Nearest-neighbor molecule

Table 28.1 In the Pauling model, the four subunits of hemoglobin are assumed to be arranged in a tetrahedron. Each sphere represents one bound ligand molecule. Nearest-neighbor ligand interactions are indicated by continuous lines. The table shows the count of these interactions. 

One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and that in the liquid. Skapski [11] has made the following simplified analysis. Considering only nearest-neighbor interactions, if n, and denote the number of nearest neighbors in the interior of the liquid and the surface region, respectively, then, per molecule... 

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

This region has been divided into two subphases, L and S. The L phase differs from the L2 phase in the direction of tilt. Molecules tilt toward their nearest neighbors in L2 and toward next nearest neighbors in L (a smectic F phase). The S phase comprises the higher-ir and lower-T part of L2. This phase is characterized by smectic H or a tilted herringbone structure and there are two molecules (of different orientation) in the unit cell. Another phase having a different tilt direction, L, can appear between the L2 and L 2 phases. A new phase has been identified in the L 2 domain. It is probably a smectic L structure of different azimuthal tilt than L2 [185]. 

Such attractive forces are relatively weak in comparison to chemisorption energies, and it appears that in chemisorption, repulsion effects may be more important. These can be of two kinds. First, there may be a short-range repulsion affecting nearest-neighbor molecules only, as if the spacing between sites is uncomfortably small for the adsorbate species. A repulsion between the electron clouds of adjacent adsorbed molecules would then give rise to a short-range repulsion, usually represented by an exponential term of the type employed... 

Enthalpies of mixing have their origin in the forces that operate between individual molecules. Intermolecular forces drop off rapidly with increasing distance of separation between molecules. This means that only nearest neighbors need be considered in the model. 

Each lattice site is defined to have z nearest neighbors, and 0i and 02 > respectively, can be used to describe the fraction of sites which are occupied by solvent molecules and polymer segments. The following inventory of interactions can now be made for the mixture ... 

In the solid state, the Ceo molecules crystallize into a cubic structure with a lattice constant of 14.17A, a nearest neighbor Ceo-Ceo distance of 10.02A [41], and a mass density of 1.72 g/cm (corresponding to 1.44 Ceo... 

The remaining AOs are the four H 1, two C 1, and four C 2p orbitals. All lie in the molecular plane. Only two combinations of the C 2s and H U orbitals meet the molecular symmetry requirements. One of these, nearest-neighbor atoms. No other combination corresponds to the symmetry of the ethylene molecule. 

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Clearly, proximity and orientation play a role in enzyme catalysis, but there is a problem with each of the above comparisons. In both cases, it is impossible to separate true proximity and orientation effects from the effects of entropy loss when molecules are brought together (described the Section 16.4). The actual rate accelerations afforded by proximity and orientation effects in Figures 16.14 and 16.15, respectively, are much smaller than the values given in these figures. Simple theories based on probability and nearest-neighbor models, for example, predict that proximity effects may actually provide rate increases of only 5- to 10-fold. For any real case of enzymatic catalysis, it is nonetheless important to remember that proximity and orientation effects are significant. 

In water at room temperature some molecules have more than four nearest neighbors, but at any moment the majority have four neighbors, as in ice. If the molecules tend to have a tetrahedral arrangement like that of ice, the average distance between next-nearest neighbors would be... 

Fig. 20. The structure of ice Molecules numbered 8, 7, 6 are in contact with 5, while molecules 5, 4, 3, 2 arc in contact with 1. Molecules 2, 3, 4 are among the next-nearest neighbors of 5, while molecules 0, 7, 8 are among the next-nearest neighbors of 1. [Diagram taken from E. J. W. Verivey, Rec. trav. chim. 60, 893 (1941).]...

Let us fix attention on a particular H20 molecule A in the interior of water (if we wish to identify this molecule we can suppose that it contains a nucleus of the oxygen isotope 01S) and let us consider the water molecules which happen to be nearest neighbors of this molecule at the moment. These molecules have been in contact with A for different lengths of time. Since all the molecules in the liquid wander about, there was a time when none of these molecules was in contact with A. Further, if we could now begin to watch these molecules, we should find that, after the lapse of different periods of time, they become separated from A and each is replaced by another molecule. Similar remarks can be made about the molecules which come into contact with any chosen molecule. We can now raise the question—-What is the rate of turnover of this process The rate depends on the degree of local order and disorder, which in turn depends on the strength and character of the forces between adjacent molecules. 

It was mentioned in Sec. 24 that in water at room temperature the average number of nearest neighbors for any HjO molecule is about 4.5, indicating that the number continually fluctuates between 4 and 5, or between 4 and 6. Whenever the number of neighbors of a particular molecule falls from 5 to 4, presumably any one of the five neighbors may... 

In the pure solvent let each particle have z nearest neighbors in contact with it. Let us ask how, removing two adjacent solvent particles from the interior, we may insert a solvent molecule. When a particular site is to be occupied by the B-half of the molecule BC, there are clearly z choices for the position of its C-half. This is true for each of the nB solute particles, provided that the solution is so dilute that they do not compete for the available sites to an appreciable extent. From the independent oiientations of nB solute particles, the quantity Wc/ receives the factor z if the molecules are heteronuclear and receives the factor (z/2)n if the molecules are homonuclear. 

Let Fig. 58a represent a molecule in the pure solvent, with four nearest neighbors and let Fig. 585 represent the simple substitution of a solute particle for the central molecule. The displaced molecule is to be put on the surface of the liquid. In water, if there were no other disturbance of the liquid, this would lead to the value 18 cm3/mole for the solute. Next let Fig. 58c represent the si lua-tion where the number of nearest neighbors has been increased by unity. In this case no solvent molecule has to be placed on the surface of the liquid and if there were no other disturbance of the surrounding liquid, the observed molal volume for the solute would clearly be zero. 

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