Chemistry Reference
In-Depth Information
Figure 2.8
Part of a poly(ethylene oxide) chain using wire frame and space-filling
models. Rotation about a carbon atom means that the following bond
sweeps out a cone with an included angle of 2y. The hydrogen atoms are
white and the oxygen atoms are grey.
defined as the positive direction) as in the opposite (to the left being defined as
negative). Therefore, the mean value of the vector will be zero. However, we
can solve the problem of the directionality by using the square of each and then
taking the square root of the average value. Hence, we define the end-to-end
dimension as the root mean square value of N bonds of length b:
R
R
2
1
=
2
¼
b
N
p
ð
2
:
36
Þ
Figure 2.8 shows part of a poly(ethylene glycol) chain and illustrates the type
of problem encountered with real chains. First, there are fixed bond angles (in
this case the tetrahedral angle), and second, there is the physical volume
occupied by each atom so that the walk cannot, for example, cross itself. Both
of these constraints mean that the random walk is more open than the simple
model. We may express this expansion by introducing a factor, C
N
, known as
the characteristic ratio:
R
2
¼
C
N
Nb
2
ð
2
:
37
Þ
C
N
b
2
is ''the effective bond length'', b
0
. For the case of just the bond-angle
restriction on the random walk:
C
N
¼
1
þ
cosy
1
cosy
ð
2
:
38
Þ
and so for the tetrahedral angle y
¼
180-109.5
o
and C
N
¼
2. Experimentally, we
can think of this characteristic ratio as describing the coil expansion due to
constraints imposed on the random walk such as fixed bond angles, restricted
