Weights in Terms of Moments

In previous sections we have seen that average molecular weights are arithmetic means of distributions of molecular weights. An alternative and generally more useful definition can be given in terms of moments of distribution that facilitate generalizations beyond the two averages we have considered to this point. This approach also clarifies the estimation of parameters related to the breadth and symmetry of the distribution. 

Problem 4.2 Write general equations for statistical moments for (a) number distribution and (b) weight distribution of molecular weights. 

Weight distributions are usually encountered during analysis of polymer samples, while number distributions are more useful in consideration of polymerization kinetics. 

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Table 2-2 lists various average molecular weights in terms of moments of the number and weight distributions, where the quantity of polymer species with particular sizes are counted in terms of numbers of moles or weights, respectively. 

The number- and weight-average molecular weights can be expressed in terms of moments as ... 

Approximate reaction networks have become customary for modeling reactions in which the species are too numerous for a full accounting or chemical analysis. Lumped components or continuous distributions commonly take the place of single components in process models for refinery streams (Wei and Kuo 1969 Weekman 1969 Krambeck 1984 Astarita 1989 Chou and Ho 1989 Froment and Bischoff 1990). Polymerization processes are described in terms of moments of the distributions of molecular weight or other properties (Zeman and Amundson 1965 Ray 1972, 1983 Ray and Laurence 1977). Lumped components, or even hypothetical ones, are also prevalent in models of catalyst deactivation (Szepe and Levenspiel 1968 Butt 1984 Pacheco and Petersen 1984 Schipper et al. 1984 Froment and Bischoff 1990). 

A special average that can be estimated by measurements of the polymer solution intrinsic viscosity is the viscosimet-ric average molecular weight, which in terms of moments... 

The molecular weight distribution can be described in terms of moments or by a continuous function, either empirical or based on a model of the polymerization reaction. Other features of molecular structure include tactidty and branching. 

Finally we define a quantity known as the kth moment of the distribution. In terms of molecular weight. 

The breadth of the molecular weight distribution is often discussed in terms of the dispersity (Z>) and is expressed in terms of the moments as shown in eq. 15 ... 

Having shown that the weighting coefficient (A) of the term giving the contribution of an ionic structure to the molecular wave function is related to the dipole moment of the molecule, it is logical to expect that equations could be developed that relate the ionic character of a bond to the electronegativities of the atoms. Two such equations that give the percent ionic character of the bond in terms of the electronegativities of the atoms are... 

When a chain has lost the memory of its initial state, rubbery flow sets in. The associated characteristic relaxation time is displayed in Fig. 1.3 in terms of the normal mode (polyisoprene displays an electric dipole moment in the direction of the chain) and thus dielectric spectroscopy is able to measure the relaxation of the end-to-end vector of a given chain. The rubbery flow passes over to liquid flow, which is characterized by the translational diffusion coefficient of the chain. Depending on the molecular weight, the characteristic length scales from the motion of a single bond to the overall chain diffusion may cover about three orders of magnitude, while the associated time scales easily may be stretched over ten or more orders. 

Carrying out the Fourier transformation, one obtains the coordinate space representation for the third Zemach moment, in terms of the weighted convolution of two nuclear charge densities p r) [18]... 

Many of the averages and mean droplet diameters in current use may be expressed in terms of the moments of the various weighted size distributions. For example, the first moment of the surface-weighted size distribution is given by the expression... 

Because the inequalities among the various mean diameters are usually strengthened when the drop sizes are widely dispersed, the ratio of some higher-order moment to a lower-order moment is often useful as a measure of the dispersion of the drop sizes. For example, the coefficient of variation for the surface-weighted size distribution is a function of the ratio of the weight-weighted mean drop size to the volume-surface mean drop size. The variance of the drop-size distribution may also be expressed in terms of the moments of the unweighted size distribution. 

Various levels of initial and final configurations take different parts in radiative transitions. Therefore, it is convenient to introduce the concepts of emissive and receptive zones, which characterize the participation of the configurations in particular transitions [298]. The zones as the weighted distributions of the level energies (with the weight of each level equal to the total strength of all lines originating from it) may be expressed in terms of their moments. The initial moments of emissive (em) and receptive (rec) zones are defined, respectively, as... 

For conversions lower than xgei the average molar mass of the polymer exhibits a continuous increase. The first two moments of the molar mass distribution are the number-average molar mass, Mn, and the mass-average molar mass, Mw, respectively. Mn is defined in terms of the number contribution of every species to the whole population. The weight factor used to define this average is the molar fraction. Mw is defined in terms of the mass contribution of every species to the whole mass, so that the mass fraction is the weight factor used in its definition. 

Using the moments of distribution, we can easily express weight fraction in terms of mole fraction as follows ... 

In this case, an eigth order approximation was used in order to include some of the higher moments of the distribution. KnU,m) represents the weight for the nth moment of the size distribution. The values for Kn are given in Appendix II. A similar expression can be obtained in terms of the D32 average if Equation 24 is applied to Q(aav,m) in equation 19. 

The exact calculation of the weight of liquid lifted, in terms of the surface tension and density, is difficult and requires usually special solutions of the fundamental equation of Capillarity, for figures which often are not figures of revolution. The pull may reach a maximum some distance before the object is completely detached and the measurement of this maximum is considered more satisfactory than that of the pull at the moment of detachment.7 In most cases, however, the pull is applied by means of a torsion balance, and the upward motion of the object cannot be checked after the maximum pull is past, so that the detachment takes place almost immediately the maximum pull is reached. 

The arithmetic mean of a weight distribution (the count is in terms of the weight c, rather than number of molecules n, of each species) is likewise given by the ratio of the first to the zeroth moment of the particular distribution about zero. (The... 

Table 2-1 lists averages of the number and weight distributions in terms of these moments. 

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