258
4 Thermodynamics and Statistical Physics
4.72 Using Planck’s formula for blackbody radiation show that Stefan’s constant
σ =
2
15
π
5 k
4
h 3 c 2 = 5.67 × 10
−8 W.m
−2
.K
−4
4.73 A blackbody has its cavity of cubical shape. Determine the number of modes
of vibration per unit volume in the wavelength region 4,990–5,010 A
◦ .
[Osmania University 2004]
4.74 A cavity kept at 4,000 K has a circular aperture 5.0 mm diameter. Calculate (a)
the power radiated in the visible region (0.4–0.7 μm) from the aperture (b) the
number of photons emitted per second in the visible region
4.75 Planck’s formula for the black body radiation is
u λ dλ =
8π hc
λ 5
1
e hc/λkT − 1
dλ
Express this formula in terms of frequency.
4.76 Estimate the temperature T E of the earth, assuming that it is in radiation
equilibrium with the sun (assume the radius of sun R s = 7 × 10
8 m, the
earth-sun distance r = 1.5 × 10
11 m, the temperature of solar surface T s =
5,800 K)
4.77 Calculate the solar constant, that is the radiation power received by 1 m
2
of earth’s surface. (Assume the sun’s radius R s = 7 × 10
8 m, the earthsun distance r = 1.5 × 10
11 m, the earth’s radius R E = 6.4 × 10
6 m,
sun’s surface temperature, T s = 5,800 K and Stefan-Boltzmann constant
σ = 5.7 × 10
−8 W
m 2 − K
4 ).
4.78 A nuclear bomb at the instant of explosion may be approximated to a blackbody of radius 0.3 m with a surface temperature of 10
7 K. Show that the bomb
emits a power of 6.4 × 10
20 W.
4.3 Solutions
4.3.1 Kinetic Theory of Gases
4.1 Consider a two-body collision between two similar gas molecules of initial
velocity ν 1 and ν 2 . After the collision, let the final velocities be ν 3 and ν 4 .
The probability for the occurrence of such a collision will be proportional to
the number of molecules per unit volume having these velocities, that is to
the product f (ν 1 ) f (ν 2 ). Thus the number of each collisions per unit volume
per unit time is c f (ν 1 ) f (ν 2 ) where c is a constant. Similarly, the number of
inverse collisions per unit volume per unit time is c
f (ν 3 ) f (ν 4 ) where c
is
also a constant. Since the gas is in equilibrium and the velocity distribution is
unchanged by collisions, these two rates must be equal. Further in the centre
4 Thermodynamics and Statistical Physics
4.72 Using Planck’s formula for blackbody radiation show that Stefan’s constant
σ =
2
15
π
5 k
4
h 3 c 2 = 5.67 × 10
−8 W.m
−2
.K
−4
4.73 A blackbody has its cavity of cubical shape. Determine the number of modes
of vibration per unit volume in the wavelength region 4,990–5,010 A
◦ .
[Osmania University 2004]
4.74 A cavity kept at 4,000 K has a circular aperture 5.0 mm diameter. Calculate (a)
the power radiated in the visible region (0.4–0.7 μm) from the aperture (b) the
number of photons emitted per second in the visible region
4.75 Planck’s formula for the black body radiation is
u λ dλ =
8π hc
λ 5
1
e hc/λkT − 1
dλ
Express this formula in terms of frequency.
4.76 Estimate the temperature T E of the earth, assuming that it is in radiation
equilibrium with the sun (assume the radius of sun R s = 7 × 10
8 m, the
earth-sun distance r = 1.5 × 10
11 m, the temperature of solar surface T s =
5,800 K)
4.77 Calculate the solar constant, that is the radiation power received by 1 m
2
of earth’s surface. (Assume the sun’s radius R s = 7 × 10
8 m, the earthsun distance r = 1.5 × 10
11 m, the earth’s radius R E = 6.4 × 10
6 m,
sun’s surface temperature, T s = 5,800 K and Stefan-Boltzmann constant
σ = 5.7 × 10
−8 W
m 2 − K
4 ).
4.78 A nuclear bomb at the instant of explosion may be approximated to a blackbody of radius 0.3 m with a surface temperature of 10
7 K. Show that the bomb
emits a power of 6.4 × 10
20 W.
4.3 Solutions
4.3.1 Kinetic Theory of Gases
4.1 Consider a two-body collision between two similar gas molecules of initial
velocity ν 1 and ν 2 . After the collision, let the final velocities be ν 3 and ν 4 .
The probability for the occurrence of such a collision will be proportional to
the number of molecules per unit volume having these velocities, that is to
the product f (ν 1 ) f (ν 2 ). Thus the number of each collisions per unit volume
per unit time is c f (ν 1 ) f (ν 2 ) where c is a constant. Similarly, the number of
inverse collisions per unit volume per unit time is c
f (ν 3 ) f (ν 4 ) where c
is
also a constant. Since the gas is in equilibrium and the velocity distribution is
unchanged by collisions, these two rates must be equal. Further in the centre