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(A) For gas in container $A$,the compression is isothermal:$P_{A,final} V_{final} = P_{initial} V_{initial}$$P_{A,final} (V/2) = P_{initial} V$$P_{A,f

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$2^{\gamma-1}$

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(A) For gas in container $A$,the compression is isothermal:$P_{A,final} V_{final} = P_{initial} V_{initial}$$P_{A,final} (V/2) = P_{initial} V$$P_{A,final} = 2 P_{initial}$For gas in container $B$,the compression is adiabatic:$P_{B,final} V_{final}^{\gamma} = P_{initial} V_{initial}^{\gamma}$$P_{B,final} (V/2)^{\gamma} = P_{initial} V^{\gamma}$$P_{B,final} = P_{initial} (V / (V/2))^{\gamma} = P_{initial} (2)^{\gamma} = 2^{\gamma} P_{initial}$The ratio of the final pressure of gas in $B$ to that of gas in $A$ is:$\frac{P_{B,final}}{P_{A,final}} = \frac{2^{\gamma} P_{initial}}{2 P_{initial}} = 2^{\gamma-1}$

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