Solid ideal systems

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

This expression can be used to describe both pore and solid diffusion so long as the driving force is expressed in terms of the appropriate concentrations. Although the driving force should be more correctly expressed in terms of chemical potentials, Eq. (16-63) provides a qualitatively and quantitatively correct representation of adsorption systems so long as the diffusivity is allowed to be a function of the adsorbate concentration. The diffusivity will be constant only for a thermodynamically ideal system, which is only an adequate approximation for a limited number of adsorption systems. 

Intermetallics also represent an ideal system for study of shock-induced solid state chemical synthesis processes. The materials are technologically important such that a large body of literature on their properties is available. Aluminides are a well known class of intermetallics, and nickel aluminides are of particular interest. Reactants of nickel and aluminum give a mixture with powders of significantly different shock impedances, which should lead to large differential particle velocities at constant pressure. Such localized motion should act to mix the reactants. The mixture also involves a low shock viscosity, deformable material, aluminum, with a harder, high shock viscosity material, nickel, which will not flow as well as the aluminum. 

Black-body radiation is the radiation emitted by a black-colored solid material, a so-called black body, that absorbs and also emits radiation of all wavelengths. A black body emits a continuous spectrum of radiation, the intensity of which is dependent on its wavelength and on the temperature of the black body. Though a black body is an idealized system, a real solid body that absorbs and emits radiation of aU wavelengths is similar to a black body. The radiation intensity of a black body, at... 

Though some real industrial reactions may never yield to simple analysis, this should not deter us from studying idealized systems. These satisfactorily represent many real systems and in addition may be taken as the starting point for more involved analyses. Here we consider only the greatly simplified idealized systems in which the reaction kinetics, flow characteristics, and size distribution of solids are known. 

In the case of non—eutectic systems, the solid phase shows nearly ideal mixing, so that the surfactant components distribute themselves between the micelle and the solid in about the same relative proportions (i.e., both the mixed micelle and mixed solid are approximately ideal). However, in the case of the eutectic type system, the crystal is extremely non-ideal (almost a single component), while the micelle has nearly ideal mixing. As seen in earlier calculations for ideal systems, even though the total surfactant monomer concentration is intermediate between that of the pure components, the monomer concentration of an individual component decreases as its total proportion in solution decreases. As the proportion of surfactant A decreases in solution (proportion of surfactant B increases) from pure A, there is a lower monomer concentration of A. Therefore, it requires a lower temperature or a higher added electrolyte level to precipitate it. At some... 

Different kinds of ideal systems arc distinguished by the form of p,(T, p). In a mixture oT perfect gases. u,(T.p) varies logarithmically with pressure, while for a liquid or solid solution, one can. to a first approximatinn, regard //, as independent of pressure. 

Figure 14.16 (Solid + liquid) phase equilibria for jtiCgHfi + JC2l,4-C6H4(CH3)2 at p = 0.1 MPa, an example of a nearly ideal system.

The most common example of dispersive mixing of particulate solid agglomerates is the dispersion and mixing of carbon black into a rubber compound. The dispersion of such a system is schematically represented in Fig. 3.22. However, the break up of particulate agglomerates is best explained using an ideal system of two small spherical particles that need to be separated and dispersed during a mixing process. 

However, we would like to point here not to the differences between the equilibrium tunneling mechanism and the above examples of mechanisms of the nonequilibrium type in low-temperature chemical conversions, but, on the contrary, to a simplifying assumption which relates them but which has to be rejected in a number of cases—and that is the subject matter of this chapter. In the above models the solid matrix itself was considered, in essence, from a special point of view, namely, as an ideal system, devoid of defects, which is in mechanical equilibrium. In other words, the fact that the systems in question are significantly out of equilibrium with respect to their mechanoenergetic state was ignored. This property of the experimentally studied samples was the result of both their preparation conditions and the ionizing radiation. 

