Solutions Manual for Polymer Science and Technology by Joel R. Fried – Step-by-Step Exercise Solutions

SOLUTION MANUAL




This text is associated with Fried/Polymer Science and Technology, Third Edition (9780137039555)

, SOLUTIONS TO PROBLEMS IN POLYMER SCIENCE AND TECHNOLOGY,
3RD EDITION

TABLE OF CONTENTS
Chapter 1 1
Chapter 2 5
Chapter 3 14
Chapter 4 24
Chapter 5 28
Chapter 7 36
Chapter 11 40
Chapter 12 51
Chapter 13 52



CHAPTER 1
1-1 A polymer sample combines five different molecular-weight fractions, each of equal weight. The
molecular weight s of these fractions increase from 20,000 to 100,000 in increments of 20,000.
Calculate M n , M w , and M z . Based upon these results, comment on whether this sample has a
broad or narrow molecular-weight distribution compared to typical commercial polymer samples.
Solution
Fraction # Mi (10-3) Wi Ni = Wi/Mi (105)
1 20 1 5.0
2 40 1 2.5
3 60 1 1.67
4 80 1 1.25
5 100 1 1.0
 300 5 11.42
5
5
M n   Wi N   43,783
i 1 1.142 104
5
W M i i 300, 000
M  i1
  60,000
w 5
5
W i
i 1
5


M 
 WM
i1
i i2


4 108  16 108  36 108  64 108 100 108
 73,333
z 5
3105
W M
i1
i i

Mz 60,000
  1.37 (narrow distribution)
Mn 43,783

1-2 A 50-gm polymer sample was fractionated into six samples of different weights given in the table
below. The viscosity-average molecular weight, M v , of each was determined and is included in the table.
Estimate the number-average and weight-average molecular weights of the original sample. For these
calculations, assume that the molecular-weight distribution of each fraction is extremely narrow and can


1
This text is associated with Fried/Polymer Science and Technology, Third Edition
(9780137039555) .

,be considered to be monodisperse. Would you classify the molecular weight distribution of the original
sample as narrow or broad?

Fraction Weight Mv
(gm)
1 1.0 1,500
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
Solution
Let M i  M v
Fraction Wi Mi Ni = Wi/Mi WiMi
(106)
1 1.0 1,500 667 1500
2 5.0 35,000 143 175.000
3 21.0 75,000 280 627,500
4 15.0 150,000 100. 2,250,000
5 6.5 400,000 16.3 2,600,000
6 1.5 850,000 1.76 1,275,000
 50.0 1208 7,929,000
6
50.0
M n   Wi N   41,322
i 1 1.21103
6

W M i i 7,930, 000
M  i1
 158,600
w 6
50.0
W i
i 1
Mw 158, 600
  3.84 (broad distribution)
Mn 41,322


1-3 The Schultz–Zimm [11] molecular-weight-distribution function can be written as
b1
a
W M   M b exp aM 
b 1

where a and b are adjustable parameters (b is a positive real number) and  is the gamma function (see
Appendix E) which is used to normalize the weight fraction.
(a) Using this relationship, obtain expressions for M n and M w 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 M n .

Solution



Mn  
 0
WdM

 W
0
M  dM
let t = aM


2
This text is associated with Fried/Polymer Science and Technology, Third Edition
(9780137039555) .

,  ab1  b ab1 1  b 1
0 WdM    b 1 0 t a exp t  d t a 
  b 1 ab1 0
t exp  t  dt 
  b 1
  b 1  1
1  b1
 ab1  b1
ab1 ab1 1
0 W M  dM   b 1 0 t a exp t  d t a    b 1 ab 0 t exp t  dt  b 1 ab b 
a b  a
bb b
1 b
Mn 
ab a

 0

WMdM  
ab1
 b1       ab1   b  2 
 

Mw 
 WMdM   b 1 0 t a exp t d t a
  b  1 ab2
0 WdM
0


b 1b 1  b 1
ab  1 a

(b) Derive an expression for Mmax, the molecular weight at the peak of the W(M) curve, in terms of M n .

Solution
dW a b1
 bM b1 exp aM   M b aexp aM   0

dM  b  1   
bM ba  aM b
b
(i.e., the maximum occurs at M n )
a  M  Mn
a




(c) Show how the value of b affects the molecular weight distribution by graphing W(M) versus M on the
same plot for b = 0.1, 1, and 10 given that M n = 10,000 for the three distributions.


Solution
b
a
10,000
b 0.1 1 10
a 110-5 110-4 110-3
b1
W a M b exp aM  dM
  b 1

where   b 1   aM  exp aM  dM .
b

0

Plot W(M) versus M

Hint:  xn exp ax dx    n 1 an1  n! an1 (if n is a positive interger).
0




3
This text is associated with Fried/Polymer Science and Technology, Third Edition
(9780137039555) .