Ring-Opening Polymerizations - Principles of Polymer Chemistry - page 251

where [M*] is the concentration of the propagating oxonium ions. Such ions could be oxonium,
sulfonium, and others. When, however, there is propagation-depropagation equilibrium, it can be
expressed as follows:

K P
K Dp

M n þ M nþ 1 M Ð

The rate expression can be written as propagation-depropagation

M ½ M R p ¼ d ½ M = d t ¼ K p ½ M K DP ½

At condition of equilibrium, if we designate the monomer concentration [M] C , and the polymeri-
zation rate is zero, we can write

K p ½ M C ¼ K DP

Hirota and Fukuda [ 1 ] described the quantitative dependence of the degree of polymerization on
various reaction parameters for an equilibrium polymerization. The equilibrium can be described as

K I

M I þ M Ð

where, I is the initiating species. It is assumed that the equilibrium constants for the initiation and
propagation are independent of the size of the propagating species. The concentration of the
propagating chains [M*] of size

n c

at equilibrium then can be written as:

½ M ¼K I ½I C ½ M C ðK p ½ M C Þ n 1

The total concentration of molecules size N can be expressed as follows

X

M n K I ½ ½N¼ ½ I C ½ M C =ð 1 K p ½ M C Þ

The total concentration of monomer segments that are incorporated into the polymer can also be
expressed as follows:

X

M ¼K I ½

2

½ W ¼ ½ I C ½ M C =ð 1 K p ½ M C Þ

n

This allows us to express the average degree of polymerization that is [ W ]/[ N ] as follows:

1

DP ¼

1 K p ½ M C

We can describe the rate of polymerization in terms of d[ M ]/d t as