Space-filling coefficient

The packing coefficient p should not be confused with the space-filling coefficient k in molecular crystals, which is calculated on the assumption that atoms contact each other at van der Waals radii, much larger than bond radii (see Sect. 4.3). In fact, in crystals comprising separate molecules, chains or layers (e.g. graphite), each atom participates in two different types of contacts, described by metallic/covalent and van der Waals radii, respectively. In this case the term radius should not be taken literally. Thus, the crystal structures of P, As, Sb and Bi (No = 3) have p = 0.23 but k 0.7. 

Archie [23] examined electrical resistivity of various sand formations having pore spaces filled with saline solutions of different salt concentrations. Based upon his own experimental results, he obtained a simple relationship for the conductivity of beds of sand (assuming the sand itself is nonconductive) containing saline solution in terms of the porosity. In terms of diffusion coefficients his expression is... 

Silyl groups, which tend to increase lipid solubility, may be used as a substitute for alkyl branching in space-filling groups. Direct substitution of a silicon atom for a carbon atom increases the hydrophobic character of a compound, even without the addition of more alkyl groups. Table III lists the relative partition coefficients of two pairs of carbon and silicon compounds in octanol-water, a system used to approximate the lipid-water system within an organism. As may be seen from the table, the silicon compounds are two to five times more soluble in the octanol phase this effect falls off with an increasing number of carbon atoms in the parent structure (36). 

As discussed by Andersen [9, 10] for muffin-tin orbitals, the locally regular components y defined in each muffin-tin sphere are cancelled exactly if expansion coefficients satisfy the MST equations (the tail-cancellation condition) [9, 384], The standard MST equations for space-filling cells can be derived by shrinking the interstitial volume to a honeycomb lattice surface that forms a common boundary for all cells. The wave function and its normal gradient evaluated on this honeycomb interface define a global matching function %(cr). 

In a space-filling cellular model, the SM variational functional can be expanded in a local basis in each atomic cell. Variation of the expansion coefficients of the trial orbital function ifr = J2l lYl in ceH T/x induces the variation... 

The first ideas on the nature of liquid crystals in hpids were derived from X-ray studies by Luzzati [1]. A crucial discovery was his demonstration of the liquid character of the hydrocarbon chains, which are thus space-filling. This was evident after it was foimd that the Upid bilayer thickness decreases with temperature with a large linear thermal coefficient about 10"3/°C. Such an effect is consistent only with a highly disordered chain conformation. Also the X-ray scattering characteristics were found to be very similar to those of liquid paraffins. 

Use a molecular mechanics program to generate energy-minimized structures and estimate the hydrodynamic radii of benzene and monochlorobenzene. For diatomic molecules, covalent and van der Waals radii are useful to calculate molecular size. From a molecular mechanics viewpoint, space-filling molecular models illustrate the van der Waals radius of each atom in the molecule. Use these hydrodynamic radii to calculate liquid-phase diffusion coefficients via the Stokes-Einstein equation. 

Preliminary analysis indicates a layer thickness of 3.09 nm and a complex permittivity of (2.34 + 0.25 i), corresponding to a refractive index, n, of 1.53 and absorption coefficient, k, of 0.08. These values are in agreement with figures produced by ellipsometric analysis of multilayer films (7 to 29 layers) of the same material [9]. The film thickness also coincides with the molecular length of 3.0 nm, measured from a precision space-filled model, suggesting vertical alignment of the molecules within the layers. 

It should be added, however, that an important factor in these reactions is the overcrowding that takes place when five or six aryl groups are introduced into ethane. It is impossible to construct space-filling models of hexaaryl-ethanes and indeed it now appears that such compounds do not in fact exist. The dimer of triphenylmethyl, which was formerly thought to be hexaphenylethane (Ph C—CPha) in fact has the quinonoid structure (1). Formation of this from triphenylmethyl is much less favorable than formation of Ph3CCPh3 (see the corresponding NBMO coefficients in Fig. 4.5) so far as the n electrons are concerned, but the steric effects outweigh this difference. 

Tel. 703-461-7078, fax 703-451-6639, e-mail eslone masoni.gmu.edu Structure building, manipulation. Stick, ball-and-stick, space-filling, and dot surface display. MM2 geometry optimization. Charge, log P, and molar refraction calculations. Interfaces to AMPAC and other modeling programs. Databases of octanol/water partition coefficients for 2500 compounds and 1500 3D structures. 2D-to-3D strucmre conversion. Structural database and searching. 

We stated earlier that standard practice is to write equations with the lowest whole-number coefficients possible. This is because equations are often interpreted on the particulate level. Thus, 2 H2(g) + 02(g) 2 H20(g) means two molecules of hydrogen react with one oxygen molecule to form two water molecules. The symbols in the equation represent the particles, shown here as space-filling models ... 

Space filling crystal structures are analyzed to find how molecules use the available space, using macro-coordinates such as molecular volumes, positions of molecular centers of mass, packing coefficients and crystal density. 

The term PDC is defined as polycrystalline diamond compact. The term TSP is defined as thermally stable polycrystalline diamond. TSP materials are composed of manufactured polycrystalline diamond which has the thermal stability of natural diamond. This is accomplished through the removal of trace impurities and in some cases the filling of lattice structure pore spaces with a material of compatible thermal expansion coefficient. 

This is defined as the increase in volume of unit volume of a substance when its temperature is raised by one degree. It is important in that the coefficient of expansion of LPG in its liquid form is relatively high, so that when filling a storage vessel adequate space must always be provided to allow for possible thermal expansion of the liquid. 

This is not surprising since at a given speed the coarseness of a track (the average spacing of the asperities) influences the friction only on a logarithmic scale. Also the observed dependence of the friction coefficient on load of soft mbber compounds on smooth surfaces disappears for harder black or silica-filled treads compounds on rough surfaces. 

Let us consider the influence of a solid-liquid interface advancing at a constant velocity on the solid-liquid fractionation of an element i. In the case of unidirectional solidification, it is convenient to consider that liquid crosses the immobile interface with an absolute constant velocity v, while a solid-liquid fractionation coefficient K is applied to the fractionation of element i. Let us assume that the interface is at x=0, the medium being solid for x<0. Liquid fills the half-space 0[Pg.442]

Hence, the temperature coefficient of k1 having been measured, for an absolute calculation of k only kt) and bo must be known, and not the heat of adsorption, X. At the moment we are concerned with b0. A simple statistical estimate can be based on the assumption that in the absence of adsorption energy the adsorption space is filled at a proportion given by the ratio of the molecular adsorption volume (liquid volume Fm) to the molecular gas volume... 

The NFE behaviour has been observed experimentally in studies of the Fermi surface, the surface of constant energy, F, in space which separates filled states from empty states at the absolute zero of temperature. It is found that the Fermi surface of aluminium is indeed very close to that of a spherical free-electron Fermi surface that has been folded back into the Brillouin zone in a manner not too dissimilar to that discussed earlier for the simple cubic lattice. Moreover, just as illustrated in Fig. 5.7 for the latter case, aluminium is found to have a large second-zone pocket of holes but smaller third- and fourth-zone pockets of electrons. This accounts very beautifully for the fact that aluminium has a positive Hall coefficient rather than the negative value expected for a gas of negatively charged free carriers (see, for example, Kittel (1986)). 

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