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An ideal monatomic 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 $2 \mathrm{~V}$, the final pressure is $P_a$. The ratio $\frac{P_a}{P_i}$ is
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Verified Answer
The correct answer is:
$2^{-2 / 3}$
$2^{-2 / 3}$
For isothermal process :
$$
\begin{aligned}
& P V=P_i 2 V \\
& P=2 P_i
\end{aligned}
$$
For adiabatic process
$$
\mathrm{PV}^\gamma=\mathrm{P}_{\mathrm{a}}(2 \mathrm{~V})^\gamma
$$
$(\because$ for monatomic gas $\gamma=5 / 3)$
[From (i) ]
or, $\quad 2 \mathrm{P}_{\mathrm{i}} \mathrm{V}^{\frac{5}{3}}=P_a(2 \mathrm{~V})^{\frac{5}{3}}$
$$
\begin{aligned}
& \Rightarrow \frac{P_a}{P_i}=\frac{2}{2^{\frac{5}{3}}} \\
& \Rightarrow \frac{P_a}{P_i}=2^{\frac{-2}{3}}
\end{aligned}
$$
$$
\begin{aligned}
& P V=P_i 2 V \\
& P=2 P_i
\end{aligned}
$$
For adiabatic process
$$
\mathrm{PV}^\gamma=\mathrm{P}_{\mathrm{a}}(2 \mathrm{~V})^\gamma
$$
$(\because$ for monatomic gas $\gamma=5 / 3)$
[From (i) ]
or, $\quad 2 \mathrm{P}_{\mathrm{i}} \mathrm{V}^{\frac{5}{3}}=P_a(2 \mathrm{~V})^{\frac{5}{3}}$
$$
\begin{aligned}
& \Rightarrow \frac{P_a}{P_i}=\frac{2}{2^{\frac{5}{3}}} \\
& \Rightarrow \frac{P_a}{P_i}=2^{\frac{-2}{3}}
\end{aligned}
$$
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