4.2 Problems
251
where W is the number of accessible states.
Probability for finding a particle in the nth state at temperature T
P(n, T ) =
e
−E n /kT
Σ
∞
n=0 e −E n /kT
(4.35)
Stirling’s approximation
n! =
√
2πn n
n e
−n
(4.36)
4.2 Problems
4.2.1 Kinetic Theory of Gases
4.1 Derive the formula for the velocity distribution of gas molecules of mass m at
Kelvin temperature T .
4.2 Assuming that low energy neutrons are in thermal equilibrium with the surroundings without absorption and that the Maxwellian distribution for velocities is valid, deduce their energy distribution.
4.3 In Problem 4.1 show that the average speed of gas molecule < ν >=
√
8kT /π m.
4.4 Show that for Maxwellian distribution of velocities of gas molecules, the root
mean square of speed < ν
2
>
1/2
= (3kT /m)
1/2
4.5 (a) Show that in Problem 4.1 the most probable speed of the gas molecules
ν p = (2kT /m)
1/2
(b) Show that the ratio ν p :< ν >:< ν
2
>
1/2 ::
√
2 :
√
8/π :
√
3
4.6 Estimate the rms velocity of hydrogen molecules at N T P and at 127
◦ C
[Sri Venkateswara University 2001]
4.7 Find the rms speed for molecules of a gas with density of 0.3 g/l of a pressure
of 300 mm of mercury.
[Nagarjuna University 2004]
4.8 The Maxwell’s distribution for velocities of molecules is given by N (ν)dν =
2π N (m/2πkT )
3/2
ν
2 exp(−mν
2
/2kT )dν
Calculate the value of < 1/ν >
4.9 The Maxwell’s distribution of velocities is given in Problem 4.8. Show that the
probability distribution of molecular velocities in terms of the most probable
velocity between α and α + dα is given by
251
where W is the number of accessible states.
Probability for finding a particle in the nth state at temperature T
P(n, T ) =
e
−E n /kT
Σ
∞
n=0 e −E n /kT
(4.35)
Stirling’s approximation
n! =
√
2πn n
n e
−n
(4.36)
4.2 Problems
4.2.1 Kinetic Theory of Gases
4.1 Derive the formula for the velocity distribution of gas molecules of mass m at
Kelvin temperature T .
4.2 Assuming that low energy neutrons are in thermal equilibrium with the surroundings without absorption and that the Maxwellian distribution for velocities is valid, deduce their energy distribution.
4.3 In Problem 4.1 show that the average speed of gas molecule < ν >=
√
8kT /π m.
4.4 Show that for Maxwellian distribution of velocities of gas molecules, the root
mean square of speed < ν
2
>
1/2
= (3kT /m)
1/2
4.5 (a) Show that in Problem 4.1 the most probable speed of the gas molecules
ν p = (2kT /m)
1/2
(b) Show that the ratio ν p :< ν >:< ν
2
>
1/2 ::
√
2 :
√
8/π :
√
3
4.6 Estimate the rms velocity of hydrogen molecules at N T P and at 127
◦ C
[Sri Venkateswara University 2001]
4.7 Find the rms speed for molecules of a gas with density of 0.3 g/l of a pressure
of 300 mm of mercury.
[Nagarjuna University 2004]
4.8 The Maxwell’s distribution for velocities of molecules is given by N (ν)dν =
2π N (m/2πkT )
3/2
ν
2 exp(−mν
2
/2kT )dν
Calculate the value of < 1/ν >
4.9 The Maxwell’s distribution of velocities is given in Problem 4.8. Show that the
probability distribution of molecular velocities in terms of the most probable
velocity between α and α + dα is given by