Question

Which substance in each of the following pairs would have the greater entropy? Explain. (a) at \(75^{\circ} \mathrm{C}\) and 1 atm: \(1 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}(1)\) or \(1 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) (b) at \(5^{\circ} \mathrm{C}\) and 1 atm: \(50.0 \mathrm{g} \mathrm{Fe}(\mathrm{s})\) or \(0.80 \mathrm{mol} \mathrm{Fe}(\mathrm{s})\) (c) 1 mol \(\mathrm{Br}_{2}\left(1,1 \text { atm }, 8^{\circ} \mathrm{C}\right)\) or \(1 \mathrm{mol} \mathrm{Br}_{2}(\mathrm{s}, 1 \mathrm{atm},\) \(\left.-8^{\circ} \mathrm{C}\right)\) (d) \(0.312 \mathrm{mol} \mathrm{SO}_{2}\left(\mathrm{g}, 0.110 \mathrm{atm}, 32.5^{\circ} \mathrm{C}\right)\) or \(0.284 \mathrm{mol}\) \(\mathrm{O}_{2}\left(\mathrm{g}, 15.0 \mathrm{atm}, 22.3^{\circ} \mathrm{C}\right)\)

Short answer

For each pair, the substance with greater entropy is: (a) 1 mol H2O(g), (b) 0.80 mol Fe(s), (c) 1 mol Br2(l), (d) 0.312 mol SO2(g).

Step-by-step solution

Understand entropy and factors

Entropy is essentially the measure of the disorder or randomness of a system. The greater the disorder, the greater the entropy. The factors that increase entropy include increasing temperature, the number of particles and transitioning from a solid to a liquid or a gas.

Analyze Part (a)

In part (a), we have 1 mol H2O in liquid (l) state and 1 mol H2O in gaseous (g) state at the same temperature and pressure. It's important to know that at the same temperature and pressure, gases have more entropy than liquids due to higher disorder. Therefore, 1 mol H2O(g) will have more entropy.

Analyze Part (b)

In part (b), the comparisons are between 50.0 g Fe(s) and 0.80 mol Fe(s). The one with more moles or particles will have more entropy as the increase in the number of particles increases disorder. Therefore, 0.80 mol Fe(s), which is approximately 44.8 grams, will have more entropy than 50.0 g Fe(s).

Analyze Part (c)

In part (c), we compare the entropy of 1 mol Br2 in the liquid state at 8°C and 1 mol Br2 in the solid state at -8°C. As a general rule, transitioning from solid to liquid or gas increases the entropy. Therefore, 1 mol Br2 (l) has greater entropy than 1 mol Br2(s).

Analyze Part (d)

In part (d), we compare 0.312 mol SO2(g) at 0.110 atm and 32.5°C and 0.284 mol O2(g) at 15.0 atm and 22.3°C. The gas with the higher number of moles and the higher temperature will have more entropy, which is 0.312 mol SO2(g).

Key concepts

Entropy

Entropy is a fundamental concept in chemistry and physics that describes the amount of disorder or chaos within a system. Imagine you have a clean room. When it's orderly and neat, the entropy is low. As the room becomes messy with items scattered everywhere, its entropy increases.

The same idea applies to molecules in a system. The more spread out and free-moving the molecules are, the greater the entropy. Entropy is crucial for understanding energy distributions, chemical reactions, and even predicting whether processes can happen spontaneously. An important point is, higher entropy generally means more disorder.

Disorder in Systems

Disorder in systems is directly related to entropy. When molecules or particles move from an ordered arrangement to a more random and spread-out configuration, the system achieves higher entropy. This shift from order to disorder is often seen when substances change phase, from solid to liquid, or from liquid to gas.

Why does disorder matter? Consider a solid, where molecules are tightly packed in a fixed structure. It's a very ordered state. But when that solid melts into a liquid or evaporates into a gas, the molecules are free to move. This freedom creates more possible arrangements, thus increasing disorder and entropy.

• Solids have the least disorder and lower entropy.
• Liquids have moderate disorder and higher entropy than solids.
• Gases have the highest disorder and entropy.

Understanding this concept helps explain why some reactions will naturally shift in certain directions.

States of Matter and Entropy

Different states of matter—solid, liquid, and gas—have distinct levels of entropy due to the arrangement and movement of their molecules.
In solids, molecules are packed closely in a fixed lattice structure, minimizing movement and thus, entropy. With liquids, molecules have more freedom to flow around each other, increasing entropy compared to solids.

Finally, gases have molecules that are completely free to move in all directions, giving them the highest entropy. You can think of gas molecules like children playing in an open field compared to soldiers standing at attention (solids) or people walking in a controlled line (liquids). Each state transitions into the next with added energy, raising entropy. This is why we observe melting (solid to liquid) and evaporation (liquid to gas) as processes where entropy increases. Each phase change represents a leap in disorder and, thus, a leap in entropy.

Factors Affecting Entropy

Several key factors influence the entropy of a system. These include temperature, state of matter, and the number of particles present.

• Temperature: Increasing temperature typically increases entropy because added heat boosts energy, leading to more energetic and disordered molecular motion.
• Phase Changes: Transitioning from solid to liquid or liquid to gas raises entropy, due to greater molecular freedom in the new state.
• Number of Molecules: More molecules mean more possible arrangements or configurations, thereby increasing the entropy of the system.
• Pressure and Volume: Lower pressure and higher volume usually correspond with higher entropy, as molecules can spread out freely.

All these factors contribute to the overall entropy and determine how substances behave in different conditions.