Problem 1 - Chemistry and Chemical Reactivity

Question

Which substance has the higher entropy? (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 \(25^{\circ} \mathrm{C}\) 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 Al \(^{3+}\) ions in the crystalline lattice are replaced with \(\left.\mathrm{Cr}^{3+} \text { ions }\right)\) (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) CO₂(g) at 0°C, (b) water at 50°C, (c) ruby, (d) N₂(g) at 1 bar.

Step 1: Understand Entropy

Entropy is a measure of disorder or randomness in a system. Higher entropy indicates greater disorder.

Step 2: Analyze Consideration (a)

Compare dry ice (solid CO₂) with gaseous CO₂. Gases have higher entropy than solids because gas particles move freely and occupy more positions.

Step 3: Analyze Consideration (b)

Compare liquid water at 25°C and 50°C. Higher temperature increases the motion of molecules, thereby increasing entropy.

Step 4: Analyze Consideration (c)

Compare pure alumina and ruby. Ruby has higher entropy due to the disorder introduced by substituting some Al³⁺ ions with Cr³⁺ ions.

Step 5: Analyze Consideration (d)

Compare N₂ gas at 1 bar and 10 bar. Lower pressure allows gas molecules more space to move, increasing entropy.

Key Concepts

Solid vs Gas Entropy

In the context of entropy, one of the most striking comparisons is between solids and gases. Solids, like dry ice (solid CO₂), have a much more ordered structure. This ordered arrangement means that the particles in a solid are confined to specific positions.
In contrast, gases are much less ordered. Gaseous CO₂ at room temperature, for example, has molecules that are free to move in all directions. This freedom of movement allows for a greater number of possible arrangements of the molecules. As a result, gases possess much higher entropy compared to their solid counterparts.
When dry ice sublimates to gaseous CO₂, there is a significant increase in entropy due to the transition from a highly ordered state (solid) to a highly disordered state (gas).

• Solids have lower entropy due to ordered molecular arrangements.
• Gases have higher entropy because molecules move freely.
• Transition from solid to gas increases entropy significantly.

Temperature and Entropy

Entropy is directly related to temperature due to its influence on molecular motion. When the temperature of a substance increases, so does the energy of its molecules. This increase in energy allows the molecules to move more vigorously, thereby increasing the disorder within the substance.
For instance, liquid water at 50°C will have higher entropy than at 25°C. Why is this? At higher temperatures, water molecules jiggle and vibrate more rapidly, occupying more possible positions and forming more random configurations. This is why heating a substance elevates its entropy.
It is essential to remember:

• Higher temperature leads to higher entropy.
• Increased molecular motion promotes disorder.
• Entropy reflects the number of configurations or positions particles can occupy.

Crystal Lattice Substitutions

Crystal lattice substitutions introduce disorder within crystalline solids, thereby affecting entropy. A pertinent example of this is the case of ruby compared to pure alumina, (\( \mathrm{Al}_2 \mathrm{O}_3 \)). Ruby has a similar structure to alumina, but some (\( \mathrm{Al}^{3+} \)) ions in its crystal lattice are substituted with (\( \mathrm{Cr}^{3+} \)) ions.
This substitution creates irregularities in the crystal lattice, distorting the uniformity that characterizes pure crystals. These irregularities introduce additional possible configurations in which the ions can be arranged, thus increasing the entropy of ruby compared to pure alumina.

• Substitutions create lattice irregularities.
• Increased configurations mean increased entropy.
• Crystal structures become less uniform due to substitutions.

Pressure and Entropy

Pressure has a noticeable impact on the entropy of gases. To illustrate, consider a mole of nitrogen gas (\( \mathrm{N}_2 \)) at different pressures while maintaining constant temperature.
At a lower pressure, such as 1 bar, the gas molecules have more space to move about freely. This increased freedom translates into a higher number of potential positions or states, thereby increasing entropy.
In contrast, at a higher pressure of 10 bar, the same gas is compressed to occupy a smaller volume. The reduced amount of space limits the movement and positioning of the gas molecules, leading to lower entropy.

• Lower pressure allows more molecular movement space.
• Higher pressure constraints molecular movement, reducing entropy.
• Gases with more freedom to move have higher entropy.