UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Using Zeroth law of thermodynamics, introduce the concept of temperature. Explain how the isotherms of two different systems can be drawn.

Write down the expressions for the Fermi-Dirac distribution and the Bose-Einstein distribution. Plot the distributions as a function of the energy.

A body of constant heat capacity C_p and a temperature T_i is put into contact with a reservoir at temperature T_f. Equilibrium between the body and the reservoir is established at constant pressure. Determine the total entropy change and prove that it is positive for either sign of \left[(T_f-T_i)/T_f\right]. Consider \left|\frac{T_f-T_i}{T_f}\right|<1.

One mole of gas obeys van der Waals equation of state. If its molar internal energy is given by u=cT-a/V (in which V is the molar volume, a is one of the constants in the equation of state and c is a constant), calculate the molar heat capacities C_v and C_p.

A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions p(T,V)=aT^{1/2}+bT^3+cV^{-2} C_v(T,V)=dT^{1/2}+eT^2V+fT^{1/2} where a through f are constants which are independent of T and V. Find the differential of the internal energy dU(T,V) in terms of dT and dV.

Assume that the Earth's atmosphere is pure nitrogen in thermodynamic equilibrium at a temperature of 300\ \mathrm{K}. Calculate the height above sea level at which the density of the atmosphere is one-half its sea level value. (Molecular weight of \mathrm{N}_2 is 28\mathrm{gm/mole})

A compressor designed to compress air is used instead to compress helium. It is found that the compressor overheats. Explain this effect, assuming that the compression is approximately adiabatic and the starting pressure is same for both the gases. \left[\gamma_{\mathrm{He}}=\frac{5}{3},\gamma_{\mathrm{Air}}=\frac{7}{5}\right]

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder in two parts. Initially, the piston is clamped in the centre, with one litre of air at 200\ \mathrm{K} and 2\ \mathrm{atm} pressure on one side and one litre of air at 300\ \mathrm{K} and 1\ \mathrm{atm} pressure on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position. Compute the final pressure and temperature.

What do you understand by negative temperature? Write and explain various restrictions on a system for the concept of negative temperature to be meaningful.

Calculate the efficiency of an engine having compression ratio 13.8 and expansion ratio 6 and working on diesel cycle. Given \gamma = 1.4.

Explain the characteristics of the following thermodynamic processes for a perfect gas:

(i) Isothermal process

(ii) Adiabatic process

(iii) Isobaric process

(iv) Isochoric process Obtain the expression for the work done by the gas during the above processes.

The melting point of tin is 232^{\circ}\mathrm{C}, its latent heat of fusion is 14\ \mathrm{cal/g} and the specific heat of solid and molten tin are 0.055 and 0.064\ \mathrm{cal/g}\,{}^{\circ}\mathrm{C} respectively. Calculate the change in entropy when 1.0\mathrm{gm} of tin is heated from 100^{\circ}\mathrm{C} to 300^{\circ}\mathrm{C}.

Eight indistinguishable balls are to be arranged in six distinguishable boxes. Calculate the total number of ways in which the above can be done.

Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}

Using the expression for internal energy U = 3N \frac{\hbar\omega}{e^{\hbar\omega/k_B T}-1}, show that Einstein specific heat capacity is given by; C = 3R \left(\frac{\hbar\omega}{k_B T}\right)^2 \frac{e^{\hbar\omega/k_B T}}{\left(e^{\hbar\omega/k_B T}-1\right)^2} Also show that Einstein specific heat capacity given above is proportional to e^{-\hbar\omega/k_B T} at very low temperature.

Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T=0 and for T_1>T_2>0. Now from the plot propose two alternative definitions of the Fermi level.

Calculate the critical constants for \mathrm{CO_2} for which the Van der Waals constants are given by a=0.0072 and b=0.002. Also calculate the Boyle's temperature of \mathrm{CO_2}. The unit of pressure is atmosphere and the unit of volume is that of a gm-mole of the gas at NTP.

Starting from Maxwell-Boltzmann distribution for a free particle in 3-dimension, obtain the expression for root mean square (rms) speed of a particle. Calculate the rms speed of nitrogen \mathrm{N}_2 molecule at room temperature 27^\circ\mathrm{C}.

State the first law of thermodynamics for a diffusively interacting system. The temperature of 10\ \mathrm{g} of air is raised by 2^\circ\mathrm{C} at constant volume. Calculate the increase in its internal energy. Given: C_v=0.172\ \mathrm{cal\ g^{-1}\ {}^\circ C^{-1}}.

(i) The energy level of a quantum harmonic oscillator with frequency \nu is given by E_n=\left(n+\frac{1}{2}\right)h\nu,\quad \text{where } n=0,1,2,\ldots Calculate its partition function.

(ii) Calculate the partition function of a two level system.