10.6 Bose–Einstein Statistics
577
ideal gas contribution of P ≈ 3.3 × 10 5 Pa to the total pressure in the container at a
temperature of 1000 K.
As the chemical potential μ is identically zero for a photon gas, the Gibbs energy
vanishes (i.e., G = Nμ = 0), and hence the entropy S associated with radiation may
be obtained from the relation
G = U − T S + P V = 0
as
S =
U + P V
T
=
4
3
U
T
=
1
3 C V .
(10.6.47a)
Another means of checking the consistency of zero chemical potential is to utilize
the Helmholtz energy A, which is given by
A(T , V ) ≡ A = U − T S =
4 −
16
3
σ V
c
T
4
= −
4σ V
3c
T
4 ,
(10.6.47b)
to determine the chemical potential using the thermodynamic relation
μ =
∂A
∂N
T ,V
= 0 .
Indeed, expression (10.6.47b) for the Helmholtz energy is consistent with the
chemical potential being identically zero, as we see that A is independent of N
for thermal photons.
That the chemical potential vanishes identically for a photon gas in thermal
equilibrium at temperature T requires us to look more carefully into some of the
general results that were obtained for the grand ensemble in Chap. 4. For example,
in discussing fluctuations in the number of particles, we saw in Eq. (4.2.43) that the
relative standard deviation in the number of particles took the form
σ N
N
2
=
k B T
V
κ T ,
in which κ T , defined as
κ T ≡ −
1
V
∂V
∂P
T
,
is the isothermal compressibility of the substance. However, as we have seen from
Eq. (10.6.44) that the pressure for a thermal photon gas depends only upon the
temperature, T , and is independent of V , the partial derivative (
∂P
∂V ) T is zero.
Normally,
∂P
∂V
T
and (
∂V
∂P ) T are related by Eq. (B.1.12) as reciprocals of one