UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Discuss the significance of Saha's ionisation formula in classification of stars.

Establish Van der Waals equation of state for a real gas. Deduce expressions for critical constants and show that critical coefficient is independent of the nature of gas.

State Dulong-Petit's law and discuss its limitations in explaining heat capacities of solids. How were these efficiencies overcome by Debye ?

The coefficient of viscosity of helium at 27^\circ\mathrm{C} is 2 \times 10^{-5}\ \mathrm{kg\,m^{-1}\,s^{-1}}. Calculate

(i) the average speed and

(ii) the diameter of a helium molecule, if it is assumed that the gas obeys Maxwell-Boltzmann distribution. Given Boltzmann constant k_{\mathrm B} = 1.38 \times 10^{-23}\ \mathrm{J\,K^{-1}} and mass of helium atom = 6.67 \times 10^{-27}\ \mathrm{kg}.

In Leh, temperature of ice on a cold winter night is measured as -20^\circ\mathrm{C}. Calculate the change in entropy when 1\ \mathrm{kg} of ice is converted into steam at 100^\circ\mathrm{C}. Given specific heat capacity of ice is 500\ \mathrm{cal\,kg^{-1}\,K^{-1}}, latent heat of ice is 3.36\times10^5\ \mathrm{J\,kg^{-1}}, latent heat of steam is 2.26\times10^6\ \mathrm{J\,kg^{-1}} and J=4.2\ \mathrm{J\,cal^{-1}}.

The vapour pressure, in mm of Hg, of a substance in solid state is given by the relation \ln p=23.03-\dfrac{3754}{T}, where T is in Kelvin. The vapour pressure, in mm of Hg, of the substance in liquid state is given by the relation \ln p=19.49-\dfrac{3063}{T}. Calculate

(i) the coordinates of the triple point, and

(ii) the latent heat of vaporisation at the triple point. Take Gas constant R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}.

A thermally insulated ideal gas is compressed quasi-statically from an initial state with volume V_0 and pressure P_0 to a final state of volume V_f and pressure P_f. Show that the work done on the gas in the process is given by W=\frac{C_V}{R}\left(P_fV_f-P_0V_0\right) where C_V and R having standard meanings.

N particles obeying Classical Statistics are distributed among three states having energies \varepsilon_1 = 0, \varepsilon_2 = k_{\mathrm B}T and \varepsilon_3 = 2k_{\mathrm B}T, where k_{\mathrm B} is Boltzmann constant. If the total equilibrium energy of the system is 1000k_{\mathrm B}T, calculate the value of N.

Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.

Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.

What are bosons ? Based on Bose-Einstein statistics, derive the expression for Bose-Einstein condensation.

Show that if an ideal gas is compressed isothermally its compressibility is \frac{1}{p}, whereas if it is compressed adiabatically its compressibility is \frac{1}{\gamma p} where \gamma = C_p / C_v.

Derive the gas equation when it undergoes adiabatic process.

A motor car tyre has a pressure of 2\text{ atmospheres} at the room temperature of 27^\circ\text{C}. If the tyre suddenly bursts, find the resulting temperature.

Find the efficiency of a Carnot's engine working between 127^\circ\text{C} and 27^\circ\text{C}.

Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.

Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.

The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.

(i) Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.

(ii) Give schematic plot of \dfrac{C_V}{3N k_B} versus \dfrac{T}{\theta_E} and comment on the validity of expressions in (i) in comparison with experiments.

Explain the thermodynamic behaviour of an ideal Fermi gas. What are fermions ?

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.