Page 179 - Carrahers_Polymer_Chemistry,_Eighth_Edition

142 Carraher’s Polymer Chemistry

The corresponding weight-average molecular weight distribution, W , can be calculated from the
n
relationship
nN
W = n (4.98)
n N
o
as follows

−
nN (1 p p n− 1
2
)
−
W = o = n (1 p p n− 1 (4.99)
2
)
n N
o
The relationship shown in Equations 4.98 and 4.99 shows that high values of p (0.99) are essen-
tial in producing high N and W values. The number-average molecular weight M calculated from
n n n
Equations 4.96 and 4.97 are as follows:
mN m
M n = o = 1 p− (4.100)
N
Where m = the molecular weight of the mer and
+
m (1 p )
+
M w = − = M n (1 p ) (4.101)
1 p
Thus, the index of polydispersity, M /M becomes 1 + p, as shown below:
w n
+
−
M w = m (1 p )/(1 p ) = 1 p
+
−
M n m /(1 p ) (4.102)

Thus, when p = 1, the index of polydispersibility for the most probable distribution for stepwise
polymerizations is 2.
Because the value of p is essentially 1 in some stepwise polymerizations employing very pure
monomers, the products obtained under normal conditions will have very high molecular weights

and are found to be difficult to process. The value of p can be reduced by using a slight excess of
one of the reactants or by quenching (stopping) the reaction before completion. Thus, if a reaction is
quenched when the fractional conversion p is 0.995, the average DP becomes 200.
When more than 1 mol of B is used with 1 mol of A, the ratio of A/B or “r” may be substituted in

the modified Carothers’ equation as follows:
r
+
total nA at p n [(11/ )] [1(1/ )]
+
r
DP = = = (4.103)
−
−
r
n
total nA at rp 2 [1 p−+ (1 rp / )]/2 1 p−+ [(1 rp )/ ]
r

Multiplying the top and bottom by r gives
+
r + 1 (1 r )
DP = = (4.104)
−
+
−
−
+
r (1 p ) (1 rp ) (1 r ) 2rp

For the formation of nylon-66, if r = 0.97 and p is about 1, the DP is equal to
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