Average number

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

By line 5, the reaction has reached 80% completion and the number average value of the degree of polymerization is 5. Although we have considered this slowly evolving polymer in terms of the extent of reaction, another question starts to be worrisome How long is this going to take ... 

The number average degree of polymerization for these mixtures is easily obtained by recalling the definition of the average from Sec. 1.8. It is given by the sum of all possible n values, with each multiplied by its appropriate weighting factor, provided by Eq. (5.24) ... 

The number average degree of polymerization is given by dividing the number of repeat units by the number of chains, or... 

An equivalent way of looking at the conclusion of item (2) is to recall that Eq. (5.40) gives the (number average) number of monomers of both kinds in the polymer and multiply this quantity by the average molecular weight of the two kinds of units in the structure (88 + 112)/2 = 100. 

The phenomena we discuss, phase separation and osmotic pressure, are developed with particular attention to their applications in polymer characterization. Phase separation can be used to fractionate poly disperse polymer specimens into samples in which the molecular weight distribution is more narrow. Osmostic pressure experiments can be used to provide absolute values for the number average molecular weight of a polymer. Alternative methods for both fractionation and molecular weight determination exist, but the methods discussed in this chapter occupy a place of prominence among the alternatives, both historically and in contemporary practice. 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

The solute molecular weight enters the van t Hoff equation as the factor of proportionality between the number of solute particles that the osmotic pressure counts and the mass of solute which is known from the preparation of the solution. The molecular weight that is obtained from measurements on poly disperse systems is a number average quantity. 

Table 9.3 lists the intrinsic viscosity for a number of poly(caprolactam) samples of different molecular weight. The M values listed are number average figures based on both end group analysis and osmotic pressure experiments. Tlie values of [r ] were measured in w-cresol at 25°C. In the following example we consider the evaluation of the Mark-Houwink coefficients from these data. 

In addition, the intercept obtained by extrapolating this asymptote back to sin (0/2) = 0 equals (2M )". Note that both Mand are number averages when this asymptotic limit is used. This is illustrated schematically in Fig. 10.15 and indicates that even more information pertaining to polymer characterization can be extracted from an analysis of the curvature in Zimm plots. 

The number-average molecular weight of most commercially available acetal resins is between 20,000 and 90,000. Weight-average molecular weight may be estimated from solution viscosities. 

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