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516 14MOLECULAR INTERACTIONS
where the integral over θ is of the form of Integral T.3 with k = 1 and
a = π.�e total mass of the sphere is mtot = ρV = (4/3)ρπa3, hence the
moment of inertia can be expressed as
I = mtot ×
3
4ρπa3
× ρ × a
5
5
× 4
3
× 2π = 2
5mtota
2
A rigid rotor with the same mass as the sphere has moment of inertia
I = mtotR2g , hence Rg = (2/5)1/2a . Note that this is the radius of gyration
associated with rotation about this axis.
P14D.6 �e radius of gyrationRg is foundby equatingmtotalR2g to themoment of inertia
I, hence R2g = I/mtotal.�e moment of inertia is given by I = ∑
N
j m jR2j , where
m j is the mass of unit j and R j is its distance from the centre of mass. If all N
units have the same mass m then I = m∑ j R2j and mtotal = Nm. Hence
Rg =
1
mtotal
∑
j
m jR2j =
1
Nm
×m∑
j
R2j =
1
N
∑
j
R2j
P14D.8 As explained in�e chemist’s toolkit 18 in Topic 7E on page 273, the frequency of
a harmonic oscillatorwithmassm and force constant kf is ν = (1/2π)(kf/m)1/2.
�e force constant is given by [14D.12c–621], kf = kT/Nl 2. �e mass is taken
as the mass of one monomer, which is given by m = M/NNA where M is the
molar mass of the macromolecule and N is the number of monomers in the
chain. Combining these expressions gives
ν = 1
2π
( kf
m
)
1/2
= 1
2π
( kT/Nl
2
M/NNA
)
1/2
= 1
2πl
(RT
M
)
1/2
where R = kNA is used.
�emonomer of polyethene –[CH2CH2]n– is taken to be CH2CH2.�e length
of each CH2CH2 unit is estimated as the length of a two C–C bonds: one C–C
bond in the centre and half a bond length either side where the unit connects
to carbons in adjacent units. From Table 9C.2 on page 362 a C–C bond length
is approximately 154 pm, so the monomer length l is taken as 2 × (154 pm) =
308 pm. Noting that 1 J = 1 kgm2 s−2 and 1Hz = 1 s−1, the vibration frequency
is therefore
ν = 1
2πl
(RT
M
)
1/2
= 1
2π × (308 × 10−12 m)
((8.3145 JK−1mol−1) × ([20 + 273.15] K)
65 kgmol−1
)
1/2
= 3.16... × 109 Hz = 3.2 GHz
�e expression shows that the frequency increases with temperature and de-
creases with increasing molar mass.�e T1/2 dependence re�ects the thermal