Density of sites

Of particular interest is the long-term behavior of voting-rule systems, which turns out to very strongly depend on the initial density of sites with value cr = 1 (= p). While all such systems eventually become either stable or oscillate with period-two, they approach this final state via one of two different mechanisms either through a percolation or nucleation process. Figure 3.60 shows a few snapshots of a Moore-neighborhood voting rule > 4 for p = 0.1, 0.15, 0.25 and 0.3. 

The gas phase partition function Qg s of the atom is the same however, since the atoms are immediately immobilized on a two-dimensional surface, we need to take the configuration of the adsorbed atoms into account in the transition state. Again we consider a surface containing M sites each with an area of a. The density of sites per area is Nq = M/A = 1 /a. The M sites are not necessarily free as some could be occupied already hence, the number of free sites will be M and 0 = M /M = (1-0a)-If we have N atoms adsorbed on these sites (we again use for the transition state Airmobile), the partition function for this system is given by... 

Figure 5.12 shows that the NO oxidation without catalyst is very slow, whereas thermodynamics shows that 100% conversion can be obtained up to about 200°C (473 K). Alumina alone has a very low activity. Higher the amount of cobalt, higher is the density of sites and higher the N02 production. Let us note that the reaction can be done without platinum. 

From the relation between Re/Rh and particle size it follows that the density of sites with the high Re/Rh value is about a factor 100 higher on crystals of approximately 200 A than on crystals of approximately 20 A. If the crystal is indeed free of defects, we cannot imagine any type of... 

At this point, the problem has been reduced to solving the surface chemistry equations to calculate the E-function. It is convenient to introduce dimensionless notations by normalizing all surface concentrations to the total density of sites Ns -... 

The reaction rate for the catalyst is the product of the density of sites and the turn over frequecy. 

Information about the number of surface sites required for an oxidation reaction, or activation of the reactant molecule, can be obtained by examination of the variation of the TOF with surface vanadia coverage. In general, reactions requiring only one surface site will exhibit a TOF that is independent of the surface vanadia coverage (surface density of sites) and reactions requiring multiple surface sites will exhibit a TOF that increases with the surface vanadia coverage (surface density of sites). From such an analysis, the number of surface vanadia sites required for various oxidation reactions is presented in Table 3. 

It is not a requirement that the number of sites of type n balance in a given reaction. If surface sites are not conserved, the density of sites T is not necessarily constant. The production rate r (mol/m2-s) for each surface phase is... 

General Crystal Surfaces. General surfaces possess a high density of sites where atoms from the vapor can be incorporated in the crystal and are therefore expected... 

The growth of spherical precipitates under diffusion-limited conditions has been observed in a number of systems, such as Co-rich particles growing in Cu supersaturated with Co (see Chapter 23). In these systems, the particles are coherent with the matrix crystal and the interfaces possess high densities of coherency dislocations, which are essentially steps with small Burgers vectors, The interfaces therefore possess a high density of sites where atoms can be exchanged and the particles operate as highly efficient sources and sinks. 

Having identified the existence of separate oxidation sites, it is desirable to determine the densities of sites which contribute to the different selectivities among different catalysts. Since the products of the thermal desorption experiments described above are directly correlated with the active sites, it is possible to measure the number of active sites by measuring the quantities of desorbed products, provided that each and every active site produces only one molecule of reaction product. To achieve this condition, it must be established that the adsorbate is fully equilibrated with the surface, that there is no multiple reaction per site during equilibration, that there is no readsorption and reaction of desorbed products, and that all reaction products are quantitatively determined. 

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

Chiche et a/.[56] have studied the oligomerization of butene over a series of zeolite (HBeta and HZSM-5), amorphous silica alumina and mesoporous MTS-type aluminosilicates with different pores. The authors found that MTS catalyst converts selectively butenes into a mixture of branched dimers at 423 K and 1.5-2 MPa. Under the same reaction conditions, acid zeolites and amorphous silica alumina are practically inactive due to rapid deactivation caused by the accumulation of hydrocarbon residue on the catalyst surface blocking pores and active sites. The catalytic behaviour observed for the MTS catalyst was attributed to the low density of sites on their surface along with the absence of diffusional limitations due to an open porosity. This would result in a low concentration of reactive species on the surface with short residence times, and favour deprotonation and desorption of the octyl cations, thus preventing secondary reaction of the olefinic products. 

Second, the neutral face, belonging to the (2116) family, contains four- and fivefold coordinated ions, aligned along rows 0.365 nm apart. Within each row, the four- and fivefold coordinated sites are located in pairs only 0.289 nm apart, and the distance between the pairs is 0.574 nm. The density of sites is 7.65 Cr ions/100 A2. The presence of fourfold coordinated ions suggests that these faces should be more reactive and more prone to reconstruction phenomena than the (0112) faces. 

Finally, the (1120) face is characterized by the highest density of sites (10.88 Cr ions/100 A2). These Cr3+ sites are all fivefold coordinated and present in pairs 0.265 nm apart, and the distance between the pairs is 0.415 nm. Because the Cr3+ ions are greatly shielded by the surrounding oxygen ions (which are located at an upper level), this surface appears quite homopolar, and it is expected to be less reactive or even unreactive with probe molecules. 

In the inset of Fig. 6c, the calculations were repeated for the same values of the parameters, but a stronger double layer (N= 2X1017 sites/m2, Ku = 1.0 M). The ion-dispersion forces have in this case only a minor effect, and the interaction can be well approximated by the traditional Poisson-Boltzmann approach, with slightly modified parameters (density of sites, dissociation constant). 

Cs+ selectivity can be controlled by varying the density of sites within catalyst pellets and the diameter of these pellets. The density of sites determines the reactant requirements and the pellet size controls the required path length of diffusing molecules. Such modifications affect the value of x, causing the performance of these catalysts to move along the curve in Fig. 20. The shape of the selectivity curve, however, depends only on the intrinsic readsorption rate constant (/Sr) and on the kinetic dependence of chain growth pathways on reactant pressure. 

An important aspect of the photorefractive effect is that the optical response of the material is nonlocal. In Figure 7, the position of the space charge field is displaced to the right of the initial excitation, in the direction of the applied electric field. In the case of a sinusoidal intensity pattern the phase shift between the optical excitation of charges and the electric field their movement produces is a parameter characteristic of a photorefractive material. It depends on the balance between the processes of drift and diffusion of mobile charges and on the number density of sites able to capture the mobile charges. 

The surface of chromia appears to be an ideal case for study at the present. By activation at increasing temperatures, one can vary the number of active sites from none to some maximum number. At a low density of sites, one can hope that the sites are well separated and noninteracting. One can compare chemisorption of various gases with specific catalytic activities at various levels of site densities and hope to gain information about site heterogeneity. This chapter reports a first approach to this objective. 

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