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488 14MOLECULAR INTERACTIONS
which is rearranged to give
µ = (9ε0k
NA
× Pm(T2) − Pm(T1)
1/T2 − 1/T1
)
1/2
= (9 × (8.8542 × 10−12 J−1 C2m−1) × (1.3806 × 10−23 JK−1)
6.0221 × 1023mol−1
×(71.43 × 10−6 m3mol−1) − (75.74 × 10−6 m3mol−1)
1/(421.7 K) − 1/(320.0 K)
)
1/2
= 3.23... × 10−30 Cm = 0.9690 D
�e value of α is found using this value of µ together with one of the data points;
both give the same answer. Rearranging [14A.11–590], Pm = (NA/3ε0)(α +
µ2/3kT), for α and using the data for 320.0 K gives
α = 3ε0Pm
NA
− µ2
3kT
= 3 × (8.8542 × 10−12 J−1 C2m−1) × (75.74 × 10−6 m3mol−1)
6.0221 × 1023mol−1
− (3.23... × 10−30 Cm)2
3 × (1.3806 × 10−23 JK−1) × (320.0 K)
= 2.552 × 10−39 C2m2 J−1
E14A.6(b) �e relationship between relative permittivity and molar polarization is given
by the Debye equation, [14A.10–590], (εr − 1)/(εr + 2) = ρPm/M. Rearranging
gives
εr − 1 =
ρPm
M
(εr + 2) hence εr − 1 = εr (
ρPm
M
) + 2ρPm
M
hence εr (1 −
ρPm
M
) = 1 + 2ρPm
M
hence εr =
1 + 2ρPm/M
1 − ρPm/M
Taking the molar mass asM = 85.0 gmol−1 gives
εr =
1 + 2ρPm/M
1 − ρPm/M
= 1 + 2 × (1.92 g cm−3) × (32.16 cm3mol−1)/(85.0 gmol−1)
1 − (1.92 g cm−3) × (32.16 cm3mol−1)/(85.0 gmol−1)
= 8.97
E14A.7(b) �e relationship between the refractive index nr at a speci�ed wavelength and
the relative permittivity εr at the same wavelength is given by [14A.13–592],
nr = ε1/2r , hence εr = n2r . In addition the relationship between relative per-
mittivity and the polarizability α is given by the Clausius–Mossotti equation,
[14A.12–590], (εr − 1)/(εr + 2) = ρNAα/3Mε0. In using this equation it is
assumed that there are no contributions from permanent electric dipole mo-
ments to the polarization, either because themolecules are nonpolar or because