- $2^{\gamma-1}$
Explanation:
According to the P-V diagram shown for the container A (which is going through isothermal process) and for container B (which is going through adiabatic process).
Both the process involves compression of the gas.
(I) Isothermal compression (gas A) (during 1 → 2)
$\text{P}_{1}\text{V}_{1}=\text{P}_{2}\text{V}_{2}$
$\Rightarrow\text{P}_{0}(2\text{V}_{0})^\gamma=\text{P}_{2}(\text{V}_{0})^\gamma$
$\Rightarrow\text{P}_{0}(2\text{V}_{0})=\text{P}_{2}(\text{V}_{0})$
(ii) Adiabatic compression, (gas B) (during 1 → 2)
$\text{P}_{1}\text{V}_{1} ^\gamma=\text{P}_{2}\text{V}_{2}^\gamma$
$\text{P}_{0}(2\text{V}_{0})^\gamma=\text{P}_{2}(\text{V}_{0})^\gamma$
$\text{P}_{2}=\Big(\frac{2\text{V}_{0}}{\text{V}_{0}}\Big)^\gamma\text{P}_{0}=(2)^\gamma\text{P}_{0}$
Hence $\frac{(\text{P}_{2})_\text{B}}{(\text{P}_{2})_\text{A}}=$ Ratio of final pressure $=\frac{(2)^\gamma\text{P}_{0}}{2\text{P}_{0}}=2^{\gamma-1}$
where, $\gamma$ is ratio of specific heat capacities for the gas.
