Stoichiometric compounds, calculating

Chapter S examines various models used to describe solution and compmmd phases, including those based on random substitution, the sub-lattice model, stoichiometric and non-stoichiometric compounds and models applicable to ionic liquids and aqueous solutions. Tbermodynamic models are a central issue to CALPHAD, but it should be emphasised that their success depends on the input of suitable coefficients which are usually derived empirically. An important question is, therefore, how far it is possible to eliminate the empirical element of phase diagram calculations by substituting a treatment based on first principles, using only wave-mecbanics and atomic properties. This becomes especially important when there is an absence of experimental data, which is frequently the case for the metastable phases that have also to be considered within the framework of CALPHAD methods. 

Thermodynamic modelling of solution phases lies at the core of the CALPHAD method. Only rarely do calculations involve purely stoichiometric compounds. The calculation of a complex system which may have literally 100 different stoichiometric substances usually has a phase such as the gas which is a mixture of many components, and in a complex metallic system with 10 or 11 alloying elements it is not unusual for all of the phases to involve solubility of the various elements. Solution phases will be defined here as any phase in which there is solubility of more than one component and within this chapter are broken down to four types (1) random substitutional, (2) sublattice, (3) ionic and (4) aqueous. Others types of solution phase, such as exist in polymers or complex organic systems, can also be modelled, but these four represent the major types which are currently available in CALPHAD software programmes. 

Again, providing the various H and S terms are temperature-independent, the solution remains exact and provides a rapid method of calculating the liquidus temperature. Equation (9.10) is generally applicable to any phase boundary between a solution phase and stoichiometric compound, so could equally well be used for solid-state solvus lines. 

The ZSA phase diagram and its variants provide a satisfactory description of the overall electronic structure of stoichiometric and ordered transition-metal compounds. Within the above description, the electronic properties of transition-metal oxides are primarily determined by the values of A, and t. There have been several electron spectroscopic (photoemission) investigations in order to estimate the interaction strengths. Valence-band as well as core-level spectra have been analysed for a large number of transition-metal and rare-earth compounds. Calculations of the spectra have been performed at different levels of complexity, but generally within an Anderson impurity Hamiltonian. In the case of metallic systems, the situation is complicated by the presence of a continuum of low-energy electron-hole excitations across the Fermi level. These play an important role in the case of the rare earths and their intermetallics. This effect is particularly important for the valence-band spectra. 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

A chemical formula tells the numbers and the kinds of atoms that make up a molecule of a compound. Because each atom is an entity with a characteristic mass, a formula also provides a means for computing the relative weights of each kind of atom in a compound. Calculations based on the numbers and masses of atoms in a compound, or the numbers and masses of molecules participating in a reaction, are designated stoichiometric calculations. These weight relationships are important because, although we may think of atoms and molecules in terms of their interactions as structural units, we often must deal with them in the lab in terms of their masses—with the analytical balance. In this chapter, we consider the Stoichiometry of chemical formulas. In following chapters, we look at the stoichiometric relations involved in reactions and in solutions. 

As early as 1943, Sommer (101) reported the existence of a stoichiometric compound CsAu, exhibiting nonmetallic properties. Later reports (53, 102, 103,123) confirmed its existence and described the crystal structure, as well as the electrical and optical properties of this compound. The lattice constant of its CsCl-type structure is reported (103) to be 4.263 0.001 A. Band structure calculations are consistent with observed experimental results that the material is a semiconductor with a band gap of 2.6 eV (102). The phase diagram of the Cs-Au system shows the existence of a discrete CsAu phase (32) of melting point 590°C and a very narrow range of homogeneity (42). 

Thermochemical data on the separate phases in equUibrium are needed to constmct accurate phase diagrams. The Gibbs energy of formation for a pure substance as a function of temperature must be calculated from experimentally determined temperature-dependent thermodynamic properties such as enthalpy, entropy, heat capacity, and equihbrium constants. By a pure substance, one generally means a stoichiometric compound in which the atomic constituents ate present in an exact, simple reproducible ratio. 

The Gibbs energy of stoichiometric compound, ApBg, is calculated using the equation ... 

Calculating Densities/Concentrations in Stoichiometric Compounds or Dilute Solutions... 

When calculating the mass density, molar concentration, or molar volume of a specific individual species that is present in combination with other species (e.g., in a compound or solution), further work is needed. If the material s composition can be expressed in terms of a single stoichiometric compound or formula unit, the approach is still fairly straightforward— it just requires application of the compound stoichiometry. Similarly, dilute solutions, where the solute species is present in very low concentrations relative to the host solvent, can be handled in a relatively straightforward manner by assuming that the host material s density is not affected by the presence of the solute species. 

As an example of the first case (dealing with one species in a stoichiometric compound), consider how to calculate the molar concentration of oxygen atoms in Si02 ... 

The KCl-NiCl2 system displays one near-stoichiometric compound KCl.NiCl2 which melts peritectlcally at -931 K. Figure 31.7 shows the calculated and experimentally determined phase diagrams ([14]) the agreement is good. 

APW method is used, the averaging is achieved by replacing the logarithmic derivative of the nonmetal wave function at the nonmetal muffln-tin radius in the Hamiltonian matrix elements for the stoichiometric compound by the average of the logarithmic derivatives of the nonmetal and the vacancy wave functions in the case of the substoichiometric phases. However, these calculations proved to be unsatisfactory in many respects, mainly because they did not take into account the changes in the local symmetry of the metal atoms adjacent to the vacancy. 

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