Page 138 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Polycondensation Polymers 101

Thus,
A A 1
DP n = 0 = 0 = (4.21)
A t A 0 (1− p ) 1− p

The relationship given in Equation 4.21 is called the Carothers equation because it was fi rst
found by Carothers while working with the synthesis of polyamides (nylons). For an essentially
quantitative synthesis of polyamides where p is 0.9999, the DP is approximately 10,000, the value
n
calculated using Equation 4.21.

DP n = 1 = 1 = 1 = 10,000 (4.22)
−
1− p 1 0.0000 0.0001

Thus, the Carothers equation allows calculation of maximum DP as a function of extent of

polymerization, and the purity of reactants. This value is sufficient to produce polyesters that will
give strong fibers. The high value of p is decreased, as is the DP, if impurities are present or if some

competing reaction, such as cyclization, occurred. Since the values of k at any temperature can be
2
2
determined from the slope (2kA ) of when 1/(1 − p) is plotted against t, DP at any time t can be
o n
determined from the expression
(DP n ) = 2 [A ] + constant (4.23)
kt
2
2
0
Much longer times are required to effect formation of high polymer polyesters in uncatalyzed

esterifications than for acid or base-catalyzed systems. For catalyzed systems, since the added acid
or base is a catalyst, its apparent concentration does not change with time; thus, it is not included in
the kinetic rate expression. In such cases the reaction follows the rate expression
A
Rate of polycondensation =− d[ ] = [ ][ ] (4.24)
B
kA
dt
For [A] = [B] we have
A
− d[ ] = kA 2 (4.25)
[]
dt
and rearrangement gives
A
− d[ ] = kt (4.26)
A
[] 2

which on integration and subsequent substitution gives

kt = 1 − 1 = 1 − 1 (4.27)
A A A (1 p− ) A
t 0 0 0
Rearrangement gives

Akt = 1 − 1= DP n − 1 (4.28)
0 (1 p− )

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K10478.indb 101 9/14/2010 3:38:10 PM