The Number and Weight Distribution Functions

We need to express Nx in terms of x and p and to do so we simply use a set of substitutions. Equations 5-15 and 5-16 show these for the number and weight distributions, respectively. 

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Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

Equations (5.44) and (5.47) describe the dilTerential number and weight distribution functions, respectively, for linear step polymerizations at the extent of reaction p. These distributions are also... 

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue distribution. 

This is the statistical weight-distribution function for a linear polycondensation reaction at the extent of reaction p. The number-distribution and weight-distribution functions are illustrated in Figs. 1 and 2 for values of p. 

Equations (1.52) and (1.53) are expressions of the number and weight fractions for the most-probable distribution of molecules expected for linear condensation polymerization. The most-probable weight fraction distribution W vr(p) is compared with experimental data in Fig. 1.18. While the number fraction for the most-probable distribution is a monotonic function, the weight fraction has a maximum. The position of the maximum in... 

Subsequently we show some important properties of the moment generating function. This include the number and weight average of the degree of polymerization and other averages, as well as the distribution function. There is a potential confusing situation in nomenclature. In polymer chemistry, the term cumulative distribution is used for what is called in mathematics the cumulative distribution function or simply distribution function. Further, polymer chemistry uses the term differential distribution for the mathematical term probability density function. 

Assuming that a polymer has a molar-mass distribution defined by (m) = ae ", where n m)Am represents the fraction of molecules with molar masses between m and m + Am, show that a = b and calculate (i) the number- and weight-average molar masses, and hence the polydispersity index and (ii) the molar mass at the mode (maximum) of the distribution function. 

The program then requests specification as to type of polymer (line 320). If the "polymer" component is a collection of oligomers, the number of unique species is sought (line 360). The values for the mole (or weight) fraction, functionality and molecular weight of each species is then entered (lines 380-650). The number, site, and mass expectation values of the functionality and molecular weight (lines 650-810) are computed. The necessary site and mass distribution functions are also computed (lines 820-850). 

The mechanical properties exhibited by a polymer after irradiation are a complex function of molecular weight and molecular weight distribution and the number and type of new structures formed. Thus, it is difficult to draw structure/radiation resistance conclusions from the change in the mechanical properties alone. However, the changes in mechanical properties are direct indications of the ultimate usefulness of the polymer in a radiation environment. 

The understanding of the macromolecular properties of lignins requires information on number- and weight-average molecular weights (Mn, Mw) and their distributions (MWD). These physico-chemical parameters are very useful in the study of the hydrodynamic behavior of macromolecules in solution, as well as of their conformation and size (1). They also help in the determination of some important structural properties such as functionality, average number of multifunctional monomer units per molecule (2, 3), branching coefficients and crosslink density (4,5). 

Figure 3. The stair step curve is the detected sample cumulative distribution function (CDF) for the combined detectors. We have weighted the two detectors equally so that the height of an IMB detection is 12/8 the height of a Kamiokande detection (note we have included the count at. 686 seconds rejected by the Kamiokande group as being too close to their threshold). If millions of counts had been seen, the CDF would be smooth and directly proportional to the number luminosity emitted by the supernova.

Figure 7 shows calculated octane numbers from hexadecane cracking as a function of gasoline yield. Calcined and steamed zeolites are represented by open and closed symbols, respectively. The calculated octane number reflects changes in the gasoline molecular weight distribution and, to a lesser extent, composition effects. 

Given the Schulz-Flory distribution, the mole and weight fractions of polymer with j monomer units (based on polymer) as a function of that number j are... 

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