Problem 21 - Chemistry: An Atoms-Focused Approach

Question

Which component in each of the following pairs has the greater entropy? a. 1 mole of \(\mathrm{S}_{2}(g)\) or 1 mole of \(\mathrm{S}_{8}(g)\) b. 1 mole of \(S_{2}(g)\) or 1 mole of \(S_{8}(s)\) c. 1 mole of \(\mathrm{O}_{2}(g)\) or 1 mole of \(\mathrm{O}_{3}(g)\)

Step-by-Step Solution

Answer

(a) 1 mole of S₂(g) or 1 mole of S₈(g) (b) 1 mole of S₂(g) or 1 mole of S₈(s) (c) 1 mole of O₂(g) or 1 mole of O₃(g) Answer: (a) S₈(g) has greater entropy. (b) S₂(g) has greater entropy. (c) O₃(g) has greater entropy.

Step 1: (a) Comparing entropy: 1 mole of S₂(g) vs 1 mole of S₈(g)

Both of these substances are in the gaseous phase, so we need to compare their molecular masses. In S₂(g), there are two sulfur atoms in a molecule, whereas in S₈(g), there are eight sulfur atoms in a molecule. Since lower molecular masses have less entropy, 1 mole of S₂(g) will have a lower entropy compared to 1 mole of S₈(g). Therefore, S₈(g) has the greater entropy.

Step 2: (b) Comparing entropy: 1 mole of S₂(g) vs 1 mole of S₈(s)

In this pair, we have one substance in a gaseous phase and the other in a solid phase. Gaseous phases have a higher entropy than solid or liquids, so 1 mole of S₂(g) will have a greater entropy compared to 1 mole of S₈(s). Therefore, S₂(g) has the greater entropy.

Step 3: (c) Comparing entropy: 1 mole of O₂(g) vs 1 mole of O₃(g)

Both of these substances are in the gaseous phase, so we need to compare their molecular masses. In O₂(g), two oxygen atoms make up a molecule, whereas in O₃(g), three oxygen atoms make up a molecule. Since lower molecular masses have less entropy, 1 mole of O₂(g) will have a lower entropy compared to 1 mole of O₃(g). Therefore, O₃(g) has the greater entropy.

Key Concepts

Molecular Mass

Molecular mass refers to the total mass of all the atoms present in a molecule. It plays a crucial role in determining the entropy of substances. Entropy is a measure of disorder or randomness in a system. Generally, molecules with higher molecular mass have more complex structures. This complexity leads to a higher number of possible arrangements for the molecules, which in turn suggests higher entropy.
For instance, in the comparison of sulfur molecules \( \mathrm{S}_{2}(g) \) and \( \mathrm{S}_{8}(g) \, \), \( \mathrm{S}_{8}(g) \) has greater entropy due to its higher molecular mass. Its complexity allows more freedom and randomness in its movements, contributing to greater entropy overall. Thus, when comparing substances in similar phases, always factor in the molecular mass for entropy determination.

Gaseous Phases

The gaseous phase is a state of matter where molecules move freely and fill the container they are in. Molecules in the gas phase have higher entropy than in solid or liquid phases. This is due to their increased freedom of motion and ability to occupy space in various ways.
For example, when comparing \( \mathrm{S}_{2}(g) \) and \( \mathrm{S}_{8}(s) \, \), the former is in a gaseous state while the latter is in a solid state. The freedom of movement in gases makes \( \mathrm{S}_{2}(g) \) have a higher entropy than \( \mathrm{S}_{8}(s) \). In general, the gaseous phase allows for the highest degree of disorder among the states of matter, hence having higher entropy.

Solid Phases

In the solid phase, molecules are closely packed in structured arrangements. This organization limits their movement. As such, solids have lower entropy compared to gases and liquids because their molecular configurations are more restricted.
Take, for instance, the molecule of \( \mathrm{S}_{8}(s) \, \) which is more organized and has less disorder compared to the same molecule in a gaseous state, like \( \mathrm{S}_{2}(g) \). The rigid structure of solids doesn't allow for as many possible arrangements, thus leading to lower entropy. When considering entropy, it's critical to recognize the phase of the substance, as this greatly influences its degree of randomness and disorder.

Comparative Analysis of Substances

A comparative analysis of substances in terms of entropy requires considering both molecular mass and phase. It's important to consider what will cause a substance to have greater or lesser entropy. When two substances are in the same phase, the one with the greater molecular mass will typically have higher entropy.
A relevant example is the comparison between \( \mathrm{O}_{2}(g) \) and \( \mathrm{O}_{3}(g) \, \). Here, both are gases, but \( \mathrm{O}_{3}(g) \) with more atoms has a higher molecular mass. This results in \( \mathrm{O}_{3}(g) \) having a more complex structure and, therefore, a higher entropy than \( \mathrm{O}_{2}(g) \).

• Phase Impact: Gaseous states = Higher Entropy; Solids/Liquids = Lower Entropy.
• Molecular Mass Impact: Higher molecular mass = Higher Entropy.

By understanding these principles, you can better analyze and predict which substances exhibit greater entropy.