An ideal monoatomic gas with pressure $P$, volume $V$ and temperature $T$ is expanded isothermally to a volume $2\, V$ and a final pressure $P_i$. If the same gas is expanded adiabatically to a volume $2\,V$, the final pressure is $P_a$ . The ratio $\frac{{{P_a}}}{{{P_i}}}$ is
AIEEE 2012, Medium
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For isothermal process:
$P V=P_{t}, 2 V$
$P=2 P_{i}\,\,\,\,\,\,\,\,...(i)$
For adiabatic process $\mathrm{PV}^{\gamma}=\mathrm{P}_{\mathrm{a}}(2 \mathrm{V})^{\gamma}$
$(\because \text { for monatomic gas } \gamma=5 / 3)$
or, $\quad 2 \mathrm{P}_{\mathrm{i}} \mathrm{V}^{\frac{5}{3}}=P_{a}(2 \mathrm{V})^{\frac{5}{3}} \quad[\text { From }(\mathrm{i})]$
$\Rightarrow \frac{P_{a}}{P_{i}}=\frac{2}{2^{\frac{5}{3}}}$
$\Rightarrow \quad \frac{P_{a}}{P_i}=2^{\frac{-2}{3}}$
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