Solids site fractions

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 ... 

We observe that the solid volume fraction Qs is closely related to the determinant of the microstretch Us and that the sum of volume fractions is equal or less to one depending on whether the pores are completely filled by the fluid inclusion or not 8f I 8S - I — f r < 1, where (3V is the volume fraction of the bare sites of matter in pores. Here, we suppose that the solid matrix is unsaturated, so fiv > 0. 

Figure 1. The fractional change in the minimum interaction energy as an adsorbed atom moves from a site center to the mid-point of a site edge is plotted against the ratio of crgg, the atomic diameter of an adsorbed atom, to (rsa, the diameter of an atom of the solid. The fractional change in energy, —E/em(0), was obtained by linear extrapolation of the theoretically computed change in eT near the site center...

In relations (11) and (12), i is a parameter which indicates the length of the chain of normal alkanes it is equal to the number of carbon atoms in the molecule, a and Pi are constants, C2, C3 and Cp are the site fractions occupied in the molecule by the condensed aromatic carbon atoms common to two cycles, three cycles and condensed atoms in angular position. The Cas parameter represents the fraction of the sites for the substituted aromatic carbons belonging to condensed cycles, which must be taken into account for the substituted derivatives of naphthalene. Table 1 shows the values of the parameters adjusted to the experimental data of the liquid-fluid and solid-fluid equilibria as a whole. 

The distribution of solid-phase fractions of Cu, Zn, and Cd of a few typical surface soils of South Australia, carried out following the fractionation scheme, is presented in Figures 11.3 and 11.4. The trace elements in these soils are dominantly (on an average 40% of Cu, 52.4% of Zn, and 33.4% of Cd) associated with the alumnosilicate mineral lattices, identified as residual fraction in the scheme, followed by the fraction associated with organic sites (on average, 32.4% of Cu, 28.0% of Zn, and 28.5% of Cd)... 

Atomic or ionic dijfusivity. In contrast to the defects, for an atom or ion in a regular site, other atoms. The probability of a site being vacant is simply equal to the mole or site fraction, denoted by A (lambda), of vacancies in that solid. Thus the frequency of successful jumps for diffusion of atoms by a vacancy mechanism is given by... 

The equilibrium concentrations of hydrogen atoms and molecules in oxides are perhaps not widely different between different oxides. As a first estimate, the entropy of the reaction from one mole of H2 gas to (H2)i or 2Hj in the solid may be expected to be -120 J moH K 1, implying that at very high temperatures (entropy controlled) we would have occupied site fractions of the order of 1 ppm of the interstitial sites for (H2)i and 0.1% for Hj. The enthalpy of dissolution, which determines how the concentrations develop with decreasing temperature, contains the bonding of the H species in the lattice or to existing defects and - in the case of atoms H - the breaking of the bond (435 kj mohi) of the H2 molecule. 

This is suitable for treatment of water containing considerable amounts of dissolved solids. The fraction of water rejected can be on the higher side (depends on the treated water quality desired) and one should check if there is any problem at site for disposal of the wastewater generated due to backwashing of sand filter, active... 

A final concentration-like unit that is sometimes needed is the site fraction, which is essentially a unitless occupancy factor that can typically be obtained from crystallographic information and defect models of a solid. A site fraction gives the fraction of sites (e.g., crystalline lattice sites) of a particular type in a material which are occupied by a particular species i. Thus, it is a ratio of the number of sites of a particular type (/ ) that are occupied by a certain species i divided by the total number of sites of that particular type in the material ... 

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 fact that Al belongs to a solid solntion rtrodiftes the expressions of the elerrrentary steps reactivities in which metal is irrvolved becarrse its concentration is not constant any more. On the assumption that the alloy constitutes a perfect solution, this introduced a rrew variable the corrtposition of alloy at t time, which we characterize by its mole fraction xi (we neglect the vacancy site fraction). 

Figure 20. A typical solid state fractional degradation curve. The curve is characterised by three areas (I) the induction phase, where there is growth of reactive sites (ii) the acceleration phase where the rate of reaction reaches a maximum (iii) the lag phase where the reaction finally terminates. a,/2 corresponds to the reaction half life.

In a review of the subject, Ubbelohde [3] points out that there is only a relatively small amount of data available concerning the properties of solids and also of the (product) liquids in the immediate vicinity of the melting point. In an early theory of melting, Lindemann [4] considered that when the amplitude of the vibrational displacements of the atoms of a particular solid increased with temperature to the point of attainment of a particular fraction (possibly 10%) of the lattice spacing, their mutual influences resulted in a loss of stability. The Lennard-Jones—Devonshire [5] theory considers the energy requirement for interchange of lattice constituents between occupation of site and interstitial positions. Subsequent developments of both these models, and, indeed, the numerous contributions in the field, are discussed in Ubbelohde s book [3]. 

