Solid-solution interactions equations

Compartmental soil modeling is a new concept and can apply to both modules. For the solute fate module, for example, it consists of the application of the law of pollutant mass conservation to a representative user specified soil element. The mass conservation principle is applied over a specific time step, either to the entire soil matrix or to the subelements of the matrix such as the soil-solids, the soil-moisture and the soil-air. These phases can be assumed in equilibrium at all times thus once the concentration in one phase is known, the concentration in the other phases can be calculated. Single or multiple soil compartments can be considered whereas phases and subcompartments can be interrelated (Figure 2) with transport, transformation and interactive equations. 

In this way and by numerical evaluation, Driessens (2) proved that the experimental activities could be explained on the basis of substitutional disorder, according to Equation (27), within the limits of experimental error. It seems, therefore, that measurements of distribution coefficients and the resulting activities calculated by the method of Kirgintsev and Trushnikova (16) do not distinguish between the regular character of solid solutions and the possibility of substitional disorder. However, the latter can be discerned by X-ray or neutron diffraction or by NMR or magnetic measurements. It can be shown that substitutional disorder always results in negative values of the interaction parameter W due to the fact that... 

A statistical thermodynamic equation for gas adsorption on synthetic zeolites is derived using solid solution theory. Both adsorbate-adsorbate and adsorbate-adsorbent interactions are calculated and used as parameters in the equation. Adsorption isotherms are calculated for argon, nitrogen, ammonia, and nitrous oxide. The solution equation appears valid for a wide range of gas adsorption on zeolites. 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

Amandykov and Gurov [155,184] have extended Kikuchi s approach to the case when the TSM is used (QCA, R — 1). According to these authors, the expression for the correlation cofactor depends on the parameter of the AC interaction with the neighboring species. Therefore, this behavior exerts a significant effect on the concentration dependence of the diffusion coefficients. The potential of atoms interaction at any distances has been applied to the construction of the kinetic equation describing the diffusion decomposition of solid solutions [185]. 

The theoretical model [130] was developed as an extension of the classical theory of dipolar broadening in dilute solid solutions in the absence of exchange interactions [16]. It was suggested in [130] how to determine the dipolar part in the line width by subtraction of the calculated input of Heisenberg exchange interaction of pairs of exchange-coupled ions. Equations were written for the three cubic lattices as a function of ion s concentration for various numbers of cationic sites included in a sphere of radius Rc with the assumption that clustering effects were absent. The results were compared well with experimental data on Cr3+ in MgO powders. 

Equation 11.32 is used to model a single-phase liquid in a ternary system, as well as a ternary substitutional-solid solution formed by the addition of a soluble third component to a binary solid solution. The solubility of a third component might be predicted, for example, if there is mutual solid solubility in all three binary subsets (AB, BC, AC). Note that Eq. 11.32 does not contain ternary interaction terms, which ate usually small in comparison to binary terms. When this assumption cannot, or should not, be made, ternary interaction terms of the form xaXbXcLabc where Labc is an excess ternary interaction parameter, can be included. There has been httle evidence for the need of terms of any higher-order. Phase equilibria calculations are normally based on the assessment of only binary and ternary terms. 

Different cubic equations of states have so far been used to model the solid-liquid-fluid phase equilibrium and the solubility of an organic solid solute in the fluid or liquid phase. These equations are listed elsewhere (11), along with the procedures for evaluating the binary parameters and adjustable interaction constants in them. These cubic equations of state (EOS) can be summarized as having the following general form ... 

Even though little thermodynamic data are available for low temperature SSAS systems, the equations and techniques presented in this paper can be used to estimate the importance of solid-solution aqueous-solution interactions on the chemical evolution of natural waters. 

The solubility of mixtures of solids is a complex question which is discussed by Banneijee (6). If two solids do not form a solid solution equation (20) applies separately to each. Assuming low solubilities, the unlike molecules should exhibit negligible interaction in the aqueous phase and the water phase activity coefficients, yi should be the same as for pure solid. Under these conditions the solubihty of the mixture should be the sum of the (small) solubilities of the pure solids. If the solids form a solid solution, the equilibrium equations take the form ... 

The second category includes mixtures of high solid concentrations. Unlike dilute systems, particle-particle interactions cannot be neglected. Also, it is very difficult to describe the boundary conditions. Because solution of equation 10 is basically a boundary value problem, no rigorous solution is available for concentrated mixtures, except for ordered arrays. To overcome these problems, various approaches have been considered. In this chapter, approximate solutions based on Maxwell s theory and empirical formulas are discussed. 

We have seen in this chapter that the behavior of both gaseous and solid solutions can be described with equations of state fitted by regression analysis to observed data. The more rigorous (and complicated) of these equations can be extrapolated with care beyond the field of experimental data. The equations of state most commonly used have appropriate mathematical forms to describe observed P-T-V-X behavior. This is because they are based on reasonable, if simplistic, models of molecular interaction. 

It was discovered four decades later that these equations fit real data so well because they had the form of a cluster expansion such as (15.37) above. The first term in a virial equation always represents ideal behavior PV/RT = 1) in the second term, Bj represents the non-ideal contribution from pairwise interactions of molecules B3 gives the interactions of triples, and so on. The virial coefficients can be calculated from known interaction potentials, or, alternatively, can be used to estimate these potentials from observed P -V -T behavior. We observed in Chapter 16 that virial equations fit the non-ideal behavior of gaseous solutions very well, and in Chapter 15 we saw that the Margules equations used to describe non-ideal behavior within solid solutions also have the form of virial equations. It should not be too surprising that expressions with this general form also work well with aqueous electrolyte solutions. 

A) in hydrocarbon solutions, NMR spectra indicated a monomer-dimer equilibrium with the energy of dissociation = 12.8 kcal mol" (equation 109). From an electron diffraction study, [(Me3Si)2CH]2Sn in the gas phase has a V-shaped monomeric structure, with C-Sn-C = 97(2)°. The related stannylene (77) is monomeric in the solid state, with the shortest Sn- Sn distance of 7.4 A. The dark-red air- and moisture-sensitive compound (77) exists in a dimer-monomer equilibrium in solution. Two crystal modifications of [2,4,6-(F3C)3C6H2l2Sn have been reported one is monomeric and the other is a weakly associated dimer with a Sn-Sn distance of 3.639(1) A. There are Sn-F interactions (equation 110). 

The solubility and the enhancement factor are dependent on the interactions in the supercritical phase, but are also dependent on the properties in the solid phase. Equating the fugacities of the solid compound in both phases, and using the convention that component 2 is the solid solute. 

Liquid Solution Behavior. The component activity coefficients in the liquid phase can be addressed separately from those in the solid solution by direct experimental determination or by analysis of the binary limits, since y p = 1. Because of the large amount of experimental effort required to study a ternary composition field and the high vapor pressures encountered in the arsenide and phosphide melts, a direct experimental determination of ternary activity coefficients has been reported only for the Ga-In-Sb system (26). Typically, the available binary liquidus data have been used to fix the adjustable parameters in a solution model with 0,p determined by Equation 7. The solution model expression for the activity coefficient has been used not only to represent the component activities along the liquidus curve, but also the stoichiometric liquid activities needed in Equation 7. The ternary melt solution behavior is then obtained by extending the binary models to describe a ternary mixture without additional adjustable parameters. In general, interactions between atoms in different groups exhibit negative deviations from ideal behavior... 

Suddenly, as we peer beyond the casual solution chemistry equation, rather complex solid state chemistry is interacting with solution and gas phases processes. 

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