Question

An ideal gas expands isothermally from an initial volume \( V_i \) and pressure \( P_i \) to a final volume \( V_f \). If the same gas is allowed to expand adiabatically from the same initial volume and pressure to the final volume \( V_f \), then

• A. \( P_{\text{adia}} = 0 \)
• B. \( P_{\text{iso}} = 0 \)
• C. \( P_{\text{adia}} < P_{\text{iso}} \)
• D. \( P_{\text{adia}} > P_{\text{iso}} \)

Hint:

In an adiabatic expansion, the pressure is always higher than in an isothermal expansion for the same final volume.

Answer 1

The Correct Option is D

Step 1: Isothermal Expansion.
During an isothermal process, temperature is constant, and the gas adheres to the ideal gas law. Pressure diminishes as volume expands. The concluding pressure for an isothermal process is determined by \( P_{\text{iso}} \).

Step 2: Adiabatic Expansion.
An adiabatic expansion involves no heat transfer with the environment. As the gas expands, its internal energy drops, causing a temperature reduction. Therefore, for an identical volume, the pressure after an adiabatic expansion will exceed that after an isothermal expansion, as the gas has performed work on its surroundings. Accordingly, \( P_{\text{adia}} > P_{\text{iso}} \).

Final Answer: \[\boxed{\text{(4) } P_{\text{adia}} > P_{\text{iso}}}\]