Surface site statistics

Quite different site densities are obtained if these assumptions are changed. Perez et al.13 have calculated the surface site statistics using a computer model which can simulate incomplete layers by removing atoms from complete shells. The atoms removed are those which have the smallest number of first and second nearest neighbours. Many more types of site are considered in the models used by Perez et al. However, one of the most interesting results of their calculations is to demonstrate that for all sites, apart from B2 sites, there are very pronounced oscillations in number as the particle size is increased. Figure 2 shows the variation in the number of B2, B3, and B4 sites, and Figure 3 shows the ratio of B3/B4 sites as a function of particle size. Any reaction which is controlled by this ratio will show activity maxima for particle diameters of 0.8 and 2.0 nm. On the other hand B2 and B2 sites are the ones most likely to catalyse structure insensitive reactions. 

More complex model reactions (selective butadiene hydrogenation) Apart from being accessible to surface spectroscopy, model catalysts also have the advantage that the nanoparticle morphology and surface structure can be accurately measured. This advantage allows the determination of the relative abundance of specific surface sites and calculation of surface site statistics, as shown, for example, in Table II. Knowledge of the exact number and type of available surface sites then allows calculation of more accurate (and perhaps more meaningful) turnover frequencies of catalytic reactions. 

The more complex selective 1,3-butadiene (BD) hydrogenation was also examined [56, 57]. Butadiene hydrogenation produces 1-butene, tranx-2-butene, cti-2-butene, and n-butane, with 1-butene as the desired product. Pd-Al Oj model catalysts with mean particle diameters of 2-8 nm were applied to examine size effects. The abihty to accurately determine the relative abundance of specific surface sites (such as terrace, edge, interface atoms, etc. cf. surface site statistics in Table 1, 2 of [51]) is a tremendous advantage of model catalysts. Knowledge of the exact number and type of available surface sites allows the calculation of more accurate turnover frequencies. 

Van Hardeveld R, Hartog E. 1969. Statistics of surface atoms and surface sites on metal crystals. Surf Sci 15 189-230. 

RuO2(110) exemplifies Langmuirian behaviour where the catalyst surface consists of equivalent sites statistically occupied by the reactants. This contrasts markedly with catalytic oxidation at metal surfaces, where oxygen transients, high surface mobility and island structures are dominant. The difference is in the main attributed to differences in surface diffusion barriers at metal and oxide surfaces. 

Suspension Model of Interaction of Asphaltene and Oil This model is based upon the concept that asphaltenes exist as particles suspended in oil. Their suspension is assisted by resins (heavy and mostly aromatic molecules) adsorbed to the surface of asphaltenes and keeping them afloat because of the repulsive forces between resin molecules in the solution and the adsorbed resins on the asphaltene surface (see Figure 4). Stability of such a suspension is considered to be a function of the concentration of resins in solution, the fraction of asphaltene surface sites occupied by resin molecules, and the equilibrium conditions between the resins in solution and on the asphaltene surface. Utilization of this model requires the following (12) 1. Resin chemical potential calculation based on the statistical mechanical theory of polymer solutions. 2. Studies regarding resin adsorption on asphaltene particle surface and... 

A weakening of the critical metal-oxygen bonds occurs as a consequence of the protonation of the oxide ions neighboring a surface metal center and imparting charge to the surface of the mineral lattice. The concentration (activity) of D should reflect that three of such oxide or hydroxide ions have to be protonated. If there is a certain numer of surface-adsorbed (bound) protons whose concentration (mol nr2) is much lower than the density of surface sites, S (mol 2), the probability of finding a metal center surrounded with three protonated oxide or hydroxide ions is proportional to (CJ/S)3. Thus, as has been derived from lattice statistics by Wieland et al. (1988), the activity of D is related to (C )3, and the rate of proton-promoted dissolution, Rh (mol nrr2 lr1), is proportional to the third power of the surface protonation ... 

The Langmuir isotherm is based on the assumption of localized adsorption. This means that an adsorbed molecule has such a high statistical preference for a certain surface site as to possess a negligible translational entropy in the adsorbed state. Localized adsorption is thus... 

