An ideal gas with pressure P, volume V and temperature T is expanded isothermally to a volume 2V and a final pressure `P_i,` If the same gas is expanded adiabatically to a volume 2V, the final pressure `P_a.` The ratio of the specific heats of the gas is 1.67. The ratio `(P_a)/(P_1)` is .......

Explanation

To solve the problem, we will analyze both the isothermal and adiabatic processes for the ideal gas and derive the required ratio of pressures \( \frac{P_a}{P_i} \).

### Step 1: Isothermal Expansion
In an isothermal process, the product of pressure and volume remains constant.

Given:
- Initial pressure = \( P \)
- Initial volume = \( V \)
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