Sphere-exclusion approach

The excluded volume of a solute molecule is the volume that is not available (because of exclusion forces or for other reasons) to the centers of mass of other similar solute molecules. As an example, let us consider the excluded volume of a spherical particle of radius, R. The position of a sphere is fully described by coordinates of its center. It is apparent from Fig. 3.7(a) that the center of one solid sphere cannot approach the center of another solid sphere closer than two radii (2i2). Hence, the volume excluded by one sphere equals 7r(2ii), that is, eight times its actual volume. The excluded volume of asymmetric particles cannot be calculated so easily. This is because the distance between their centers of mass when they are in contact depends on their orientation. Nevertheless, it has been... 

The behaviour of some of these methods is illustrated using a two-dimensional example in Figure 12.30. If the most dissimilar compound is chosen as the first molecule in the maximum-dissimilarity cases then the MaxSum method tends to select compounds at the extremities of the distribution. Hiis is also the initial behaviour of the MaxMin approach, but it then starts to sample from the middle. The sphere exclusion methods typically start somewhere in the middle of the distribution and work outwards. 

The environment is often seen as being composed of environmental spheres, speciflcally the atmosphere, the hydrosphere, the terrestrial environment, and the biosphere (Fig. 2). Environmental chemicals can occur in all these spheres. If a chemical occiu exclusively or predominantly in one of these spheres, its environmental fate can be reduced to the fate in that particular phase. For example, some very volatile chemicals occur exclusively in the atmosphere, whereas some very water-soluble chemicals are restricted to the aqueous phase. They are sometimes referred to as air pollutants or water contaminants and a distinct sphere-specific approach to such chemicals fate and transport may be appropriate. Most chemicals, however, do occirr in more than one of the environmental spheres in notable amounts, and a single-compartment perspective is insufficient to capture their environmental behavior. 

Classification exclusively in terms of a few basic mechanisms is the ideal approach, but in a comprehensive review of this kind, one is presented with all reactions, and not merely the well-documented (and well-behaved) ones which are readily denoted as inner- or outer-sphere electron transfer, hydrogen atom transfer from coordinated solvent, ligand transfer, concerted electron transfer, etc. Such an approach has been made on a more limited scale. Turney has considered reactions in terms of the charges and complexing of oxidant and reductant but this approach leaves a large number to be coped with under further categories. 

Fig. 5. Probabilities pn of observing n water-oxygen atoms in spherical cavity volumes v. Results from Monte Carlo simulations of SPC water are shown as symbols. The parabolas are predictions using the flat default model in Eq. (11). The center-to-center exclusion distance d (in nanometers) is noted next to the curves. The solute exclusion volume is defined by the distance d of closest approach of water-oxygen atoms to the center of the sphere. (Hummer et al., 1998a)... 

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). 

The geometric approach tries to calculate the volume of an arrangement of atom spheres exactly. The inclusion-exclusion principle can be appUed First, the volumes of all spheres are added, then each intersection of two spheres is subtracted, each intersection of three spheres is added, and so on. However, the calculation of... 

It should be noted, that the term radius implies that atoms are considered as hard spheres, which touch each other when atoms are bonded, but can not penetrate or deform each other. This image seems to contradict the concepts of quantum mechanics, as the electron clouds of atoms have no clear-cut boundaries and must overlap to form a chemical bond. However, one must keep in mind that (i) the repulsion which prevents closer approach of atoms, is due mainly to Pauli exclusion which forbids electrons with similar spins to occupy the same space, and (ii) the electron cloud of an atom (other than H or He) comprises two distinct parts the dense core and the much more diffuse valence electron clouds, with a steep jump of electron density between the former and the latter (Fig. 1.5). For these reasons, the repulsive force increases very steeply with the decrease of d, and the overlap of valence shells of chemically bonded atoms is limited in practice to area where the electron density does not exceed ca. 10 % of the maximum [37]. Furthermore, since core electrons are unaffected by chemical bonding, the atomic radius can be regarded in the first approximation as the constant of a given element, invariant against the composition and structure of the solid. 

Why the VSEPR approach should be so successful has been muchly discussed whether the electron pairs are truly similar in energy and whether they repel by either simple electrostatic forces or by the Pauli Exclusion principle (i5). In his comparison of the VSEPR "points-on-a-sphere" formalism with results of Molecular Orbital computations of potential energy surfaces, Bartell concluded that "the VSEPR model somehow captures the essence of molecular behavior" (76). 

Depending on the size of the ligand, the inner cavity of these structures had a diameter of 2-4 nm, along with 96 inward-facing hydroxyl groups. Addition of tetramethoxysilane resulted in sol-gel condensation exclusively inside the host, thus producing silica nanoparticles of defined size and shape—the size and shape of the host s inner cavity. Despite the amorphous nature of silica nanoparticles, the polydispersities observed approached unity. No side reactions, such as condensation outside the spheres or... 

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