1.Derive the pressure exerted by an ideal gas $P = \frac{1}{3} n m v_{rms}^2$ starting from elastic collision of gas molecules with a container wall.
Sol: Momentum change per collision $\Delta p = 2 m v_x$. Force $F = \frac{N m v_x^2}{L}$. Isotropic distribution $v_x^2 = v_y^2 = v_z^2 = \frac{1}{3} v_{rms}^2 \implies P = \frac{F}{L^2} = \frac{1}{3} \rho v_{rms}^2 = \frac{1}{3} n m v_{rms}^2$.
2.A vertical cylinder closed at both ends is divided into two equal halves by a frictionless piston of mass $m$. Each half contains 1 mole of air at temperature $T_0$. If cylinder is turned upside down, find final temperature $T$ required to restore piston to middle position.
Sol: Initially $P_1 - P_2 = \frac{mg}{A}$. When inverted, to restore middle position $P_2' - P_1' = \frac{mg}{A} \implies \frac{R T}{V_0} - \frac{R T_0}{V_0} = \frac{2 mg}{A} \implies T = T_0 + \frac{2 mg V_0}{R A}$.
