Number distribution function

Numerical results for the some model polydisperse systems have been reported in Refs. 81-83. It has been shown that the effect of increasing polydispersity on the number-number distribution function is that the structure decreases with increasing polydispersity. This pattern is common for the behavior of two- and three-dimensional polydisperse fluids [81] and also for three-dimensional quenched-annealed systems [83]. 

Nanocluster diameter (nm) XRD 2.6 From number distribution function"... 

Many droplet size distributions in random droplet generation processes follow Gaussian, or normal distribution pattern. In the normal distribution, a number distribution function/(D) may be used to determine the number of droplets of diameter D ... 

The normal distribution function, also referred to as the Flory-Schulz distribution, relates the fraction of an x-mer (a polymer molecule consisting of x repeat units) in the entire assembly of molecules to its formation probability. It can be defined either as a number distribution function or as a weight distribution function. The number of moles of an x-mer (Nx) is given by the normal number distribution as follows ... 

Figure 2a, b. Antineutrino number distribution function for the 1.64 Hq core evolved with the soft EOS. The maximum is normalized to unity, as only the shape of the curve is being considered. Also shown is the shape of the expected positron number spectrum that would be produced by electron antineutrino capture on protons taking into account detector characterestics. 

The foregoing analysis was expressly elaborated to treat a diverse range of microscopic kinetic processes. Each termination process, however, was regarded as acting alone. It is possible to handle any combination of termination mechanisms by simple addition of the number distribution functions for the first order and second order events ... 

With the data given in Prob. 1, plot the number distribution function An/ (nT A log d) and the mass distribution function Am/(mT A log d) as a function of the logarithm of the particle diameter. Assume all particles within a size interval are spheres having a diameter equal to the midpoint of the size interval. The density of the particles equals 1 g/cm3. 

Equation (5-22) is the differential number distribution function for equilibrium step-growth polymerizations in homogeneous systems. 

We conclude, then, that the probability that the original M] unit was part of a sequence of H, such units is P" (l — P ). But the probability that the sequence contains n Mj units is also the fraction of all M] sequences which contain n, units. That is to say, it is the number distribution function W(Mi, n,) for Mj sequence lengths ... 

This last equation is the statistical number-distribution function for a linear polycondensation reaction at the extent of reaction p. 

Similar methods can be used to construct universal plots for molar mass distributions of linear and hyperbranched condensation polymers. The number distribution function n p, N) for linear condensation polymers is obtained from the number fraction distribution [Eq. (1.66)] ... 

Z (x, r,t) single number distribution function denoting the number of particles per unit volume of the particle phase space at time t (general) r,t) average single particle number density function using particle diameter as inner coordinate (i— —3)... 

U An aerosol initially has a number distribution function iioidp). 

Equations (5.102) and (5.103) give the differential number distribution functions for polymerization of nonstoichiometric mixtures of A—A and B —B. In Eq. (5.102) the first term represents nnAA> the mole fraction of type IIAA species the second and third terms represent, respectively, Jt-iibb and nuAB-... 

An expression for weight distribution function can be derived from that of number distribution function Eq. (5.102). Let N be the total moles of all species at Pa. PB extents of reaction. The weight fraction of type IIAA species will thus be given by... 

X number of Mi units is also the fraction of all Mi sequences which contain x units. That is to say, it is the number distribution function n,c(Mi) for Mi-sequence lengths ... 

Equation (6.173) represents the number distribution function. The corresponding weight distribution function Wx, by direct analogy to that for step-growth polymerization (p. 254), will be written as... 

By using the function ns Dp) we implicitly assume that the number distribution is no longer a discrete function of the number of molecules but a continuous function of the diameter Dp. This assumption of a continuous size distribution is valid beyond a certain number of molecules, say, around 100. In the atmosphere most of the particles have diameters smaller than 0.1 pm and the number distribution function n (Dp) usually exhibits a narrow spike near the origin (Figure 8.4). 

Expressing the aerosol distributions as functions of In Dp or log Dp instead of Dp is often the most convenient way to represent the aerosol size distribution. Formally, we cannot take the logarithm of a dimensional quantity. Thus, when we write In Dp, we really mean In (Dp/1), where the reference particle diameter is 1 pm and is not explicitly indicated. We can therefore define the number distribution function neN r Dp) as... 

The Continuous Coagulation Equation Although (13.59) and (13.60) are rigorous representations of the coagulating aerosol population, they are impractical because of the enormous range of k associated with the equation set above. It is customary to replace Nk(t) (cm-3) with the continuous number distribution function n(v,t) (pm 3 cm-3), where v = kv is the particle volume with v the volume of the monomer. If we let vq = g v, then (13.60) becomes, in the limit of a continuous distribution of sizes... 

Evans el al. (1977) have shown how the loss of configurational entropy can be calculated assiuning, somewhat unrealistically, that the segments behave as disconnected gas molecules with the appropriate number distribution function over the available space. Results obtained in this way for several distributions are summarized in Table 12.4. Also included in this table are the corresponding expressions for spheres obtained by a Deijaguin integration. These allow the elastic repulsion for spheres to be calculated from... 

Keywords Characteristic drop diameter Cumulative volume fraction Discrete probability function (DPF) Drop size distribution Empirical drop size distribution Log-hyperbolic distribution Log-normal distribution Maximum entropy formalism (MEF) Nukiyama-Tanasawa distribution Number distribution function Probability density function (pdf) Representative diameter Root-normal distribution Rosin-Rammler distribution Upper limit distribution Volume distribution... 

In Fig. 1 the space structure, the interatomic distance and coordination number distribution function for the most regular diamond-like silicon dioxide structure is presented. This cluster was constructed as 3x3x3 extended crystobalite cells and it containings 708 atoms. This cluster has been used as a reference for highly regular structures. 

Fig. 1. The space structure, the interatomic distance and coordination number distribution function for the most regular diamond-like silicon dioxide structure.

Graphical overlay of distribution curves can serve for comparison of a set of samples and finding even subtle differences among them. Note Instead using the weight fraction it is possible to use the mole fraction. The frequency functions are then called the number distribution functions and subscripts w and n can be used in order to differentiate between the weight distribution and number distribution, respectively. 

In the foregoing, we have considered the heterodisperse system as composed of a number of distinct homodisperse fractions. However, as a rule, the distributions are continuous. This is approximated by taking - Mi(=dM) infinitesimally small n(M) is the number of particles having a molar mass between M and M + dM. The function n M) is called the (differential) number distribution function. Note that n(M) dM is a number and because M is expressed in kg mol", n M) has the dimension mol kg". In a similar way, w(M) dM is the mass of the particles having a molar mass between M and M + dM w(M) is the (differential) mass distribution function, having the dimension mole. 

What is for the given molar mass distribution the maximum variation in MJM 7 Give a graph for the number distribution function n(M) in case of the maximum value for MJM. ... 

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