Chemistry Reference
In-Depth Information
EXAMPLE 1-1
For a polyethylene chain, l is the C—C bond distance (1.54
10
2
10
m or 0.154 nm) and
θ
3
is 180
minus the tetrahedral bond angle
180
2
109
28
0
5
70
32
0
. Cos
θ
then is 0.33
5
d
2
l
2
. The chain end-to-end distance calculated using this model
and
h
i
5
2
σ
is thus
expanded by a factor of
p
for this reason. And the corresponding C
N
value is 2.
1.14.2.3
Hindered Rotation Chains
When some conformations are preferred over others (e.g., in
Fig. 1.6
), the chain
dimensions are further expanded over those calculated, and
Eq. (1-29)
becomes
1
1
cos
θ
1
h
cos
φi
1
d
2
l
2
h
i
5
σ
(1-31)
1
cos
θ
1
2
h
cos
φi
2
where
is the statistical mechanical (not arithmetic) average value of the
cosine of the rotation angle
h
cos
φi
φ
with probabilities determined by Boltzmann factors,
exp(
)/kT). The effect of Boltzmann factors is that rotation angles with high
energy contribute less to the average as such rotation angles do not occur fre-
quently, especially at low temperatures. For free rotation, all values of
2
E(
φ
φ
are
φ
equally probable, cos
is zero, and
Eq. (1-31)
reduces to
Eq. (1-30)
.Ina
completely planar zigzag conformation, all rotamers are trans and
0. Then
h
cos
φi
5
1, and the model breaks down in this limit. Nevertheless, it does show
that the chain becomes more and more extended the closer the rotation angle is to
zero. The values of
φ
5
can be calculated if the functional dependence of
potential energy of a sequence of bonds on the bond angle is known. For small
molecules, this can be deduced from infrared spectra.
Figure 1.6
showed this rela-
tion approximately for a normal paraffin.
The value of
h
cos
φi
will depend on temperature (Boltzmann factors),
of course, since the molecule will have sufficient torsional motion to overcome
the energy barriers hindering rotation when the temperature is sufficiently
high.
h
cos
φi
EXAMPLE 1-2
To illustrate how to calculate hcos φi using Boltzmann factors, let's assume that each
CaC bond in a polyethylene molecule obeys the torsional potential curve shown in
Fig. 1.6
(i.e., each bond is either in the trans, g
state, not in any other rotational angles).
Given that at 140
C (413 K), the difference between the torsional potentials between the
trans and gauche states (i.e.,
,org
2
1
Δ
e) is about 2100 J/mol. Fraction of the bonds in the trans
state is given by
