An ideal gas with pressure $P$, volume $V$ and temperature $T$ is expanded isothermally to a volume $2V$ and a final pressure $P_i$. The same gas is expanded adiabatically to a volume $2V$, the final pressure is $P_a$. In terms of the ratio of the two specific heats for the gas $\gamma$, the ratio $\frac{P_a}{P_i}$ is