\(P V=P_{t}, 2 V\)

\(P=2 P_{i}\,\,\,\,\,\,\,\,...(i)\)

For adiabatic process \(\mathrm{PV}^{\gamma}=\mathrm{P}_{\mathrm{a}}(2 \mathrm{V})^{\gamma}\)

\((\because \text { for monatomic gas } \gamma=5 / 3)\)

or, \(\quad 2 \mathrm{P}_{\mathrm{i}} \mathrm{V}^{\frac{5}{3}}=P_{a}(2 \mathrm{V})^{\frac{5}{3}} \quad[\text { From }(\mathrm{i})]\)

\(\Rightarrow \frac{P_{a}}{P_{i}}=\frac{2}{2^{\frac{5}{3}}}\)

\(\Rightarrow \quad \frac{P_{a}}{P_i}=2^{\frac{-2}{3}}\)