4.2 Problems
255
4.42 Calculate the change of melting point of naphthalene per atmospheric change
of pressure, given melting point = 80
◦ C, latent heat = 35.5 cal/g, density of
solid = 1.145 g/cc and density of liquid = 0.981 g/cc
[University of Calcutta]
4.43 The total energy of blackbody radiation in a cavity of volume V at temperature
T is given by u = aV T
4 , where a = 4σ/c is a constant.
(a) Obtain an expression for the entropy S in terms of T, V and a.
(b) Using the expression for the free energy F, show that the pressure P =
1
3
u.
4.44 Given that the specific heat of Copper is 387 J/kg K
−1 , calculate the atomic
mass of Copper in amu using Dulong Petit law.
4.2.3 Statistical Distributions
4.45 Calculate the ratio of the number of molecules in the lowest two rotational
states in a gas of H 2 at 50 K (take inter atomic distance = 1.05 A
◦ )
[University of Cambridge, Tripos 2004]
4.46 Consider a photon gas in equilibrium contained in a cubical box of volume
V = a
3 . Calculate the number of allowed normal modes of frequency ω in the
interval dω.
4.47 Show that for very large numbers, the Stirling’s approximation gives
n! ∼ =
√
2π n n
n e
−n
4.48 Show that the rotational level with the highest population is given by
J max(pop) =
√
I 0 kT
−
1
2
4.49 Assuming that the moment of inertia of the H 2 molecule is 4.64×10
−48 kg-m
2 ,
find the relative population of the J = 0, 1, 2 and 3 rotational states at 400 K.
4.50 In Problem 4.49, at what temperature would the population for the rotational
states J = 2 and J = 3 be equal.
4.51 Calculate the relative numbers of hydrogen atoms in the chromosphere with
the principal quantum numbers n = 1, 2, 3 and 4 at temperature 6,000 K.
4.52 Calculate the probability that an allowed state is occupied if it lies above the
Fermi level by kT , by 5kT , by 10 kT .
4.53 If n is the number of conduction electrons per unit volume and m the electron
mass then show that the Fermi energy is given by the expression
E F =
h
2
8m
3n
π
2/3
255
4.42 Calculate the change of melting point of naphthalene per atmospheric change
of pressure, given melting point = 80
◦ C, latent heat = 35.5 cal/g, density of
solid = 1.145 g/cc and density of liquid = 0.981 g/cc
[University of Calcutta]
4.43 The total energy of blackbody radiation in a cavity of volume V at temperature
T is given by u = aV T
4 , where a = 4σ/c is a constant.
(a) Obtain an expression for the entropy S in terms of T, V and a.
(b) Using the expression for the free energy F, show that the pressure P =
1
3
u.
4.44 Given that the specific heat of Copper is 387 J/kg K
−1 , calculate the atomic
mass of Copper in amu using Dulong Petit law.
4.2.3 Statistical Distributions
4.45 Calculate the ratio of the number of molecules in the lowest two rotational
states in a gas of H 2 at 50 K (take inter atomic distance = 1.05 A
◦ )
[University of Cambridge, Tripos 2004]
4.46 Consider a photon gas in equilibrium contained in a cubical box of volume
V = a
3 . Calculate the number of allowed normal modes of frequency ω in the
interval dω.
4.47 Show that for very large numbers, the Stirling’s approximation gives
n! ∼ =
√
2π n n
n e
−n
4.48 Show that the rotational level with the highest population is given by
J max(pop) =
√
I 0 kT
−
1
2
4.49 Assuming that the moment of inertia of the H 2 molecule is 4.64×10
−48 kg-m
2 ,
find the relative population of the J = 0, 1, 2 and 3 rotational states at 400 K.
4.50 In Problem 4.49, at what temperature would the population for the rotational
states J = 2 and J = 3 be equal.
4.51 Calculate the relative numbers of hydrogen atoms in the chromosphere with
the principal quantum numbers n = 1, 2, 3 and 4 at temperature 6,000 K.
4.52 Calculate the probability that an allowed state is occupied if it lies above the
Fermi level by kT , by 5kT , by 10 kT .
4.53 If n is the number of conduction electrons per unit volume and m the electron
mass then show that the Fermi energy is given by the expression
E F =
h
2
8m
3n
π
2/3