Sphere, diameter radius

Figure 3.20 The effective hard sphere diameter, r0, calculated from Equation (3.65) for 100 nm radius particles with ( = 50 mV...

Commonly [17], when the length-to-diameter ratio of a cylindrical catalyst is close to 1, the cylindrical catalyst can be simplified as a sphere, the radius of which, Rp, is calculated by 3 Kp/.S p. The one-dimensional, key-component based reaction-diffusion models of methanation system are as follows ... 

Solution This weakness of inverse-sixth-power van der Waals forces between small particles is discussed at length in the main text. Its thermal triviality is easily seen. Begin with [—(16/9)](R6/z6)AHam for the energy of interaction between two spheres of radius R and center-to-center separation z and ask what the Ah am would have to be for the magnitude of this energy to be comparable with kT, (16/9)(R6/z6)AHam = kT or Anam = (9/16)(z6/R6)kT. Even if the center-to-center separation z were equal to 4R, spheres separated by a distance equal to their diameter, R6/z6 would be 46 = 4096. Anam would have to be a ridiculous 4096 x (9/16) kT = 2304 kT for there to be thermally significant attraction. 

The first term is simply the volume of a sphere of radius ro, or eight times the volume of the sphere of diameter ro which represents a molecule. In the second integral, we may expand in power series, since is relatively... 

Ewald sphere, sphere of reflection A geometrical construction used for predicting conditions for diffraction by a crystal in terms of its reciprocal lattice rather than its crystal lattice. It is a sphere, of radius 1/A (for a reciprocal lattice with dimensions d = X/d). The diameter of this Ewald sphere lies in the direction of the incident beam. The reciprocal lattice is placed with its origin at the point where the incident beam emerges from the sphere. Whenever a reciprocal lattice point touches the surface of the Ewald sphere, a Bragg reflection with the indices of that reciprocal lattice point will result. Thus, if we know the orientation of the crystal, and hence of its reciprocal lattice, with respect to the incident beam, it is possible to predict which reciprocal lattice points are in the surface of this sphere and hence which planes in the crystal are in a reflecting position. 

In any sphere packing, it is possible to partition the given volume into occupied and available space. The former is the union of all the exclusion spheres, and the latter is its complement, namely, the volume available for the placement of the center of an additional sphere. The exclusion region of a sphere of diameter concentric sphere of radius a. Exclusion spheres can overlap. At a high enough density, the available space is in general composed of disconnected cavities. 

In general, in such a local sphere around the solute, the composition of the solvent might be different from the bulk composition xA. Let xA (and xB = 1 — xA) be the composition of the solvent in this local sphere. We shall refer to xA as the local composition of the solvent around the solute. We shall discuss further the meaning of the word local in the next section, but in the meantime we assume that we can choose some sphere of radius Ra centered at the center of s, where Ra is on the order of a few molecular diameters. 

The SPT aims at providing an approximate expression for P0(r) or, equivalently, for G(A). Before presenting this expression, we note that an exact expression is available for P0(r) at very small r. If the diameter of the HS particles is a, then in a sphere of radius r< a/2, there can be at most one center of a particle at any given time. Thus, for such a small r, the probability of finding the sphere occupied is 47ir3p/3. Since this sphere may be occupied by at most one center of an HS, the probability of finding it empty is simply... 

The scaled particle theory was extended to mixture of hard spheres by Lebowitz et al. (1965). In a one-component system of hard spheres of diameter a, placing a hard particle of radius RHs produces a cavity of radius r= RHs + a/2. When there is a mixture of hard spheres of diameters a the radius of a cavity produced by a hard sphere of radius RHs depends on the species i, i.e., Y = Rhs + ai/2. 

Problem 2-29. Interfacial Tension-Capillary Effects. A rigid sphere of radius a rests on a flat surface, and a small amount of liquid surrounds the point of contact, making a concave-planar lens whose diameter is small compared with a. The angle of contact 9C with each of the solid surfaces is zero (see Problem 2-28), and the tension in the air-liquid surface is a. Show that there is an adhesive force of magnitude Anaa acting on the sphere. (The fact that this adhesive force is independent of the volume of liquid is noteworthy. Note also that the force is repulsive when 9C = n.)... 

The contact resistance of the sphere-flat contact shown in Fig. 3.23 is discussed in this section. The thermal conductivities of the sphere and flux tube are /c, and k2, respectively. The total contact resistance is the sum of the constriction resistance in the sphere and the spreading resistance within the flux tube. The contact radius a is much smaller than the sphere diameter D and the tube diameter. Assuming isothermal contact area, the general elastoconstriction resistance model [143] becomes ... 

A large petroleum storage tank is 100 ft in diameter. The free surface is really a very small part of a sphere with radius 4000 mi (the radius of the earth). If one drew an absolutely straight line from the liquid surface at one side of the tank to the liquid surface directly across the diameter on the other side, how deep into the fluid would that line go j... 

A solid sphere of radius sphere and density Psphere falls throngh an incompressible Newtonian fluid which is quiescent far from the sphere. The viscosity and density of the flnid are p-auid and pauid, respectively. The Reynolds number is 50, based on the physical properties of the fluid, the diameter of the sphere, and its terminal velocity. The following scaling law characterizes the terminal velocity of the sphere in terms of geometric parameters and physical properties of the fluid and solid ... 

Fig. 7. Scattering curves from models of uniform scattering density, (a) A sphere of radius 6.5 nm and Rq 5.0 nm as used in Fig. 4. (b) A hollow sphere with an outer radius of 6.5 nm as in (a) and a ratio of inner/outer radii of 0.5. (c) A straight, cylindrical rod of length 59 nm and diameter 3.4 nm. (d) The cylindrical rod as in (c) is now bent into a circle of radius 9.4 nm.

Fig. 15. Modelling of j3-haemocyanin (H. pomatia) constructed from 160 spheres of radius 2.5 nm with overall dimensions of 39 tun (height) and 34 nm (diameter). For clarity, the 20 subunits at the inner side of the bottom are not drawn. The spheres corresponding to the one-twentieth dissociation product are shown connected by lines to indicate three adjacent units [119]. The comparison between the theoretical...

For a homogeneous sphere with radius R, Rg=Rg, oo which may be calculated by the relation R =3I5 R. Thus, a homogenous sphere of 100 nm diameter is characterized by Rg=38.73 nm. The validity of the Guinier-law Equation (17) requires that Rgqlatex systems with sufficient accuracy. 

By interpreting b as the excluded volume of a mole of spherical molecules, we can obtain an estimate of molecular size. The centers of spherical particles are excluded from a sphere whose radius is the diameter of those spherical particles (i.e. twice their radius) that volume times the Avogadro constant is the molar excluded volume b... 

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