Initial pressure and volume of a gas are ' $P$ ' and ' $V$ ' respectively. First its volume is expanded to '…
Initial pressure and volume of a gas are ' $P$ ' and ' $V$ ' respectively. First its volume is expanded to ' 4 V ' by isothermal process and then again its volume is reduced to ' V ' by adiabatic process then its final pressure if $\left(\gamma=\frac{3}{2}\right)$
P
$\quad 2 \mathrm{P}$
3 P
4 P
Solution
After isothermal expansion,
$\begin{aligned}
& P_1 V_1=P_2 V_2 \\
& P V=P_2 \times 4 V \\
\therefore \quad & P_2=\frac{P}{4}
\end{aligned}$
After adiabatic compression,
$\begin{array}{lll}
& P_2 V_2^\gamma=P_3 V_3^\gamma & \\
& \frac{P}{4}(4 V)^\gamma=P_3(V)^\gamma & \ldots\left(\text { given } V_3=\dot{V}\right) \\
\therefore \quad & P_3=\frac{P}{4} 4^\gamma=\frac{P}{4} 4^{3 / 2} & \ldots\left(\text { given }, \gamma=\frac{3}{2}\right) \\
\therefore \quad & P_3=\frac{P}{4} \times 8=2 P &
\end{array}$