Problem 1 - Chemistry and Chemical Reactivity

Question

Which substance has the higher entropy in each of the following pairs? (a) dry ice (solid \(\mathrm{CO}_{2}\) ) at \(-78^{\circ} \mathrm{C}\) or \(\mathrm{CO}_{2}(\mathrm{g})\) at \(0^{\circ} \mathrm{C}\) (b) liquid water at \(25^{\circ}\) C or liquid water at \(50^{\circ} \mathrm{C}\) (c) pure alumina, \(\mathrm{Al}_{2} \mathrm{O}_{3}(\mathrm{s}),\) or ruby (Ruby is \(\mathrm{Al}_{2} \mathrm{O}_{3}\) in which some of the Al \(^{3+}\) ions in the crystalline lattice are replaced with \(\mathrm{Cr}^{3+}\) ions.) (d) one mole of \(\mathrm{N}_{2}(\mathrm{g})\) at 1 bar pressure or one mole of \(\left.\mathrm{N}_{2}(\mathrm{g}) \text { at } 10 \text { bar pressure (both at } 298 \mathrm{K}\right)\)

Step-by-Step Solution

Answer

(a) CO\(_2\)(g) at 0°C, (b) liquid water at 50°C, (c) ruby, (d) N\(_2\)(g) at 1 bar.

Step 1: Comparing States of Matter

Entropy is a measure of disorder. Gases have higher entropy than solids because they have more freedom of movement. For (a), compare dry ice (solid CO\(_2\)) to gaseous CO\(_2\). The CO\(_2\) gas at 0°C has higher entropy than solid CO\(_2\) at -78°C because gas particles have more freedom to move and occupy more space.

Step 2: Effect of Temperature on Entropy

Entropy increases with temperature as particles move more vigorously. For (b), compare liquid water at 25°C and 50°C. The water at 50°C has higher entropy as the increased temperature enhances molecular motion, leading to higher disorder.

Step 3: Comparing Composition and Structure

A more complex or impure structure generally has higher entropy due to increased disorder. For (c), compare pure alumina, Al\(_2\)O\(_3\), and ruby. Ruby has Cr\(^{3+}\) substitution, introducing defects and increasing disorder, so it has higher entropy.

Step 4: Effect of Pressure on Entropy

Entropy decreases with increasing pressure because the particles are more confined. For (d), compare N\(_2\)(g) at 1 bar and 10 bar. The N\(_2\) gas at 1 bar has higher entropy as it has more volume to occupy, leading to higher disorder.

Key Concepts

States of Matter

Entropy is a fundamental concept that relates to how particles arrange themselves in different states of matter. When we talk about states of matter, we primarily refer to solids, liquids, and gases. The entropy of a substance is closely linked to its state.

• Solids: In a solid, particles are tightly packed and have very fixed positions within a structured lattice. This orderliness results in low entropy.
• Liquids: Liquids have particles that are more spaced out compared to solids, allowing them to move around more freely. This increases their entropy.
• Gases: Gases have particles that are extremely far apart and can move freely. Gases exhibit the highest entropy of the three states.

Thus, when comparing substances in different states, like solid and gaseous carbon dioxide (as in the exercise), the gaseous state will naturally have higher entropy.

Temperature and Entropy

As temperature rises, so does the entropy of a substance. This is because increased temperature provides more energy for the particles within a material to move around. Higher temperatures lead to:

• Increased Molecular Motion: The particles vibrate, rotate, and translate more vigorously with increased temperature.
• Higher Disorder: More motion equates to more disorder, thereby increasing entropy.

For instance, consider liquid water at two different temperatures: 25°C and 50°C. The water at 50°C will have higher entropy than at 25°C due to increased thermal motion, resulting in greater disorder.

Molecular Structure

The complexity and purity of a molecular structure directly impact entropy. Substances with more intricate or varied structures have higher entropy due to a higher degree of disorder. Consider pure alumina ( Al _2 O _3) which has a regular crystalline structure, compared to a ruby, where some aluminum ions are replaced with chromium ions. These substitutional defects:

• Introduce irregularities, destabilizing the orderly arrangement.
• Provide more ways the particles can organize themselves.

Thus, these structural changes increase the complexity and overall disorder, leading to higher entropy in the ruby compared to pure alumina.

Pressure and Entropy

Entropy is inversely related to pressure. As pressure increases, particles are forced closer together into a smaller volume. This confinement decreases the number of ways they can arrange and move, reducing entropy. Think of nitrogen gas at two different pressures: 1 bar and 10 bar. At the lower pressure of 1 bar:

• The gas molecules have more space to occupy, leading to higher entropy.
• They experience less restriction in their movement.

Overall, when pressure is higher, the molecules are more constricted, leading to a reduction in entropy.