Two identical containers $A$ and $B$ with frictionless pistons contain the same ideal gas at the same temperature and pressure. Gas in container $A$ is compressed isothermally to half its original volume, while gas in container $B$ is compressed adiabatically to half its original volume. What is the ratio of the final pressures of gas in container $B$ to that in container $A$, assuming the gas is monatomic?

1. A.$2^{2/3}$
2. B.$2^{5/3}$
3. C.$2^{1/3}$
4. D.$2$