For a liquid, which would you expect to be larger, \(\Delta S_{\text {fusion }}\) or \(\Delta S_{\text {evaporation }}\) ? Why?

Entropy (S) is a measure of the disorder or randomness in a system. The second law of thermodynamics states that any natural process will tend to increase the total entropy of the universe. In the context of phase transitions, it means that as a substance moves from a more organized phase to a less organized phase, its entropy increases.

In the process of fusion, a solid becomes a liquid, whereas in the process of evaporation, a liquid transitions to a gas. Recall from step 1 that entropy increases when a substance moves from a more organized phase to a less organized phase. Let's assess the order in each phase: 1. Solid: Particles are closely packed and have a well-defined structure. 2. Liquid: Particles are more spaced apart than in a solid and have less defined structure. 3. Gas: Particles are very spread out and have no defined structure. Comparing the phases, we can see that going from a solid to a liquid (fusion) introduces some disorder, but going from a liquid to a gas (evaporation) introduces much more disorder.

Given the increase in disorder for each phase transition, we can now identify which entropy change is larger. Since evaporation introduces a larger increase in disorder compared to fusion, we can conclude that the entropy change for evaporation, \(\Delta S_{\text {evaporation}}\), is larger than the entropy change for fusion, \(\Delta S_{\text {fusion}}\).

The larger entropy change for evaporation is due to the substantial increase in disorder when a substance transitions from a liquid (with some degree of structure) to a gas (with no structure). In the gas phase, particles are much more spread apart and have more freedom to move around, leading to a higher degree of randomness. On the other hand, while fusion also results in an increase in entropy, the increase is not as significant because the disorder between a solid and a liquid is not as substantial as between a liquid and a gas. In conclusion, \(\Delta S_{\text {evaporation}}\) is larger than \(\Delta S_{\text {fusion}}\) because the increase in disorder during evaporation is greater than that during fusion.