Attraction, point centers

Ruggiero Giuseppe Boscovich (1711-1787 CE), a Jesuit priest, replaced corpuscles with force-atoms (1758 CE) or point-centers of alternating attractive and repulsive forces. The views of Father Boscovich were similar to those of Newton, the Hindu atomists of the Nyaya-Vaisheshika, and the Arab followers of the Kalam (49). 

Fig. 2-1 Models of intermolecular potentials, (a) Forceless mass points (b) elastic hard spheres (c) elastic hard spheres with superposed central attractive forces (d) molecules with central finite repulsive and attractive forces (e) square-well model (f) point centers of inverse-power repulsion or attraction.

Two models, which are of more interest for their mathematical trac-tability than for their physical plausibility, are point centers of inverse power repulsion and point centers of inverse power attraction. Models with both central finite repulsion and attraction are more general cases of these interactions. 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

Thus, the thermodynamic basis functions (P, U, and can be readily determined for dilute gases from Eqs. (4.74), (4.80), and (4.85), respectively, for various interaction potentials such as hard-sphere, point centers of attraction or repulsion, and Lennard-Jones (L-J) interaction potentials. We now determine second virial coefficients for simple in-termolecular interactions that admit analytical results. 

Prove the formula for the second virial coefficient for point centers of attraction or repulsion. 

The empirical potentials for the molecules were obtained on the assumption of single attraction centers. This assumption is probably good for H2, fair for CH4 and N2, and very poor for Cl2. Even for molecules such as CH4 which are relatively spherical in shape, the fact that some atoms are near the outer surface rather than the center has an important effect. The closest interatomic distances are emphasized by the i 6 dependence of the potential. This point has been considered by several authors who worked out examples showing the net intermolecular potential for several models. 

The inner layer (closest to the electrode), known as the inner Helmholtz plane (IHP), contains solvent molecules and specifically adsorbed ions (which are not hilly solvated). It is defined by the locus of points for the specifically adsorbed ions. The next layer, the outer Helmholtz plane (OHP), reflects the imaginary plane passing through the center of solvated ions at then closest approach to the surface. The solvated ions are nonspecifically adsorbed and are attracted to the surface by long-range coulombic forces. Both Helmholtz layers represent the compact layer. Such a compact layer of charges is strongly held by the electrode and can survive even when the electrode is pulled out of the solution. The Helmholtz model does not take into account the thermal motion of ions, which loosens them from the compact layer. 

Many sweets (confections) must be colored, a strong point in their attractiveness for consiuners. The commonly colored products are candies (starch jellies, candy cream centers, pan-coated candies, and hard candies), tablets, wafers, oil-based coatings, and chewing gmns. 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

We have demonstrated that as in the case of the spherical mass the attraction field increases linearly approaching the spheroid surface and it is equal to zero at the center. At the same time, the potential has a maximum at this point and then decreases gradually as a parabolic function and reaches a minimum on the surface of the spheroid. 

Thus, the pressure has a maximum at the center and the decreases as a parabolic function and it is equal to zero at the pole. Next, consider the distribution of pressure in the channel A, where both the attraction and centrifugal forces act on any particle. Inasmuch as a difference of a pressure at terminal points of both channels is the same and a >, it is natural to assume that the attraction field in the channel A is smaller and suppose that the correction factor is equal to the ratio of axes, bja. Correspondingly, a condition of equilibrium is... 

In 1749, D Alembert pointed out that there is a connection between the theory of precession and a figure of the earth. As is well known, precession is caused by the fact that the resultant force of attraction due to celestial bodies, such as Sun and Moon, does not pass through the center of the earth. Correspondingly, there are couple of forces, which tend to turn Earth in such way that the plane of an equator would go through an attracting body and produce a precession. If the earth had a spherical form, then due to spherical symmetry the resultant force passes through the center. However, the spheroidal form does not have such symmetry. Points of the equator or polar axes are exceptions, since the resultant force passes through the earth s center. For all other points this condition is not met. Besides, the position of the resultant force depends also on the distribution of a density inside the earth. Let... 

If the substituents are nonpolar, such as an alkyl or aryl group, the control is exerted mainly by steric effects. In particular, for a-substituted aldehydes, the Felkin TS model can be taken as the starting point for analysis, in combination with the cyclic TS. (See Section 2.4.1.3, Part A to review the Felkin model.) The analysis and prediction of the direction of the preferred reaction depends on the same principles as for simple diastereoselectivity and are done by consideration of the attractive and repulsive interactions in the presumed TS. In the Felkin model for nucleophilic addition to carbonyl centers the larger a-substituent is aligned anti to the approaching enolate and yields the 3,4-syn product. If reaction occurs by an alternative approach, the stereochemistry is reversed, and this is called an anti-Felkin approach. 

Inert Gases. The calculation of 7 should be relatively straightforward for crystals of inert gases, in which only one kind of interaction may be expected. These crystals have a face-centered cubic structure. If each atom is treated as a point source of attractive and repulsive forces, only the forces between the nearest pairs of atoms are considered, the zero point energy is neglected, and no re-arrangement of atoms in the surface region is permitted, then the calculated 7 still depends on the equation selected to represent the interatomic potential U. 

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