Question

Using Appendix \(C\), compare the standard entropies at \(25^{\circ} \mathrm{C}\) for the following pairs of substances: (a) \(\mathrm{CuO}(s)\) and \(\mathrm{Cu}_{2} \mathrm{O}(\mathrm{s}) ;\) (b) \(1 \mathrm{~mol} \mathrm{~N}_{2} \mathrm{O}_{4}(g)\) and \(2 \mathrm{~mol} \mathrm{NO}_{2}(g) ;\) (c) \(\mathrm{SiO}_{2}(\mathrm{~s})\) and \(\mathrm{CO}_{2}(\mathrm{~g}) ;\) (d) \(\mathrm{CO}(\mathrm{g})\) and \(\mathrm{CO}_{2}(\mathrm{~g})\). Fur each pair, explain the difference in the entropy values.

Short answer

In summary, the comparison of standard entropies at 25°C for the given pairs are: (a) Cu₂O(s) has higher entropy than CuO(s) due to its more complex molecular structure. (b) 2 mol of NO₂(g) has higher entropy than 1 mol of N₂O₄(g) due to the higher number of molecules. (c) CO₂(g) has higher entropy than SiO₂(s) because gases have more disorder than solids. (d) CO₂(g) has higher entropy than CO(g) due to the presence of an extra oxygen atom, increasing molecular complexity.

Step-by-step solution

Find the standard entropies from Appendix C

Look up the standard entropies for the given pairs of substances in Appendix C at 25°C. (a) CuO(s) and Cu₂O(s) (b) 1 mol N₂O₄(g) and 2 mol NO₂(g) (c) SiO₂(s) and CO₂(g) (d) CO(g) and CO₂(g)

Compare entropy values and identify reasons for the differences

Now, compare the standard entropy values for each pair of substances and identify the factors causing the differences. (a) CuO(s) and Cu₂O(s) CuO(s): \(S^{\circ} = 42.6 \frac{J}{mol \cdot K}\) Cu₂O(s): \(S^{\circ} = 93.8 \frac{J}{mol \cdot K}\) Cu₂O(s) has a higher entropy than CuO(s) because it has more atoms in its formula unit, which leads to more complex molecular structure and increased disorder in the system. (b) 1 mol N₂O₄(g) and 2 mol NO₂(g) N₂O₄(g): \(S^{\circ} = 304.3 \frac{J}{mol \cdot K}\) 2 mol NO₂(g): \(S^{\circ} = 2 \times 240.0 \frac{J}{mol \cdot K} = 480.0 \frac{J}{mol \cdot K}\) For the second pair, the difference in entropies is due to the difference in the number of molecules. Two moles of NO₂ gas result in a higher entropy than one mole of N₂O₄ gas, as the system becomes more disordered with the higher number of molecules. (c) SiO₂(s) and CO₂(g) SiO₂(s): \(S^{\circ} = 41.5 \frac{J}{mol \cdot K}\) CO₂(g): \(S^{\circ} = 213.6 \frac{J}{mol \cdot K}\) The significant difference in entropies between SiO₂(s) and CO₂(g) is due to the difference in their states. CO₂ being a gas, has higher disorder than SiO₂, which is a solid. (d) CO(g) and CO₂(g) CO(g): \(S^{\circ} = 197.6 \frac{J}{mol \cdot K}\) CO₂(g): \(S^{\circ} = 213.6 \frac{J}{mol \cdot K}\) Both substances are gases, but the entropy of CO₂(g) is higher due to the presence of an extra oxygen atom, which adds to the molecular complexity and increases the disorder in the system.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It describes how thermal energy is converted to and from other forms of energy and how it affects matter. The four laws of thermodynamics lay the foundation for heat transfer, energy transformations, and the relationships between properties of substances.

At the heart of thermodynamics lie concepts such as temperature, energy, and entropy. Entropy, in particular, is a central idea, representing the amount of disorder or randomness in a system. A higher entropy usually means there are more possible arrangements of a system, indicating a higher degree of uncertainty or disorder. Entropy changes give us profound insights into the direction of spontaneous processes and the feasibility of chemical reactions.

Entropy Values

In thermodynamics, the entropy value of a substance is a quantitative measure of the amount of disorder within a system. Standard entropy, denoted as S°, refers to the entropy content of a substance at a standard state, often considered to be 1 bar of pressure and 25°C for substances. These values are crucial for predicting the direction of chemical reactions and for calculating the Gibbs free energy change, which can indicate whether a process will occur spontaneously.

The values of standard entropy are affected by various factors including the complexity of the molecule (more atoms typically mean higher entropy), the phase of the matter (gases have higher entropy than liquids or solids due to their greater freedom of movement), temperature, and the number of particles, as seen in the comparison between 1 mol of N₂O₄(g) and 2 mol of NO₂(g), where two moles of NO₂ gas manifest more disorder and thus, a higher entropy than one mole of N₂O₄ gas.

Chemical States of Matter

Matter can exist in different chemical states, primarily solid, liquid, and gas, which significantly influence entropy. Solids have the lowest entropy because their particles are typically arranged in a fixed, orderly fashion. Liquids have higher entropy than solids because their particles can flow and move more freely, leading to more randomness. Gases have the highest entropy among the three classical states because their particles move independently and occupy the volume of their container, contributing to a substantial increase in disorder.

The differences between the chemical states of matter play a critical role in standard entropy comparisons. For instance, the comparison between SiO₂(s), which is a solid, and CO₂(g), which is a gas, clearly shows that CO₂, in its gaseous state, has a significantly higher entropy. This owes to the freedom of motion and the wide range of possible microstates that gas molecules can have compared to those in a rigid solid lattice.