Step-Growth Polymerization and Step-Growth Polymers - Principles of Polymer Chemistry - page 401

One way is by continual removal of the by-products. In such situations, the reactions take place at
nonequilibrium conditions and there is no K 4 .

7.1.2 Kinetic Considerations

The rate of polymerization can be expressed as the rate of disappearance of one of the functional

groups

. In reactions of polyesterification, this can be the rate of disappearance of carboxyl groups,

t

d[CO 2 H]/d .

R P ¼ d ½ CO 2 H
d

OH Þ 2 ½

¼ k 3 ½ C ð OH

t

OH Þ 2

½ ð

represent carboxyl, hydroxy, and protonated
carboxyl groups, respectively. Also, it is possible to write an equilibrium expression for the proton-
ation reaction of the acid as follows:

In the above equation [CO 2 H], [OH], and C

OH Þ 2 ½ A

K ¼ k 1

k 2 ¼ ½

C ð

½ ½

COOH HA

This equation can be combined with the above rate expression:

d ½ COOH

¼ k 1 k 3 ½

COOH ½ OH ½ HA

¼ k 3 K½ COOH ½ OH ½ HA

A

d t k 2 ½

If there is no catalyst present and the dicarboxylic acid acts as its own catalyst, HA is replaced by
[COOH] and the expression becomes:

d ½ COOH

2

¼ k½ ½

COOH OH

d t

A

In the above expression k 1 , k 2 , k 3 , and the concentration of the ½ ions have been replaced by an

experimentally determined rate constant, .

In most step-growth polymerization reactions, the concentrations of the two functional groups are
very close to stoichiometric. This allows writing the above rate equation as follows:

k

d ½ M

3

¼ k½ M

d t

represents the concentration of each of the reacting species. They can be
hydroxy and carboxylic acid groups in a polyesterification reaction, or amino and carboxylic acid
groups in polyamidation reaction, and so on.
The above equation can also be written as follows:

In this equation, M

3

k t ¼ ½ =½

d d M M

after integrating of the above, we get:

2

2 kt ¼ 1 =½ M þ constant