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.
For an isothermal process, the temperature remains constant, and the gas follows the ideal gas law. The pressure decreases as the volume increases. The final pressure for an isothermal process can be calculated using \( P_{\text{iso}} \).

Step 2: Adiabatic Expansion.
In an adiabatic expansion, there is no heat exchange with the surroundings. As the gas expands, its internal energy decreases, leading to a decrease in temperature. Consequently, for the same volume, the pressure after an adiabatic expansion will be higher than that after an isothermal expansion because the gas has done work on the surroundings. Hence, \( P_{\text{adia}} > P_{\text{iso}} \).

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