Question

For each pair of substances listed here, choose the one having the larger standard entropy value at \(25^{\circ} \mathrm{C}\). The same molar amount is used in the comparison. Explain the basis for your choice. (a) \(\operatorname{Li}(s)\) or \(\operatorname{Li}(l),(b) C_{2} H_{5} O H(l)\) or \(\mathrm{CH}_{3} \mathrm{OCH}_{3}(l)\) (Hint: Which molecule can hydrogen- (d) \(\mathrm{CO}(g)\) or \(\mathrm{CO}_{2}(g),\) (e) \(\mathrm{O}_{2}(g)\) bond?), (c) \(\operatorname{Ar}(g)\) or \(\operatorname{Xe}(g)\), or \(\mathrm{O}_{3}(g),(\mathrm{f}) \mathrm{NO}_{2}(g)\) or \(\mathrm{N}_{2} \mathrm{O}_{4}(g) .\)

Short answer

(a) \(\operatorname{Li}(l)\); (b) \(\mathrm{CH}_3\mathrm{OCH}_3(l)\); (c) \(\operatorname{Xe}(g)\); (d) \(\mathrm{CO}_2(g)\); (e) \(\mathrm{NO}_2(g)\).

Step-by-step solution

Analyze Solid vs. Liquid States

For the pair \( \operatorname{Li}(s) \) and \( \operatorname{Li}(l) \), consider the states of matter. Generally, liquids have higher entropy than solids due to increased randomness and movement freedom. Therefore, \( \operatorname{Li}(l) \) has a larger standard entropy value at \( 25^{\circ} \mathrm{C} \) than \( \operatorname{Li}(s) \).

Analyze Molecular Structure and Hydrogen Bonding

Compare \( C_2 H_5 OH(l) \) and \( \mathrm{CH}_3 \mathrm{OCH}_3(l) \). The ethanol molecule (\( C_2 H_5 OH \)) can form hydrogen bonds due to its hydroxyl group, leading to more ordered interactions, while dimethyl ether (\( \mathrm{CH}_3 \mathrm{OCH}_3 \)) cannot hydrogen bond as effectively. Thus, \( C_2 H_5 OH(l) \) has a lower entropy due to stronger intermolecular forces, making \( \mathrm{CH}_3 \mathrm{OCH}_3(l) \) the one with larger entropy.

Compare Atomic Size and Entropy

For \( \operatorname{Ar}(g) \) and \( \operatorname{Xe}(g) \), xenon is larger than argon, resulting in greater freedom of movement and thus higher entropy. Therefore, \( \operatorname{Xe}(g) \) has a greater standard entropy than \( \operatorname{Ar}(g) \).

Analyze Bonding Complexity

Consider \( \mathrm{CO}(g) \) and \( \mathrm{CO}_{2}(g) \). The \( \mathrm{CO}_{2}(g) \) molecule is more complex with more atoms, which implies a higher number of possible arrangements and thus a higher entropy. Hence, \( \mathrm{CO}_{2}(g) \) has a larger standard entropy.

Assess Molecular Complexity and Intermolecular Forces

Examine \( \mathrm{NO}_{2}(g) \) and \( \mathrm{N}_{2} \mathrm{O}_{4}(g) \). The dimer \( \mathrm{N}_{2} \mathrm{O}_{4} \), having more atoms and stronger intermolecular forces, is less random than the individual molecule \( \mathrm{NO}_{2}(g) \). Therefore, \( \mathrm{NO}_{2}(g) \) has a higher entropy.

Key concepts

Standard Entropy Values

Standard entropy values are a measure of the amount of disorder or randomness in a system. At a given temperature, such as \(25^{\circ} \mathrm{C}\), substances with higher standard entropy can have more possible microscopic arrangements. Entropy is closely related to the states of matter, molecular structure, and intermolecular forces.

Entropy is an intrinsic property that is extensive, meaning it depends on the amount of substance. When comparing entropy values, it's important to consider all the intrinsic and extrinsic factors that influence the disorganization of the particles.

For chemists, standard entropy values are crucial as they help predict the spontaneity of reactions and the stability of substances.

States of Matter

The state of matter is a key factor affecting the standard entropy of a substance. There are three principal states: solid, liquid, and gas.

- **Solids**: Particles are tightly packed in a regular pattern. This restricts their movement, resulting in lower entropy. - **Liquids**: More movement and some randomness are possible compared to solids, providing higher entropy. - **Gases**: Particles in gases move freely and occupy more space. This maximum freedom leads to the highest entropy among the three states.

For example, when comparing the entropy of Li in solid and liquid states, the liquid form has a greater degree of molecular motion and disorder, hence a higher entropy.

Molecular Structure

The molecular structure significantly influences a substance's entropy. The complexity and arrangement of atoms within a molecule can alter how its entropy is measured.

- **Complex molecules** typically possess more intricate structures. This allows for a higher number of possible microstates, leading to greater entropy. - In contrast, **simple molecules** with fewer atoms tend to have fewer possible arrangements, resulting in lower entropy.

For instance, between carbon monoxide (CO) and carbon dioxide (CO $_2$), CO $_2$ has a more complex arrangement due to the additional oxygen atom. This complexity gives CO $_2$ greater entropy because of the increased number of particle arrangement possibilities.

Intermolecular Forces

Intermolecular forces affect the interaction and arrangement of molecules, thereby influencing entropy.

- **Hydrogen Bonds**: These are strong interactions that can create more ordered systems, reducing entropy. Substances like ethanol, which can form hydrogen bonds, have lower entropy compared to similar molecules that do not. - **Van der Waals Forces**: Weaker interactions that allow more freedom of movement and higher entropy.

In the case of ethanol ( C $_2$ H $_5$ OH) vs. dimethyl ether ( CH $_3$ OCH $_3$), dimethyl ether cannot form hydrogen bonds as effectively. This results in higher entropy owing to its lower interconnectedness and higher randomness.