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Consider two containers $A$ and $B$ containing monoatomic gases at the same Pressure $(P)$, Volume $(V)$ and Temperature $(T)$. The gas in $A$ is compressed isothermally to $\frac{1}{8}$ of its original volume while the gas $B$ is compressed adiabatically to $\frac{1}{8}$ of its original volume. The ratio of final pressure of gas in $B$ to that of gas in $A$ is ...........
$8$
$8^{\frac{3}{2}}$
$\frac{1}{8}$
$4$
Solution
Isothermal process, $T =$ constant
$PV =n R T=\text { constant }$
$P _1 V_1= P _2 V_2$
$PV = P _{ A }( V / 8)$
$P _{ A }=8 P$
Adiabatic process, PV $\gamma=$ constant $\gamma$ for monoatomic gas is $\frac{5}{3}$.
$P _1 V _1^\gamma= P _2 V _2^\gamma$
$\frac{ P _{ B }}{ P }=\left(\frac{ V _1}{ V _2}\right)^\gamma=\left(\frac{ V }{ V / 8}\right)^{\frac{5}{3}}$
$P _{ B }=32 P$
$\frac{ P _{ B }}{ P _{ A }}=\frac{32 P }{8 P }=4$
