Chemistry Reference
In-Depth Information
independent processes that govern the size of the macromolecules that are pro-
duced. If we ignore all these important complications we can write the following
expressions for the rates of initiation (
R
i
), propagation (
R
p
), termination (
R
t
), and
transfer to monomer (
R
tr
,
M
):
R
i
5
ð
K
½
ZX
n
½
BA
Þ
k
i
½
M
(11-44)
M
"
R
p
5
k
p
½
M
½
(11-45)
M
"
R
t
5
k
t
½
(11-46)
M
"
½
R
tr
;M
5
k
tr
½
M
(11-47)
where [M
"
] represents
P
i
5
1
½
AM
i
and it is assumed that an overall rate con-
stant can represent
the reactions of all species of carbocations and their
counterions.
It is necessary to invoke the steady-state assumption (
R
i
5
R
t
;
d
[M
"
]
/dt
5
0)
to make this model mathematically tractable. With this assumption,
5
Kk
i
½
ZX
n
½
BA
½
M
M
"
(11-48)
k
t
and
2
k
p
½
M
½
ZX
n
½
BA
Kk
i
R
p
5
(11-49)
k
t
If the molecular growth is controlled by chain transfer to monomer, then
X
n
5
R
p
=
R
tr
5
k
p
=
k
tr
(11-50)
This is more likely under practical polymerization conditions than the alterna-
tive case in which termination reactions limit the size of the macromolecules and
X
n
5
R
p
=
R
t
5
k
p
½
M
=
k
t
(11-51)
We pause here to note that the steady-state assumption that is so helpful in
simplifying the analysis of free-radical kinetics (Section 8.3.4) will not apply to
many cationic polymerizations of vinyl monomers, because propagation through
free carbenium ions is so much faster than any of the other reactions in the
kinetic chain.
Despite all these qualifications,
Eq. (11-49)
gives a form with which the kinet-
ics of cationic systems can perhaps be studied. This is
d
½
dt
5
M
a
b
c
2
R
p
5
k
obs
½
M
½
catalyst
½
cocatalyst
(11-52)
where the superscripts are determined experimentally. It is obvious from the pre-
ceding discussion, however, that the composite nature of
k
obs
in
Eq. (11-52)
and
the rate constants in
Eq. (11-49)
make it possible to postulate many mechanisms
to fit any single set of kinetic data.
