Interaction energy between an adsorbate

It was pointed out in Chapter 1 that it is usually assumed that the overall interaction energy between an adsorbate molecule and the adsorbent is given by the summation of the pairwise interactions. Furthermore, if the assemblage of discrete force centres in the solid can be treated as a continuum, the summation can be replaced by integration (Hill, 1952). In this case, the non-specific Lennard-Jones interaction energy between a single molecule and a semi-infinite slab of solid takes the 9-3 form (Steele, 1974) ... 

Considering the complexity of the electronic states, the calculation of chemisorption or physisorption interaction energies, between an adsorbate and a cluster may be performed applying one of the following criteria ... 

Here we illustrate an explicit solvation approach for estimating the effect of water on the relative stability of surface intermediates. We calculate the interaction energy between an adsorbate and a shell of water molecules according to ... 

We hypothesize this is a general phenomenon To a first approximation, the interaction energy between an adsorbate and the surface at a mixed site is simply an interpolation between the interaction energies of the individual components. Consequently, other combinations of metals are also expected to have N adsorption energies close to the optimum and thus higher activity than their constituents. The... 

The interaction energy between an adsorbate and an electrical field e can be written as... 

In the last section, we summarized the different contributions to the potential energy for the interactions between an adsorbate molecule (or atom) and an atom on the solid surface. To calculate the interaction energy between the adsorbate molecule and all atoms on the surface, pairwise additivity is generally assumed. The task is then to sum the interactions, pairwise, with all atoms on the surface, by integration. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

There is a long history of calculations of adsorption potentials for simple gases adsorbed on the exposed low index Miller planes of ionic crystals, especially alkali halides (see the review [26] and references therein). The total interaction potential energy between an adsorbed molecule and the surface of a solid is generally expressed as a sum of dispersion, repulsion, induction, and electrostatic contributions (see, e.g.. Ref. [27]) ... 

In Eigure 4.3, in turn, typical plots of a type III isotherm as a function of both the pressure and its logarithm are presented. It differs from type II in that there is no slope decrease as the quantity adsorbed reaches a monolayer amount. Figure 4.3c is a pictorial representation of the physical situation, showing multilayer adsorption the interaction energy of an adsorbate molecule with the surface is comparable to or lower than the interaction between adsorbate molecules in successive layers, so that the first layer is not complete (sometimes far from complete) when multilayer adsorption starts. In extreme cases, such as water on glass (McHaffie and Lenher 1925), the adsorption starts only when the pressure is near Py. This isotherm type is... 

The Kelvin-equation-based methods were found to perform reasonably well for macro-porus and some mesoporous materials. However, it was foimd that the classical approach does not hold trae for micropores, in which case the intermolecular attractive forces between the sorbate and sorbent molecules predominate over bulk fluid forces such as surface tension. The potential energy fields of neighboring sorbent surfaces are known to overlap when the pores are only a few molecular dimensions wide. This results in a substantial increase in the interaction energy of an adsorbed molecule [12], which is not accounted for by simple classical thermodynamic models such as the Kelvin equation. 

Loops and tails of an isolated adsorbed polymer chain assume a number of different configurations and they substantially determine the configurational entropy of the adsorbed polymer, while the interaction energy between trains and the surface determines the enthalpy of adsorption. 

In numerical calculations of adsorption energies, however, expression (7) is mostly used. It is assumed that the two last terms of Eq. (11) are counterbalanced by the contribution of the repulsion forces (see Sec. IV,4). Expression (7) gives the interaction energy between two atoms. In order to evaluate the adsorption energy, the interaction energies of the adsorbed atom with all individual atoms of the adsorbent should be calculated and added together. This addition is allowed, as the dispersion forces are, at a first approximation, additive. If a molecule instead of an atom is adsorbed, the summation should be made for all atoms of the molecule. In the latter case we may sometimes expect deviations from the additive law. Many molecules show different polarizabilities in different directions. If the position of an adsorbed molecule is fixed, the angles of its various axes of polarizability with respect to the surface enter into the calculations (25). If, however, the molecule rotates freely, which is often the case in physical adsorption, this correction is not necessary. 

The interaction energy between a free surface of a solid and an adsorbate molecule is rather lower than in a micropore, as a consequence of the overlap of the adsorption potential from the neighboring walls. This overlap leads to a strong adsorption of the gas by the micropore and, then, to an enhancement of the heat of adsorption. 

Depending on the type of interaction between an adsorbed particle and a solid state surface there are cases, where adsorption enthalpies can be calculated using empirical and semi-empirical relations. In the case of atoms with a noble-gas like ground-state configuration and of symmetrical molecules the binding energy (EB) to a solid surface can be calculated as a function of the polarizability (a), the ionization potential (IP), the distance (R) between the adsorbed atom or molecule and the surface, and the relative dielectric constants (e) (Method 9) [58-61] ... 

Consider, for instance, the case that each subsystem is "binary" in that it consists of two sites, 1 and 2, with different adsorption energies and, hence, different subsystem partition functions, called qj and q2 if only one molecule adsorbs. Let the maximum number of molecules on the double site be two and let there be an interaction energy between the molecules if there are two of them. The binary sites are Independent of each other. This model is a very primitive way to introduce surface heterogeneity and lateral interaction simultaneously. The elaboration runs as follows. In [1.5.38] q(0) = 1. q(l) = qj + and q 2) = qj where w is the interaction parameter as used in the FFG model. 

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