Course Code: BPHE-106/PHE-06

Assignment Code: BPHE-106/PHE-06/TMA/2023

Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it.

1. a) Define degree of freedom of a molecule. Write the formula for the number of degrees of freedom of a molecule. Calculate the degree of freedom for a rigid diatomic molecule.

b) The average energy of a helium molecules is \(2.89 \times 10^{-21} \mathrm{~J}\). Calculate their most probable speed and average speed.

c) Write the van der Waals’ equation of state. Using this equation, obtain the critical constants and show that

\[
\frac{R T_{C}}{p_{C} V_{C}}=\frac{8}{3}
\]

d) Derive an expression for survival equation for distribution of free paths. Also, plot survival equation.

e) Derive Einstein formula for mean square displacement of a Brownian particle.

2. a) Describe the construction and working of a Platinum resistance thermometer. Write its two principal merits.

b) Define the followings with example: (i) Intensive variables (ii) Extensive variables (iii) Adiabatic boundary (iv) Open system (v) Isolated system.

c) Obtain the expression for isothermal compressibility \(\left(\beta_{\mathrm{T}}\right)\) and coefficient of volume expansion \((\alpha)\) for a van der Waals’ gas.

d) Obtain an expression for work done by an ideal gas in an adiabatic process. Two litre of an ideal gas at a pressure of \(5 \mathrm{~atm}\) expands adiabatically to two times its initial volume. Calculate the work done by the gas. Given \(\gamma=1.4\).

e) Derive an expression for Clausius- Clayperon equation.

3. a) Draw a \(T-S\) diagram of a Carnot cycle and derive an expression of efficiency of a heat engine working between \(T_{1}\) and \(T_{2}\).

b) A freezer operates between \(-15^{\circ} \mathrm{C}\) and \(27^{\circ} \mathrm{C}\). Calculate the maximum value of coefficient of performance \((\omega)\) for this refrigerator. With this \(\omega\), how much electrical energy would be required to freeze \(0.8 \mathrm{~kg}\) of water, initially at \(0^{\circ} \mathrm{C}\). Given specific latent heat of fusion \(=334 \mathrm{~kJ} \mathrm{~kg}^{-1}\).

c) Using Maxwell’s relations, derive first and second \(T d S\) equations.

d) What is Joule-Thomson effect? Derive an expression of Joule-Thomson coefficient for a van der Waals’ gas. 4. a) The thermodynamic probability for a Boson system is given by

\[
W=\pi \frac{\left(g_{i}+N i-1\right) !}{\left(g_{i}-1\right) ! N_{i} !}
\]

Using this relation, derive an expression for the Bose-Einstein distribution function.

b) Establish the Boltzmann relation \(S=k_{\mathrm{B}} \ln W\).

c) What is Gibbs paradox? Derive the Sackur-Tetrode equation for the entropy for an ideal monatomic gas.

d) The expression for Planck’s law for energy density is given by

\[
u_{v} d v=\frac{8 \pi h}{c^{3}} \frac{v^{3} d v}{\exp \left[\frac{h v}{k_{\mathrm{B}} T}-1\right]}
\]

Using this expression, obtain (i) Wien’s law and (ii) Stefan-Boltzmann law.