Chemistry Reference
In-Depth Information
Morawetz et al. also presented another interesting contribution to the calcula-
tion of C. They started out from Kuhn's random flight theory of a coiled flexible
chain which allows one to calculate the probability that one end group is directly
neighboring the second end group of the same chain. This probability is, in turn,
identical with C in Stoll's work. The largest ring studied by Stoll et al. (24 ring
members) was considered, and the average end-to-end distance of an aliphatic
chain of 24 atoms was taken from Flory's statistical calculations [
12
]. Using
Eq. (
7.4
), where N
A
is Avogadro's number and \h
2
[ the mean square end-to-end
distance, a C value of 0.1 was calculated, whereas C = 0.007 was calculated from
the experimental data of Stoll et al. Among several arguments presented to explain
this conspicuous difference, one argument is of general importance. Kuhn's model
was designed for long flexible polymer chains under theta conditions, and thus, is
not good model for oligomers and ignores the role of solvation.
3
=
2
Þ
3
=
2p\h
2
[
C ¼ 1000
=
N
A
ð
ð
7
:
4
Þ
\h
2
[ ¼ K
ch
b
0
2
¼ nl
2
ð
7
:
5
Þ
with n = number of skeletal bonds and l = bond length
d C½
=
dt ¼ K
intra
½
M
i
½
with i ¼ 1
;
2
;
3
;
...
;
n
ð
7
:
6
Þ
and with C
i
and M
i
= i-meric cyclic and linear oligomers
K
inter
X
M
j
M
i
j
þ
2K
inter
M
j
SM
j
ð
7
:
7
Þ
d M
i
½
=
dt ¼ K
intra
½
M
i
½
d C½
=
dt
0
¼ EM M
i
½
7
:
8
Þ
The weak point in the kinetic concept of Morawetz et al. was the assumption
that cyclization of oligomers and polymers do not need to be considered. This
short-coming was revised in the work of Mandolini et al. [
13
-
15
] who demon-
strated that the cyclization factor C
1
of the monomer (M
1
) depends on the cycli-
zation factors of the oligomers and vice versa. However, the main purpose of their
work was different and defined as follows: ''We now describe a more refined
approximation treatment, where the formation of both, linear and cyclic oligomers
with DP's up to 12 is taken into account. The procedure involves the micro-
computer-assisted numerical integration of the proper system of differential rate
equations by the simple Euler method [
16
]
''
.
Mandolini et al. started out from the rate equations (
7.6
) and (
7.7
). These
equations were simplified by a change of the time scale (t
0
= ft) so that the
numerical values of the rate constants called EM (effective molarity), which in the
case of the monomer EM
1
is identical with the cyclization constant C in Stoll's
work. EM
1
was arbitrarily selected, and the consequences for the EMs of the
higher oligomers were computed (and listed in tables). EM
2
was taken from the
literature, and the higher EMs were calculated from Eqs. (
7.4
) and (
7.5
), where
Kuhn's random flight model comes into the play. In Eq. (
7.5
) the mean square
