Molecular Weights in Terms of Moments

In the previous section we have seen that average molecular weights are arithmetic means of distributions of molecular weights. An alternative and generally more 

A moment in mechanics is generally defined as Uj = Qd, where Uj is the jth moment, about a specified line or plane a of a vector or scalar quantity Q (e.g., force, weight, mass, area), d is the distance from Q to the reference line or plane, and j is a number indicating the power to which d is raised. [For example, the first moment of a force or weight about an axis is defined as the product of the force and the distance of the fine of action of the force jfrom the axis. It is commonly known as the torque. The second moment of the force about the same axis (i.e., i = 2) is the moment of inertia.] If Q has elements Qi, each located a distance di from the same reference, the moment is given by the sum of the individual moments of the elements  

Problem 4.1 Write general equations for statistical moments for (a) number distribution and (b) weight distribution of molecular weights. 

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

In the previous section we have seen that average molecular weights are arithmetic means of distributions of molecular weights. An alternative and generally more useful de nition of average molecular weights is, however, obtained in terms of moments of distribution (Rudin, 1982). 

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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. 

The product r o is the characteristic relaxation time xq of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(x), Xo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectrum ... 

These are molecular descriptors defined in terms of the weighted absolute central moment of first order, which is a statistical quantity used to measure variability of a distribution around a center. They are defined as... 

Now, the average molecular weights of the MWD can be more simply defined in terms of the moments (Eq. 1.12). The number average molecular weight is simply... 

Once this data table has been completed, it is possible to compute the molecular weight averages or moments of the distribution. The most common averages defined in terms of the molecular weight at each time slice and either the number of molecules n, or the area of each time slice are... 

Another approach to the SEC-viscometry data is that of Kirkland et al. (20). The intrinsic viscosity is a fundament property of the polymer sample in solution, and thus polymers may be characterized in terms of their intrinsic viscosity distribution (IVD) without attempting to convert this into a molecular weight distribution. Moments of the IVD may be calculated similar to those for the MWD (21). The advantage is that the intrinsic viscosity distribution is di-... 

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