BPHCT-135 (THERMAL PHYSICS AND STATISTICAL MECHANICS)

(Valid from 1st January, 2023 to 31 st December, 2023)

PART A
1. a) Calculate the temperature at which root mean square speed of nitrogen molecules exceeds their most probable speed by \(200 \mathrm{~ms}^{-1}\). Take \(m_{N_2}=28 \mathrm{~kg} \mathrm{kmol}^{-1}\).

b) Using the relation
\[
d N_v=4 \pi N\left(\frac{m}{2 \pi k_{\mathrm{B}} T}\right)^{3 / 2} v^2 \exp \left[-\left(\frac{m v^2}{2 k_{\mathrm{B}} T}\right)\right] d v
\]
for the number of molecules in a Maxwellian gas having speeds in the range \(v\) to \(v+d v\), obtain an expression for (i) average speed, and (ii) root mean square speed.

c) Derive the survival equation for distribution of free paths. Hence, plot distribution of free paths as a function of \(\frac{x}{\lambda}\).

d) Calculate the diffusion coefficient of hydrogen molecules at \(27^{\circ} \mathrm{C}\) when pressure is \(3 \mathrm{~atm}\). Assume that it behaves as a Maxwellian gas. Take \(r_{\mathrm{H}_2}=1.37 \times 10^{-10} \mathrm{~m}\) and \(k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{JK}^{-1}\).

e) Define Brownian motion. Write its four observed characteristics.

2. a) What do you understand by (i) isobaric (ii) isochoric (iii) isothermal, and (iv) cyclic processes? Represent these processes on \(p-V\) diagrams.

b) Prove that for a \(p V T\)-system
\[
\frac{d V}{V}=\alpha d T-\beta_T d p
\]
where \(\alpha\) is the isobaric coefficient of volume expansion and \(\beta_T\) is isothermal compressibility.

c) Write the differential form of first law of thermodynamics. Show that for an ideal gas, it can be written as \(\delta Q=C_V d T+p d V\). Using this result for one mole of an ideal gas which undergo quasi-static adiabatic expansion, obtain the expression \(T V^{\gamma-1}=K\), where \(\gamma\) is the ratio of heat capacity at constant pressure to that at constant volume.

d) Two moles of an ideal gas at STP is expanded isothermally to thrice its volume. It is then made to undergoes isochoric change to attain its original pressure. Calculate the total work done in these processes. Take \(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\).

PART B

3. a) Write Kelvin-Planck and Clausius statements of the second law of thermodynamics. Show that these two statements are equivalent.

b) Calculate the change in entropy when \(20 \mathrm{~g}\) of ice at \(0^{\circ} \mathrm{C}\) is converted into steam. [Given: latent heat of fusion of ice \(=80 \mathrm{cal} \mathrm{g}^{-1}\), latent heat of fusion of steam \(=540 \mathrm{cal} \mathrm{g}^{-1}\) ].

c) Using Maxwell’s relations, deduce first and second TdS-equations. Also, obtain the first \(T d S\)-equation in terms of volume expansivity \((\alpha)\) and isothermal compressibility \(\left(\beta_T\right)\).

d) State Stefan-Boltzmann’s law of black body radiation. Plot spectral energy density of a black body with wavelength at different temperatures and discuss the results of these plots.

4. a) Derive Boltzmann entropy relation \(S=k_{\mathrm{B}} \ln W\), where \(W\) is a thermodynamic probability.

b) Using the expression of thermodynamic probability of a Fermi-Dirac system, derive the expression for the distribution function and plot it as a function of energy at temperatures (i) \(T=0 \mathrm{~K}\) and (ii) \(T>0 \mathrm{~K}\).

c) A box of volume \(1 \mathrm{~cm}^3\) contains \(4 \times 10^{21}\) electrons. Calculate Fermi energy of these electrons. [Take: \(m_e=9.1 \times 10^{-28} \mathrm{~g}\) and \(h=6.62 \times 10^{-28} \mathrm{erg}\) s].

d) Write the expression for \(N\) distinguishable particles partition function for an ideal gas and hence obtain expressions for heat capacity at constant (i) volume, and (ii) pressure.