UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What led van der Waals to modify the ideal gas equation? Using the concepts of critical temperature T_c, pressure P_c and volume V_c, show that the critical constant for a real gas is 8/3.

Find out the expressions for van der Waals constants a and b.

Write down the expression for the Bose-Einstein distribution function and explain the meaning of the symbols used.

Consider a gas of photons in equilibrium and contained in a volume V at a temperature T. Using the Bose-Einstein statistics, calculate the total energy of the photon gas.

Write down the expressions for Bose-Einstein and Fermi-Dirac distribution functions, and show how Fermi-Dirac distribution leads to the explanation of Pauli's exclusion principle.

Show that the elemental quantity of heat \delta Q is not a total differential.

Consider one mole of an ideal gas whose pressure changes with volume as P=\alpha V, where \alpha is a constant. If it is expanded such that its volume increases m times, find the change in internal energy, work done by the gas and heat capacity of the gas.

Calculate the change in pressure for a change in freezing point of water equal to -0.91^\circ\mathrm{C}. Given, the increase of specific volume when 1 gm of water freezes into ice is 0.091 cc/gm and latent heat of fusion of ice is 80 cal/gm.

Derive an expression for the thermal efficiency of a reversible heat engine operating on the Diesel cycle with an ideal gas of constant heat capacity as the working medium.

1 kmol of an ideal gas is compressed isothermally at 400 K from 100 kPa to 1000 kPa in a piston and cylinder arrangement. Calculate the entropy change of the gas, the entropy change of the surroundings and the total entropy change resulting from the process if the process is mechanically reversible and the surroundings consist of a heat reservoir at 400 K.

Calculate the number of different arrangements of 10 indistinguishable particles in 15 cells of equal a priori probability, considering that one cell contains only one particle.

Consider the following statement: ``The Fermi energy of a given material is the energy of that quantum state which has the probability equal to \frac{1}{2} of being occupied by the conduction electrons.'' Is the above statement correct? Give reasons for your answer.

Explain why the distribution of speeds of molecules emerging through a small hole in an effusive molecular beam source is not a Maxwellian distribution.

N particles are distributed among three states having energies E = 0, E = kT and E = 2kT. If the total equilibrium energy of the system is 1000 kT, what is the value of N?

(ii) -\left(\dfrac{\partial f}{\partial E}\right) is symmetric about the Fermi level.

Let f be the Fermi-Dirac distribution function, then show that— (i) -\left(\dfrac{\partial f}{\partial E}\right) is a maximum at the Fermi level;

State Gibbs phase rule. Show that for a 1-component closed thermodynamic system having two phases, the condition for equilibrium between the phases is that their specific Gibbs functions are equal.

Derive an expression for the specific heat of a solid on the basis of Debye's model. Show that it converges to Dulong and Petit's law at high temperatures.

Energy distribution for n_i particles in classical statistical mechanics is given by n_i = g_i e^{-\alpha -\beta \varepsilon_i} where \alpha and \beta are constants. g_i is the single particle states in the i\text{ th} level. Using equipartition theorem, show that correct thermodynamic interpretation is \beta = \frac{1}{kT} (Use \int_0^\infty e^{-x} x^{1/2} \, dx = \frac{\sqrt{\pi}}{2} and \int_0^\infty e^{-\beta x} x^{3/2} \, dx = \frac{3\sqrt{\pi}}{4\beta^{5/2}})

Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S = k \log \Omega, where \Omega is the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system \Delta S \ge \frac{\Delta Q}{T} where the equality sign applies for internally reversible processes and the inequality for internally irreversible processes.