Site density calculations, determination

Consider some specific TST examples. First, confidence in the TST method is strengthened if site densities calculated are the same as those determined by other methods. Glasstone, Laidler, and Eyring in their 1941 book ( ) report 15 TST calculations for several different kinds of reactions—adsorption, desorption, and various other unimolecul an bimolecular reactions. They assumed a site density of 10 cm in each case. In 12 of the 15 cases this assumption was justified. This is one kind of confirmation of TST validity. 

Here AN is the difference in water number, and it can be computed by integrating the deviation of the oxygen site local density to the bulk one, i.e., Pq (r) — Pq, over the system volume. Note that the system volume should be sufficiently large so that the local density of water at the edge of system recovers to bulk water density. The solvation free energy can finally be calculated by injecting the local site density profile determined by Eq. (59) into above equation. 

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

Surface site densities used in the computation of the oxide site concentrations presented in this paper were determined by either rapid tritium exchange or acquired from published values (18). Reported total site densities for hydrous metal oxides show relatively little variation generally they range by less than a factor of 3. Since [M], [SOM], [H] and x are known or can be determined from experimental data, uncertainties in estimates of the total site concentration are directly translated into uncertainties in the calculated partitioning coefficient. 

To calculate L values using Table I we need forward reaction rates, the number of molecules converted per unit area per second. These reaction rates are most easily obtained at low conversions. But frequently—often for practical purposes—high conversions are reported. We have developed a method for the determination of site densities in a certain class of systems even though the conversion and back reaction are both large. Also, the method can be used under certain conditions even if product isomerization is involved. We shall describe the theory here and in Section III,B apply it to the isomerization of 1-butene to cis- and trans-2-butene over silica-alumina. Although in this article we do not use this method to analyze any other systems, we present it in some detail because it may have potential for further use. 

For Example 9 the order is 0.7, suggesting a model for which the logL value is between that for Step 2 (0.5 order, log L = 15) and Step 1 (1.0 order, log L = 22). Thus, the rate-determining step may be a reaction on a partially filled surface. Since the log L value calculated in this way for 0.7 order is rather large, some surface mobility and/or rotation is indicated. The zero-order reactions of Examples 11 and 19 are clearly surface reactions for which expected site densities are obtained. For Example 11 Tottrup (25) suggested that the rate-determining step is C-O scission in adsorbed CO. 

In Section II,B,8 we discussed the question of determining site densities using high-conversion data. We developed a method applicable in the inter-conversion of three isomers when there is a common surface complex for the three possible reactions. We have tested this method using the conversion of 1-butene to cis- and rrans-2-butene over silica-alumina, a system that, according to Hightower and Hall, proceeds through a common surface complex (111). Their conclusion has been confirmed experimentally (112) and by semiempirical quantum-chemical calculations (113). 

Wang and Rikvold [60] have applied ab initio total-energy density-functional methods in combination with supercell models to calculate the c(2 x 2) structure of bromide adsorbed on Au(lOO) and Ag(lOO) surfaces. The preferred bonding sites have been determined. The calculations have shown that bromide favorably binds the bridge site on the Au(lOO) surfaces. These results explain experimental observations that adsorption of bromide on the Au(lOO) and Ag( 100) surfaces proceeds via different bonding configurations. 

The best value for the effective molecular cross-sectional area, o(Kr), of krypton in the BET monolayer at 77 K has been under discussion for many years. In their original work on krypton adsorption, Beebe et al. (1945) recommended the value 0.195 nm2 for o(Kr) and this empirical value is still used by many investigators. For the adsorption of krypton on graphitized carbon, Ismail (1990, 1992) gives preference to the value molecular area calculated from the liquid density and determined by X-ray scattering. This, of course, implies that Kr and N2 molecules undergo localized adsorption on the same sites. For ungraphitized carbons, Ismail (1992) recommends cr(Kr) = 0.214 nm2. 

In this equation the Rate is the molar TOF of the reaction, moles of product formed/mole of metal catalyst/unit time. The terms in [ ] are the STO measured site densities given in moles of site/mole of metal. The specific site TOFs, A, B and C, have units of moles of product/mole of site/unit time. Of these factors, the site densities are available from an STO characterization of the catalyst and the Rate is determined for the specific reaction nm over the STO characterized catalyst. When a series of at least three STO characterized catalysts is used for the same reaction, run under the same conditions, the specific site TOFs can be calculated from the simultaneous equations expressed as in Eqn. 3.6. When this approach was used in the hydrogenation of cyclohexene over a series of seven Pt/CPG catalysts specific site TOF values for the Mr and MH sites were found to be 2.1, 18.2 and 5.2 moles of product/mole of site/second, respectively.21 Not surprisingly, that site with the weakly held hydrogen was the most active and that on which the hydrogen was strongly held was the least active. 

Other methods—indirect, but not utilizing chemisorption— have been used to determine site densities. In 1958 the magnetic moment of surface atoms was taken to be a measure of the number of catalytically active atoms ( ). The number of surface free valencies was used for the same purpose in 1964 ( ). In 1965 Mellor and coworkers ( ) oxidized KI over silica-alumina by chemical analysis the number of surface atoms capable of oxidation was determined and taken to be the site density. The number of hydrogen vibrations on the ZnO surface which catalyzed the dehydrogenation of isopropanol was used to calculate the site density (91). 

Number of active sites per volume of catalyst. For the calculation of site densities of Me-N-C catalysts, often two assumptions have to be made (1) the density is similar to other carbon-based catalysts (0.4 g/cm ) and (2) each metal atom is related to an active site. In some cases, authors determine the exact mass density and/or number of active sites, so that the value becomes more accurate. In all other cases, it is usually overestimated as not all metal atoms are associated with active sites. 

Particle-speciflc surface area It is necessary to know the area of the solid in contact with the liquid in order to estimate surface site concentrations (e.g., fi om specific surface site densities) or if speciation calculations are carried out on the basis of surface specific units, the experimental data pertaining to the surface must be transformed from molar concentrations using the specific surface area. For the constant capacitance model, the whole treahnent can be done on a mass-specific basis. In principle for this model, the specific surface area only has to be involved if surface-specific site densities can be evaluated. The specific surface area is usually measured by gas adsorption. For in situ methods, other probe molecules are used (e.g., EGME method). Furthermore, microscopic methods can be used to determine the shape and size of particles from particle size distributions, which can, for example, be obtained with setups for microelectrophoresis or with acoustophoretic methods, specific surface area can be calculated for either known or assumed particle geometries. Problems in... 

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