Packing radii

This expression has turned out to be remarkably useful in correlating the heats of formation of the salts of tetrahedral ions, provided suitable values are assumed for the ionic radii R+ and R. Kapustinskii and his coworkers recognized that these quantities are not necessarily equal to the packing radii of the ions in the actual structure of the crystal consequently, they have come to be known as thermochemical radii. The thermochemical radius and heat of formation for a tetrahedral ion are normally determined from Equation 1 and the known heats of formation of two of its salts. 

Some indication of the periodicity in properties of elements in the sequence of atomic numbers will be evident in the discussion of valence, covalent radii, and packing radii of atoms in the following pages. 

In the crystal and in the liquid, the hydrogen molecules may be described as in contact with one another. The observed distance between the centers of two nonbonded atoms in contact is given reasonably well by the sum of the packing radii of the two atoms. Distances between the centers (the nuclei) of two atoms connected to one another by a chemical bond are given reasonably well by the sum of their covalent-bond radii. (See the tables following plate 57.)... 

Packing radii give the effective size for nonbonded contacts between atoms packed together in a crystal or liquid. They are also called Van der Waals radii, after the Dutch physicist J. D. van der Waals (1837-1923). 

Each atomic species is assigned a bond radius and a packing radius these are collected in Thble 12.1. These quantities are considered invariant for our purposes. The packing radii, / w, are considered simply to represent the bulk of each atom, without implications thus far on the actual contact distances in crystals (see 12.3.2.1). 

Fig. 12.9. Radial distribution (D(R) = TV(contacts)/(47tl d/ )) for N... H and O... H contacts, with non-bonded energy curves shown by ordinate scale on the right. The vertical solid line shows the sum of packing radii. M curves [14]. G curves [46]...

The connection between molecular mechanics and crystal structures came about in the attempt to quantify the non-bonded interactions. These were first taken oyer from intermolecular interaction potentials of rare-gas-type molecules. They start from the premise, contained in the van der Waals equation of state for real gases, that atoms are not localized at points, i.e. not at their respective nuclei. They occupy a volume of space and can be assigned, at least as a first step, more or less definite radii, by custom called van der Waals radii, which were initially estimated for many types of atom mainly from packing radii in crystals. Mutual approach of non-bonded atoms to distances less than the sum of these radii leads to strong repulsive forces. The empirical atom-atom potentials that were introduced to describe the balance between atom-atom attractions and repulsions were assumed to be characteristic of the atom types and independent of the molecules they are embedded in. They were assumed to hold equally for interactions between non-bonded atoms in... 

The nature of spheron-spheron interactions is such that maximum stability is achieved when each spheron ligates about itself the maximum number of neighbors, to produce a nucleus with a closest-packed structure. A simple argument (12) leads to the conclusion that the spherons in a nucleus are arranged in concentric layers. The packing radius of a spheron varies from 1.28 f for the dineutron to 1.62 f for the helion. The radius (to nucleon density half that of the inner region) of the largest nucleus is 6.8 f... 

It will be shown subsequently that the thermodynamically favoured vesicles have their hydrocarbon chains maximally extended. Then / = 4, and Ri takes the minimum possible value for a given Oj. This value will be called the critical packing radius R. To a first approximation the outer amphiphile surface area Ox may be put equal to Oq, and the value of may then be readily obtained from eqn (7.11). As an example of... 

The concentration of vesicles has a near-Gaussian profile which peaks at an outer radius Ri close to, but slightly smaller than, the critical packing radius R. At the peak, the surface area per amphiphile is slightly greater (by 1 %) than that of the bilayer Oq, though the difference is diminished as the total amphiphile concentration increases. 

We note that the elasticity discussed above is only for planar bilayers under compression or tension and does not extend to the bending of bilayers. On the contrary, our analysis has shown that the bending of a bilayer is favoured down to the critical packing radius (assuming that the lipids can freely rearrange by lateral movement and/or flip-flop), and that bending elasticity sets in only for radii smaller than this critical value. The elasticity of a fluid bilayer is therefore seen to be profoundly different from that of a classical elastic plate or shell. 

The equilibrium distance is called Rq. It should not be confused with the packing radius, called R, which is a radius for each atomic species such that interatomic distances below the sum of the corresponding radii never or hardly ever occur - it is thus essentially an experimental quantity, based on observations of thousands of crystal structures. Nor should Rq be confused with R/, which is the distance from the nucleus at which the electron density falls below a selected threshold (say, 0.02 e A ). These latter quantities may be numerically similar, and one may even be used in place of another for convenience (we use here / w for Rf, for example) but they are nevertheless conceptually distinct. 

Fig. 12.4. The parameters of an interatomic interaction curve (energy versus distance, see 12.3.2.1). The packing radius falls within the range shown (see text for explanation of symbols)...

The shock wave impacts from same explosives with different masses follow the geometry similarity law in the space before the shock waves meet the boundaries or obstacles. For an explosive with packed radius rl, the super pressure of shock wave front at 7 1 is AP and if the second explosive with rl, the super pressure of shock wave front at Rl is AP. These two explosives are similar in geometries of packing. The geometric similarity rate is of practical importance for the design of engineering. The experiments can be smdied with small amount of explosives and measure all parameters in free field. The conditions of explosion with large amount explosives can be calculated/predicted based on the experiments. It helps to reduce the experiment numbers and lower the cost of experiments. 

For the explosion of liquid explosives in air, the fundamental physical quantities, which influenced the pressure of shock wave fronts, are detonation heat Qy, packed density po of an explosive, packed radius r, distance to explosive s center R, air pressure Pa, and its origin density pa- After ignoring the viscosity and thermal conduction of air, the super pressure of air shock wave is a function of aU parameters (Eq. 2.65). 

The super pressure of air shock waves, packed radius, packed density, and detonation heat of explosives are displayed below, separately. 

Radial distance from the axis of the column, m Inner radius of the column, m Velocity to concentration time scale ratio Packing radius, m... 

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

One of the major factors in determining the structures of the substances that can be thought of as made up of cations and anions packed together is ionic size. It is obvious from the nature of wave functions that no ion has a precisely defined radius. However, with the insight afforded by electron... 

FIG. 14-61 Liq uid distribution in a 6-in column packed with 1/4-in broken-stone packing. Increments of radius represent equal-annual-area segments of tower cross section. Central-point inlet. Water rate = 500 lb/(b-fr). Air rate = 810 lb/(b-ft ). To convert pounds per bour-square foot to kilograms per second-square meter, multiply by 0.00L356 to convert inches to centimeters, multiply by 2.54. (Data from Baker, Chilton, and Vernon, in Shetxuood and Pigford, Absorption and Extraction, 2d ed., McGraw-Hill, New York, 1952. )... 

The development of the function describing (tm) for a capillary column is similar to that for the packed column but (r), the column radius, replaces (dp), the particle diameter. 

For a packed column of radius (rp) and length (Ip), the permissible extra-column... 

Thus, for a packed column length (L), radius (r), with a mobile phase volume equivalent to 60% of the column volume,... 

In a packed column the HETP depends on the particle diameter and is not related to the column radius. As a result, an expression for the optimum particle diameter is independently derived, and then the column radius determined from the extracolumn dispersion. This is not true for the open tubular column, as the HETP is determined by the column radius. It follows that a converse procedure must be employed. Firstly the optimum column radius is determined and then the maximum extra-column dispersion that the column can tolerate calculated. Thus, with open tubular columns, the chromatographic system, in particular the detector dispersion and the maximum sample volume, is dictated by the column design which, in turn, is governed by the nature of the separation. 

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