Question

Mixing identical gases results in

• A. no change in entropy
• B. decrease in entropy
• C. increase in entropy
• D. infinite entropy

Hint:

Entropy of mixing is zero for identical gases (Gibbs Paradox).
Entropy increases only when mixing different, distinguishable gases because the number of distinct microstates increases.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the change in thermodynamic entropy when two samples of the same identical gas are mixed at constant temperature and pressure.

Step 2: Key Formula or Approach:

• The entropy of mixing ($\Delta S_{mix}$) for two different ideal gases (distinguishable molecules) is: \[ \Delta S_{mix} = -n R \left( x_1 \ln x_1 + x_2 \ln x_2 \right) > 0 \] where $x_1$ and $x_2$ are the mole fractions.

• For identical gases (indistinguishable molecules), this formula does not apply. This is known as the Gibbs Paradox.

Step 3: Detailed Explanation:

• The Gibbs Paradox:
If we have two containers of the same gas (e.g., Nitrogen in both compartments) at the same temperature and pressure, and we remove the partition, the molecules mix.
From a macroscopic and microscopic perspective, because the molecules of the same gas are completely identical and indistinguishable, there is no physical change in the thermodynamic state of the system.
No work can be extracted from this process, and no heat is transferred.
Since there is no change in the number of accessible microstates that represent a physically distinguishable state, the entropy of the system remains unchanged.
Therefore, the entropy of mixing for identical gases is exactly zero: \[ \Delta S_{mix} = 0 \]

Step 4: Final Answer:
Mixing identical gases results in no change in entropy.