Moments of molecular weight

M0 total number of polymer molecules XP (zeroth moment of molecular weight distribution). 

The determination of is of particular value in studying mechanochemistry. One advantage is that the methods for number average molecular weight are generally independent of chemical composition. Compositional variations are a sizable problem in determinations for other moments of molecular weight. 

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

There are other types of molecular weights based on higher-order moments to the distribution (see Table 1.23), but M and M, are the most widely nsed qnantities. In... 

A RIS model with neighbor dependence is used to calculate mean-square dipole moments and their temperature coefficients for PDMS chains over a wide range of molecular weight. Chain conformational energies required in the calculations are obtained from a previous analysis of the random-coil dimensions of PDMS chains in the limit of large x (S 116). 

The mean-square dipole moments of POE and POMg are determined from dielectric constant measurements on dilute solutions in benzene. The values obtained are in good agreement with those predicted using the RIS models for these chains. In addition, the unperturbed dimensions of POMg are calculated as a function of molecular weight using the RIS theory. 

More recently Christiansen 4 has made a careful study of the polythionic acids and especially of certain physical properties—K-radiations, electrical moments, and molecular weights by ebuHioscopic methods—of the p-toluoyl-tri-, -penta- and -hexa-thionates, and suggests that none of the foregoing formulae is adequate, but that the properties and reactions of the acids are best explained by the assumption that the... 

Fig. 9. Mass fractions, line-widths or second moment of three components for bulk-crystals as a function of molecular weight using a logarithmic scale. The same data were reported in Ref.66 using an ordinary scale of molecular weight...

The possibility that the compounds might be polymerization isomers has been eliminated by cryoscopic determinations of molecular weights in benzene solution, which showed both isomers to be monomolecular. Dipole moments in such solutions were found to be 9.5 D. for the cis isomer and 2.41 D. for the trans isomer. The latter is not zero since the four ethyl groups lie on the same side of the plane containing the platinum, chlorine, and sulfur atoms. Interconversion of the isomers does not occur read-... 

We have seen that average molecular weights arc arithmetic means of distributions of molecular weights. An alternative and generally more useful definition is in terms of moments of the distribution. This facilitates generalizations beyond the two averages we have considered to this point and clarilies the estimation of parameters related to the breadth and symmetry of the distribution. 

In the case of polydisperse polymers, the limiting compliance increases strongly with the broadness of the distribution of molecular weights. The limiting comphance is not, however, a simple function of the polydispersity index, because its value depends on the shape of the distribution itself. There is indeed no simple correlation with any molectilar weight moments (averages), and moleciilar models will be really helpfid to describe the elasticity of the melt. 

Fig. 45. Effect of molecular weight on the bending moment-displacement curves of notched PE...

Figure 45 shows the efTect of molecular weight on the bending moment-displacement curves of PE samples. Figure 46 shows the corresponding damage zone of the samples taken out just before maximum load is reached. The most pronounced difference is that the concentrated craze line along the midplane ahead of the notch disappears... 

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). 

Figure 4. Plot of dipole moment versus molecular weight for series of symmetrical and dipolar dendritic macromolecules.

---