Physical Chemistry Third Edition

13.4 The Reaction Kinetics of Polymer Formation 591

We assume that the reaction is first order in the diacid, first order in the dialcohol, and

first order in the catalyst. We also assume that the reverse reaction and the uncatalyzed

reaction can be neglected. The forward rate differential equation is

dc

dt

−kf[cat]c^2 −k′c^2 (13.4-2)

where [cat] is the concentration of the catalyst. We let

k′kf[cat] (13.4-3)

We assume that the volume of the system does not change significantly, and since the

catalyst is not consumed,k′is a constant at constant temperature. Equation (13.4-2) is

the same as Eq. (11.2-8) except for the symbols used. Transcription of Eq. (11.2-11)

with the proper replacement of symbols gives the solution:

1

c



1

c 0

+k′t (13.4-4)

We define the variablep, the fraction of carboxyl groups that have reacted:

p

c 0 −c

c 0

 1 −

c

c 0

(13.4-5)

Equation (13.4-4) can be written in the form

p

k′c 0 t

1 +k′c 0 t

(13.4-6)

which is equivalent to

1

1 −p

 1 +k′c 0 t (13.4-7)

Exercise 13.15

Verify Eqs. (13.4-6) and (13.4-7).

Exercise 13.16

Construct an accurate graph ofpas a function oft, assuming thatk′c 0  1 .00 s−^1.

Thedegree of polymerization,x, is defined as the number of monomer units in a

polymer molecule. Thenumber-average degree of polymerizationis denoted by〈x〉n

and is given by

〈x〉n

c 0

c



1

1 −p

(13.4-8)

This value is a mean value with each molecule given equal weight (importance) in the

averaging. Equation (13.4-4) can be expressed in terms of〈x〉n:

〈x〉n 1 +k′c 0 t (13.4-9)