Question

For each of the following pairs, indicate which substance possesses the larger standard entropy: (a) \(1 \mathrm{~mol}\) of \(\mathrm{P}_{4}(g)\) at \(300^{\circ} \mathrm{C}, 0.01 \mathrm{~atm},\) or \(1 \mathrm{~mol}\) of \(\mathrm{As}_{4}(g)\) at \(300^{\circ} \mathrm{C}, 0.01 \mathrm{~atm} ;\) (b) \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm},\) or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm} ;\) (c) \(0.5 \mathrm{~mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) volume, or \(0.5 \mathrm{~mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) volume; (d) \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C}\)

Short answer

(a) 1 mol As4(g) has the larger standard entropy. (b) 1 mol of H2O(g) has larger standard entropy than H2O(l). (c) 0.5 mol CH4(g) has the larger standard entropy compared to N2(g). (d) 100 g Na2SO4(aq) has the larger standard entropy compared to Na2SO4(s).

Step-by-step solution

(a) Comparing P4(g) and As4(g)

Both P4(g) and As4(g) are in the gas phase at the same temperature and pressure. Since the atomic mass of As is greater than P, the molecules of As4(g) are more massive and have a larger number of energy levels, leading to higher entropy. Hence, 1 mol As4(g) has the larger standard entropy.

(b) Comparing H2O(g) and H2O(l)

Here, we are comparing water in a gas phase to water in the liquid phase. Both substances are at the same temperature and pressure. Gases generally have higher entropy than liquids due to higher energy levels and molecular disorder. Therefore, 1 mol of H2O(g) has larger standard entropy compared to H2O(l).

(c) Comparing N2(g) and CH4(g)

Both N2(g) and CH4(g) are in the gas phase at the same temperature and volume. CH4(g) has a more complex molecular structure than N2(g), with four hydrogen atoms attached to the carbon atom. This complexity leads to a higher number of molecular energy levels in CH4(g), resulting in higher entropy. Hence, 0.5 mol CH4(g) has the larger standard entropy compared to N2(g).

(d) Comparing Na2SO4(s) and Na2SO4(aq)

In this case, we are comparing sodium sulfate in the solid state to sodium sulfate in the aqueous state. Both are at the same temperature. In the aqueous state, ions are more dispersed, and the solution has a higher level of disorder compared to the solid-state. Therefore, 100 g Na2SO4(aq) has the larger standard entropy compared to Na2SO4(s).

Key concepts

Entropy Comparison

Entropy is a measure of disorder or randomness in a system. It represents the number of ways a system's particles can be arranged, maintaining its energy level. A greater entropy value indicates a higher level of disorder and vice versa. In the context of comparing entropies, students often face pairs of substances in different physical states at identical conditions. For instance, when you have gases like \( P_4(g) \) and \( As_4(g) \) at the same temperature and pressure, the entropy comparison is influenced by factors like the mass of the atoms and molecular structure.

Heavier atoms and larger molecules typically translate to higher entropy because they have more complex internal structures, allowing for a greater number of microstates. In an educational setting, it's crucial to emphasize that entropy isn't just related to the state of matter but also to the specifics of the substances being compared, ensuring that students grasp the nuances of the concept.

Physical States of Matter

Understanding the physical states of matter is pivotal when discussing entropy. Typically, gases have the highest entropy, followed by liquids, and then solids. This is due to the differing levels of molecular motion and freedom in each state. In gases, particles are far apart and move randomly at high speeds, leading to a large number of possible arrangements.

Therefore, when comparing the entropy of water in gas \( H_2O(g) \) and liquid phases \( H_2O(l) \) under the same conditions, \( H_2O(g) \) will have higher entropy owing to the greater freedom of movement. Learning platforms should highlight these differences and provide visual representations or analogies to aid comprehension, like comparing the particles to people in a crowded room versus an open field.

Molecular Complexity

Molecular complexity pertains to the number of atoms in a molecule and how they are arranged. More complex molecules have a higher entropy because there are more ways to distribute energy among the bonds and motions within the molecule. For example, when comparing nitrogen gas \( N_2(g) \) and methane \( CH_4(g) \) at the same conditions, \( CH_4(g) \) has a complex tetrahedral structure with four hydrogen atoms bonded to a central carbon atom.

Consequently, \( CH_4(g) \) possesses a larger number of vibrational, rotational, and translational motions. These additional degrees of freedom increase the entropy. Educational materials should convey these complexities using models or diagrams to elucidate how molecular structure impacts entropy. This can help students better visualize and understand why certain molecules have higher entropies than others.