1000 Solved Problems in Modern Physics

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/ccand density of liquid= 0 .981 g/cc

[University of Calcutta]

4.43 The total energy of blackbody radiation in a cavity of volumeVat temperature

Tis given byu=aV T^4 , wherea= 4 σ/cis a constant.

(a) Obtain an expression for the entropySin terms ofT,Vanda.

(b) Using the expression for the free energyF, show that the pressureP=^13 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 StatisticalDistributions .............................

4.45 Calculate the ratio of the number of molecules in the lowest two rotational

states in a gas ofH 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 frequencyωin the

interval dω.

4.47 Show that for very large numbers, the Stirling’s approximation gives

n!∼=

√

2 πnnne−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 theJ= 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
statesJ=2 andJ=3 be equal.

4.51 Calculate the relative numbers of hydrogen atoms in the chromosphere with
the principal quantum numbersn= 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 bykT,by5kT,by10kT.
4.53 Ifnis the number of conduction electrons per unit volume andmthe electron
mass then show that the Fermi energy is given by the expression

EF=

h^2

8 m

(

3 n

π

) 2 / 3