All the previous theoretical considerations have been established assuming an ideal system without any boundary conditions. It should be pointed out however that in practice, all the studied systems, especially in SHE chemistry, have finite dimensions (time and volume). As only ideal system were considered, edge effects, pseudo-colloid formation, sorption phenomena, redox processes with impurities or surfaces, medium effects have not been taken into account. All these effects, representing the most important part from the deviation to ideality, cannot be predicted with formal thermodynamics and/or kinetics. Thus, radiochemists who intend to perform experiments at the scale of one atom must be aware that the presence of any solid phase (walls of capillary tubes, vessels, etc.) can perturb the experimental system. It is important to check that these edge effects are negligible at tracer level before performing experiments at the scale of the atom [11]. The following section describes experimental techniques used in SHE chemistry. 

Naming becomes more difficult if the molecular complexity of a compound changes when it is fused or vaporized. Aluminum chloride is an ionic solid but vaporizes to dimeric molecules the solid is then aluminum trichloride whereas the vapor is dialuminum hexachloride. Similar considerations hold also for FeCl3, P2O5, and a number of additional compounds. Likewise, phosphorus(V) chloride is an appropriate name for PCI5 in the vapor state, but an ideal system of nomenclature should find some way of indicating that the solid consists of equal quantities of PCl and PClJf ions. It is a little difficult to decide just how much information about the structure of a compound must be included in its name before this name is to be considered adequate. 

Figure 14 A model calculation of the 2D-IR spectra of a idealized system of two coupled vibrators. The frequencies of these transitions were chosen as 1615 cm-1 and 1650 cm-1, the anharmonicity as A = 16 cm the coupling as = 7 cm and the homogeneous dephasing rate as T2 = 2 ps. The direction of both transitions as well as the polarization of the pump and the probe pulse were set perpendicular. The spectral width of the pump pulses was assumed 5 cm-1. The figure shows (a) the linear absorption spectrum and (b) the nonlinear 2D spectrum. In the 2D spectra, light gray colors and solid contour lines symbolize regions with a positive response, while negative signals are depicted in dark gray colors and with dashed contour lines.

In this system, the continuous injection of the feed and the continuous collection of A and B from the extract and the raffinate is an ideal system which provides the maximum efficiency of the adsorbent. A and B are recovered diluted in the desorbent but can be obtained pure after distillation. From a practical point of view, the true moving bed concept would be extremely difficult to implement. Although recycling the liquid would be feasible, the circulation of a solid packing from the bottom to the top of the fixed column... 

In addition to the physical state of reactants, it should be remembered that the ideal behavior is encountered only in the gaseous state. As the polymerization processes involve liquid (solution or bulk) and/or solid (condensed or crystalline) states, the interactions between monomer and monomer, monomer and solvent, or monomer and polymer may introduce sometimes significant deviations from the equations derived for ideal systems. The quantitative treatment of thermodynamics of nonideal reversible polymerizations is given in Ref. 54. 

Interstage Mixing EfFiciencies Mixing or stage efficiencies rarely achieve the ideal 100 percent, in which solute concentrations in overflow and underflow liquor from each thickener are identical. Part of the deficiency is due to insufficient blending of the two streams, and attaining equilibrium will be hampered further Iw heavily flocculated solids. In systems in which flocculants are used, interstage effi-... 

Recent papers by Othmer ( 7) and Caram and Scriven (8) have pointed out that uniqueness is characteristic of ideal systems whereas for non-ideal systems a solution may occur at the global minimum (most stable equilibrium point) but it also may occur at a nonunique local minimum. For applications in aquatic chemistry the problem of nonuniqueness is particularly important in the interpretation of solid precipitation and dissolution processes. 

As an example, we briefly describe here the observations made for Laj. Cax-Fe03. In this solid solution system, a single perovskite-brownmillerite intergrowth structure ( = 3) is foimd at composition x = % [224]. The intergrowth structure exhibits ideal stoichiometry, i.e. LaCa2Fe30g, at reduced oxygen partial pressures near the minimum observed in the electrical conductivity [199]. For any other composition, a disordered intergrowth is observed. 

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