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 need to find the ratio of the final pressures \( \frac{P_a}{P_i} \) after an ideal gas is expanded isothermally and adiabatically. Let's break down the solution step by step.

### Step 1: Isothermal Expansion
For the isothermal expansion of the gas, we use the ideal gas law, which states that \( PV = nRT \). Since the temperature is constant during an isothermal process, we can say that \( P_1 V_1 = P_2 V_2 \).

Given:
- Initial pressure \( P = P_1 \)
- Initial volume \( V = V_1 \)
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