UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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

A gas of N spinless Bose particles of mass m is enclosed in a volume V at a temperature T.

(i) Find an expression for the density of single-particle state D(\varepsilon) as a function of the single-particle energy \varepsilon. Sketch the result.

(ii) Write down an integral expression which implicitly determines \mu(T). Referring to your sketch in (i), determine in which direction \mu(T) moves as T is lowered.

There are g cells of energy \varepsilon. Show that the number n of bosons of energy \varepsilon distributed among these cells is given by n = \frac{g}{e^{(\varepsilon - \mu)/kT} - 1} What is \mu and how will you find it?

10\text{ g} of water at 60\text{ }^\circ\text{C} is mixed with 30\text{ g} of water at 20\text{ }^\circ\text{C}. Will the entropy of the system increase or decrease? Calculate the change.

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.

Discuss the principle of adiabatic demagnetization process to achieve low temperatures. Determine the fall in temperature produced by adiabatic demagnetization of a paramagnetic material at initial temperature of 3\ \mathrm{K} when the magnetic field is switched off from 10{,}000 oersted to zero. Given: heat capacity at constant magnetic field =0.2\ \mathrm{J\ g^{-1}\ K^{-1}} and Curie constant per gram mole per \mathrm{cm^3} =0.042\ \mathrm{erg\ K^{-1}\ g^{-1}\ Oe^{-2}}.

Calculate (i) the internal energy of the electron gas per unit volume and (ii) the molar specific heat at constant volume for sodium at 100\text{ K} containing one free electron per atom. Given that the density of sodium = 0{\cdot}97 \times 10^3\text{ kg m}^{-3} and the atomic weight of sodium = 23.

Obtain the Clausius-Clapeyron equation. Using this equation, show that for the phase boundary of the liquid and vapour phases, p–T relation can be written as p=p_0e^{-L/kT}. Here it has been assumed that the latent heat L is independent of temperature, that vapour is treated as an ideal gas and that V_{\mathrm{vapour}}=V\gg V_{\mathrm{liquid}} and that p\to p_0 as T\to\infty.

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}.

Deduce Clausius-Clapeyron equations based on reversible cycle. Show that the specific heat of steam is negative. What is the significance of negative specific heat?

Write down the distribution law obeyed by electron gas and apply the same to derive Richardson-Dushman equation.

Obtain an expression for the specific heat capacity of a solid on the basis of Einstein's theory. How far do the results from this theory agree with experimental data ?

What is Gibbs' phase rule? Find the values of degrees of freedom when

(i) only the liquid \mathrm{CO_2} is in equilibrium with the gaseous \mathrm{CO_2}.

(ii) water is in the vapour-liquid saturation region.

(iii) water is in a single-phase region.

(iv) water is at the triple point.

Consider a quantum – mechanical gas of non-interacting spin zero bosons, each of mass m which are free to move within volume V. (i) Find the energy and heat capacity in the very low temperature region. Discuss why it is appropriate at low temperatures to put chemical potential equal to zero. (ii) Show how the calculation is modified for a photon (mass = 0) gas. Prove that the energy is proportional to \text{T}^4.

Einstein's molar specific heat capacity of a solid is given by C_V = 3R\left(\frac{\theta_E}{T}\right)^2\frac{e^{\theta_E/T}}{\left(e^{\theta_E/T}-1\right)^2}, where \theta_E=\frac{\hbar\omega}{k_B}. Obtain the expressions for the cases:

(i) when T \gg \theta_E

(ii) when T \ll \theta_E What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature? The molar specific heat capacity of a solid at constant volume is 2.77\ \mathrm{J\,K^{-1}} at 36.8\ \mathrm{K}. Determine the Debye temperature of the solid.

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.

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.

Explain the effect of pressure on the melting and boiling points of a substance using Clapeyron's latent heat equation. Calculate under what pressure, water will boil at 120^\circ\mathrm{C}, if the change in specific volume when 1 gram of water is converted into steam is 1676\ \mathrm{cm^3}. Latent heat of steam =540\ \mathrm{cal/g}, 1 atmospheric pressure =10^6\ \mathrm{dynes/cm^2}.

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