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Q.
A monoatomic gas at pressure $P$ and volume $V$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be:
Constant entropy means process is adiabatic
$ PV ^\gamma=$ constant
$ V _2=\frac{ V _1}{8} $
$ P _1 V _1^\gamma= P _2 V _2^\gamma $
$P _1 V _1^\gamma= P _2\left(\frac{ V _1}{8}\right)^{5 / 3} $
$ P _1 V _1^{5 / 3}=\frac{ P _2 V _1^{5 / 3}}{32} $
$ P _2=32 P _1$