576
10 Quantum Statistics
Given that σ/c 1.8915 × 10 −16 Pa K −4 , radiation pressure will thus be negligible
under terrestrial conditions. However, in stellar interiors, where temperatures can
reach several million degrees, radiation pressure can, and does, play a significant
role.
We may obtain an explicit expression for the dependence of the number of
thermal photons from our expression for N by converting the discrete sum over
photon energy states into an integral over a continuous frequency distribution having
the density of photon states given by Eq. (10.6.36). This gives N as
N =
8πV
(hcβ) 3
∞
0
x 2
e x − 1
dx
=
16πk 3
B
h 3 c 3 ζ(3) V T
3 ,
(10.6.46a)
or, upon evaluating the constant, as
N = 2.0286 × 10
7 V T
3 .
(10.6.46b)
In particular, let us note that the number of photons in a photon gas at thermal
equilibrium at temperature T depends linearly upon the volume V of its black-body
container.
Example 10.5 Comparison of the pressures of a thermal photon gas and an ideal
classical gas at a temperature of 1000 K.
Let us consider a tungsten-walled container of volume V = 10 m 3 at temperature
T = 1000 K. Thus, for thermal photons, the radiation pressure is given by
Eq. (10.6.45) as
P = 2.5220 × 10
−16 T
4 ,
giving P radiation (T = 1000 K) 2.5 × 10 −4 Pa [1 J m −3 ≡ 1 Pa] or 2.5 × 10 −9 bar,
and the number of photons in the enclosure, given by Eq. (10.6.46b) will be of order
2 × 10 17 . As the radiation pressure, P rad , is independent of the system volume (i.e.,
the volume of the container), we shall compare it with the pressure exerted by an
ideal gas that represents what might be the laboratory situation generated by starting
with an ideal gas in a container of volume V at room temperature (say T = 300 K),
then heated to T = 1000 K, so that both N and V are fixed. The resultant ideal gas
pressure will thus be
P 1000 K = P 300 K
1000
300
k B 3.33 bar
for P 300 K = 1 bar, corresponding (for V = 10 m 3 ) to N 2.4 × 10 21 particles.
Thus, the radiation contributes a partial pressure P rad ≈ 3 × 10 −4 Pa, versus the
10 Quantum Statistics
Given that σ/c 1.8915 × 10 −16 Pa K −4 , radiation pressure will thus be negligible
under terrestrial conditions. However, in stellar interiors, where temperatures can
reach several million degrees, radiation pressure can, and does, play a significant
role.
We may obtain an explicit expression for the dependence of the number of
thermal photons from our expression for N by converting the discrete sum over
photon energy states into an integral over a continuous frequency distribution having
the density of photon states given by Eq. (10.6.36). This gives N as
N =
8πV
(hcβ) 3
∞
0
x 2
e x − 1
dx
=
16πk 3
B
h 3 c 3 ζ(3) V T
3 ,
(10.6.46a)
or, upon evaluating the constant, as
N = 2.0286 × 10
7 V T
3 .
(10.6.46b)
In particular, let us note that the number of photons in a photon gas at thermal
equilibrium at temperature T depends linearly upon the volume V of its black-body
container.
Example 10.5 Comparison of the pressures of a thermal photon gas and an ideal
classical gas at a temperature of 1000 K.
Let us consider a tungsten-walled container of volume V = 10 m 3 at temperature
T = 1000 K. Thus, for thermal photons, the radiation pressure is given by
Eq. (10.6.45) as
P = 2.5220 × 10
−16 T
4 ,
giving P radiation (T = 1000 K) 2.5 × 10 −4 Pa [1 J m −3 ≡ 1 Pa] or 2.5 × 10 −9 bar,
and the number of photons in the enclosure, given by Eq. (10.6.46b) will be of order
2 × 10 17 . As the radiation pressure, P rad , is independent of the system volume (i.e.,
the volume of the container), we shall compare it with the pressure exerted by an
ideal gas that represents what might be the laboratory situation generated by starting
with an ideal gas in a container of volume V at room temperature (say T = 300 K),
then heated to T = 1000 K, so that both N and V are fixed. The resultant ideal gas
pressure will thus be
P 1000 K = P 300 K
1000
300
k B 3.33 bar
for P 300 K = 1 bar, corresponding (for V = 10 m 3 ) to N 2.4 × 10 21 particles.
Thus, the radiation contributes a partial pressure P rad ≈ 3 × 10 −4 Pa, versus the