While it is inherently probable that product formation will be most readily initiated at sites of effective contact between reactants (A IB), it is improbable that this process alone is capable of permitting continued product formation at low temperature for two related reasons. Firstly (as discussed in detail in Sect. 2.1.1) the area available for chemical contact in a mixture of particles is a very small fraction of the total surface (and, indeed, this total surface constitutes only a small proportion of the reactant present). Secondly, bulk diffusion across a barrier layer is usually an activated process, so that interposition of product between the points of initial contact reduces the ease, and therefore the rate, of interaction. On completion of the first step in the reaction, the restricted zones of direct contact have undergone chemical modification and the continuation of reaction necessitates a transport process to maintain the migration of material from one solid to a reactive surface of the other. On increasing the temperature, surface migration usually becomes appreciable at temperatures significantly below those required for the onset of bulk diffusion within a product phase. It is to be expected that components of the less refractory constituent will migrate onto the surfaces of the other solid present. These ions are chemisorbed as the first step in product formation and, in a subsequent process, penetrate the outer layers of the... 

Genuine iron(V) is a very rare oxidation state. In the preparation of iron oxides (and of other solid-state materials), the intended iron(V) disproportionates mostly into an iron(lll) fraction and two parts of an iron(Vl) fraction [276]. The only example of an iron(V) oxide for which the Mbssbauer parameters are known [185] is La2Li-FeOs- A low isomer shift of = —0.41 mm s was observed at room temperature with practically zero quadmpole splitting [277], which was taken as a proof that iron is accommodated in octahedral FeOg sites surrounded by six Li ions. Although repeatedly cited, it seems that the spectra have never been published, and the data must therefore be considered with care. 

Figure 11. The clinopyroxene-liquid partition coefficient for 4+ ions entering the M2-site shown as a function of the ionic radins of the trace cation. Changes in clinopyroxene composition along a solid solntion lead to small changes in the dimensions of M2 (ro Jw 2>), which can lead in turn to changes in the relative fractionation between 4+ ions of similar ioiuc radii, such as and Th" (shown as vertical lines). We contrast the partitioning behavior of a diopside-rich clinopyroxene = 1 044 A) and a...

Fig. 6 Simulational cooling curves of disorder parameters (solid lines) and mixing parameters (dashed lines) for 32-mers with different sets of energy parameters in a 64-sized cubic box (the concentration is fixed at 0.150). The mixing parameter is defined as the mean fraction of neighboring sites occupied by the solvent for each chain unit [84]... 

Fig. 15 Cooling curves of crystallinity (solid line) and demixing parameter of comonomers (dashed line). The latter is defined as the mean fraction of neighboring sites occupied by other comonomers around each comonomer. The cooling program is a stepwise increase of Ep/(k T) from zero with a step length of 0.002 and a step period of 300 Monte Carlo cycles, a The slightly alternating copolymer with a comonomer mole fraction 0.36 b the heterogeneous copolymer with a comonomer mole fraction of 0.36 [52]...

Fig. 1. (left panel) [Eu/Fe] as a function of [Fe/H]. Gray-scale indicates predicted distribution of stellar fraction. The r-process site is assumed to be SNe of 8 — IOMq. The average stellar distributions are indicated by thick-solid lines with the 50% (solid lines) and 90% confidence intervals (thin-solid lines). The current observational data are given by large circles, with other previous data (small circles). 

Figure 3. The effect of degree of polymerization on surface coverage (fractional site occupancy) at various polymer concentrations. The solid lines represent the present model and the symbols correspond to the theory of Scheutjens and Fleer. The parameter values are the same as in Figure 2.

One type of point defect that cannot be entirely eliminated from a solid compound is the substituted ion or impurity defect. For example, suppose a large crystal contains 1 mole of NaCl that is 99.99 mole percent pure and that the 0.01% impurity is KBr. As a fraction, there is 0.0001 mole of both K+ and Br ions, which is 6.02 X 1019 ions of each type present in the 1 mole of NaCl Although the level of purity of the NaCl is high, there is an enormous number of impurity ions that occupy sites in the lattice. Even if the NaCl were 99.9999 mole percent pure, there would still be 6.02 X 1017 impurity cations and anions in a mole of crystal. In other words, there is a defect, known as a substituted ion or impurity defect, at each point in the crystal where some ion other than Na+ or Cl- resides. Because K+ is larger than Na+ and Br is larger than Cl-, the lattice will experience some strain and distortion at the sites where the larger cations and anions reside. These strain points are frequently reactive sites in a crystal. 

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