatively easily through the crystal lattice. As illustrated in Fig. 20.21b, this is accompanied by movement of an atom in the opposite direction indeed, the existence of vacancies was originally postulated to explain solid-state diffusion in metals. In order to jump into a vacancy an adjacent atom must overcome an energy barrier. The energy required for this is supplied by thermal vibrations. Thus the diffusion rate in metals increases exponentially with temperature, not only because the vacancy concentration increases with temperature, but also because there is more thermal energy available to overcome the activation energy required for each jump in the diffusion process. 

Note that equations 8.105 and 8.106 effectively define a simple discrete diffusion process in one dimension the presence of a threshold condition also makes the diffusion process a nonlinear one (see below). 

It is important to realize that the migration in an electric field depends on the magnitude of the concentration of the charged species, whereas the diffusion process depends only on the concentration gradient, but not on the concentration itself. Accordingly, the mobility rather than the concentration of electrons and holes has to be small in practically useful solid electrolytes. This has been confirmed for several compounds which have been investigated in this regard so far [13]. 

However, while it is generally accepted that the rate of radical-radical reaction is dependent on how fast the radical centers of the propagating chains (Pp and Pj ) come together, there remains some controversy as to the diffusion mechanism(s) and/or what constitutes the rate-determining step in the diffusion process. The steps in the process as postulated by North and coworkers30 3" arc shown conceptually in Scheme 5.5. 

The differences in rate for the two positions of naphthalene show clearly that an additional-elimination mechanism may be ruled out. On the other hand, the magnitude of the above isotope effect is smaller than would be expected for a reaction involving rate-determining abstraction of hydrogen, so a mechanism involving significant internal return had been proposed, equilibria (239) and (240), p. 266. In this base-catalysed (B-SE2) reaction both k and k 2 must be fast in view of the reaction path symmetry. If diffusion away of the labelled solvent molecule BH is not rapid compared with the return reaction kLt a considerable fraction of ArLi reacts with BH rather than BH, the former possibility leading to no nett isotope effect. Since the diffusion process is unlikely to have an isotope effect then the overall observed effect will be less than that for the step k. ... 

The internal structure of the catalyst particle is often of a complex labyrinth-like nature, with interconnected pores of a multiplicity of shapes and sizes, In some cases, the pore size may be less than the mean free path of the molecules, and both molecular and Knudsen diffusion may occur simultaneously. Furthermore, the average length of the diffusion path will be extended as a result of the tortuousity of the channels. In view of the difficulty of precisely defining the pore structure, the particle is assumed to be pseudo-homogeneous in composition, and the diffusion process is characterised by an effective diffusivity D, (equation 10.8). 

The concentration fluctuations arising from the diffusion process in a static solution with a large amount of supporting electrolyte are followed by... 

If a local concentration of solute is placed at the midpoint of a tube filled with either a liquid or a gas, the solute will slowly diffuse to either end of the tube. It will first produce a Gaussian distribution with a maximum concentration at the center and finally, when the solute reaches the end of the tube, end effects occur and the solute will continue to diffuse until there is a constant concentration throughout the length of the tube. This diffusion effect occurs in the mobile phase of a packed LC column but the end effects are never realized. The diffusion process is depicted in figure 2. 

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