Potential energy-distance function

A useful way to represent intmnolecular (and inte-particle) forces is via the potential energy-distance function, V(r), which is related to the intemolecu-lar force ... 

The Derjarguin, Landau, Verwey, Overbeck (DLVO) theory [23-25] has established the potential energy-distance relationship between two particles as a function of the characteristics of both the particles and the suspending solution. In natural systems, this approach requires compilation [26-28] of the major key physicochemical parameters that characterize the colloid material, including (a) colloid shapes... 

Figure 7. The potential-energy-distance relations (Morse functions with axo = 0.4) at different values of electrode potential.

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Figure 7-9 shows two mathematical functions used to describe the potential energy cuiwe for the variation of the distance between two bound atoms within a molecule. 

Figure 7-9. Variation of the potential energy of the bonded interaction of two atoms with the distance between them. The solid line comes close to the experimental situation by using a Morse function the broken line represents the approximation by a harmonic potential.

An alternative way to eliminate discontinuities in the energy and force equations is to use a switching function. A switching function is a polynomial ftmction of the distance by which the potential energy function is multiplied. Thus the switched potential o (r) is related to the true potential t> r) by v r) = v(r)S(r). Some switching functions are applied to the entire range of the potential up to the cutoff point. One such function is ... 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Fig. 1.2 The potential energy (r) of two isolated atoms as a function of the distance r between their centres. The curve for the potential energy (z) of a molecule as a function of its distance z from the surface of a solid is similar in general shape to this curve.

Here 0p and 0 correspond to the terms in r" and respectively in Equation (1.8) as already pointed out, these contributions are always present, whereas the electrostatic energies 0, and may or may not be present according to the nature of the adsorbent and the adsorptive. In principle. Equation (1.16) could be used to calculate the numerical value of the interaction potential as a function of the distance z of any given molecule from the surface of a chosen solid. In practice, however, the scope has to be limited to systems composed of a simple type of gas molecule and... 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

If we were to calculate the potential energy V of the diatomic molecule AB as a function of the distance tab between the centers of the atoms, the result would be a curve having a shape like that seen in Fig. 5-1. This is a bond dissociation curve, the path from the minimum (the equilibrium intemuclear distance in the diatomic molecule) to increasing values of tab describing the dissociation of the molecule. It is conventional to take as the zero of energy the infinitely separated species. 

The semiempirical methods combine experimental data with theory as a way to circumvent the calculational difficulties of pure theory. The first of these methods leads to what are called London-Eyring-Polanyi (LEP) potential energy surfaces. Consider the triatomic ABC system. For any pair of atoms the energy as a function of intermolecular distance r is represented by the Morse equation, Eq. (5-16),... 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

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