Thermodynamics site fractions

In some thermodynamic models there are also potential minima associated with different site occupations, even though the composition may not vary, e.g., a phase with an order/disorder transformation. This must be handled in a somewhat different fashion and the variation in Gibbs energy as a function of site fraction occupation must be examined. Although this is not, perhaps, traditionally recognised as a miscibility gap, there are a number of similarities in dealing with the problem. In this case, however, it is the occupation of sites which govern the local minima and not the overall composition, per se. 

We wish here to obtain the thermodynamic equations defining the liquidus surface of a solid solution, (At BB)2, ). It is assumed that the A and atoms occupy the sites of one sublattice of the structure and the C atoms the sites of a second sublattice. For the specific systems considered here Sb and play the role of C in the general formula above. It is also assumed that the composition variable is confined to values near unity so that the site fractions of atomic point defects is always small compared to unity. This apparently is the case for the solid solutions in the two systems considered. Then it can be shown theoretically (Brebrick, 1979), as well as experimentally for (Hgj CdJ2-yTe)l(s) (Schwartz et al, 1981 Tung et al., 1981b), that the sum of the chemical potentials of A and C and that of and C in the solid are independent of the composition variable y ... 

If in addition to a thermodynamic driving force, a system has kinetic mechanisms available to produce a phase transformation (e.g., diffusion or atomic structural relaxation), the rate and characteristics of phase transformations can be modeled through combinations of their cause (thermodynamic driving forces) and their kinetic mechanisms. Analysis begins with identification of parameters (i.e., order parameters) that characterize the internal variations in state that accompany the transformation. For example, site fraction and magnetization can serve as order parameters for a ferromagnetic crystalline phase. 

All real crystals above 0 K contain point defects which are thermodynamically inherent [21,22]. In a monatomic crystal, the simplest defects are the vacancy, a lattice site that is empty, and the interstitial atom, an atom on an interstitial site in the lattice. The equilibrium concentration of these defects is thermally controlled and has an exponential dependence on temperature. For example, the site fraction of vacancies, c in a pure monatomic crystal is given by ... 

The solution thermodynamics is formulated in terms of reduced thermodynamic fxmctions for the mixture. A new parameter similar to the Flory X is introduced X12. To accoxmt for molecules of different size and shape, a site fraction is introduced for intermolecular interactions 0. The polymer is divided into r isometric parts, with size chosen so that one isometric unit corresponds to one solvent molecule. For an oligomeric solvent, more than one isometric unit can be assigned to the solvent. Each isometric unit is characterized by s intermolecular contact sites. Corresponding to each intermolecular contact site there is an energy of interaction -tj. For a two-component liquid mixture, an exchange interaction parameter is defined as ... 

The first two types of stmcture elements are normal elements of the solid, while the others are native point defects. In general, a given solid contains several types of defects, which will be as many components in the thermodynamic sense of the term, and will form a solution with the normal elements. In practice, the problem boils down to the superposition of the equilibria of a base of the vector space. Usually, the defects are very dilute in comparison to the normal elements, so that they can be considered to be solvents with constant activity and the activities of the defects can be considered equal to their site fractions. 

The application of (3) to (6) gives the expression for the thermodynamic mole fraction of a component in the phase. However, in the multi-site case, this presents a problem. Consider a phase in which elements A and B mix randomly on site 1 and elements C and D mix randomly on site 2, A phase of some particular composition can be represented as a block diagram ... 

A MC study of adsorption of living polymers [28] at hard walls has been carried out in a grand canonical ensemble for semiflexible o- 0 polymer chains and adsorbing interaction e < 0 at the walls of a box of size C. A number of thermodynamic quantities, such as internal energy (per lattice site) U, bulk density (f), surface coverage (the fraction of the wall that is directly covered with segments) 9, specific heat C = C /[k T ]) U ) — U) ), bulk isothermal compressibility... 

Moreover, the use of heat-flow calorimetry in heterogeneous catalysis research is not limited to the measurement of differential heats of adsorption. Surface interactions between adsorbed species or between gases and adsorbed species, similar to the interactions which either constitute some of the steps of the reaction mechanisms or produce, during the catalytic reaction, the inhibition of the catalyst, may also be studied by this experimental technique. The calorimetric results, compared to thermodynamic data in thermochemical cycles, yield, in the favorable cases, useful information concerning the most probable reaction mechanisms or the fraction of the energy spectrum of surface sites which is really active during the catalytic reaction. Some of the conclusions of these investigations may be controlled directly by the calorimetric studies of the catalytic reaction itself. 

In all above mentioned applications, the surface properties of group IIIA elements based solids are of primary importance in governing the thermodynamics of the adsorption, reaction, and desorption steps, which represent the core of a catalytic process. The method often used to clarify the mechanism of catalytic action is to search for correlations between the catalyst activity and selectivity and some other properties of its surface as, for instance, surface composition and surface acidity and basicity [58-60]. Also, since contact catalysis involves the adsorption of at least one of the reactants as a step of the reaction mechanism, the correlation of quantities related to the reactant chemisorption with the catalytic activity is necessary. The magnitude of the bonds between reactants and catalysts is obviously a relevant parameter. It has been quantitatively confirmed that only a fraction of the surface sites is active during catalysis, the more reactive sites being inhibited by strongly adsorbed species and the less reactive sites not allowing the formation of active species [61]. 

Stimpfl M., Ganguly J., and Molin G. (1999) Fe +-Mg order-disorder in orthopyroxene equilibrium fractionation between the octahedral sites and thermodynamic analysis. Contrib. Mineral. Petrol. 136, 297-309. 

The thermodynamic incompatibility of many of the solid phases present with each other as well as their local environment, results in formation of secondary minerals. Although the secondary materials may comprise only a small volume fraction of the waste, they (1) tend to increase in amount with time, as weathering processes proceed, (2) typically form at grain surfaces and are thus physically liable to react with percolating gas or liquids, and (3) may exhibit sites suitable for sorption or crystallo-chemical incorporation of trace elements (see Donahoe, 2004). Frequently observed secondary minerals include jarosite and ettringite the former is known to sorb ions such as Mn and As, whereas ettringite can form solid solutions, in which SO4 is replaced by Cr04 (Kumarathasan et al. 1990). 

Differences in Afor different AB5Hn compounds compared with A for CeCosHs are listed in Table III. The values of these numbers (see Table III), calculated using the fractional site occupations for the 0 phase, can be compared with the experimentally determined entropy differences listed in Table I. The calculated configurational entropy differences (see Table III) agree satisfactorily with the experimental data (see Table I) currently available for seven ABsHn compounds. Structures of some ABsHn compounds deduced from neutron diffraction data (4) are listed in Table I. For compounds whose structures have not been determined, the occupation numbers listed in Table III are in best agreement with the thermodynamic data. 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

Symmetry is represented by the elements of a (mathematical) group and thus cannot change continuously. The a-0 phase transition therefore occurs at a distinct temperature. Let us now assume that we have identified an extensive thermodynamic variable which can distinguish states between the a and 0 phases. We call it an order parameter (/ ). For a quantitative description of order-disorder or continuous displacive processes, the order parameter is normalized (0< s 1). For example, if we regard the classic 0-0 brass transition, tj is defined as (2/Cu -1), where /Cu is the fraction of Cu atoms which occupy the (0,0,0) sites of the (Cu,Zn) bcc structure. 

The NMR determinations of the site-specific hydrogen isotope ratios at natural deuterium abundance permitted one to assess primary and secondary thermodynamic fractionation factors in exchange reactions avoiding the synthesis of selectively labelled reagents and their degradations691. 

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