Factor radius

Properties included in this subsection are divided into 10 categories (1) physical constants critical properties, normal melting and boiling points, acentric factor, radius of gyration, and dipole moment (2) vapor... 

In the flocculations or coacervations which are discussed in this paragraph indications of this kind are indeed very clearly present. The required concentrations of the neutral salt are here not necessarily very high, on the contrary they are frequently relatively low. In so far as lyotropic influences are also found in the coacervations or flocculations now in question, they are not due to a binding of the water as mentioned above, but to the fact that the interaction between the ionised groups of the colloid and the ions of the salt is influenced by the same factors (radius, polarisability) which determined the order of the ions in the lyotropic series. 

Fig. 8.13 illustrates the manner in which stress concentrates in inside corners by plotting the ratio of the inside radius (R) to the wall thickness (WT against the stress concentration factor. At ratios below 0.5, the stress concentration factor rises dramatically. At the ratio of 0.25, it has reached 2.25. That should be regarded as an absolute minimum. Below 0.25, the stress concentration factor reaches toward astronomical levels. Even a little radius is better than none at all. In addition to the reduction in stress concentration factor, radiused corners also improve the flow of plastic in the mold resulting in a more uniform melt and a shorter molding cycle. 

It is to be noted that not only is the correction quite large, but for a given tip radius it depends on the nature of the liquid. It is thus incorrect to assume that the drop weights for two liquids are in the ratio of the respective surface tensions when the same size tip is used. Finally, correction factors for r/V < 0.3 have been determined, using mercury drops [37],... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

Figure Bl.8.1. The atomic scattering factor from a spherically synnnetric atom. The volume element is a ring subtending angle a with width da at radius r and thickness dr.

Because the neutron has a magnetic moment, it has a similar interaction with the clouds of impaired d or f electrons in magnetic ions and this interaction is important in studies of magnetic materials. The magnetic analogue of the atomic scattering factor is also tabulated in the International Tables [3]. Neutrons also have direct interactions with atomic nuclei, whose mass is concentrated in a volume whose radius is of the order of... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

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... 

Decreasing a particle s radius by a factor of 2, for example, decreases its mass by a factor of 2, or 8. Instead of an 80-g sample, a 10-g sample will now contain 10 particles. 

This can be accomplished by decreasing the radius of the particles by a factor of... 

Decreasing the radius by a factor of approximately 5 allows you to decrease the sample s mass from 80 g to 0.6 g. 

Unless extremely high potentials are to be used, the intense electric fields must be formed by making the radius of curvature of the needle tip as small as possible. Field strength (F) is given by Equation 5.1 in which r is the radius of curvature and k is a geometrical factor for a sphere, k = 1, but for other shapes, k < 1. Thus, if V = 5000 V and r = 10 m, then, for a sphere, F = 5 x 10 V/m with a larger curvature of, say, Iff m (0.1 mm), a potential of 500,000 V would have to be applied to generate the same field. In practice, it is easier to produce and apply 5000 V rather than 500,000 V. 

For spherical particles of radius R moving through a medium of viscosity 17, Stokes showed that the friction factor is given by... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Like e, t is the product of two contributions the concentration N/V of the centers responsible for the effect and the contribution per particle to the attenuation. It may help us to become oriented with the latter to think of the scattering centers as opaque spheres of radius R. These project opaque cross sections of area ttR in the light path. The actual cross section is then multiplied by the scattering efficiency factor optical cross... 

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