An ideal monatomic gas with pressure $P$, volume $V$ and temperature $T$ is expanded isothermally to a…
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
$2^{-1 / 3}$
$2^{1 / 3^6}$
$2^{2 / 3}$
$2^{-2 / 3}$
Solution
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}
$