Occupancy of the Sites

We generalize here the simple model of section 2.8.1 with respect to condition (1). Instead of single occupancy of each site we allow multiple occupancy. The sites are still independent in the sense that what happens on one site does not affect any other site. Until now we have assumed that there are no interactions (direct or indirect) between adsorbed molecules. Here, the lack of interaction still applies between any pair of sites or any pair of molecules on different sites. However, ligands within the same site can interact with each other. These interactions will be taken into account explicitly in Chapter 3. 

Let m be the maximum occupancy of any single site. We denote by q(i) the PF of a site containing i adsorbed particles q(i) contains both the internal PFs of the i ligands and any interactions between the ligands and the site and among the ligands themselves. 

As in the simple model, we assume again that the M sites are identical, independent, and distinguishable, and the total number of adsorbed ligands is N. For any configuration of the system, let = 0, 1. m) be the number of sites containing exactly i molecules. 

By configuration of the system we mean a particular distribution of N molecules on the M sites characterized by the vector a = ( o,, , m)- (Note that we can also talk 

For a specific configuration a the number of ways we can distribute N particles on M sites is 

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The site preference of several transition-metal ions is discussed in References 4 and 24. The occupation of the sites is usually denoted by placing the cations on B-sites in stmcture formulas between brackets. There are three types of spinels normal spinels where the A-sites have all divalent cations and the B-sites all trivalent cations, eg, Zn-ferrite, [Fe ]04j inverse spinels where all the divalent cations are in B-sites and trivalent ions are distributed over A- and B-sites, eg, Ni-ferrite, Fe Fe ]04 and mixed spinels where both divalent and trivalent cations are distributed over both types of sites,... 

Recently, similar but not identical, lattice models of water have been proposed by Fleming and Gibbs 61) and by Bell ezK In both models molecules are restricted to occupation of the sites of a body centered cubic array. The fundamental tetra-hedrality of the water-water interaction is accounted for in that four noncontiguous nearest neighbor points of the total of eight nearest neighbor points of a... 

A clear example of this effect is found in the Li4 3,Al,Si04 solid solutions (Garcia, Torres-Trevino and West, 1990), scheme 1, Fig. 2.2. Single phase solid solutions form over the entire range x = 0-0.5. At X = 0, in stoichiometric Li4Si04, all the Li sites are full and the conductivity is low. As x increases, one particular set of Li sites in the crystal structure starts to empty and is completely empty at x = 0.5, i.e. at Li2.5Alo.5Si04. The effect on the ionic conductivity of this variation is dramatic, as shown in Fig. 2.3. A broad conductivity maximum occurs around x = 0.25, at which composition the mobile Li" sites are half-full (Mc = 0.5, where is the occupancy of the site by ions). To either side, the conductivity becomes very small as x 0 (n,. 1) and x -> 0.5 (tie 0) respectively. ... 

In this section we characterize our system only by the pattern of occupations of the sites and overlook the conformational states. Hence, we write... 

T = (1 - 2p ) takes the values +1 or -1 depending on the occupancy of the site, n, is the total number of 7-type clusters and the sum runs over all such clusters that can be formed by combining sites for the entire crystal. 

The Mean-Field Approximation. The rate of a reaction when there are lateral interactions does not only depend on the reactants and temperature, but also on the occupation of the sites surrounding the sites where the reactants are found. As a consequence exact reactions rate equations contain probabilities of the occupation of clusters with many sites. We have already seen this for CO desorption in eqn. (6). To use this equation we have to express the 5-site probability on the right-hand-side in terms of 1-site probabilities i.e., the coverages). The simplest way to do this is to approximate a multisite probability as a product of 1-site probabilities. This is called a mean-field approximation. For the 5-site probability in eqn. (6) this would mean... 

