1.A Carnot engine operates between temperatures $T_1 = 500\text{ K}$ and $T_2 = 300\text{ K}$. Calculate its efficiency $\eta$ and work done per cycle if heat absorbed from hot reservoir is $1000\text{ J}$.
Sol: $\eta = 1 - \frac{T_2}{T_1} = 1 - \frac{300}{500} = 0.40 \text{ or } 40\%$. Work done $W = \eta Q_1 = 0.40 \times 1000 = 400\text{ J}$.
2.One mole of a ideal gas with $\gamma = 1.4$ expands adiabatically from pressure $P_1 = 4\text{ atm}$ and volume $V_1 = 1\text{ L}$ to volume $V_2 = 32\text{ L}$. Find final pressure $P_2$.
Sol: $P_1 V_1^\gamma = P_2 V_2^\gamma \implies P_2 = P_1 \left(\frac{V_1}{V_2}\right)^\gamma = 4 \left(\frac{1}{32}\right)^{1.4} = 4 \times (2^{-5})^{1.4} = 4 \times 2^{-7} = \frac{4}{128} = \frac{1}{32}\text{ atm} = 0.03125\text{ atm}$.
