Question

For each of the following pairs, which substance has the greater value of S? a. \(\mathrm{N}_{2} \mathrm{O}(\text { at } 0 \mathrm{K})\) or He (at 10 \(\mathrm{K} )\) b. \(\mathrm{N}_{2} \mathrm{O}(g)\) (at \(1 \mathrm{atm}, 25^{\circ} \mathrm{C} )\) or He(g) (at 1 atm, \(25^{\circ} \mathrm{C} )\) c. \(\mathrm{NH}_{3}(s)\) (at 196 \(\mathrm{K} ) \longrightarrow \mathrm{NH}_{3}(l)(\text { at } 196 \mathrm{K})\)

Short answer

a. He (at 10 K) has a greater value of S. b. N2O (at 1 atm, \(25^{\circ} \mathrm{C}\)) has a greater value of S. c. NH3 in its liquid state (at 196 K) has a greater value of S.

Step-by-step solution

Analyze factors affecting entropy in Case a

In case a, we have \(\mathrm{N}_{2} \mathrm{O}(\text { at } 0 \mathrm{K})\) and He(at \(10 \mathrm{K}\)). Since entropy (S) generally increases with temperature, He at 10 K will have higher entropy than N2O at 0K: Result for case a: Helium (He) at 10 K has greater value of S.

Analyze factors affecting entropy in Case b

In case b, we have \(\mathrm{N}_{2} \mathrm{O}(g)\) (at \(1 \mathrm{atm}, 25^{\circ} \mathrm{C})\) and He(g) (at 1 atm, \(25^{\circ} \mathrm{C}\)). Both substances are in the gaseous state and at the same temperature and pressure. Thus, we can compare their entropies based on the molecular complexity. Greater molecular complexity is associated with greater entropy. Nitrous oxide (N2O) is more complex than helium (He), a monoatomic gas: Result for case b: Nitrous oxide (N2O) at 1 atm and \(25^{\circ} \mathrm{C}\) has a greater value of S.

Analyze factors affecting entropy in Case c

In case c, we have the phase transition \(\mathrm{NH}_{3}(s)\) (at 196 \(\mathrm{K}\)) \(\longrightarrow \mathrm{NH}_{3}(l)(\text { at } 196 \mathrm{K})\). When comparing the same substance but in different states at the same temperature, we can determine which state has a greater entropy by considering the order within the states. Entropy increases as the order in a system decreases. The solid state has more order compared to the liquid state: Result for case c: Ammonium (NH3) in its liquid state at 196 K has a greater value of S. To summarize: a. He (at 10 K) has a greater value of S. b. N2O (at 1 atm, \(25^{\circ} \mathrm{C}\)) has a greater value of S. c. NH3 in its liquid state (at 196 K) has a greater value of S.

Key concepts

Entropy and Temperature

Entropy is a fundamental concept in thermodynamics, representing the degree of disorder or randomness in a system. The relationship between entropy and temperature is pivotal in understanding how entropy affects different substances.
At absolute zero (0 K), a perfect crystal is in a state with minimum entropy, often near zero, as there is minimal molecular movement. However, as temperature increases, even slightly like from 0 K to 10 K, molecules gain energy, and their vibrations increase, leading to greater disorder.
For example, in the case of \(_2O (at 0 K)\) vs. \(He (at 10 K)\), the helium atoms at 10 K would have more thermal energy than nitrous oxide at 0 K, despite helium being monoatomic. This additional energy translates to increased movement and disorder, thereby raising the entropy of helium more than nitrous oxide at absolute zero.

• Higher temperature typically leads to higher entropy due to increased molecular motion.
• At 0 K, substances generally have minimum entropy.

Gaseous State Entropy

Gases tend to have high entropy values compared to liquids and solids because their particles are far apart and move freely. When comparing gases, the entropy can be influenced by their molecular complexity.
Complex molecules have more modes of movement (rotational, vibrational), which contribute to higher entropy. For instance, in comparing \(_2O(g)\) with \(He(g)\) both under the same conditions, \(_2O \) has a greater entropy.
Nitrous oxide is a triatomic molecule, whereas helium is a single atom. The greater connectivities in \(N_2O\) provide more states and positions the molecules can occupy. Thus, it imparts higher degrees of freedom compared to monoatomic helium, increasing \(N_2O's\) entropy.

• Gases have high entropy because particles are spread out.
• More complex molecules have higher entropy due to additional movement modes.

Phase Transition Entropy

Entropy changes significantly during phase transitions. During these transitions, the structure of the substance and the arrangement of its particles undergo substantial changes.
For example, when \(NH_3(s)\) transitions to \(NH_3(l)\) at 196 K, the molecules become less ordered. Solids have fixed positions and structured arrangements, contributing to lower entropy. Liquids, in turn, allow for freer movement of molecules as they partially overcome the forces keeping them fixed.
In this context, moving from a solid to a liquid state involves an increase in entropy due to the transition to a less ordered state.

• Phase transitions prompt significant changes in entropy.
• Solid-to-liquid transitions increase entropy as order decreases.