Hard-sphere model limitations

N. Sutin, Brookhaven National Laboratory Strictly speaking, the outer-sphere and inner-sphere designations refer to limiting cases. In practice, reactions can have intermediate outer-sphere or inner-sphere character this occurs, for example, when there is extensive interpenetration of the inner-coordination shells of the two reactants. Treating this intermediate situation requires modification of the usual expressions for outer-sphere reactions — particularly those expressions that are based upon a hard-sphere model for the reactants. 

Let US consider the repulsive force model. The repulsive force is proportional to the inverse power n) of the distance (r) force l/r . Based on this consideration alone the pentagonal bipyramidal structure seems to be more stable for small values of n, the capped trigonal prism for intermediate values and the capped octahedron for large values of n, upto the limit of hard-sphere model. It is obvious, however, that such analysis cannot be applied for systems where all the hgands are not equal, i. e. MX Y 7- , and the ligands X and Y are very different from each other. 

Just as the ideal gas forms a convenient point of reference in discussing the properties of real gases, so does the hard-sphere fluid in discussing the properties of liquids. This is especially true at low densities, where the role of intermolecular forces in real systems is not so important. In this limit, the hard-sphere model is useful in developing the theory of solutions, as will be seen in chapter 3. 

Its exponential behavior makes this term the dominant one when short atom-atom distances between the interacting molecules are produced. Consequently, this term prevents molecules from getting closer than some limiting distance this is the physical principle behind the hard-sphere model and Kitaigorodsky s close-packing principles [2]. Moreover, the shortest atom-atom distances that one can find in different intermolecular interactions for the same pair of atoms always fall in a restricted range, a fact that allows one to define atomic radii. They differ in ionic and neutral crystals due to the different electronic structure of ionic and neutral species, as easily shown when comparing the contours at 90% probability in electron density maps for isolated atoms and their ions. 

The motion of ions in a buffer gas is governed by diffusive forces, the external electric field and the electrostatic interactions between the ions and neutral gas molecules. Ion-dipole or ion-quadrupole interactions, as well as ion-induced dipole interactions, can lead to attractive forces that will slow the ion movement, mainly due to clustering effects. The interaction potential can be calculated according to different theories, and three such approaches—the hard-sphere model, the polarization limit model, and the 12,4 hard-core potential model— were introduced here. Under... 

The hard-sphere model, as described in Chapter 2, only accounted in a qualitative fashion for the mass, momentum, and energy transport that underlie the phenomena of diffusion, viscous flow, and heat conduction. For systems other than monatomic gases, the model was of limited utility. Thus, its failure to predict reasonable values of the Arrhenius -factor is to be expected. 

With regard to the imperfection of gases, we are limited to the forces between two molecules only (two body forces) which give the expressions of the second coefficient of the virial. Researchers have endeavored to calculate the third, fourth and fifth coefficient of the virial. Here the three body forces are involved for the third coefficient, four body forces for the fourth and compact packing of spheres for the fifth coefficient. As for the second coefficient, the authors initially stuck to the hard-sphere model without attraction force (see section 7.3.3.1 and Figure 7.6), and as in the case of the second coefficient, they obtained coefficients practically independent of temperature, which allowed Hirshfelder and Roseveare to propose a state equation in the form ... 

This section was called reahstic interatomic potentials for good reason. We did improve upon the hard-sphere model by adding a qualitative correction, namely, the long-range attraction, and by having a more reasonable, somewhat softer repulsion. But the potentials that we have so far discussed stiU have a major limitation they were taken to depend only on the distance R between the centers of the two interacting molecules. We were still discussing particles without internal chemical structure. The anisotropic shape of molecules means that a molecule does not look the same from all possible directions of approach. 

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