Number density average

It is evident that most studies reported to date have used number density, average size or weight per eent as eontrol variables. Often these variables are inferred from other measurements, ineluding density, solution supersaturation, refraetive index ete. Inferential teehniques have been shown to be partieularly suitable for industrial seale applieations where laser seattering deviees for on-line size distribution measurement are not yet praetieal for industrial eontrol purposes, although substantial progress is being made to that end. Even when usable, however, these measurement deviees are often eharaeterized by noise and require operation at very low solids eoneentration. 

We note, as already mentioned, that the definition of Ruc that was used in Ref. 27 is different from the definition given in Eq. (50) that is, in Ref. 27 a volume fraction averaged radius was calculated instead of the number density averaged radius, Eq. (50). For this reason the values of Kcq and 2K -f- K are slightly different from the ones found in Ref 27. 

Disturbing Event Probability of Disturbing Event Actual Change in Number Density Average Change in Number Density... 

Ignoring the effect of molecular interactions, which is a gross assumption, the macroscopic response Xap measured in a laboratory axis frame will be the molecular property multiplied by the number density, averaged over all possible orientations of... 

For homogeneous systems, the average number density is n = (N) / V=v Let us define a local number... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

The constants K depend upon the volume of the solvent molecule (assumed to be spherica in slrape) and the number density of the solvent. ai2 is the average of the diameters of solvent molecule and a spherical solute molecule. This equation may be applied to solute of a more general shape by calculating the contribution of each atom and then scaling thi by the fraction of fhat atom s surface that is actually exposed to the solvent. The dispersioi contribution to the solvation free energy can be modelled as a continuous distributioi function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

The experiments with a beam of silver particles were conducted at room temperature. The energy of dissociation of diatomic molecules of silver is 1.78 eV, the heat of evaporation of silver molecules is 95 kcal/mol [46], and the heat of evaporation of an uniatomic silver is 64 kcal/mol. Mass-spectrometric studies [46] of silver vapour above a metallic silver showed that the ratio of number densities of ions Ag /Ag2 is equal to two. In other studies [47], a considerably larger value of this ratio was found. At 1037 - 1147 C molecular mass of silver particles in vapours was found to be 278 90 [46], i.e., an average number of atoms in a molecule of silver is 2.56. 

Fig. 6.7. Evolution of the sample averaged (R< ) as a function of MC time. The initial value of e(N) = C = 1.0 was changed to the values indicated after 600 MC steps. The indicated melt value corresponds to a comparable system with explicit chains with repulsive Lennard-Jones interactions and a number density of 0.85 cr-3 (from [45])...

In the description of MPC dynamics, the size of the collision cell was not specified. Given the number density h = N/V of the system, the cell size will control how many particles, on average, participate in the multiparticle collision event. This, in turn, controls the level of coarse graining of the system. As originally formulated, it was assumed that on average particles should free stream a distance comparable to or somewhat greater than the cell length in the... 

The number density of crystals (v) increased linearly with an increase of t for all samples, which indicates a steady nucleation process. Figure 29 shows the plot of log/ against AT-2 for different 1.1 obeyed the well-known equation, I = Io exp(- C/AT2) where Iq and C are constants. These straight lines were parallel to each other. This indicates that the slope of the straight line C is almost constant irrespective of l. We obtained the average of C ((C)),... 

Another procedure for calculating the W value has been developed by La Verne and Mozumder (1992) and applied to electron and proton irradiation of gaseous water. Considering a small section Ax of an electron track, the energy loss of the primary electron is S(E) Ax, where S(E) is the stopping power at electron energy E. The average number of primary ionizations produced over Ax is No. Ax where o. is the total ionization cross section and N is the number density of molecules. Thus, the W value for primary ionization is 0)p = S(E)/No.(E). If the differential ionization cross section for the production... 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

Thus, the source terms for each environment S(c) and Sk ((/)) will be closed. Of particular interest are the local nucleation rates /(c ). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations /((c)). This results in a rapid increase in the local particle number density mo due to the creation of a very large number of nuclei. As discussed below, this will have significant consequences on the local rate of aggregation. 

Calculate the total number density and (kinetic) energy density (in eV cm-3) of cosmic-ray protons from Eq. (9.5) and deduce their average energy. 

Nuclides, reaction with monomers, 14 248 NuDat database, 21 314 Nukiyama-Tanasawa function, 23 185 Null-background techniques, in infrared spectroscopy, 23 139-140 Number-average molecular weight, 20 101 of polymers, 11 195, 196 Number density, of droplets, 23 187 Number of gas-phase transfer units (Nq), packed column absorbers, 1 51 Number of overall gas-phase transfer units (Nog), packed column absorbers, 1 52 Number of transfer units (Nt, NTU), 10 761... 

Table 2. Composition of electrically and color neutral mixed phases, corresponding quark number chemical potentials and average baryon number densities pB = n/3 in unities of nuclear matter saturation density po = 0.17/fm3. The various components are defined in Tab. 1.

The value of the peaks and troughs in the pair distribution function represent the fluctuation in number density. The peaks represent regions where the concentrations are in excess of the average value while the troughs represent a deficit. As the volume fraction is increased, the peaks and troughs grow, reflecting the increase in order with concentration. We... 

Membrane-like microstructures are generally several micrometers thick, while the lateral dimensions of the structures and the surrounding package are on the order of a few hundred micrometers. If the layered thin-film structure would be directly transferred to a 3-d geometry model, an enormous number of finite elements would be created, as the smallest structure size determines the mesh density. Averaging the structural information and properties over the different layers in the cross section of the membrane is a good method to avoid such problems. The membrane is, therefore, initially treated as a quasi-two-dimensional object. 

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