Occupied sites

The BET treatment is based on a kinetic model of the adsorption process put forward more than sixty years ago by Langmuir, in which the surface of the solid was regarded as an array of adsorption sites. A state of dynamic equilibrium was postulated in which the rate at which molecules arriving from the gas phrase and condensing on to bare sites is equal to the rate at which molecules evaporate from occupied sites. 

Nitric acid treatment lowered the methane uptake by about ten percent. This could be due to oxygen occupying sites within pores, but may be the result of weaker interaction between methane and an oxide surface as is observed for silica. Reduction of these treated carbons with hydrogen restored their original methane uptake. Clearly for methane storage, there is no advantage in modifying the carbon surface by nitric acid treatment. 

As examples of the successful application of the PIMC method outlined in Sec. IV D 1, we focus here on studies of crystal properties where either Ar, Ne atoms or N2 molecules occupy sites on an fee lattice. 

In addition to the treatment costs, it should be remembered that provision of treatment plant occupies site space, which may be more profitably used, and that the penalties for infringements of the consent standards are increasing. 

The quantities of main consumables used. Electricity, water and gas will be included in this item but of more importance to the plant layout are those items that occupy site space. Fuel oil is such an item, and it is... 

It is clear that if all the program counters arc unoccupied (i.e. are in the state I 0 >), nothing at all happens all terms in the Hamiltonian start out with an annihilation operator, and all states thus remain in the state 0 > for all time. If we assume that only one of the sites 0,1,... fc sites is occupied, however, we see that only one site will always be occupied, though not necessarily the same site at different times. If we think of the occupied site, say the first site 0, as a cursor, the Hamiltonian effectively moves the cursor along the program counter sites while the operators Ai operate on the register n. Feynman shows how, by the time the cursor arrives at the final site fc, the n register has been multiplied by the entire set of desired operators Tj, T2,..., A -... 

By far the most abundant phosphate mineral is apatite, which accounts for more than 95% of all P in the Earth s crust. The basic composition of apatite is listed in Table 14-2. Apatite exhibits a hexagonal crystal structure with long open channels parallel to the c-axis. In its pure form, F , OH , or Cl occupies sites along this axis to form fluorapatite, hydroxyapatite, or chlor-apatite, respectively. However, because of the "open" nature of the apatite crystal lattice, many minor substitutions are possible and "pure" forms of apatite as depicted by the general formula in Table 14-2 are rarely found. 

The free Sy molecule is of Cs symmetry but in its various solid allotropes it occupies sites of Ci symmetry [154]. In any case, in these point groups all fundamental modes are infrared and Raman active and no degeneracies occur. Four allotropes of Sy (a, p, y, S) have been identified by Raman spec-... 

In the monoclinic crystals of cyc/o-decasulfur the molecules occupy sites of C2 symmetry although the molecular symmetry is close to D2 [151, 158]. In-... 

Cyc/o-Undecasulfur Su was first prepared in 1982 and vibrational spectra served to identify this orthorhombic allotrope as a new phase of elemental sulfur [160]. Later, the molecular and crystal structures were determined by X-ray diffraction [161, 162]. The Sn molecules are of C2 symmetry but occupy sites of Cl symmetry. The vibrational spectra show signals for the SS stretching modes between 410 and 480 cm and the bending, torsion and lattice vibrations below 290 cm [160, 162]. For a detailed list of wavenumbers, see [160]. The vibrational spectra of solid Sn are shown in Fig. 23. 

The molecule S12, like Se, is of Dsd symmetry but in the soHd state it occupies sites of the much lower C211 symmetry [163]. Due to the low solubihty and the thermal decomposition on melting only solid state vibrational spectra have been recorded [2,79]. However, from carbon disulfide the compound Si2-CS2 crystallizes in which the S12 molecules occupy sites of the high Sg symmetry which is close to 03a [163]. The spectroscopic investigation of this adduct has resulted in a revision [79] of the earher vibrational assignment [2] and therefore also of the earlier force constants calculation [164]. In Fig. 24 the low-temperature Raman spectra of S12 and Si2-CS2 are shown. 

The site balance specihes that the number of empty plus occupied sites is a constant, Sq. Equality of the reaction rates plus the site balance gives four independent equations. Combining them allows a solution for while eliminating the surface concentrations [S], [AS], and [PS]. Substitute the various reaction rates into the site balance to obtain... 

This form suggests a Hougen and Watson mechanism in which silane and hydrogen atoms occupy sites that must also be used by the silicon being deposited. The primary disposition reaction can be complemented by dopant reactions involving compounds such as AsHs, PH3, and B2H6, which deposit trace amounts of the dopant metals in the silicon lattice. 

We can think of a heterogeneous catalyst as a collection of active sites (denoted by ) located at a surface. The total number of sites is constant and equal to N (if there is any chance of confusion with N atoms, we will use the symbol N ). The adsorption of the reactant is formally a reaction with an empty site to give an intermediate I (or more conveniently R if we explicitly want to express that it is the reactant R sitting on an adsorption site). All sites are equivalent and each can be occupied by a single species only. We will use the symbol 6r to indicate the fraction of occupied sites occupied by species R, making N6r the number of occupied sites. Hence, the fraction of unoccupied sites available for reaction will be 1 - 0r The following equations represent the catalytic cycle of Fig. 2.7 ... 

Frequently, adsorption proceeds via a mobile precursor, in which the adsorbate diffuses over the surface in a physisorbed state before finding a free site. In such cases the rate of adsorption and the sticking coefficient are constant until a relatively high coverage is reached, after which the sticking probability declines rapidly. If the precursor resides only on empty surface sites it is called an intrinsic precursor, while if it exits on already occupied sites it is called extrinsic. Here we simply note such effects, without further discussion. 

Snyder and Soczewinski created and published, at the same time, another model called the S-S model describing the adsorption chromatographic process [19,61]. This model takes into account the role of the mobile phase in the chromatographic separation of the mixture. It assumes that in the chromatographic system the whole surface of the adsorbent is covered by a monolayer of adsorbed molecules of the mobile phase and of the solute and that the molecules of the mobile phase components occupy sites of identical size. It is supposed that under chromatographic process conditions the solute concentrations are very low, and the adsorption layer consists mainly of molecules of the mobile phase solvents. According to the S-S model, intermolecular interactions are reduced in the mobile phase but only for the... 

The number of occupied sites can be reduced by a factor of two if the mapping procedure discards every second carbon atom in the chains. C-C bonds no longer appear discretely in the model of the chain. This change in the lattice is depicted in Fig. 4.2.A and 4.2.B. The light and dark sites in Fig. 4.2.A describe a diamond lattice, where the light sites alternate with the dark sites. Only the dark sites are retained in the lattice depicted in Fig. 4.2.B. The lattice depicted in Fig. 4.2.B is well known. It is identical to the fee (face centered... 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

It has been proposed that the precursor state [81, 82] for the adsorption-desorption reaction consists of weakly physisorbed CO. This can be CO sitting on an occupied site (COad-CO) or on an sterically unfavorable Pt site. According to Ertl [81], the desorption process occurs through a trapping mechanism on such sites if the surface is saturated by chemisorbed CO the desorption channel involves either a COad-CO potential well or a Pt-CO attractive well which is sterically weakened by the presence of pre-absorbed CO . 

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

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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