Chemistry Reference
In-Depth Information
(I is the second moment of mass about the center of mass, cf. Section 2.4.1.)
Then from the above equations,
X
m
i
r
i
X
m
i
r
g
=
(1-21)
and the average value of r
g
will be
D
E
1
=
2
X
m
i
r
i
X
m
i
r
g
i
1
=
2
h
r
g
i
5
h
=
(1-22)
5
where the
means an average.
The radius of gyration is directly measurable by light scattering (
Section 3.2
),
neutron scattering, and small angle X-ray scattering experiments. The end-to-end
distance is not directly observable and has no significance for branched species
that have more than two ends. A unique relationship exists between r
g
and d for
high-molecular-weight linear macromolecules that have random coil shapes:
hi
p
6
r
g
5
d
=
(1-23)
The end-to-end distance is more readily visualized than the radius of gyra-
tion and is more directly applicable in the molecular explanation of rubber
elasticity. The derivations in the following section therefore focus on d rather
than r
g
.
1.14.2
Root Mean Square End-to-End Distance of Flexible
Macromolecules
1.14.2.1
Freely Oriented Chains
The simplest calculation is based on the assumption that a macromolecule com-
prises
bonds
of fixed length l. (If the bonds differ in length, an average value can be used in
this calculation. Here we assume that all bonds are equivalent.) All angles
between successive bonds are equally probable. Such a chain is illustrated in
Fig. 1.8
where each bond is represented by a vector l
i
. The end-to-end vector in a
given conformation is
σ
1
1 (sigma
1) elements of equivalent size which are joined by
σ
1
X
σ
d
l
i
(1-24)
5
i
5
1
Now it is convenient to recall the meaning of the dot product of two vectors.
For vectors a and b, the dot (or scalar) product is equal to the product of their
lengths and the cosine of the angle between them. That is,
θ
a
b
5
ab cos
(1-25)
where
is the bond angle (which is the supplement of the valence angle, for C—
C bonds) and a and b are the respective bond lengths. The dot product is a scalar
θ
