Solid homogenous diffusion

For solid homogeneous diffusion (quadratic driving force), it is (6)... 

The breakthrough curve for solid homogeneous diffusion (linear driving force) combined with film mass transfer can be derived ... 

In most cases the study of equilibria between solid solutions and aqueous solutions containing their ions is extremely difficult, since solid state diffusion is virtually absent at ordinary temperatures. Most ionic solid solutions can be made homogeneous only at temperatures above 500°C, where solid state diffusion is relatively fast. 

High temperatures are generally needed in solid state synthesis to improve reaction rates and to facilitate solid state diffusion. Solid state diffusion is typically very slow. Thus, mechanical grinding steps are important to homogenize the sample and encourage complete reaction. It is important to realize, however, that some phases decompose at elevated temperatures. For example, Ba CujOy is unstable above about 1050°C (1) and the related phase, BajYCUjOg is only stable to 860°C in one atmosphere of oxygen (2). Thus, efforts to prepare these phases require a balance between the heat put in to speed the reaction kinetics and the stability limits of the desired phase. 

In this equation, ,/ and mi/2 are the masses of the two isotopes making up RZ1, and the terms are condensation coefficients for the two isotopes, which are determined experimentally and are typically close to 1. Equation (7.2.1) is valid if a is independent of the evolving composition of the evaporating liquid, and the diffusive transport rate is fast enough to keep the liquid homogeneous. The last condition is violated in solids, where diffusion is very slow relative to the evaporation rate, so solids do not undergo Rayleigh distillation. 

The thermodynamics of formation and transformation of a solid phase into another are characterized by two aspects, both of them explaining the difficulty to produce solids of homogeneous composition. The more important of these is nucleation The other is the tendency of certain components of the solid to diffuse to, or away from, surfaces. These aspects, however, cannot be considered in isolation. Chemical reactions involve the breaking of bonds and formation of new ones. This involves kinetically limited processes. In many cases, diffusion brings about additional kinetic limitations. The final result is the combination of the effects of all these processes. 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

Historically, stabilized (and partially stabilized) zirconia ceramics were prepared from powders in which the component oxides are mechanically blended prior to forming and sintering. Because solid state diffusion is sluggish, firing temperatures in excess of 1800°C are normally required. Furthermore, the dopant was nonuniformly distributed, leading to inferior electrical properties. Trace impurities in the raw materials can also lead to enhancement of electronic conductivity in certain temperature ranges, which is also undesirable. To overcome these problems, several procedures have been developed to prepare reactive (small particle size) and chemically pure and homogeneous precursor powders for both fully stabilized and partially stabilized material. Two of these are alkoxide synthesis and hydroxide coprecipitation. 

The number of transfer units for each mechanism can be estimated from known parameters and mass transfer correlations (4). For example, for a column with particles 0.01 cm in diameter, a superficial velocity of 0.01 cm/sec, and a solute bulk diffusivity of 7 x 10-7 cm2/sec, the estimated number of transfer units in a packed bed of length L for the four mechanisms, axial dispersion, external fluid film mass transfer, pore diffusion, and solid homogeneous particle diffusion,are... 

A requirement for making YAG crystals is a homogeneous, high-purity starting material for use in the Czochralski technique. This means that the application of mechanical mixing and solid-state diffusion techniques are limited. Accordingly, a variety of synthesis techniques have been developed, most of which are sol-gel based. 

The phenomenon of compensation is not unique to heterogeneous catalysis it is also seen in homogeneous catalysts, in organic reactions where the solvent is varied and in numerous physical processes such as solid-state diffusion, semiconduction (where it is known as the Meyer-Neldel Rule), and thermionic emission (governed by Richardson s equation ). Indeed it appears that kinetic parameters of any activated process, physical or chemical, are quite liable to exhibit compensation it even applies to the mortality rates of bacteria, as these also obey the Arrhenius equation. It connects with parallel effects in thermodynamics, where entropy and enthalpy terms describing the temperature dependence of equilibrium constants also show compensation. This brings us the area of linear free-energy relationships (LFER), discussion of which is fully covered in the literature, but which need not detain us now. 

Nonuniform composition occurs not only in a linear plane front solidification process, but can occur in almost any nucleation-and-growth processes (solid/liquid or solid/solid). In such cases, compositional partitioning between the growing nuclei and the parent matrix phase leads to a characteristic cored microstructure, as shown in Figure 6.33. Coring can have undesirable effects on materials properties, and so it is often removed by heat treating the microstructure at a temperature just below the transformation temperature to allow solid-state diffusion to homogenize the composition. 

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