Interaction between two hard

Another difficulty exists. It has been found that in the gas phase or in dipolar aprotic solutions, small anions (cations) have higher HOMO s (lower LUMO s) than the larger one. It follows that interaction between two hard ions is then favoured by both charge control and frontier control. Therefore, the equivalences suggested by Klopman ... 

For the interaction between two hard spheres 1 and 2 having radii a and 02. and constant surface potentials i/ oi and ij/o2, respectively, at separation H between the two spheres, the interaction energy (R) is given by [3,6,8,10]... 

Fig. 1.8 Sketch of the depletion interaction between two hard spheres...

They are caused by interactions between states, usually between two different electronic states. One hard and fast selection rule for perturbations is that, because angidar momentum must be conserved, the two interacting states must have the same /. The interaction between two states may be treated by second-order perturbation theory which says that the displacement of a state is given by... 

For more general interactions - i.e. not necessarily between two hard-spheres - we introduce a differential scattering cross section a v,8), defined by b db d(j> a u,8)dU. Boltzman s equation (equation 9.32) becomes... 

Equation 11.9 certainly implies that the local hardness should be used. It is known that this depends only on the functional dependence of the kinetic energy and electron repulsion terms upon the value of p [3]. However, it is difficult to calculate local values. In spite of this uncertainty, Equation 11.9 or its equivalent has often been used to calculate the interaction between two chemical systems [4]. 

In Scheme 1-71, the reversible reaction is shifted to the right when the anion, X, is larger and the cation, M+, is smaller. For example, this shift to the right is 100% in the presence of Na+, PI v, and only 35% in the presence of Na+, F (Hamon Astruc 1988). The equilibrium takes place as an exchange reaction between the two ion pairs. Reactions of this type are based on the symbiotic-effect premise The interaction between a hard cation and a hard anion or between two soft ions is stronger than that between two ions of different types. 

Rigorous perturbational treatments of the interaction between two molecules belong to the field of intermolecular forces, and I shall not attempt a comprehensive review, since the topic has been reviewed by Stamper129 in the previous volume in this series. However, several authors have devised perturbation schemes with a view to their application in problems of reactivity, which is a departure from conventional theory of intermolecular forces, where the possibility of making and breaking of bonds is usually excluded, on the reasonable grounds that the problem is quite hard enough anyway. 

FIGURE 7.7 Schematic Baxter (—), Van der Waals attraction (—), and steric stabilization (.) potential curves describing the interaction between two DMDBTDM A aggregates. Rc is the radius of the polar core, Rhs is the hard-sphere radius, and (8-Rhs) represents the distance of the effective attractive interaction. (From L. Martinet, Organisation Supramoleculaire des Phases Qrganiques de Malonamides du Precede d Extraction DIAMEX. PhD thesis. Rapport CEA-R-6105, 2005. With permission.)... 

N identical charged hard spherical particles with a radius of 670 A located in a three-dimensional cubic box were considered in the simulations. The pair interaction between two particles was represented by the Debye-Hiickel screened potential ( D-hOX/)). 

A general scheme, based on a rigorous statistical mechanical formulation, for obtaining the interaction between two colloidal particles in a fluid has been outlined. The implementation of the theory is in its early stages. In the DLVO theory and the theory of HLC, it is assumed that the various contributions can be added together. In the MSA, the hard core and electrostatic terms will be additive. However, it is only at low electrolyte concentration that the effect of dipole orientation and the repulsive contribution of the double layer overlap will be additive. There is no reason to believe (or disbelieve) that the van der Waals term should also be additive. 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

FIGURE 14.1 Interaction between two parallel dissimilar hard plates 1 and 2 at separation h. 

FIGURE 14.3 Interaction between two charged hard spheres 1 and 2 of radii ai and U2 at a separation R between their centers. H = R — — 02) is the closest distance between their... 

Before considering the interaction between two ion-penetrable membranes, we here treat the interaction between two similar ion-impenetrable hard plates 1 and 2 carrying surface charge density cr at separation h in a salt-free medium containing counterions only (Fig. 18.1) [2]. We take an x-axis perpendicular to the plates with its origin on the surface of plate 1. As a result of the symmetry of the system, we need consider only the region 0 < x < h 2. Let the average number density and the valence of counterions be o and z, respectively. Then we have from electroneutrality condition that... 

FIGURE 18.1 Schematic representation of the electrostatic interaction between two parallel identical hard plates separated by a distance h between their surfaces. 

Isolated atoms show spherical symmetry, and it is natural to model atoms by spheres of some suitably defined radii. The potential energy of interaction between two atoms rises very sharply at short internuclear distances during atomic collisions, not unlike the potential energy increase in the collisions of hard, macroscopic bodies. In a somewhat crude, approximate sense, atoms behave as hard balls. This analogy can be used for a simple molecular model where atoms are represented by hard spheres. Once a choice of atomic radii is made, the approximate atomic surfaces can be defined as the surfaces of these spheres. 

The term tooth wear is commonly used to describe the loss of tooth hard tissue due to non-carious causes [1], This encompasses a variety of both chemical and mechanical causes of both intrinsic and extrinsic origin. The term tooth wear is preferred over some of the more precise definitions of individual hard tissue loss mechanisms, because it acknowledges the fact that wear is usually a multifactorial process one mechanism may dominate, but the overall wear is commonly due to the interaction between two or more wear mechanisms. In dentistry, the terms erosion, abrasion, attrition and abfraction are widely used to describe particular mechanisms of hard tissue loss. 

A collision between atoms, molecules, or ions is not like one between two hard billiard balls. Whether or not chemical species collide depends on the distance at which they can interact with one another. For instance, the gas-phase ion-molecule reaction CH4+ + CH4 CH5+ + CH3 can occur with a fairly long-range contact. This is because the interactions between ions and induced dipoles are effective over a relatively long distance. By contrast, the reacting species in the gas reaction CH3 + CH3 C2H are both neutral. They interact appreciably only through very short-range forces between induced dipoles, so they must approach one another very closely before we could say that they collide. ... 

Equation (3.28) requires all the filled orbitals, followed by a Mulliken (or other) population analysis, for both the reactant molecule and its cation. It has the advanage that the charge, qk N), can also be useful. It will be recalled that only soft-soft interactions between two reactants are controlled by the Fukui function. That is, electron transfer, or covalency, is dominant. For hard-hard interactions, the charges on each atom dictate where reaction will occur. 

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