8. (a) Starting from the expression N = \sum_k <n_k> where <n_k> is the average number of particles in the k^{\text{th}} quantum state, derive an expression for the average number of particles in the ground state of an ideal Bose gas.

(b) Utilize the above expression to elaborate the concept of the Bose-Einstein condensation and discuss that the phenomenon explains qualitatively the properties in the low-temperature phase of liquid ^4\text{He}.

Show that for Fermi-Dirac distribution of electrons, the number of electrons N_i in the energy state \varepsilon_i are given by N_i = \frac{g_i}{A \exp(\varepsilon_i / kT) + 1} where g_i represents the number of quantum states in the energy level \varepsilon_i. Further state under what conditions this distribution law goes over to Maxwell-Boltzmann statistics. Show by drawing curves how the Fermi-Dirac distribution function varies with the energy at T = 0 and also at the other finite temperatures.

A gas expands isothermally from the pressure p_1 and volume V_1 to the pressure p_2 and the volume V_2. Calculate (i) the change in internal energy, (ii) the change in entropy, (iii) the change in enthalpy. What will be the corresponding quantities when the gas expands adiabatically ?

Show that the entropy of one gram molecule of an ideal gas is given by S = C_p \ln V + C_v \ln p + S_0

Two bodies A and B have the same mass m each and are made of the same material with specific heat C per unit mass. A is at a temperature T and B is at a temperature 4T. What will be the entropy change of the universe when the bodies are brought into diathermic contact with each other?

Describe the Otto cycle and obtain an expression for the efficiency of the cycle, Show that the efficiency is lower than that of a Carnot cycle operating between the highest and the lowest temperature of Otto cycle.

Define Fermi energy. For an ideal Fermi gas of N particles at absolute zero temperature, show that the total energy of is 3/5 N E_F, where E_F is the Fermi energy.

100 particles at a temperature T and distributed among three energy levels E_0 = 0, E_1 = kT and E_2 = 2kT. What is the total energy of the system ?

Discuss the phenomenon of Bose-Einstein condensation. Obtain the expression for the condensation temperature. Briefly comment on observation of Bose-Einstein condensate.

Describe Carnot cycle and show that efficiency is given by \eta = \frac{Q_1 - Q_2}{Q_1} = \frac{T_1 - T_2}{T_1} where the symbols have their usual meaning.

Derive the Bose-Einstein distribution for an ideal gas.

Obtain van der waals' equation of state for real gases. What is the value of critical coefficient for an ideal gas ? Show that the value of the critical coefficient for van der Waals' gas is independent of the type of gas.

Derive an expression for the Maxwellian distribution of velocities for the molecules of an ideal gas.

Calculate the increase in entropy when 1 kg ice melts at zero degree centigrade. The Latent heat of fusion of ice is 3.36 \times 10^5\text{ joules/kg}. (Assume that the melting is an isothermal reversible process)

How can one obtain a temperature and identify the elements in stellar bodies using Saha's thermal ionisation equation ?

Write a short note on Debye's theory of specific heat.

How one can obtain the various types of information about the physical conditions on the stellar surface using thermal ionization equation to interpret the stellar spectra.

Define the efficiency of a Carnot Engine. An engine is designed to have an efficiency of 25\% and to absorb heat at a temperature of 267^\circ\text{ C}. Find the maximum temperature at which it can exhaust heat.

A tank whose capacity is 0.1\text{ m}^3 contains helium at a pressure of 10\text{ atmospheres} and a temperature of 20^\circ\text{C}. A rubber weather balloon is inflated with this helium.

(i) When the pressure of helium in the balloon is 1\text{ atmosphere}, its temperature is -40^\circ\text{ C}. Find the volume of the balloon.

(ii) Eventually the helium in the balloon absorbs heat from the air around ii and returns to 20^\circ\text{ C}. Find the volume of the balloon at this time.