Consider two containers $A$ and $B$ containing identical gases at the same pressure, volume and temperature. The gas in container $A$ is compressed to half of its original volume isothermally while the gas in container $B$ is compressed to half of its original value adiabatically. The ratio of final pressure of gas in $B$ to that of gas in $A$ is
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When the compression is isothermal for gas in $A$
$P_{2} V_{2}=P_{1} V_{1}$
$P_{2}=P_{1} \frac{V_{1}}{V_{2}}=P_{1} \frac{V_{1}}{V_{1} / 2}=2 P_{1}$
For gas in $\mathrm{B}$, when compression is adiabatic,
$P_{2}^{\prime} V_{2}^{\prime}=P_{1} V_{1}^{\gamma}$
$P_{2}^{\prime}=P_{1}\left(\frac{V_{1}}{V_{2}^{\prime}}\right)^{\gamma}=P_{1}\left(\frac{V_{1}}{V_{1} / 2}\right)^{\gamma}=2^{\gamma} P_{1}$
$\therefore \frac{P_{2}^{\prime}}{P_{2}}=\frac{2^{\gamma} P_{1}}{2 P_{1}}=2^{\gamma-1}$
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