Question

For each of the following pairs, predict which substance has the higher entropy per mole at a given temperature: (a) \(\mathrm{I}_{2}(s)\) or \(\mathrm{I}_{2}(g)\) (b) \(\mathrm{O}_{2}(g)\) at \(50.7 \mathrm{kPa}\) or \(\mathrm{O}_{2}\) at \(101.3 \mathrm{kPa}\) (c) 1 molof \(\mathrm{N}_{2}\) in 22.4 Lor \(1 \mathrm{~mol}\) of \(\mathrm{N}_{2}\) in \(44.8 \mathrm{~L}\). (d) \(\mathrm{CH}_{3} \mathrm{OH}(I)\) or \(\mathrm{CH}_{3} \mathrm{OH}(s)\)

Short answer

In summary: (a) \(\mathrm{I}_{2}(g)\) has higher entropy than \(\mathrm{I}_{2}(s)\). (b) \(\mathrm{O}_{2}(g)\) at \(50.7 \mathrm{kPa}\) has higher entropy than \(\mathrm{O}_{2}(g)\) at \(101.3 \mathrm{kPa}\). (c) \(\mathrm{N}_{2}\) in \(44.8 \mathrm{~L}\) has higher entropy than \(\mathrm{N}_{2}\) in \(22.4 \mathrm{~L}\). (d) \(\mathrm{CH}_{3}\mathrm{OH}(l)\) has higher entropy than \(\mathrm{CH}_{3}\mathrm{OH}(s)\).

Step-by-step solution

Pair (a) Analysis

For iodine, we have to compare the entropy of the solid phase and the gas phase. As mentioned, entropy is higher for gases than for solids.

Pair (a) Conclusion

In pair (a), Iodine gas (\(\mathrm{I}_{2}(g)\)) has the higher entropy per mole at a given temperature compared to solid iodine (\(\mathrm{I}_{2}(s)\)).

Pair (b) Analysis

In this case, we have to compare the entropy of oxygen gas at two different pressures. As pressure increases, entropy decreases.

Pair (b) Conclusion

In pair (b), oxygen gas (\(\mathrm{O}_{2}\)) at \(50.7 \mathrm{kPa}\) pressure has higher entropy per mole at a given temperature compared to oxygen gas at \(101.3 \mathrm{kPa}\).

Pair (c) Analysis

In this case, we have to compare the entropy of 1 mole of nitrogen gas in different volumes. As volume increases, entropy also increases.

Pair (c) Conclusion

In pair (c), 1 mole of nitrogen gas in 44.8 L (\(\mathrm{N}_{2}\) in \(44.8 \mathrm{~L}\)) has higher entropy per mole at a given temperature compared to 1 mole of nitrogen gas in 22.4 L (\(\mathrm{N}_{2}\) in 22.4 L).

Pair (d) Analysis

For methanol (\(\mathrm{CH}_{3}\mathrm{OH}\)), we have to compare the entropy of the liquid phase and the solid phase. As mentioned, entropy is higher for liquids than for solids.

Pair (d) Conclusion

In pair (d), liquid methanol (\(\mathrm{CH}_{3}\mathrm{OH}(l)\)) has higher entropy per mole at a given temperature compared to solid methanol (\(\mathrm{CH}_{3}\mathrm{OH}(s)\)).

Key concepts

Phase Transition

Phase transitions are changes in the state of matter. They involve transformations like solid to liquid, liquid to gas, or vice versa. These changes play a crucial role in altering the entropy of a substance.
Entropy, which measures the disorder of a system, often increases when there is a phase transition. Here's why:

• In solids, particles are tightly packed, with minimal freedom of movement. This results in lower entropy.
• Liquids allow more particle movement, leading to higher entropy compared to solids.
• Gases have particles widely dispersed and moving freely, thus possessing the highest entropy among the phases.

Consider iodine transitioning from a solid to a gas. The dramatic increase in disorder as particles move from a fixed arrangement in a solid to freely moving in a gas leads to a higher entropy in the gaseous state.

Gas Laws

Gas laws describe the behavior of gases and relate different gas properties such as pressure, volume, and temperature. These laws are crucial to understanding gas entropy.
One primary law is the **Ideal Gas Law**, expressed as \( PV = nRT \), where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the gas constant, and \(T\) is temperature.
In the context of entropy:

• An increase in volume (\(V\)) at constant pressure allows gas particles more space to move, increasing entropy.
• A decrease in pressure (\(P\)) could imply a volume increase or a drop in temperature, affecting entropy.

For nitrogen gas, increasing the volume results in greater entropy due to enhanced molecular movement.

Pressure and Volume

Pressure and volume are key factors in a gas's entropy. They are inversely related in the Ideal Gas Law, meaning a change in one affects the other.
**Understanding their relationship: **

• When volume increases, gases expand, allowing molecules more room to move freely. This leads to an increase in entropy.
• Conversely, when pressure rises, the volume decreases, limiting molecular movement and reducing entropy.

For example, oxygen gas at a lower pressure of 50.7 kPa will have higher entropy than at a higher pressure of 101.3 kPa, as molecules can disperse more freely in less restrictive environments.

Thermodynamics

Thermodynamics assesses the principles that govern energy and matter interactions, including entropy. It offers insight into how and why changes occur in physical systems.
**The four laws of thermodynamics** primarily guide these interactions:

• The **zeroth law** establishes temperature as a uniform measure.
• The **first law**, or conservation of energy, highlights that energy cannot be created or destroyed.
• The **second law** states that entropy tends to increase, driving spontaneous processes.
• The **third law** posits that absolute zero is unattainable as it requires zero entropy.

Entropy is a central concept. It predicts the direction of state changes, favoring states with higher disorder.
This understanding explains phenomena such as the persistence of methanol in a liquid state or the higher entropy of gases compared to solids.