Question

Predict which member of each of the following pairs has the greater standard entropy at \(25^{\circ} \mathrm{C}:\) (a) \(\mathrm{HNO}_{3}(g)\) or \(\mathrm{HNO}_{3}(a q)\) (b) \(\mathrm{PCl}_{3}(l)\) or \(\mathrm{PCl}_{3}(g)\), (c) \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)\) or \(\mathrm{Fe}_{3} \mathrm{O}_{4}(s),(\mathbf{d}) \mathrm{Li}(s)\) or \(\mathrm{Li}(g)\). Use Appendix \(\mathrm{C}\) to find the stan- dard entropy of each substance.

Short answer

The substances with greater standard entropy at \(25^{\circ}\mathrm{C}\) are: (\textbf{a}) HNO3(g), (\textbf{b}) PCl3(g), (\textbf{c}) Fe3O4(s), and (\textbf{d}) Li(g).

Step-by-step solution

Comparison 1: HNO3(g) vs HNO3(aq)

Using values from Appendix C: Standard Entropy of HNO3(g) = 266.4 J/mol·K Standard Entropy of HNO3(aq) = 110.7 J/mol·K So, HNO3(g) has a greater standard entropy than HNO3(aq).

Comparison 2: PCl3(l) vs PCl3(g)

Using values from Appendix C: Standard Entropy of PCl3(l) = 145.8 J/mol·K Standard Entropy of PCl3(g) = 296.9 J/mol·K So, PCl3(g) has a greater standard entropy than PCl3(l).

Comparison 3: Fe2O3(s) vs Fe3O4(s)

Using values from Appendix C: Standard Entropy of Fe2O3(s) = 87.4 J/mol·K Standard Entropy of Fe3O4(s) = 146.4 J/mol·K So, Fe3O4(s) has a greater standard entropy than Fe2O3(s).

Comparison 4: Li(s) vs Li(g)

Using values from Appendix C: Standard Entropy of Li(s) = 29.1 J/mol·K Standard Entropy of Li(g) = 152.3 J/mol·K So, Li(g) has a greater standard entropy than Li(s).

Key concepts

Gaseous State

A key concept in understanding the behavior of gases is the idea of the **gaseous state**. In this state, molecules have greater freedom of motion compared to liquids or solids. This freedom is due to the weaker intermolecular forces present in gases, allowing the molecules to move randomly and occupy a larger volume.
As a result, gas molecules tend to have higher energy and, correspondingly, higher **entropy**. Entropy is a measure of disorder or randomness. In gases, the high entropy reflects their high degree of disorder.
This is illustrated by comparing \(\mathrm{HNO}_{3}(g)\) and \(\mathrm{HNO}_{3}(aq)\). The gaseous nitric acid shows greater entropy due to its fewer restrictions and greater disorder compared to when it is dissolved in water.

Phase Transition

When substances change from one state of matter to another, they undergo a **phase transition**. This transition plays a significant role in determining the entropy of a substance.
For instance, when moving from a liquid to a gaseous state, a substance will typically see an increase in entropy. This is because the molecules in the gaseous phase are more disordered compared to their liquid state, where they are more restricted.
The case of \(\mathrm{PCl}_{3}(l)\) transforming to \(\mathrm{PCl}_{3}(g)\) exemplifies this concept. In the gaseous phase, the phosphorus trichloride atoms have more freedom to move and thus show a higher standard entropy than in the liquid phase.

Entropy Comparison

Entropy comparison is crucial for predicting the thermodynamic behavior of substances. The structure, composition, and state of a substance impact its level of entropy, which can be directly compared through standard entropy values.
Generally, substances with complex molecular structures or those in less ordered states (like gases over solids) demonstrate higher entropy. For example, \(\mathrm{Fe}_{3} \mathrm{O}_{4}(s)\) has a more intricate molecular structure than \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)\), so it possesses a higher entropy. Similarly, \(\mathrm{Li}(g)\) versus \(\mathrm{Li}(s)\) highlights how change in state elevates entropy levels due to increased molecular motion.

Appendix Use

When addressing exercises in thermodynamics, using appendices such as the one here, Appendix C, is invaluable. These sections provide the necessary data for calculating entropy, enthalpy, and other thermodynamic properties.
For example, to assess which substance in a given pair has a greater standard entropy, you would use the values listed in the appendix. These values are determined through experimental data and allow for accurate comparisons and predictions.
In our outlined solution, Appendix C was crucial in determining the greater entropy in each case. It provided specific entropy values such as 296.9 J/mol·K for \(\mathrm{PCl}_{3}(g)\) compared to 145.8 J/mol·K for its liquid form, which guided us to conclude that the gaseous form had higher entropy.