Question

An isolated ideal gas is kept at a pressure \( P_1 \) and volume \( V_1 \). The gas undergoes free expansion and attains a pressure \( P_2 \) and volume \( V_2 \). Identify the correct statement(s).
\(\left(\gamma = \frac{C_p}{C_v}\right)\)

• A. This is an adiabatic process
• B. \( P_1 V_1 = P_2 V_2 \)
• C. \( P_1 V_1^\gamma = P_2 V_2^\gamma \)
• D. This is an isobaric process

Hint:

Free expansion of an ideal gas is both adiabatic and irreversible; no work is done, and temperature remains constant because \( \Delta U = 0. \)

Answer 1

The Correct Option is A, B

Step 1: Nature of free expansion.
In a free expansion, the gas expands into a vacuum without doing any work (\( W = 0 \)) and without heat exchange (\( Q = 0 \)), since the system is isolated.
Step 2: Apply the first law of thermodynamics.
\[ \Delta U = Q - W. \] Since \( Q = 0 \) and \( W = 0 \), we have \( \Delta U = 0. \) This implies that the internal energy and hence the temperature remain constant.
Step 3: Process type.
A process in which no heat is exchanged with the surroundings is an adiabatic process.
Step 4: Final Answer.
Therefore, the process is adiabatic (Option A).