Two gases A and B having same initial state (P, V, n, T). Now gas A is compressed to \(\frac{\mathrm{V}}{8}\) by isothermal process and other gas \(B\) is compressed to \(\frac{V}{8}\) by adiabatic process. The ratio of final pressure of gas A and B is (Both gases are monoatomic, \(\gamma=5 / 3\) )

For isothermal process (gas A)
\(\begin{aligned}
& P_1 V_1=P_2 V_2 \\
\therefore \quad & P_0\left(8 V_0\right)=P_2\left(V_0\right) \\
\Rightarrow & \Rightarrow P_2=8 P_0
\end{aligned}\)
For adiabatic process (gas B)
\(\begin{aligned}
& P^\gamma=\text { constant } \\
\therefore \quad & \frac{P_2}{P_1}=(8)^\gamma \\
\therefore \quad & P_2=(8)^\gamma P_0
\end{aligned}\)
Hence, \(\frac{\left(\mathrm{P}_2\right)_{\mathrm{B}}}{\left(\mathrm{P}_1\right)_{\mathrm{A}}}=\frac{(8)^\gamma \mathrm{P}_0}{8 \mathrm{P}_0}=(8)^{\gamma-1}\)
\(\begin{aligned}
& \frac{\left(P_2\right)_B}{\left(P_1\right)_A}=8^{2 / 3} \\
& =\sqrt[3]{64}=4 \\
\therefore \quad & \frac{\left(P_1\right)_A}{\left(P_2\right)_B}=\frac{1}{4}
\end{aligned}\)
\(\ldots(\text { given }, \gamma=5 / 3)\)