Step 1: Understanding the Question:
The gas undergoes a two-step expansion process: first isothermal, then adiabatic. We need to find the final pressure after both steps.
Step 2: Key Formula or Approach:
1. Isothermal process: $P_1 V_1 = P_2 V_2$
2. Adiabatic process: $P_2 V_2^\gamma = P_3 V_3^\gamma$
3. For a polyatomic gas at moderate temperatures, the ratio of specific heats $\gamma = \frac{4}{3}$ (assuming 6 degrees of freedom).
Step 3: Detailed Explanation:
Step I: Isothermal expansion
Initial state: $(P, V)$
Final state: $(P', 3V)$
\[ P \cdot V = P' \cdot (3V) \implies P' = \frac{P}{3} \]
Step II: Adiabatic expansion
Initial state: $(P', 3V)$
Final state: $(P_{final}, 24V)$
Using $P' \cdot V_{initial}^\gamma = P_{final} \cdot V_{final}^\gamma$:
\[ \frac{P}{3} \cdot (3V)^{4/3} = P_{final} \cdot (24V)^{4/3} \]
\[ P_{final} = \frac{P}{3} \cdot \left(\frac{3V}{24V}\right)^{4/3} \]
\[ P_{final} = \frac{P}{3} \cdot \left(\frac{1}{8}\right)^{4/3} \]
Since $8 = 2^3$:
\[ P_{final} = \frac{P}{3} \cdot \left((2^3)^{1/3}\right)^{-4} = \frac{P}{3} \cdot (2)^{-4} \]
\[ P_{final} = \frac{P}{3} \cdot \frac{1}{16} = \frac{P}{48} \]
Step 4: Final Answer:
The final pressure of the gas is $P/48$.