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continues its second-order behavior at least up to a degree of polymerization of 90 corresponding to a molecular weight of $10,000. There is no change in the reactivities of the hydroxyl and carboxyl groups in spite of the large increase in molecular size (and the accompanying large viscosity increase of the medium). Similar results have been observed in many other polymerizations. Data on the degradation of polymers also show the same effect. Thus in the acid hydrolysis of cellulose there is no effect of molecular size on hydrolytic reactivity up to a degree of polymerization of 1500 (molecular weight 250,000) [Flory, 1953]. The concept of functional group reactivity independent of molecular size has been highly successful in allowing the kinetic analysis of a wide range of polymerizations and reactions of polymers. Its validity, however, may not always be quite rigorous at very low or very high conversions. 2-2c Step Polymerizations Other than Polyesteri cation: Catalyzed versus Uncatalyzed The kinetics of step polymerizations other than polyesteri cation follow easily from those considered for the latter. The number of different general kinetic schemes encountered in actual polymerization situations is rather small. Polymerizations by reactions between the



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III. I} x { PkPF+2Pk'o~W+k'-k+i8'

A and B functional groups of appropriate monomers proceed by one of the following situations [Saunders and Dobinson, 1976]: 1 Some polymerizations, such as the formation of polyamides, proceed at reasonable rates as uncatalyzed reactions 2 Other polymerizations such as those of urea, melamine, or phenol with formaldehyde (see Table 1-1) require an externally added acid or base catalyst to achieve the desired rates of reaction 3 A few polymerizations can be reasonably employed either in a catalyzed or an uncatalyzed process Polyurethane formation is an example of this type of behavior The reaction between diols and diisocyanates is subject to base catalysis However, the polymerization is often carried out as an uncatalyzed reaction to avoid various undesirable side reactions Regardless of the situation into which a particular polymerization falls, the observed overall kinetic features will be the same.

[aD(k), ab(k')]





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The polymerization rates will be dependent on both the A and B groups For the usual case where one has stoichiometric amounts of the two functional groups, the kinetics will be governed by Eq 2-30 or 2-33 The observed kinetics will also be the same whether the polymerization is carried by the reaction of A and B A B monomers or by the self-reaction of an A B monomer The derivation (Sec 2-2b) of the kinetics of catalyzed polyesteri cation assumes that the catalyzed reaction is much faster than the uncatalyzed reaction, that is, k0 ) k This assumption is usually valid and therefore one can ignore the contribution by the uncatalyzed polyesteri cation to the total polymerization rate For example, k0 is close to two orders of magnitude larger than k for a typical polyesteri cation.

For the atypical situation where k is not negligible relative to k0 , the kinetic expression for [M] or X n as a function of reaction time must be derived [Hamann et al, 1968] starting with a statement of the polymerization rate as the sum of the rates of the catalyzed and uncatalyzed polymerizations:.

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sink' L cos kL } sin kL sin k'L. (9.48)

dM k M 3 k0 M 2 dt 2-34

Now then. in general. (9.46) is not verified. But for E 1. up to first order in E. we find that (k) 1 ~ -E/2VL (9.49) ~ VL ~ k - k n + iE2 / 4L and

k0 t k M k M 0 k0 1 1 ln k0 k M k0 M 0 M M 0 2-35

-iE2 /4L'

The natural log term on the right side of Eq. 2-35 is the contribution of the uncatalyzed reaction. Its relative importance increases as k=k0 increases. (When k=k0 is very small, Eq. 2-35 converts to Eq. 2-32.) 2-2d 2-2d-1 Nonequivalence of Functional Groups in Polyfunctional Reagents Examples of Nonequivalence

2 3 A

(9.50)

There are instances where some or all parts of the concept of equal reactivity of functional groups are invalid [Kronstadt et al., 1978; Lovering and Laidler, 1962]. The assumption of equal reactivities of all functional groups in a polyfunctional monomer may often be incorrect. The same is true for the assumption that the reactivity of a functional group is

Under this approximation. [ac(k) and (9.52) From (9.51) and (9.52), we find that (9.46) does hold for In an analogous way, we can show that [a(k), a(k')J

3-76 3-77 3-78 3-79

at (k')] ~ 8(k -

+ CO2 + ] + CO2 + ]

k') _

[2 COO ] [2CO2 + 2 ] [2CO2 + 2 ]

[2CO2 + 2 ] [2CO2 + 2CO2 + 2 ]

~ k - kn -

3-80 3-81

(9.51)

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