4.3 Solutions
285
Where C = constant which can be determined as follows.
ΣN i = N = CΣe
−β E i
(11)
or C =
N
Σe −β Ei
(12)
Equation (10) then becomes
N i =
N e
−β Ei
Σe −β Ei
(13)
The denominator in (13)
Z = Σe
−β Ei
(14)
Is known as the partition function. It can be shown that the quantity
β =
1
kT
(15)
where k is the Boltzmann constant and T is the absolute temperature.
α =
N
Z
(16)
4.3.4 Blackbody Radiation
4.61 Electric power = power radiated
W = σ T
4 A
A = 2πrl = 2π × 10
−3
× 1.0 = 6.283 × 10
−3 m
2
T =
W
σ A
1/4
=
1,000
5.67 × 10 −8 × 6.283 × 10 −3
1/4
= 1,294 K
4.62 The Solar constant S is the heat energy received by 1 m
2 of earth’s surface per
second. If R is the radius of the sun and r the earth-sun distance, then the total
intensity of radiation emitted from the sun will be σ T
4 W m
−2 and from the
sun’s surface σ T
4
.4π R
2 . The radiation received per second per m
2 of earth’s
surface will be
S = σ T
4
.
4π R
2
4πr 2
Solving,
σ T
4
= S.
r
2
R 2 = 1,400
1.5 × 10
8
7 × 10 5
2
= 6.43 × 10
7 W m
−2
T =
6.43 × 10
7
σ
1/4
=
6.43 × 10
7
5.67 × 10 −8
1/4
= 5,800 K
4.63 Using the analogy between radiation (photon gas) and gas molecules, the photons move in a cavity at random in all directions, rebounding elastically from
the walls of the cavity. The pressure exerted by an ideal photon gas is