10.4 Fermi–Dirac Statistics
539
we require values for the number of conduction electrons, N ce , and the volume V of
the sample. For convenience, we shall determine the value for n ce (Na) for a molar
sample of metallic sodium at the standard temperature of 25 ◦ C (298.15 K). For this,
we need values for the molecular weight (molar mass) and the mass density, ρ(Na).
These quantities are available from any standard reference source, such as the NIST
website http://webbook.nist.gov.chemistry or the CRC Handbook of Chemistry and
Physics [5], for example: M(Na) = 22.99 g mol −1 (0.02299 kg mol
−1 in SI units)
and ρ(Na) = 0.97 g cm −3 at 298.15 K (970 kg m
−3 in SI units). The molar volume
of sodium metal at T = 298.15 K is thus obtained from V = M/ρ as
V (Na) =
0.02299 kg mol
−1
970 kg m
−3
= 2.37 0 × 10
−5 m
3 mol
−1 .
Note that the overbars on V and M denote molar amounts, unlike the overbar on N
that we have employed to distinguish the thermodynamic quantity N thermo from the
variable value of N for members of the grand ensemble.
The number density of conduction electrons in metallic sodium at T = 298.15 K
is thus given by
n ce (Na) =
N 0
V (Na)
=
6.022 × 10 23 mol
−1
2.370 × 10 −5 m 3 mol −1
= 25.40 9 × 10
27 m
−3 .
We shall also require a value for n ce (Na) at 0 K, which we may estimate from
n ce (Na) at 298.15 K by making use of the coefficient of thermal expansion, α(Na) =
7.1 × 10 −5 K −1 (from the CRC Handbook of Chemistry and Physics) to estimate
the relative change in the molar volume as V (Na)/V (Na) ≈ α(Na))T , with
T = −298.15 K, to give
V T=0 (Na) (1 − 0.0212)V T=298 (Na)
= 2.31 5 × 10
−5 m
3 mol
−1 .
This gives a conduction electron number density of
n ce (T = 0) = 25.96 × 10
27 m
−3 ,
from which the Fermi energy F (Na) is obtained via Eq. (10.4.21) as
F (Na) =
5n ce (T = 0)
8π
2
3 h 2
8m e
= 5.12 2 × 10
−19 J
539
we require values for the number of conduction electrons, N ce , and the volume V of
the sample. For convenience, we shall determine the value for n ce (Na) for a molar
sample of metallic sodium at the standard temperature of 25 ◦ C (298.15 K). For this,
we need values for the molecular weight (molar mass) and the mass density, ρ(Na).
These quantities are available from any standard reference source, such as the NIST
website http://webbook.nist.gov.chemistry or the CRC Handbook of Chemistry and
Physics [5], for example: M(Na) = 22.99 g mol −1 (0.02299 kg mol
−1 in SI units)
and ρ(Na) = 0.97 g cm −3 at 298.15 K (970 kg m
−3 in SI units). The molar volume
of sodium metal at T = 298.15 K is thus obtained from V = M/ρ as
V (Na) =
0.02299 kg mol
−1
970 kg m
−3
= 2.37 0 × 10
−5 m
3 mol
−1 .
Note that the overbars on V and M denote molar amounts, unlike the overbar on N
that we have employed to distinguish the thermodynamic quantity N thermo from the
variable value of N for members of the grand ensemble.
The number density of conduction electrons in metallic sodium at T = 298.15 K
is thus given by
n ce (Na) =
N 0
V (Na)
=
6.022 × 10 23 mol
−1
2.370 × 10 −5 m 3 mol −1
= 25.40 9 × 10
27 m
−3 .
We shall also require a value for n ce (Na) at 0 K, which we may estimate from
n ce (Na) at 298.15 K by making use of the coefficient of thermal expansion, α(Na) =
7.1 × 10 −5 K −1 (from the CRC Handbook of Chemistry and Physics) to estimate
the relative change in the molar volume as V (Na)/V (Na) ≈ α(Na))T , with
T = −298.15 K, to give
V T=0 (Na) (1 − 0.0212)V T=298 (Na)
= 2.31 5 × 10
−5 m
3 mol
−1 .
This gives a conduction electron number density of
n ce (T = 0) = 25.96 × 10
27 m
−3 ,
from which the Fermi energy F (Na) is obtained via Eq. (10.4.21) as
F (Na) =
5n ce (T = 0)
8π
2
3 h 2
8m e
= 5.12 2 × 10
−19 J