Question

Mixing two distinct ideal gases results in an increase in entropy primarily because of

• A. Increase in internal energy
• B. Increase in molecular interactions
• C. Increase in spatial uncertainty
• D. Chemical reaction between gases

Hint:

Entropy can be understood as a measure of structural uncertainty. When distinct gases mix, the molecules scatter across a larger shared space, increasing spatial uncertainty and positional chaos. This drives the increase in entropy.

Answer 1

The Correct Option is C

Concept: When two distinct ideal gases undergo an isothermal mixing process at constant pressure, they intermingle without any chemical changes. In an ideal gas model, intermolecular forces are non-existent, meaning there are no energetic alterations during mixing. The process is driven entirely by statistical mechanics and probabilities. Detailed Structural Analysis:
Let us analyze why entropy increases by evaluating the microstates of the system:
• Before mixing, the gas molecules are separated into individual compartments. We know with absolute certainty that molecules of Gas 1 are in the first compartment and molecules of Gas 2 are in the second.
• Once the partition is removed and mixing occurs, each gas expands to fill the entire combined volume.
• Because the molecules are now distributed across a larger combined space, our knowledge of any single molecule's exact location decreases. This increases the system's structural randomness and spatial uncertainty.
• Statistically, the total number of microscopic spatial configurations (\(\Omega\)) increases dramatically. According to Boltzmann's relation (\(S = k_B \ln \Omega\)), this increase in positional choices results in a positive entropy of mixing (\(\Delta S_{\text{mix}} > 0\)). Evaluating the alternative options:
• Option (1) is incorrect: Ideal gas mixing is isothermal, meaning the internal energy remains completely constant (\(\Delta U_{\text{mix}} = 0\)).
• Option (2) is incorrect: Ideal gases are defined as having zero intermolecular interactions, so there is no change in molecular interactions during mixing.
• Option (4) is incorrect: The scenario describes physical mixing without any chemical reactions occurring between the gas species. Therefore, the entropy increase is driven entirely by the increase in spatial uncertainty, matching Option (3).