Ideal behavior solid

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

Fig. 2.10. Certain high strength solids with low thermal conductivity show a loss or reduction of shear strength when loaded above the Hugoniot elastic limit. The idealized behavior of such solids upon loading is shown here. The complex, heterogeneous nature of such yield phenomena probably results in processes that are far from thermodynamic equilibrium.

A dependence of w upon composition must also be adduced in the case of the Fe-Ni solid solutions. Over the range from 0 to 56 at. per cent Ni, these solid solutions exhibit essentially ideal behavior,39 so that w 0. Since the FeNi3 superlattice appears at lower temperatures, either w is markedly different at compositions about 75 at. per cent Ni than at lower Ni contents, or w 0 for the solid solutions about the superlattice. Either possibility represents a deviation from the requirements of the quasi-chemical theories. 

Figure 8.23 (Solid + liquid) phase diagram for (. 1CCI4 +. yiCHjCN), an example of a system with large positive deviations from ideal solution behavior. The solid line represents the experimental results and the dashed line is the ideal solution prediction. Solid-phase transitions (represented by horizontal lines) are present in both CCI4 and CH3CN. The CH3CN transition occurs at a temperature lower than the eutectic temperature. It is shown as a dashed line that intersects the ideal CH3CN (solid + liquid) equilibrium line.

The data of Table III show that the surface layer of the solid particles is indistinguishable from pure fluorapatite in all equilibrations at x = 0.110, 0.190 and 0.435 and 0.595. However, some equilibrations at x = 0.763 and all at x = 0.868 do deviate significantly from the behavior of pure fluorapatite. A peculiar aspect is that the activity of fluorapatite becomes significantly larger than 1. Simutaneously, the activity of hydroxyapatite approaches unity. This would mean that at all values of x both activities would become smaller than 1, and thus an ideal behavior of the solid solutions would not explain the observed solubility behavior. 

There are many chemically reacting flow situations in which a reactive stream flows interior to a channel or duct. Two such examples are illustrated in Figs. 1.4 and 1.6, which consider flow in a catalytic-combustion monolith [28,156,168,259,322] and in the channels of a solid-oxide fuel cell. Other examples include the catalytic converters in automobiles. Certainly there are many industrial chemical processes that involve reactive flow tubular reactors. Innovative new short-contact-time processes use flow in catalytic monoliths to convert raw hydrocarbons to higher-value chemical feedstocks [37,99,100,173,184,436, 447]. Certain types of chemical-vapor-deposition reactors use a channel to direct flow over a wafer where a thin film is grown or deposited [219]. Flow reactors used in the laboratory to study gas-phase chemical kinetics usually strive to achieve plug-flow conditions and to minimize wall-chemistry effects. Nevertheless, boundary-layer simulations can be used to verify the flow condition or to account for non-ideal behavior [147]. 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

When the deviations from ideal behavior become large enough, the liquid mixture separates into two liquid phases. Cooling of this liquid mixture can result in the freezing of a solid, giving (liquid + liquid) and (solid 4-liquid)... 

At higher pressures, (liquid + liquid) equilibrium is not present and the (solid-I-liquid) equilibria line shows large positive deviations from ideal behavior, but no phase separation. It is appropriate to describe the equilibrium at these pressures by saying that the (liquid + liquid) equilibrium curve has disappeared below the (solid + liquid) line. 

Fig. 2.4. CMC of various hydrocarbon, fluorocarbon and partially fluorinated surfactants as a function of the fluorine-to-hydrogen ratio. CMCs are given relative to those of the hydrocarbon surfactants. Solid line is a suggested ideal behavior (From Ref.25 )...

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

Section 4 presents a variety of solid-gas surface processes adsorption, desorption, catalytic reaction, and surface diffusion. Non-ideal behavior of the systems is considered through the effective pair potentials of inter-molecular interactions. A wide circle of experimental data can be described on taking into account a non-ideal behavior of the surrounding medium. 

