Concentrations site fractions

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

The site fractions of the Si-containing species are one site occupied by SiH4 out of a total of 32 sites, and two sites out of 32 occupied by Si2H4. The site fraction of open sites is 29/32 = 0.906. As is seen in Eq. 11.8, it is necessary to divide the site fraction of each species by the site occupancy number ok to convert to a molar concentration. The concentration of SiH4 (number per unit area) is equal to that of Si2H4. 

In a steady-state calculation, it must be the case that the surface species concentrations (or site fractions) are not changing with time, that is,... 

From the reaction mechanism above, derive an expression for the adsorbed ammonia site fraction, 6 = [NH3(ad)]/T. Make the simplifying assumption that kip(NH3) + k-1 >> 2[Si(s)] this is equivalent to assuming that reaction 1 is at equilibrium, and that reaction 2 makes a negligible difference in the adsorbed ammonia concentration. Determine the NH3(ad) site fraction for an ammonia pressure of 124 Pa. 

Krauss and Wilcke examined the TiC -photocatalyzed oxidation of 12 PCB congeners [and 20 polycyclic aromatic hydrocarbons (PAHs)] on various soil samples (four mineral topsoil horizons, six organic horizons, and four particle-site fractions in three different soils) [107]. When the Ti02/soil mixture was irradiated in the absence of H2O, no photooxidation of the chlorobiphenyls occurred. When slurried with water, however, chlorobiphenyl concentrations decreased by 40-50% after 48 hours of irradiation, while the PAH concentrations were unchanged. By way of contrast, PAHs and PCBs doped onto quartz sand diminished by 95-100% after 8 hours of photolysis. The pollutants are clearly more accessible to hydroxyl radicals on sand than on soil. It is also clear that the photooxidation occurred in the soil and not in solution. Thus, OH is generated on one surface (Ti02), diffuses in the water to the other surface (soil), where the oxidation occurs. 

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 ... 

Here, c+ and c are the site fractions of the cation and anion vacancies, respectively, Ks is the equilibrium constant for the formation of the Schottky pair, and gs (= g+ + g, the sum of the individual defect formation energies) is the Gibbs free energy to form the pair. In the bulk of a pure crystal, the condition of electrical neutrality demands that the concentrations of each defect in the pair are equal that is ... 

The effects of small changes in the molecular structure can be observed in the case of the related diarylamidines which are the nitrogen analogs of formic acid and which represent models for nucleic acids. In tetrahydrofuran, for N,N -di-(p-F-phenyl)amidine (DFFA) three forms were observed by NMR, a solvated s-cis-form and a solvated s-trans-form which is in fast equilibrium with a cyclic dimer in which a HH-transfer takes place [24] as illustrated in Fig. 6.43. Fortunately, at low temperatures, the s-cis- and the s-trans-forms were in slow exchange. The rate constants of the HH, HD and DD reactions were determined by dynamic H and i F NMR as a function of concentration, deuterium fraction in the mobile proton sites and of temperature. The dependence of the observed rate constants of the s-trans-form on concentration is depicted in Fig. 6.44. The solid lines were calculated using Eq. (6.39) from which the rate constants in the dimer as well as the equilibrium constants of the dimer formation could be obtained. The Arrhenius... 

It is worth emphasizing once more that both [/i] and p are dimensionless, whereas p and n have the units of 1 /m. The advantage of using site fractions, instead of actual concentrations, in the mass action expression is that the left-hand side of the mass action expression [for example, Eq. (6.23)] would be dimensionless and thus equal to exp -Ag/Arr. If other units are used for concentration, the K values have to change accordingly (see Worked Example 6.2). 

E vs. Aa seems to be most sensitive to product concentrations near the external surface of the catalyst and adsorption/desorption equilibrium constants. I c.surf. I d, surf, and 6>, directly affect the vacant-site fraction on the interior catalytic surface and the rate of reactant consumption. In the previous simulations, product molar densities near the external surface of the catalyst were varied by a factor of 50 (i.e., from 0.1 to 5), and 0, was varied by a factor of 20 (i.e., from 0.05 to 1). The effectiveness factor increases significantly when either 4 c,surf, I d. surf or 6i is larger. E vs. Aa is marginally sensitive to a stoichiometric imbalance between reactants A2 and B, but I B.sur ce was only varied by a factor of 4 (i.e., from 0.5 to 2). A four-fold decrease in the molecular weight of reactant B, which produces two-fold changes in 30b, effective and 5b, does not affect E. 

These equations are related because they express the occupancy of saturable binding sites as a function of either O2 or substrate concentration. The fraction of active sites filled, as reflected in VTVmax. is analogous to Y, the degree of myoglobin saturation with oxygen [S] and PO2 are the concentrations of substrate and O2, respectively and and P50 are substrate or O2 concentrations at half-maximal saturation. 

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. 

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 ... 

Here, 0/ and a are respectively the steady-state concentrations of surface species and the relative label concentrations (isotope fractions) in them rf is the rate of a chemical reaction step y is the number of label atoms transferring from one substance to another in elementary reaction step (an analog of stoichiometric coefficient), P is a dimensionless coefficient equal to LW/VNa, where V is the gas-phase volume (crn ), L is the number of active sites per gram of catalyst (mol/g), W is the catalyst weight (g), Na is the number of gas molecules per unit volume (mol/cm ), and 0 Cf(x.j) is the operator depending on the mass transfer regime in the reactor ... 

In this equation, AG is the free energy change for the reaction, R is the gas constant, and T is the absolute temperature. For defect reactions, the use of Eqs. (7.12) and (7.13) requires that the concentration of the defects be expressed as site fractions. For low defect concentrations, the concentration of ions in their regular lattice sites is assigned a value of unity. For reactions involving gaseous species, the concentration of a gas is taken as its partial pressure. The application of the law of mass action is illustrated in the next two sections. 

The vacancy outflux is generated by the difference in equilibrium vacancy concentrations at two curved boundaries (Figure 7.1). The equation governing the vacancy concentration Cy (fraction of vacant sites) inside the shell is... 

It is important to note that the number of positive and negative charges must be compared on equal bases, namely as concentrations (numbers or moles per volume). One may in general not use site fractions since the concentrations of different types of sites are not necessarily equal. Thus, the electroneutrality condition can be expressed as... 

In the following, concentration will in most cases be expressed as the number of defects or atoms per molecule or site, i.e. in molar or site fractions, while we for some purposes need to use the number of defects per cm. As mentioned before, the value of the equilibrium constant depends on the units of concentration that are employed, but it is a simple matter to convert the values of the equilibrium constant from one system to another. 

When the concentrations of the defects are expressed in site fractions, or properly converted to mole fractions as here, Ks is related to the Gibbs molar energy, AGs, enthalpy change, AHs, and entropy change, ASs, of formation of pairs of doubly charged Schottky defects through the relation above ... 

The equilibrium constant may then be expressed as site fractions, which in our case immediately simplifies into an expression containing only volume or molar concentrations ... 

Since delocalised electrons in a band cannot be treated by classical statistics, we abandon the site fraction principle, and express all activities simply as concentrations, and denote the new equilibrium constant with a prime" ... 

Kroger-Vink notation [4]), the mobile oxygen vacancy concentration, expressed as a site fraction, ao, is the distance between equivalent sites, Vq is a characteristic lattice frequency, and vq exp y " is the jump frequency for the migrating ion. Hy and are the enthalpy and entropy of the migrating ion. 

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