Question

Sol. [Key Concept: For adiabatic expansion $\Delta\ U = {nC}_{v}\Delta\ T = \frac{P_{2}V_{2} - P_{1}V_{1}}{\gamma - 1}$ with $\gamma = \frac{5}{3}$ and $P_{2} = \frac{P}{3^{5}}$ (from ${PV}^{\gamma}$const), giving $- \frac{4}{3}\ PV$.]

$${\ P_{1}{\text{ }V}_{1}^{\gamma} = P_{2}{\text{ }V}_{2}^{\gamma} }{\gamma = \frac{5}{3} }{{PV}^{5/3} = P_{2} \times (27\text{ }V)^{5/3} }{P_{2} = \frac{P}{(27)^{5/3}} = \frac{P}{3^{5}} }{\Delta U \Rightarrow {nC}_{v}\Delta T = \frac{nR\Delta T}{(\gamma - 1)} }{= \frac{P_{2}V_{2} - P_{1}V_{1}}{(\gamma - 1)} }{\Rightarrow \frac{\frac{P}{3^{5}} \times 3^{3} \cdot \text{ }V - PV}{\frac{5}{3} - 1} \Rightarrow \frac{- PV\left\{ 1 - \frac{1}{9} \right\}}{\frac{2}{3}} }{= - PV \times \frac{8}{9} \times \frac{3}{2} }{\Rightarrow - \frac{4}{3}PV }$$