The rate equations may be established by considering all distinguishable ways by which a given sequence can react or be formed by reaction at a site. There are two types of sites in the sequence, namely, terminal ones and sites which are at least one place removed from each end. For the latter, the near-neighbor reaction environment is completely specified by the sequence itself. For a terminal site, however, we must additionally specify the occupation of the site neighboring the sequence at the end considered. This can be either a 0 or a 1. It is thus clear that the rate equation of a sequence of length j will depend on the numbers of hierarchic sequences of length / + 1, j, and less. The infinite set of rate equations is thus required for a complete solution,... 

The value of ( — Coo) depends only on the relative probability of occupation of the sites in thermal equilibrium in a constant applied field. Provided that the difference in free energy between the rites, due to the field, is ven only by the potential energy difference gEd, the ratio of the probabilities of occupation of the sites is... 

The exclusion of selenium from the proteins of accumulator plants is thought to be the basis for their selenium tolerance. Their selenium metabolism is based mainly on water-soluble nonprotein forms such as selenium methylselenomethionine (Jacobs, 1989). The garlic odor characteristic of selenium-accumulator plants reflects the volatile organic compounds dimethylselenide and dime-thyldiselenide. Plants can suffer selenium toxicity as a result of selenium competition with essential metabolites for biochemical sites, replacement of essential ions by selenium, mainly major cations, selenate occupation of the sites of essential groups such as phosphate and nitrate, or selenium substimtion in essential sulfur compounds. 

The population parameters of the 3(g) site are listed as g, i.e. they represent fractional occupancies by Ni and Sn, refined assuming full overall occupancy of the site. 

The distribution of the Ce dopant is not uniform. Of the three cation sites in the parent compound, the first two are fully occupied by Bi atoms, while the third is empty. This results in a sequence of cation occupation on the [010] face as shown below in A and B. Site a is 92% Bi, 4% Ce, site b is 88% Bi, 12% Ce, and site c, normally vacant, is 4% Ce occupied. Examination of the oxide environment about site c indicates that there is insufficient room for a Ce cation. Occupation of the site must therefore result in some local disorder, the most likely manifestation thereof being a vacancy in site a (the distance between sites a and c are too short to allow simultaneous occupation). Note also that site a is not 100% occupied. Consequently, about 96% of the time the distribution of cations (Bi/Ce) in Bi, gCe 2 3 12... 

Preserved plant and animal remains document the diet of the inhabitants. The Indian occupants of the site used the river, lakes, and swamps in the area for aquatic resources and exploited the uplands for deer and nuts. Hickory nutshells are very common in the middens acorn shells also occur throughout the occupation area, along with the remains of hazelnuts and walnuts. These nuts are available in... 

Fig. 8. Simple model of order-disorder or displacive ferroelectric phase transition. Left, ferroelectricity by relative displacement of the anion and cation sublattices (a) displacive model, where r — 0 in the HTP and the atoms are translated by r/0 in the LTP. The order parameter is r. (b) Order-disorder model in the high-temperature phase, the ions are symmetrically disordered with equal probabilities p+ — p — 1/2 over two positions r — +rQ. In the low-temperature phase, the occupancies of the sites become unequal with probabilities p p +. The order parameter is the difference Ap — p+—p. The spontaneous polarization Psocr and PsccAp for the displacive model and order-disorder model, respectively. Right, ferroelectricity by alignment of molecular dipoles (c) displacive model in the HTP, all the molecules are aligned with a = 0 in the LTP, the molecules are rotated around the center of inversion with angles +a/0, the order parameter is a. (d) Order-disorder model. The spontaneous polarization Ppx ct and PsccAp for the displacive model and order-disorder model, respectively.

Figure 3. Simulation of the kinetic scheme from Figure 2 using the constants from Table 3. The top figure represents the kinetics of Ca dissociation which has a biphasic response (fast phase 663 s and slow phase 9 s ). The middle figure represents the titration of calmodulin by Ca ". The signal rising between 0 and 2-3 Ca " ions is associated with the occupancy of the sites from the COOH terminus and the other signal is associated with the occupancy of the N-terminal sites. The bottom figure is a Scatchard representation of the direct calcium binding isotherm.

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