Problems
1.1 A polymer sample combines five different molecularweight fractions of equal weight. The molecular weights of these fractions increase from 20,000 to 100,000 in increments of 20,000. Calculate , , and Based upon these results, comment on whether this sample has a broad or narrow molecularweight distribution compared to typical commercial polymer samples.
1.2 A 50g polymer sample was fractionated into six samples of different weights given in the table below. The viscosityaverage molecular weight, , of each was determined and is included in the table. Estimate the numberaverage and weightaverage molecular weights of the original sample. For these calculations, assume that the molecularweight distribution of each fraction is extremely narrow and can be considered to be monodisperse. Would you classify the molecularweight distribution of the original sample as narrow or broad?
Fraction 
Weight (g) 

1 
1.0 
1500 
2 
5.0 
35,000 
3 
21.0 
75,000 
4 
15.0 
150,000 
5 
6.5 
400,000 
6 
1.5 
850,000 
1.3 The Schultz–Zimm ^{11} molecularweightdistribution function can be written as
where a and b are adjustable parameters (b is a positive real number) and Γ is the gamma function (see Appendix E) that is used to normalize the weight fraction.
(a) Using this relationship, obtain expressions for and in terms of a and b and an expression for M_{max}, the molecular weight at the peak of the W(M) curve, in terms of .
(b) Derive an expression for M_{max}, the molecular weight at the peak of the W(M) curve, in terms of .
(c) Show how the value of b affects the molecularweight distribution by graphing W(M) versus M on the same plot for b = 0.1, 1, and 10 given that = 10,000 for the three distributions.
Hint: (if n is a positive integer).
1.4 The following requested calculations refer to Examples 1.1, 1.2, and 1.3 in the text:
(a) Calculate the zaverage molecular weight, , of the discrete molecular weight distribution described in Example 1.1.
(b) Calculate the zaverage molecular weight, , of the continuous molecularweight distribution shown in Example 1.2.
(c) Obtain an expression for the zaverage degree of polymerization, , for the Flory distribution described in Example 1.3.