Compound energy model

Equation (9.83) is also the basis for the compound energy model. The excess energy of the mixture is here represented by any type of equation, for example a power series [15, 16], Equation (9.83) has also been derived using the conformal solution theory after Blander [14] and as an extension of the molten salts models presented by Flood, Fprland and Grjotheim [17],... 

Non-stoichiometry in solid solutions may also be handled by the compound energy model see for example a recent review by Hillert [16]. In this approach the end-member corresponding to vacancies is an empty sub-lattice and it may be argued that the model loses its physical significance. Nevertheless, this model represents a mathematically efficient description that is often incorporated in thermodynamic representations of phase diagrams. 

The difference between the compound energy model and the simple two-sublattice model can be illustrated with two ternary intermetallic phases from the Al-Mg-Zn system. One of these two phases is known to contain a constant composition of 54.5 atomic percent magnesium and an extended homogeneity range of aluminum and zinc, corresponding to the formula Mg6(Al,Zn)5. However, no crystallographic data is available for this phase. Therefore, it is appropriately thermodynamically modeled by two sublattices, in which one sublattice is exclusively occupied by Mg, while A1 and Zn are allowed to randomly mix on the second sublattice (Liang et al., 1998). 

The intermetallic phase /3-SbSn has the rock-salt structure with one lattice almost exclusively occupied by antimony and the other by tin. Write the Gibbs energy expression using the compound energy model for a ternary phase including bismuth, where it is assumed that Bi goes preferentially into the mostly Sb sublattice. 

For each of the following oxides, decide if the compound energy model would be appropriate for modeling the Gibbs energy of formation. 

This is a stoichiometric compound, not a solid solution. Each type of cation exclusively occupies its own set of lattice sites the sublattices are not alloyed. The compound energy model does not apply. The free energy of formation must be measured. 

Surprisingly, there is no complete Calphad assessment of this system available in the literature. [1989Kau] calculated 6 isotherms for the ternary system by combining available thermodynamic descriptions for the binary systems. Some of the modeling was quite primitive by present standards, particularly with respect to the a phase, which was modeled as a substitutional solid solution as opposed to the compound energy model that is used currently. Nevertheless, the calculated and experimental isothermal section for 900°C were in reasonable quahtative agreement. Sadly, the same could not be said for the sections at 600°C. 

Fer] Femandez-Guillermet, A., Du, H., Thermodynamic Analysis of the Fe-N System using the Compound-energy Model with Prediction of the Vibrational Entropy , Z. Metallkd., 85(3), 154-163 (1994) (Phase Diagram, Theory, Assessment, 75)... 

The short summary of these results is that none of these model chemistries is very accurate at modeling this process in toto. Some of them achieve good results on either the component dissociation energies or the final value of AH, but no method does well for all of them. Not even QCISD(T) at the very large 6-311+G(3df) basis set Ls adequate. A compound energy method is required to successfully address thus problem. We will see such a solution in the next chapter. ... 

The literature/X-ray data was sparse (-1000), so the information content was amplified by generating vast numbers of alternative conformations (-10 ) with a 64-node Linux cluster. Typically, we were able to generate 1 distinct conformation per compound in an average of <1 per minute. At critical points during the drug discovery process, such as when new structural data was obtained, we retrained our free energy model to improve its performance. 

Evidence relevant to the phase stability problem has been given by Massalski (1989). The so-called compound energy formalism was constructed by Hillert and Staffansson (1970) in order to describe models of the thermodynamic properties of phases with two or more sublattices showing a variation in composition, which therefore belong to the class of solution phases. A review of this formalism and a summary of its applications have been recently published by Hillert (2001) and Frisk and Selleby (2001). 

Of special importance are tautomeric equilibria of two forms in which proton jumps lead to a change of the type of conjugation. Katritzky (72KGS1011 91H329) has developed a useful approach to estimating the empirical resonance energies from the constants of tautomeric equilibria which, in their turn, are determined from the pKa values of suitable compounds properly modeling individual tautomers. 

Likewise the Hubbard model the periodic Anderson model (PAM) is a basic model in the theory of strongly correlated electron systems. It is destined for the description of the transition metals, lanthanides, actinides and their compositions including the heavy-fermion compounds. The model consists of two groups of electrons itinerant and localized ones (s and d electrons), the hybridization between them is admitted. The model is described by the following parameters the width of the s-electron band W, the energy of the atomic level e, the on-site Coulomb repulsion U of d-electrons with opposite spins, the parameter V of the... 

By contrast, the Gibbs energy of an ordered sublattice phase is modeled using what is known as the compound energy formalism. For the simplest sublattice... 

Resonance Energies from AMI, MNDO and PN3 Calculations with Energies for individual isodesmic model compounds (energies reported in kcal/mol for lowest energy conformers only) ... 

M. Farcasiu, S. C. Petrosios, P. A. Eldredge, R. R. Anderson and E. P. Ladner. Modeling coal liquefaction. 3. Catalytic reactions of polyfunctional compounds. Energy and Fuels, 8, 920-924 (1994). 

K. Aimoto, I. Nakamura and K. Fujimoto. Transfer hydrocracking of heavy oil and its model compound. Energy and Fuels, 5, 739-744 (1991). 

Most low-energy nuclear reactions proceed via formation of a compound nucleus (eq. (8.5)). In the compound nucleus model that was proposed in 1936 by Bohr it is assumed that the energy of the incident particle and its binding energy are distributed evenly or nearly evenly to all nucleons of the target nucleus. The excitation energy of the compound nucleus is... 

A different class of iron compounds, viz. model haem complexes (Por)Fe(PMc3)(L ) (Por = substituted porphyrins), were studied by Walker et using a selective double-resonance technique in connection with isotopic enrichment of Fe. The complexes display a linear correlation between S Fe and 5 P of the phosphine ligand, and the variation of iron chemical shifts could be explained in terms of electronic d-d transition energies. 

For concentrated Bi compounds another model yields a similar temperature dependence, viz. mobile excitons with concentration nj and self-trapped excitons with concentration n and an energy difference AE, representing the thermal activation energy for exciton migration through the lattice. Unfortunately it is seldom checked whether the temperature dependence of the decay time in concentrated systems refers to an intrinsic property of an isolated luminescent centre or to an activation energy for migration. This, by the way, holds also for other compounds which have been discussed above. ... 

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