Volume sphere

In practice, tliere are various ways by which ( ) can be detennined for a given sample, and tire results may be (slightly) different. In particular, for sterically stabilized particles, tire effective hard-sphere volume fraction will be different from tire value obtained from tire total solid content. 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Diam. Cu. Fl. per Gallons per 42 Gallon Barrels per Sphoro Surlaeo Sphere Vulume Diam. in Foot Cu. Ft. per Gallons per 42 Gallon Barrels j)cr Sphere Surface Sphere Volume... 

Diam. in Feet Cu. Ft. per Foot of Cylinder Gallons per Foot of Cylinder 42 Gallon Barrels per Foot of Cylinder Sphere Surface irr Sq. Ft. Sphere Volume In Cu. Ft. Diam. in Feet Cu. Ft. per Fool of Cylinder Gallons per Foot of Cylinder 42 Gallon Barrols por Fool of Cylinder Sphere Surface in Sq. Fl. Sphere Volume in Cu. Ft. 

Dian). Cn. R. per Gallons per Barrels pur Sr hcre Surface Sphere Volume I Diam. Cu FI. per Gallons per Barrels per Sphere Surface ... 

To = temperature of the solvent at which tan A goes through a maximum. These values are presented in Table II. VSE (the Stokes-Eeinstein volume) is calculated for a spherical molecule if the molecule is aspherical this calculation (VSE) is called Vapparent The Vapparent can be smaller or larger than the Stokes-Einstein volume and varies from the equivalent sphere volume obtained by solution of equations 3,4 and 5. 

An analog expression can be found for parallel diffusion in Chapter 8. By integration of this expression over the entire sphere volume the fraction / (r) of initial gas remaining at dimensionless time r becomes... 

The influence of surfactant structure on the nature of the microemulsion formed can also be predicted from the thermodynamic theory by Overbeek (17,18). According to this theory, the most stable microemulsion would be that in which the phase with the smaller volume fraction forms the droplets, since the osmotic term increases with increasing i. For w/o microemulsion prepared using an ionic surfactant, the hard sphere volume is only slightly larger than the water volume, since the hydrocarbon tails of the surfactant may interpenetrate to a certain extent, when two droplets come close together. For an oil in water microemulsion, on the other hand, the double layer may extend to a considerable extent, depending on the electrolyte concentration... 

Cut number Radius (cm) Number of spheres Volume per sphere (cm3) Area per sphere (cm2) Total area (cm2)... 

XD, in centimeters, and (-) Debye sphere volume, IVp, in cubic centimeters. The plasma frequency is given on the right-hand axis. Condensed-state... 

Special Treatment of Voids— An interesting concept of voids in a regular array of spherical particles has been presented by W. O. Smith et al (1929). Assume a rhombohedral array such as shown in Figure 30, where the spacing is d + h between the particle centers, and 6 is adjusted to the observed porosity. From geometry, the number of spheres per unit-volume is equal to V2/ d + 5)3 and the total sphere volume is this quantity multiplied by (7t/6)dz. Hence, if the observed void is ... 

Fig. 37. The ratio of the equivalent hard sphere volume fraction based on the measured intrinsic viscosity as a function of for polyfmethyl methacrylate) spheres with grafted poly( 12-hydroxy stearic add) layers such that a/L = 4.7 (Mewis et ai, 1989). Open and closed circles correspond to the low and high shear limits of suspension viscosity.

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