Volume-weighted probability density

Since Xpo follows a distribution and the relation between A.po and the hole radius r is nonlinear [Eq. (11.3)], we prefer to estimate the mean hole-volume as the mean of the number-weighted hole volume distribution. The radius distribution [the probability density function (pdf)], n(rfc), can be calculated from n (rh) = - 3 (A.) (dx/drh) [Gregory, 1991 Deng et al., 1992b] ... 

Many researchers have published SANS data on RPV steels. Plate VI (see colour section between pages 202 and 203) shows a typical example of the type of scattering curve from irradiation-induced features. The lines show the fit obtained from a maximum entropy analysis of these data and the resultant most probable size distribution of scattering features is shown in Fig. 9.13. Once a mean composition and mean magnetic character have been ascribed to the scattering features, it is possible to calculate the absolute number density of scattering features and their mean size. Their mean size is often defined in terms of their volume-weighted mean size. 

Figure 4-4 The volume-weighted probability density for the lowest-energy eigenfunction of the hydrogenlike ion. The most probable value of r occurs at rmp-...

Fig. 9. Comparison of results derived from neutron diffraction experiments and from the model for the distribution of terminal methyl groups in spherical dodecylsulphate micelles at 35 °C. The micelles are observed to have a mean aggregation number of 78. If the methyl group probability density is p(r) where r is the distance from the micelle center, the plot shows the volume weighted probability density P(r) = AnCr p r), where C is a normalization...

One of the most common ways of expressing the effectiveness of strength or stiffness of a material is as a ratio of either of the quantities to the density, i.e., weight per unit volume. Such an index does not include the cost to achieve a certain strength or stiffness, but cost comparisons are probably not valid by themselves because many factors influence cost beyond raw material cost. 

Usually a good deal of experimentation is needed before a substance can be considered to be pure. Even then, much more work and study are needed before one can decide with confidence that a given pure substance is an element or a compound. Consider the substance water. Water is probably the most familiar substance in our environment and all of us recognize it easily. We are familiar with its appearance and feel, its density (weight per unit volume), the way in... 

Density or specific gravity (relative density) is used whenever conversions must be made between mass (weight) and volume measurements. This property is often used in combination with other test results to predict oil quality, and several methods are available for measurement of density (or specific gravity). However, the density (specific gravity) (ASTM D1298 IP 160) is probably of... 

Instead of measuring the weight directly we may calculate it from the volume and the density the drop volume method has been applied by Harkins chiefly to the measurement of the tension between two liquid phases, and it probably falls little short in accuracy from the previous method. More frequently it has been j modified, especially for biochemical purposes, as a drop number method that is, a known volume of liquid iFallbwed oo nov. of a tube, and the number of drops formed is compared with that formed by a standard fluid. This method is necessarily very rough. 

You are probably well aware that the pressures at great depths under the ocean are extremely large, and the deeper you go, the greater the pressure that is exerted. Let s start small and look at a beaker filled with water. The beaker has a diameter of 10 cm, and the height of the water is 20 cm. How much pressure does the water exert on the bottom of the beaker To answer this question we simply need to recall that P = FI A. The force involved is the weight of the water contained in the beaker. The weight of the water can be calculated from the mass of the water, and the mass of the water can be calculated from the volume, using density as a conversion factor. 

An electronic parameter that often correlates with metabolic rates is the electrophilic (or nucleophilic) superdelocalizability. This quantity is a reactivity index formulated by Fukui and colleagues as an orbital-weighted electron density.145 The total electrophilic superdelocalizability, 2SE, summed over all atoms in a molecule, exhibits a parallelism with the hydrophobic parameter, log P, in several series of compounds such as PAHs and aliphatic amines, where it is probably approximating molecular volume. 

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