Similarity solutions sphere

The solute-solvent and the solvent-solvent interaction potentials are assumed to be given by Lennard-Jones potential. The Lennard-Jones interaction parameters for dissimilar solvent (i) and solute (/ ) spheres are estimated from those of the interaction of similar spheres through the combining rule ey = ( ,/ ,y) 2 and oi = (tr,-,- + ojj)/2 [121, 124]. 

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... 

In the case of a solid sphere, considered in the preceding section, we solved the thermal boundary-layer equation analytically by using a similarity transformation. An obvious question is whether we may also solve (9-257) by means of the same approach. To see whether a similarity solution exists, we apply a similarity transformation of the form... 

Now, Eq. (9-260) is identical to equation (9 234), which was found earlier for the sphere, and we have already seen that it can be solved subject to the conditions (9 261). The solution for 9 is given in (9 240). The existence of a similarity solution to (9 257) thus rests with Eq. (9 259). Specifically, for a similarity solution to exist, it must be possible to obtain a solution of (9-259) for g(q2), which remains finite for all q2 except possibly at a stagnation point where a = 0, from which a thermal wake may emanate. 

We know that a similarity solution of this equation exists for the special case of a sphere, and we thus seek again to solve (9-264) by means of similarity transformation for the general case. If we simply follow the prescription of the preceding section, we obtain the same DE and boundary conditions for the similarity function 0 (rj), but now the equation for g(q2) takes a slightly more general form,... 

A similar solution can be obtained for the flow in a cylindrical pipe where y is replaced by the radial distance from the axis of the cylinder. There are a few other simple analytic solutions of the Stokes equation, e.g. for the flow around a sphere, etc. (Lamb, 1932). 

Spheres (Fig. 4.41c). Cheng [46] reports the similarity solution for heat transfer from an isothermal sphere to be... 

In 1958, Kirkaldy (K4) published a self-similar solution for the growth, limited by the diffusion of heat or mass, of slabs, cylinders, or spheres of initially negligible size from a large body of surrounding quiescent fluid of... 

Whether or not the coordination number of the lanthanide ion in aqueous complexes remains constant as ML, MLj, etc., the form of the species is an open question. Using the fluorescence technique, Albin et al. (1984) reported the hydration number of Eu(EDA) to be seven and of Eu(EDA)2" to be three. Since EDA is a tetradentate ligand, these values lead to CN = 11 in both complexes. Lifetime measurements on the crystalline hydrate Na[Eu(EDA)2(H20), ] are consistent with the presence of two water molecules in the primary sphere (CN = 10), reflecting that the solid compounds may not be valid structural models for similar solution species. 

Figure 12.5 There is not a centered square lattice, because if you tile squares and put lattice points on the corners and the center of each square it would be possible to draw a smaller square (rotated by 45°) that only has lattice points on the corners. Hence a centered square lattice would be indistinguishable from a primitive square lattice with a smaller unit cell. Figure 12.12 Face-centered cubic, assuming similar size spheres and cell edge lengths, since there are more atoms per volume for this unit cell compared to the other two. Figure 12.13 A hexagonal lattice Figure 12.15 The solvent is the majority component and the solute the minority component. Therefore, there will be more solvent atoms than solute atoms. Figure 12.17 The samarium atoms sit on the comers of the unit cell so there is only 8 X (1 /8) = 1 Sm atom per unit ceU. Eight of the nine cobalt atoms sit on faces of the unit cell, and the other sits in the middle of the unit ceU so there are 8 X (1/2) -I- 1 = 5 Co atoms per unit cell.

For purposes of calculation the Plexiglas walls of the cylinders employed in the experiments were treated as HjO and were mixed in with the solution, giving a concentration of 349 g/U (92.6% U-23S) per liter. The homogenized units were approximated by spheres having a radius of 11.25 cm. The reflectors were all assumed to be HjO, and the effect of thickness was calculated from experimental data expressed as albedos. The material buckling of a unit was calculated to be 0.0306 cm" which, with bare and water reflected extrapolation distances of 2.6 and 5.8 cm, is consistent with experimental data for similar solutions. ... 

In the crystallization of these hydrated salts from aqueous solutions it is essential that a low pH (high level of acidity) is maintained, otherwise hydrolysis occurs and yellow impurities contaminate the products. Similarly, if the salts are redissolved in water, the solutions turn yellow/brown. The hydrolytic processes are complicated, and, in the presence of anions with appreciable coordinating tendencies, are further confused by displacement of water from the coordination sphere of the iron. However, in aqueous solutions of salts such as the perchlorate the following equilibria are important ... 

For the silica gel (Figure 3A), the solution was removed slightly less effectively, and more Cs was left (ca. 0.0020 atoms/A2). The spectral behavior is quite similar to that of boehmite, except that there is a peak due to surface Cs coordinated by only water molecules and not in contact with the surface oxygens (so-called outer sphere complexes)at 30% RH. Complete dynamical averaging among sites at frequencies greater than ca. 10 kHz occurs at 70% RH and greater. 

It is well known in crystallography that, when spheres of equal radius are packed together, the closest type of packing is one in which each sphere has 12 other spheres in contact with it. In Sec. 24 it was mentioned that in water at room temperature each molecule has, on the average, only 4.4 other molecules in contact with it. If we wanted to place one or two additional H20 molecules in contact with any H20 molecule, there would be plenty of room to do this without seriously disturbing the neighbors that are already in contact with this molecule. Similarly, if this molecule is replaced by a solute particle of the same size, the same remark could be made about placing molecules in contact with the solute particle. 

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