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508 14MOLECULAR INTERACTIONS
From above the amounts in moles of each polymer are 35ntotal and
2
5ntotal so
MW = n1M
2
1 + n2M22
n1M1 + n2M2
=
3
5ntotalM
2
1 + 2
5ntotalM
2
2
3
5ntotalM1 +
2
5ntotalM2
= 3M
2
1 + 2M22
3M1 + 2M2
= 3 × (62 kgmol−1)2 + 2 × (78 kgmol−1)2
3 × (62 kgmol−1) + 2 × (78 kgmol−1)
= 69 kgmol−1
E14D.2(b) �e rootmean square separation of the ends of a freely jointed one-dimensional
chain is given by [14D.6–617], Rrms = N1/2 l , whereN is the number ofmonomer
units and l is the length of each unit. In this case
Rrms = N1/2 l = 12001/2 × (1.125 nm) = 38.97 nm
E14D.3(b) �e contour length Rc of a polymer is given by [14D.5–617], Rc = Nl , and the
root mean square separation of the ends of a freely jointed one-dimensional
chain is given by [14D.6–617], Rrms = N1/2 l . In both cases N is the number of
monomer units and l is the length of each unit.
�emonomer of polypropene –[CH2CH(CH3)]n– is taken to beCH2CH(CH3).
�e number of monomers in the chain is given by
N =
Mpolymer
MCH2CH(CH3)
= 174 × 10
3 gmol−1
42.0774 gmol−1
= 4.13... × 103
�e length of each CH2CH(CH3) 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 in the
Resource section a C–C bond length is approximately 154 pm, so the monomer
length l is taken as 2× (154 pm) = 308 pm.�e contour length and root mean
square separation are then given by
Rc = Nl = (4.13... × 103) × (308 pm) = 1.27... × 106 pm = 1.27 µm
Rrms = N1/2 l = (4.13... × 103)1/2 × (308 pm) = 1.98... × 104 pm = 19.8 nm
E14D.4(b) �e radius of gyrationRg of a one-dimensional randomcoil is given by [14D.7a–
618], Rg = N1/2 l . Rearranging gives
N = (
Rg
l
)
2
= ( 18.9 × 10
−9 m
450 × 10−12 m
)
2
= 1.8 × 103
E14D.5(b) �e probability that the ends of a one-dimensional random coil are a distance
nl apart is given by [14D.3–616], P = (2/πN)1/2e−n
2/2N where N is the total
number of monomers in the chain and l is the length of each monomer unit.