The Langmuir adsorption isotherm can be derived [134,417] using the statistical thermodynamics techniques discussed in Chapters 8 and 9. The assumptions necessary are basically the same as were used in deriving the Langmuir adsorption isotherm in Section 11.4.1. That is, adsorption is assumed to occur on a fixed array of surface sites there is assumed to be no interaction between adsorbed species the particular sites that are filled are assumed to be random and adsorbed species are immobile, corresponding to a chemisorbed species. 

The additivity of the component current is based on the simplifying assumption that the anodic and cathodic processes are independent of each other and implies that, statistically, the electrode surface sites are indistinguishable for the representative reactions. 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

An alternative and elegant derivation of the BET equation is by a statistical mechanical treatment (Hill, 1946 Steele, 1974). The adsorbed phase is pictured as a lattice gas that is molecules are located at specific sites in all layers. The first layer is localized and these molecules act as sites for molecules in the second layer, which in turn act as sites for molecules in the third layer, and so on for the higher layers. As the surface is assumed to be planar and uniform, it follows that all surface sites are identical. It is also assumed that the occupation probability of a site is independent of the occupancy of neighbouring sites. This is equivalent to the assumption that there are no lateral interactions between adsorbed molecules. In accordance with the BET model, the probability for site occupation is zero unless all its underlying sites are occupied. Furthermore, it is assumed that it is only the molecular partition function for the first layer which differs from that for molecules in the liquid state. 

An exaggerated emphasis on heats of chemisorption has probably been harmful in the proper understanding of the role of chemisorption in surface phenomena. Thus the marked nonuniformity of all surfaces with respect to heats of chemisorption has led to rather elaborate treatments where models of surface heterogeneity (statistical distribution of energy sites) or, less successfully, specific forces of interaction between adsorbed species have been invoked to explain the non-Langmuirian adsorption isotherms. For instance the Frumkin isotherm can be obtained with a linear variation of heats of adsorption with coverage, and the Freundlich isotherm is attributed to an exponential variation of heats of adsorption. 

Hobza, P., J. Sauer, G. Morgeneyer, J. Hurych, and R. Zahradnik (1981). Bonding ability of surface sites in silica and their effect on hydrogen bonds. A quantum-chemical and statistical thermodynamics treatment. J. Phys. Chem. 85, 4061-67. 

There are two approaches to the equilibrium constants of surface reactions. Originally, the so-called intrinsic approach was developed [28]. For example, binding of a proton to a surface site was separated into two steps. In the first step the proton was transferred from the bulk of solution (aq) to the space in the vicinity of the surface site (int). This equilibrium was treated by Boltzmann statistics, so that the distribution was affected by the electrostatic potential of that space (( jnt). The second process was chemical binding with the surface site represented by so-called intrinsic equilibrium constant (Ki t) ... 

In spite of the difference in the underlying concepts and the forms of equations, Eqs. (3.3) and (3.4), both descriptions reflect the statistical sense of the rate constant. The latter statement is crucially important for better understanding of the problem existing in heterogeneous kinetics. Indeed, the above-mentioned theories are based on gas statistics and the given equations assume an equilibrium Maxwell-Boltzman distribution for gas species, which in the absence of reaction interact only via elastic collisions. If this can be considered as a satisfactory approximation for gas reactions at moderate temperatures and pressures discussed here (with some exceptions—see Section III.D), its applicability to the processes involving surface sites (i.e., elements of solid lattice) or adsorbed species is not so obvious. 

It is also important to notice that reactions similar to (12)—(16) must be taken into account from another point of view. Even in the presence of low-surface area catalysts, the total number of species in the gas phase inside the catalyst bed can be less than the number of surface sites. For instance, at 1 bar and 1,000 K the total number of species in 1cm3 of gas is 8 x 1018. It is less than the total number of sites on an MgO surface at 1 m2/g specific surface area and 1 g/cm3 bulk density ( 2 x 1019 sites/cm3). This means that even statistically gas species have a higher probability to collide with surface sites than with any other gas particle. 

---