Volume of the sphere

Next we consider replacing the sandwiched fluid with the same liquid in which solid spheres are suspended at a volume fraction unit volume of liquid-a suspension of spheres in this case-the total volume of the spheres is also 0. We begin by considering the velocity gradient if the velocity of the top surface is to have the same value as in the case of the... 

The molecular weight M of a solid-sphere is proportional to the volume of the sphere, i.e., M r therefore ... 

In the case of a crack parallel to the chains the surface energy relates to the cleavage of secondary bonds between the chains by the shear stress osin0cos0. The shear strain energy released by the sphere is equal to the product of the volume of the sphere multiplied by the shear strain energy given by the second term in Eq. 32. Thus... 

If we make a somewhat more realistic molecular model, such as that of linked springs (Hopfield),2 then just, as the volume of the sphere is dependent on a great variety of external effects, so must this structure respond at the molecular level to these external effects, but now all effects become directional. The idea of allosteric equilibrium (e.g., in hemoglo-... 

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). 

The equations for the diffusion profile can be obtained from the heat-conduction equations of Carslaw and Jaeger [Ref. 3, Eq. 9.4 (10)] by using the substitutions we have indicated. The subject is discussed from a different approach by Adams, Quan, and Balkwell (I). The profiles can then be integrated over the volume of the sphere to obtain the uptake as a function of time. 

The dimensions of the polarizability a are those of volume. The polarizability of a metallic Bphere is equal to the volume of the sphere, and we may anticipate that the polarizabilities of atoms and ions will be roughly equal to the atomic or molecular volumes. The polarizability of the normal hydrogen atom is found by an accurate quantum-mechanical calculation to be 4.5 ao that is, very nearly the volume of a sphere with radius equal to the Bohr-orbit radius a0 (4.19 a ). 

A more common way of looking at the problem is to consider the atom to be composed or layers much like an onion and to examine the probability oT finding the electron in the layer which extends from r to r + dr, as shown in Fig. 23. The volume of the thin shell may be considered to bedV. Now the volume of the sphere is... 

In this expression, 3 is a purely geometric factor generated from the ratio of the surface area to the volume of the sphere the first bracket is the maximum total rate of reaction, achieved when the concentration is everywhere at cf the second bracket is the total diffusive flux across the surface when the... 

We first consider the sum of states. Now, in Eq. (A.33) the integration over coordinates gives the volume of the container, and the integral over the momenta is the momentum-space volume for H having values between 0 and E. Equation (A.41) is the equation for a sphere in momentum space with radius j2rn, 11. Thus, the volume of the sphere is 4Tt(y/2mH)3/3 and... 

Although this result was derived from the special case of an isotropic sphere, the above relationship between the total field and the polarization also applies to anisotropic systems. Furthermore, the volume of the sphere can be made arbitrarily small to envelop a single dipole. 

The density term includes the effect of kinetic energy (E = me2 ), so that energy conservation can be written inside the volume of the sphere, and elementary thermodynamics gives ... 

The force acting on the particle can be calculated by integrating over the surface of the sphere or over the volume of the sphere... 

The largest possible value of a is that for a sphere of infinitely conducting material for which ss is effectively infinite then a = 47ra3, three times the volume of the sphere. At the opposite extreme, when ss is almost equal to sm, a [(47ra3)/3]( s - m), the spherical volume times difference in s s. 

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... 

Compare relative dimensions of a sphere, platelet, and fiber, assuming that the fiber element diameter and platelet thickness are one-tenth the sphere diameter and that the volumes of the sphere, platelet, and fiber are equal. Assume a circular cross-section. 

The edges of this dodecahedron sized a - 8.5 cm. When the volume of the rubber balloon at inflation became bigger than the volume of the sphere inscribed in the dodecahedron, the balloon was deformed by the dedecahedron faces and took a shape close to the respective shape of a bubble in a monodisperse dodecahedral foam with a definite expansion ratio. The expansion ratio of the foam was determined by the volume of liquid (surfactant solution or black ink in the presence of sodium dodecylsulphate) poured into the dodecahedron. An electric bulb fixed in the centre of the balloon was used to take pictures of the model of the foam cell obtained. The film shape and the projection of the borders and vertexes on the dodecahedron face are clearly seen in Fig. 1.10. 

The volume occupied by a molecule should then vary as The number of molecules in a given weight of polymer varies inversely with the molar mass hence the total volume of the spheres is... 

Hence, the instrument response is proportional to the volume of the sphere, F, modified by the function F. This equation results from a simple integration of the area available for conduction. Several approximations have been derived for F. . [22,24-26] and these have been compared to find the one that agrees best with experiment [27]. 

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