Chemistry Reference
In-Depth Information
l
15
l
17
l
14
l
18
l
13
l
19
l
12
l
20
l
11
l
4
l
6
θ
l
3
l
9
l
7
l
5
l
8
l
10
l
28
l
2
l
22
l
27
l
23
l
1
l
29
l
26
d
l
24
l
25
FIGURE 1.8
An unrestricted macromolecule. The bond lengths l
i
are fixed and equal and there are no
preferred bond angles.
quantity and the dot product of a vector with itself is just the square of the length
of the vector. To obtain the end-to-end scalar distance, we take the dot product of
d. Because the single chain can take any of an infinite number of conformations,
we will compute the average scalar magnitude of d over all possible conforma-
tions. That is,
*
!
U
!
+
X
σ
X
σ
d
2
h
i
5
h
d
d
i
5
l
i
l
j
(1-26)
i
5
1
j
5
1
The dummy subscripts i and j indicate that each term in the second sum is
multiplied by each term in the first sum.
If
θ
is the angle between the positive directions of any two successive bonds,
then
l
i
l
i
1
1
5
l
i
l
i
1
1
cos
θ
(1-25a)
θ
All values of
are equally probable for a chain with unrestricted rotation, and
θ
52
2
θ
since cos
cos(
)
l
i
l
i
1
1
5
l
i
l
i
1
1
h
cos
θi
5
0
(1-27)
