8.3 Dielectric Constant of a Polar Gas
425
or
L(a)
a
3
−
a 3
45
, a 1.
(8.3.21)
Even though we have obtained an expression that has a term cubic in a, we shall
retain only the linear term for the present calculation of the dielectric constant of
HCl.
If we employ the linear approximation to the Langevin function in our expression
for the polarization, we find that P z is given by
P z =
N
V
αE +
N
V
μ 2
d E
3k B T
≡ χ
(e) E.
(8.3.22)
With the electric susceptibility given by
χ
(e)
=
N
V
α +
μ 2
d
3k B T
,
(8.3.23)
we obtain for the dielectric constant ε the result
ε = 1 +
N
V
α
0
+
μ 2
d
3 0 k B T
.
(8.3.24)
Example 8.3 Dielectric constant ε of dry HCl gas.
We wish to calculate a value for the dielectric constant ε of dry HCl gas at a
temperature of T = 273 K. We know that the permanent electric dipole moment of
HCl has the value μ d = 1.11 D, and if we take one mole of HCl, we have V = V
◦
and N = N 0 . We must first convert from the chemical unit debye for the dipole
moment into the SI unit using 1D = 3.336 × 10 −30 C m, so that μ d (HCl) = 3.70 ×
10 −30 C m. The dielectric permittivity of free space has the value 0 = 8.854188 ×
10 −12 C 2 J −1 m −1 , and the Loschmidt number L 0 = N 0 /V
◦ has the value L 0 =
2.686780 × 10 25 m −3 . Finally, the isotropic polarizability of HCl is given in the
combination α// 0 , which for HCl has the value α(HCl)// 0 = 2.61 × 10 −30 m 3 .
Using these data we see that the permanent component of the dipole moment of
HCl contributes
N 0
V
◦
μ 2
d
3 0 k B T
3.677 × 10
−3
≈ 0.0037,
to ε, while the electric-field-induced component of the dipole contributes
N 0
V
◦
α
0
= (2.687 × 10
25 )(2.61 × 10
−30 ) ≈ 0.0001,