FIGURE 7.1 Solubility phase diagrams of diastereoisomeric salts, (a) Ideal behavior (b) end solid-solution behavior (c) full solid-solution behavior and (d) double salt formation. 

Frequently deviations from this ideal behavior are observed—for instance, as a result of end or even full solid-solution behavior (Figures 7.1b and 7.1c) or double salt formation (Figure 7.Id). In the former situation repeated (or fractional) crystallization is needed to obtain a diastereoiso-merically pure salt, whereas in the latter it is virtually impossible to obtain a pure salt. In addition to these physical constraints, cost and availability play a determining role in the actual application of resolving agents on an industrial scale. 

When TAGs in the liquid state are mixed, no changes in heat or volume are observed (Walstra et al., 1994). However, ideal behavior is not observed in the solid phase of milk fat (Timms, 1984 Walstra et al., 1994). As a result, the melting curve of milk fat does not equal the sum of its component TAGs (Walstra et al., 1994). Mulder (1953) proposed the theory of mixed crystal formation to explain the complex crystallization behavior of milk fat. Mixed or compound crystals contain more than one molecular species (Rossell, 1967 Mulder and Walstra, 1974). Mixed crystals form in natural fats, like milk fat, which are complex mixtures of TAGs (Mulder, 1953 Sherbon 1974 Walstra and van Beresteyn, 1975b Timms, 1980 ... 

The factor y /y <>> < represents the deviation from ideal behavior and may be a relative enhancement or reduction in the concentration of the solid. This approach is used to describe the doping distribution coefficient in Section 6.2.7. 

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. 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

A solid consists of a mixture of NaN03 and Mg(N03)2. When 6.50 g of this solid is dissolved in 50.0 g of water, the freezing point is lowered by 5.40°C. What is the composition of the solid (by mass) Assume ideal behavior. 

A solid mixture contains MgCl2 (molar mass = 95.218 g/mol) and NaCl (molar mass = 58.443 g/mol). When 0.5000 g of this solid is dissolved in enough water to form 1.000 L of solution, the osmotic pressure at 25.0°C is observed to be 0.3950 atm. What is the mass percent of MgCl2 in the solid (Assume ideal behavior for the solution.)... 

Forms of the Activity Coefficient.—The equations given above are satisfactory for representing the behavior of liquid solutes, but for solid solutes, especially electrolytes, a modified form is more convenient. In a very dilute solution the mole fraction of solute is proportional both to its concentration (c), i.e., moles per liter of solution, and to its molality (m), i.e., moles per 1000 g, of solvent hence for such solutions, which are known to approach ideal behavior, it is possible to write either... 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

A difficulty with the above scheme is that measurements carried out with various actual gases that approach ideal behavior will lead to slightly dilferent results. A better absolute standard is provided by the so-called triple point of water. As we shall see later, the coexistence conditions of water in the solid, liquid, and vapor state can occur only under a set of precisely controlled, invariant conditions determined by the physical characteristics of H2O. These conditions are completely reproducible all over the world. For consistency with the above absolute temperature scheme the triple point of water is assigned a temperature T (triple point of H2O) = 273.16 K = 7). Then any other absolute temperature is determined through the proportionality T = (P/Pt) 273.16, where P is the pressure at T and Pt is the pressure measured for He in equilibrium with water at its triple point. 

Physical adsorption (physisorption) occurs when an adsorptive comes into contact with a solid surface (the sorbent) [1]. These interactions are unspecific and similar to the forces that lead to the non-ideal behavior of a gas (condensation, van der Waals interactions). They include all interactive and repulsive forces (e.g., London dispersion forces and short range intermolecular repulsion) that cannot be ascribed to localized bonding. In analogy to the attractive forces in real gases, physical adsorption may be understood as an increase of concentration at the gas-solid or gas-liquid interface imder the influence of integrated van der Waals forces. Various specific interactions (e.g., dipole-induced interactions) exist when either the sorbate or the sorbent are polar, but these interactions are usually also summarized under physisorption unless a directed chemical bond is formed. 

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