Question

Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following:

• A. \( q = 0, \Delta T \neq 0, w = 0 \)
• B. \( q = 0, \Delta T<0, w \neq 0 \)
• C. \( q \neq 0, \Delta T = 0, w = 0 \)
• D. \( q = 0, \Delta T = 0, w = 0 \)

Hint:

In an adiabatic free expansion of an ideal gas, there is no heat transfer, no work done, and no change in temperature, as the gas expands into a vacuum.

Answer 1

The Correct Option is D

Step 1: {Understanding Free Expansion of an Ideal Gas} 
In free expansion, the gas expands without doing work and without heat exchange with the surroundings. Since the process is adiabatic, \( q = 0 \), meaning no heat is exchanged. 
Step 2: {Work Done in Free Expansion} 
Since the expansion is against a vacuum, no work is done on or by the system. Hence, \( w = 0 \). 
Step 3: {Temperature Change in Free Expansion} 
For an ideal gas undergoing free expansion, there is no change in temperature, so \( \Delta T = 0 \). Thus, the correct answer is (D). 
 

Answer 2

Step 1: Understand free expansion
Free expansion is a process in which a gas expands into a vacuum without opposing pressure.
There is no external force resisting the expansion, so the gas does no work on the surroundings.

Step 2: Consider adiabatic conditions
Adiabatic means there is no heat exchange with the surroundings.
So, \( q = 0 \)

Step 3: Work done in free expansion
Since the gas expands freely into a vacuum, external pressure is zero.
Therefore, the work done \( w = 0 \)

Step 4: Use the first law of thermodynamics
\[ \Delta U = q + w \]
Substituting the known values:
\[ \Delta U = 0 + 0 = 0 \]
For an ideal gas, internal energy depends only on temperature, so if \( \Delta U = 0 \), then:
\[ \Delta T = 0 \]

Step 5: Conclusion
For free expansion of an ideal gas under adiabatic condition:
\( q = 0, \Delta T = 0, w = 0 \)

Final Answer: \( q = 0, \Delta T = 0, w = 0 \)