1.State Law of Equipartition of Energy.
Sol: For a system in thermal equilibrium, average energy per degree of freedom per molecule is $\frac{1}{2} k_B T$.
2.Define degrees of freedom for monoatomic, diatomic, and triatomic gas molecules.
Sol: Monoatomic: $f = 3$ (translational). Diatomic (rigid): $f = 5$ ($3$ trans + $2$ rot). Non-linear triatomic: $f = 6$.
3.Write expression for RMS speed ($v_{rms}$) of a gas molecule.
Sol: $v_{rms} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 k_B T}{m}}$.
4.Define mean free path ($\lambda$).
Sol: Average distance travelled by a molecule between two successive collisions: $\lambda = \frac{1}{\sqrt{2} \pi n d^2}$.
5.Find RMS speed of Oxygen gas molecules at $27^\circ\text{C}$ ($M = 32 \text{ g/mol}, R = 8.314$).
Sol: $v_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{0.032}} = \sqrt{233831.25} \approx 483.56 \text{ m/s}$.
6.At what temperature will RMS speed of Hydrogen gas be double its value at $0^\circ\text{C}$?
Sol: $v \propto \sqrt{T} \implies \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \implies 2 = \sqrt{\frac{T_2}{273}} \implies T_2 = 4 \times 273 = 1092 \text{ K} = 819^\circ\text{C}$.
7.Calculate internal energy of $2 \text{ moles}$ of a rigid diatomic gas at $300 \text{ K}$.
Sol: $U = \frac{f}{2} n R T = \frac{5}{2} \times 2 \times 8.314 \times 300 = 12471 \text{ Joules}$.
8.Find ratio of specific heats $\gamma = C_p / C_v$ for a monoatomic gas.
Sol: $\gamma = 1 + \frac{2}{f} = 1 + \frac{2}{3} = \frac{5}{3} \approx 1.67$.
