Page 262 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Copolymerization 225

•
k [M ][M ]
• 21 2 1
[M ] = (7.12)
1
k 12 [M ]
2
The ratio of disappearance of monomers M /M is described by Equation 7.13 from Equations 7.10
2
1
and 7.11. Remember, that this is also the average composition of the growing chains and the resulting
polymer.
• • • •
d[M ] k 11 [M ][M ] + k 21 [M ][M ] [M ] (k 11 [M ] + k 21 [M ])
1
1
1
2
1
1
2
1
= = (7.13)
•
•
•
•
•
d[M ] k [M ][M ] + k [M ][M ] [M ] (k [M ] + k [M ])
2
22 2 2 12 1 2 2 22 2 12 1
•
Substitution of [M ] into Equation 7.13 gives
1
•
•

2 
d[M ] [M ]  (kk [M ][M ]/k [M ]) + k [M ] 
1 = 1 =  11 21 2 1 21 2 21 
•
•
d[M ] [M ]   (kk [M ][M ]/k [M ]) + k [M ] (7.14)
2  
2
2
12 21 2 1 21 2 22
Division by k and cancellation of the appropriate k’s gives
12

•
•


d[M ] = [M ] =  (k 11 [M ][M ]/k 12 [M ]) + [M ] 
2
1
2
2
1
1

•
•
d[M ] [M ]   ([M ][M ]/[M ]) + k [M ]/k 21   (7.15)
2
2
2 1 2 22 2
•
Substitution of r = k /k and r = k /k and cancellation of [M ] gives
2
1
21
12
2
11
22
 ( [M ]/[M ]) + 1r
d[M ] = [M ] =  1 1 2  (7.16)
1
1

d[M ] [M ]  ([M ]/[M ]) + r
2 2 1 2 2 
Multiplication by [M ] gives what are generally referred to as the “copolymerization equations,”
2
Equations 7.17 and 7.19, which gives the copolymer composition without the need to know any free
radical concentration, and which gives the composition of the growing polymer as a function of
monomer feed (Equation 7.19).
r
2
1
1
1
1
n = d[M ] = [M ] ( [M ] + [M ]) (7.17)
r
d[M ] [M ] ([M ] + [M ])
2 2 1 2 2
This equation, (Equation 7.17), is also presented in another form that allows greater ease of seeing
the relationship between the monomer feed, x, and copolymer composition. This is achieved by the
following steps.
•
Multiplying through by [M ] and [M ] gives
1 2
n = d[M ] = [M ]r 1 [M ] + [M ] [M ] (7.18)
1
2
1
1
1
r
d[M ] [M ] [M ] + [M ] [M ]
2 2 1 2 2 2
Then division of both the top and bottom by [M ] [M ] gives the second form, Equation 7.19, of
1 2
the copolymerization equation but in terms of the composition of the feed (x) on the composition of
the copolymer (n) as shown below:
2
1
1
1
n = d[M ] = r 1 ([M ] / [M ]) + 1 = r x + 1 (7.19)
x
d[M ] r ([M ] / [M ]) + 1 r / + 1
2 2 2 1 2
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