Problem 45 - Chemistry: The Central Science

Question

Use Appendix \(C\) to compare the standard entropies at \(25^{\circ} \mathrm{C}\) for the following pairs of substances: (a) \(\mathrm{Sc}(s)\) and \(\mathrm{Sc}(g) ;\) (b) \(\mathrm{NH}_{3}(g)\) and \(\mathrm{NH}_{3}(a q) ;\) (c) \(1 \mathrm{~mol} \mathrm{P}_{4}(g)\) and \(2 \mathrm{~mol}\) \(\mathrm{P}_{2}(\mathrm{~g}) ;\) (d) C(graphite) and \(\mathrm{C}\) (diamond). In each case explain the difference in the entropy values.

Step-by-Step Solution

Answer

In short, by comparing standard entropies from Appendix C, we find that: (a) Sc(g) has a higher entropy than Sc(s) due to the higher degree of disorder in the gas phase. (b) NH3(g) has a higher entropy than NH3(aq) because gases have a higher degree of disorder than aqueous solutions, with more freedom of molecular motion. (c) 2 mol P2(g) has a higher entropy than 1 mol P4(g) due to the larger number of moles and increased overall disorder. (d) C(graphite) has a higher entropy than C(diamond) because the weaker bonds in its layered structure result in a higher degree of disorder compared to the rigid 3D structure of diamond.

Step 1: Look up standard entropy values for the substances

Assuming we have access to Appendix C, look up the standard entropy values for the following substances at 25°C: Sc(s), Sc(g), NH3(g), NH3(aq), P4(g), P2(g), C(graphite), and C(diamond). We'll call these values S1, S2, S3, S4, S5, and S6 respectively, where S1 = Sc(s), S2 = Sc(g), etc.

Step 2: Compare entropy values for Sc(s) and Sc(g)

Compare the standard entropy values S1 and S2. Typically, the entropy of a substance in the gas phase would be higher than in the solid phase, so S2 > S1. This is due to the higher degree of disorder in the gas phase. Molecules in the gas phase have higher kinetic energy and are more randomly distributed, increasing the entropy.

Step 3: Compare entropy values for NH3(g) and NH3(aq)

Compare the standard entropy values S3 and S4. Usually, the entropy of a substance in the gas phase would be higher than in the aqueous phase, so S3 > S4. Similar to the above case, gases have a higher degree of disorder than aqueous solutions. In a solution, the molecules' movement is somewhat restricted by their interaction with the solvent, resulting in a lower entropy.

Step 4: Compare entropy values for 1 mol P4(g) and 2 mol P2(g)

Calculate the entropy value for 2 mol P2(g) by multiplying S6 by 2. Then compare this value with S5. The difference in their entropy values can be attributed to the number of moles and the chemical structure of the substances. More moles of P2(g) mean more molecules and a larger overall disorder, leading to a higher entropy value compared to 1 mol of P4(g), given that other factors, like temperature, remain constant.

Step 5: Compare entropy values for C(graphite) and C(diamond)

Finally, compare the standard entropy values for C(graphite) and C(diamond), S7 and S8, respectively. The difference in entropy values will be linked to their crystal structures. Graphite has a layered structure with weaker bonds between layers, while diamond has a rigid 3D structure. The weaker bonds in graphite result in a higher degree of disorder, resulting in a greater entropy value compared to diamond.

Key Concepts

Entropy Comparison

Entropy is a measure of disorder or randomness in a system. When comparing entropies, it's essential to consider the state of matter and its interaction with surroundings.
Entropy can vary greatly between different phases and molecular structures, which makes it crucial to understand these comparisons to grasp chemical processes better.
Here's a quick guide to understanding entropy comparisons:

• Gas entropies are generally higher than solids and liquids due to higher disorder.
• Aqueous solutions have lower entropies compared to gases, as solvent interactions restrict movement.
• The molecular structure and number of moles influence the overall entropy significantly.

Recognizing these patterns helps in predicting and explaining changes during chemical reactions.

Gas vs Solid Entropy

The entropy of a gas is always higher than that of a solid. This distinction arises because gases have more freedom of motion.
In gases, molecules can move independently of each other and occupy more space, leading to greater chaos or disorder.
Solids, on the other hand, have molecules tightly packed in a fixed position, greatly limiting their movement. This ordered structure results in lower entropy.

• Molecules in a gas are spread out and can move easily, leading to higher entropy.
• Solid molecules are arranged in a defined lattice, restricting movement and lowering entropy.

As a result, when a substance changes from solid to gas, such as from \(\mathrm{Sc}(s)\) to \(\mathrm{Sc}(g)\), its entropy increases significantly.

Aqueous vs Gas Entropy

In comparing a substance in an aqueous solution to its gaseous form, the gas is typically higher in entropy.
The reason lies in the interaction between the solute and solvent molecules in the aqueous state.
In an aqueous solution, the solute's movement is influenced by its interaction with water molecules, which limits its disorder.
Meanwhile, gases like \(\mathrm{NH}_3(g)\) have much less restricted movement. Their molecules are not bound by interactions with a solvent, allowing them to move freely, thus having a higher entropy.

• Gases have unconfined molecular movement, leading to higher entropy.
• Aqueous solutions have solute-solvent interactions that lower the system's entropy.

Therefore, the entropy of \(\mathrm{NH}_3(g)\) is higher than that of \(\mathrm{NH}_3(aq)\).

Molecular Structure and Entropy

The molecular structure and amount of a substance can greatly impact its entropy.
For example, in a comparison between 1 mole of \(P_4(g)\) and 2 moles of \(P_2(g)\), entropy differences arise due to molecular composition.
Each mole of \(P_2(g)\) contributes its entropy value; having more moles results in a higher entropy due to increased total disorder and complexity.
Additionally, the physical structure, such as the crystal formations in \(\mathrm{C}(graphite)\) versus \(\mathrm{C}(diamond)\), affects the entropy. Graphite, with its layered and less tightly bound molecular structure, permits more movement compared to diamond's rigid lattice.

• More moles mean greater disorder and increased total entropy.
• Structures with more freedom of movement within the molecules have higher entropy.

Thus, substances with complex, flexible structures or greater quantity generally have higher entropy than more rigid or fewer counterparts.