Problem 40 - Chemistry The Central Science In Si Units

Question

For each of the following pairs, predict which substance possesses the larger entropy per mole: (a) \(1 \mathrm{~mol}\) of \(\mathrm{O}_{2}(g)\) at \(300^{\circ} \mathrm{C}, 1.013 \mathrm{kPa}\), or \(1 \mathrm{~mol}\) of \(\mathrm{O}_{3}(g)\) at \(300^{\circ} \mathrm{C}, 1.013 \mathrm{kPa} ;\) (b) \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(100^{\circ} \mathrm{C}, 101.3 \mathrm{kPa}\), or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(100^{\circ} \mathrm{C}, 101.3 \mathrm{kPa} ;(\mathbf{c}) 0.5 \mathrm{~mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) vol- ume, or \(0.5 \mathrm{~mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) volume; \((\mathbf{d}) 100 \mathrm{~g}\) \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

O₃(g), H₂O(g), CH₄(g), and Na₂SO₄(aq) have higher entropy.

Step 1: Understand Entropy Definitions

Entropy is a measure of the randomness or disorder of a system. For gases, entropy generally increases with an increase in the number of possible arrangements, temperature, and phase transitions. For solids and liquids, solutions often have more entropy than pure solids due to the additional disorder from mixing.

Step 2: Analyze Part A - O₂ vs O₃

For part (a), we're comparing 1 mol of O₂ and O₃ gases both at the same temperature and pressure. Entropy generally increases with molecular complexity and mass. Ozone (O₃) is more complex than oxygen (O₂) due to its triatomic nature, which allows for more vibrational and rotational modes. Therefore, 1 mol of O₃(g) will have higher entropy than 1 mol of O₂(g).

Step 3: Analyze Part B - H₂O(g) vs H₂O(l)

In part (b), we're comparing water in gaseous form with water in liquid form at 100°C. Gases have higher entropy than liquids due to their higher degree of freedom and random motion. Therefore, 1 mol of H₂O(g) will have higher entropy than 1 mol of H₂O(l) at the same temperature and pressure.

Step 4: Analyze Part C - N₂ vs CH₄

For part (c), we're comparing N₂ and CH₄ gases at the same conditions. The molar entropy of a gas is influenced by its molecular weight and complexity. CH₄, being a larger and more complex molecule compared to N₂, will have higher entropy per mole due to more vibrational and rotational modes.

Step 5: Analyze Part D - Na₂SO₄(s) vs Na₂SO₄(aq)

In part (d), we're comparing solid and aqueous sodium sulfate. An aqueous solution typically has higher entropy compared to a solid due to the increased disorder from the dissolved ions. Therefore, 100 g of Na₂SO₄(aq) is expected to possess higher entropy than 100 g of Na₂SO₄(s).

Step 6: Final Step: Compile the Results

For each set of pairs, the substance with higher entropy is: (a) O₃(g), (b) H₂O(g), (c) CH₄(g), (d) Na₂SO₄(aq).

Key Concepts

Phase Transitions

During a phase transition, a substance changes from one state of matter to another, such as from liquid to gas. This process usually involves either an uptake or release of energy and is accompanied by a change in entropy as well. For instance, when water vaporizes, it transitions from a liquid state, where molecules are relatively close and ordered, to a gaseous state, where molecules are far apart and move more freely.
In this transition, entropy increases because there's a significant rise in the randomness and disorder of the system. Molecules in a gas can occupy a vast number of positions and have more freedom in movement, making their configuration more disordered than in a liquid.
Phase transitions are important to understand because they illustrate how energy and entropy are interlinked. A system tends to reach a state with maximum entropy since nature favors disorder.

Molecular Complexity

Molecular complexity influences a molecule's entropy significantly. Simply put, the more complex a molecule, the greater the entropy it can achieve. Complexity can arise from various structural features such as having more atoms, larger size, or various types of bonds.
Take ozone (\( ext{O}_3\)) and oxygen (\( ext{O}_2\)) for example. Ozone is a triatomic molecule, meaning it has three atoms, offering more vibrational and rotational modes than diatomic oxygen. This complexity allows ozone to have a larger number of possible microstates, and therefore, higher entropy.
To generalize, the more ways a molecule can internally move and reorder itself, the higher its entropy. This is why larger and more structurally diverse molecules typically possess greater entropy compared to simpler ones.

Entropy Comparison

When comparing entropy between substances, it's essential to consider factors like molecular complexity, temperature, and physical state. Entropy is a measure of the disorder or randomness of a system. Generally, more complex molecules and higher temperatures lead to higher entropy.
In the same phase at the same conditions, a larger, more complex molecule like methane (\( ext{CH}_4\)) usually has higher entropy than a smaller molecule like nitrogen (\( ext{N}_2\)) due to the larger number of possible arrangements.
Furthermore, a gaseous substance typically has higher entropy than a liquid or solid. In entropy comparison, it's crucial to analyze the system's state and its molecular structure to predict which has higher entropy.

Gaseous State

Gases are characterized by their high entropy owing to the large degree of freedom the molecules experience. In the gaseous state, molecules move independently of one another, fill any available volume, and exhibit minimal intermolecular forces.
This freedom results in numerous possible arrangements, or microstates, increasing the randomness and therefore, the entropy of the system. For example, comparing water in its gaseous and liquid state shows this clearly: the gaseous state has more entropy due to independent molecular motion.
Entropy in gases can also be affected by molecular complexity and temperature. For instance, at the same conditions, a triatomic gas will generally exhibit higher entropy than a diatomic gas, as it allows more vibrational and rotational modes for molecules.

Aqueous Solutions

In aqueous solutions, the dissolution of solids into a liquid increases system entropy. When a solid dissolves, ions or molecules become dispersed throughout the solvent, increasing the disorder or randomness.
Let's consider sodium sulfate. As it dissolves in water, the structured arrangement of ions in the solid is disrupted, allowing them to move freely and independently in the solution. This disruption and increased mobility culminate in higher entropy.
The elevated entropy in aqueous solutions compared to solids highlights the role of solvation processes in affecting a system's energy and behavior. The breakdown of lattice structures and incorporation into a solvent leads to substantial entropy changes, making aqueous solutions integral in understanding chemical reactions and processes.