UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Obtain Clausius – Clapeyron equation which applies to any first-order change of phase or any transition that occurs at constant temperature and pressure. Use Maxwell's thermodynamic relation for deriving the equation.

A gas has only two particles, a and b. With the help of a diagram, show that how these two particles can be arranged in the three quantum series 1, 2, 3 using (i) Maxwell-Boltzmann, (ii) Fermi-Dirac, and (iii) Bose-Einstein statistics.

What is Carnot's theorem? Prove that Carnot's reversible engine is the most efficient one and no other engine can be more efficient than Carnot's engine.

Derive Maxwell's thermodynamic relations using concepts of internal energy, Helmholtz function, Gibbs' function and enthalpy.

If the partition function for a perfect gas is given by Z=\frac{V}{h^3}(2\pi m k T)^{3/2}, calculate (i) average kinetic energy per molecule and (ii) specific heat of the gas.

What are the conditions for the change in temperature of a van der Waals gas passing through a porous plug? Prove that the ideal gas passing through the porous plug does not show any change in temperature.

Starting from the first law of thermodynamics, show that C_p - C_v = \left[ P + \left( \frac{\partial U}{\partial V} \right)_T \right] \left( \frac{\partial V}{\partial T} \right)_P

Define Fermi energy and show that for an electron gas at absolute zero temperature, the Fermi energy is given by E_F = \left( \frac{h^2}{2m} \right) \left( \frac{3}{8\pi} \frac{N}{V} \right)^{2/3} where the symbols have their usual meanings. Estimate the numerical value of Fermi energy for copper taking number of electrons per unit volume as 8\cdot 4 \times 10^{22}\text{ electrons/cm}^3 and find Fermi temperature T_F.

Consider N molecules of a gas obeying van der Waals' equation of state given by \left( P + \frac{a N^2}{V^2} \right) (V - Nb) = N k_B T where a is a measure of the attractive forces between the molecules and b is another constant proportional to the size of the molecules. The other symbols have their usual meanings. Show that during an isothermal expansion from volume V_1 to volume V_2 quasi-statically and reversibly, the work done is W = -N k_B T \log \left( \frac{V_2 - Nb}{V_1 - Nb} \right) + a N^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right)

Show that the probability of occupation for an electron state at the Fermi energy is equal to 0\cdot 5 for all finite temperatures.

A system having two energy levels, -\frac{1}{2}\Delta and +\frac{1}{2}\Delta with \Delta=10\mathrm{meV} is populated by 1000 particles at a low temperature close to 100\ \mathrm{K}. Obtain the average energy per particle using classical distribution law.

Schematically, show the variation of density of states, D(\varepsilon) and distribution function, f(\varepsilon,T), of particles in a non-relativistic Fermi gas at high temperatures. At a temperature T, an electron occupies a state with energy 100\mathrm{meV} above the Fermi energy (\varepsilon_{\mathrm F}) with the probability of 1\%. Find the temperature T.

The pressure on 100 g of solid copper is increased quasi-statically and isothermally at 0^\circ\mathrm{C} from 0 to 0.5 \times 10^{8}\ \mathrm{Pa}. Assuming the density and isothermal compressibility to remain at constant values of 8.96\ \mathrm{g/cm^3} and 7.16 \times 10^{-12}\ \mathrm{Pa^{-1}}, respectively, calculate the work done. Comment on the sign and magnitude of work.

Write Bose-Einstein distribution function explaining every symbol. Derive Einstein's result on specific heat of solids explaining the assumptions made in the model. Discuss the low and high temperature limits of the predicted specific heat. In which limit, the Einstein formula fails to explain the experimental data?

One mole of a gas obeys the following equation of state: \left(P + \frac{a}{v^{2}}\right)(v-b)=RT, where v is the molar volume and, a and b are constants. Show that internal energy of the gas increases as the volume increases, with the temperature remaining constant.

At 4^{\circ}\mathrm{C} temperature, the density of water is found to be maximum. Prove that heat capacity at the constant pressure (c_p) is equal to the heat capacity at constant volume (c_v) for water at 4^{\circ}\mathrm{C}.

If the temperature variation of heat capacity is known, how do you calculate the change of entropy during an isochoric process? According to Debye's theory of specific heat of a solid, the molar heat capacity of diamond crystal at constant volume varies with temperature (T) as follows: c_v=\frac{12}{5}\pi^{4}R\left(\frac{T}{\Theta}\right)^{3} where R is the molar gas constant =8.315\ \mathrm{J/mol\ K} and \Theta=2230\ \mathrm{K} for diamond. Calculate the change in entropy of diamond of 0.36 g mass when it is heated at constant volume from 0 K to 300 K.

Explain Maxwell-Boltzmann formula for distribution of velocities of gas molecules at temperature T. What will be the formula for distribution of speeds? If \bar{v}, v_{\text{rms}} and v_m denote average speed, root mean square velocity and most probable speed, show that \bar{v} : v_{\text{rms}} : v_m = \sqrt{\frac{8}{\pi}} : \sqrt{3} : \sqrt{2}

State Maxwell's distribution law of molecular speeds. Draw and explain a curve between n(c) and c in a gas at a given temperature T, where n(c) dc is the number of molecules having speed between c and c + dc. Discuss the effect of T and mass m of the molecule on the nature of the curve.

Derive the mathematical expression for the total energy of a degenerate Fermi gas at a temperature T and calculate the specific heat of the Fermi gas at this temperature.