1.State Law of Equipartition of Energy and find total internal energy of 1 mole of monoatomic and diatomic gases.
Sol: Energy associated with each degree of freedom per molecule = $\frac{1}{2} k_B T$. Monoatomic ($f=3$): $U = \frac{3}{2} R T$. Diatomic ($f=5$): $U = \frac{5}{2} R T$.
2.Write expression for rms speed $v_{rms} = \sqrt{\frac{3RT}{M}}$ and average kinetic energy per molecule $E = \frac{3}{2} k_B T$.
Sol: Pressure $P = \frac{1}{3} \rho v_{rms}^2 \implies v_{rms} = \sqrt{\frac{3P}{\rho}} = \sqrt{\frac{3RT}{M}}$. Average translational KE per molecule = $\frac{3}{2} k_B T$.
3.Define mean free path $\lambda$ of gas molecules and write its expression $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$.
Sol: Average distance travelled by a molecule between two successive collisions: $\lambda = \frac{1}{\sqrt{2} n \pi d^2} = \frac{k_B T}{\sqrt{2} \pi d^2 P}$.
