A cylindrical vessel of uniform cross-section consisting of a gas of \(\gamma=1.5\) is divided into two parts \(A\) and \(B\) using a piston. Initially the piston is kept fixed such that part \(A\) has pressure \(p\) and volume \(5 \mathrm{~V}\) and the part \(B\) has pressure \(8 p\) and volume \(V\). If the piston is let free and the gas is allowed to undergo adiabatic process, then the final volume of the gas in part \(A\) is

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.