PYQ Bank

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.

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

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 ?

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?

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 ?

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.

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

Derive the Bose-Einstein distribution for an ideal gas.

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

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

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.

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)

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.

Explain the evaporation of a liquid at a temperature below its boiling point on the basis of kinetic theory of matter.

What is the most probable distribution of speeds in a large number of molecules of a gas and indicate the steps for its derivation.

Write a short note on Clausius-Clapeyron equation.

What is an adiabatic process? Give three engineering examples of adiabatic processes which are in common use.

Write a short note on Adiabatic demagnetisation.