Problem 2 - Chemistry and Chemical Reactivity

Question

Which substance has the higher entropy in each of the following pairs? (a) a sample of pure silicon (to be used in a computer chip) or a piece of silicon containing a trace of some other elements such as boron or phosphorus (b) \(\mathrm{O}_{2}(\mathrm{g})\) at \(0^{\circ} \mathrm{C}\) or \(\mathrm{O}_{2}(\mathrm{g})\) at \(-50^{\circ} \mathrm{C}\) (c) \(\mathrm{I}_{2}(\mathrm{s})\) or \(\mathrm{I}_{2}(\mathrm{g}),\) both at room temperature (d) one mole of \(\mathrm{O}_{2}(\mathrm{g})\) at 1 bar pressure or one mole of \(\left.\mathrm{O}_{2}(\mathrm{g}) \text { at } 0.01 \text { bar pressure (both at } 298 \mathrm{K}\right)\)

Step-by-Step Solution

Answer

(a) Impure silicon; (b) \(\mathrm{O}_{2}(\mathrm{g})\) at \(0^{\circ}\mathrm{C}\); (c) \(\mathrm{I}_{2}(\mathrm{g})\); (d) \(\mathrm{O}_{2}(\mathrm{g})\) at 0.01 bar.

Step 1: Understanding Entropy

Entropy can be considered a measure of disorder or randomness in a system. Substances that are more disordered have higher entropy. Temperature, phase, and composition affect entropy.

Step 2: Identify the Influences on Entropy for Each Pair

For a given pair of substances, look at differences such as impurities, phase changes, temperature, and pressure to determine which condition is more disordered.

Step 3: Analyze Situation (a)

Pure silicon has fewer possible configurations than silicon with impurities. Impurities introduce disorder, increasing entropy. Thus, the silicon with a trace of other elements has higher entropy.

Step 4: Analyze Situation (b)

At a higher temperature, molecules have more kinetic energy and can occupy more microstates, leading to higher entropy. Therefore, \(\mathrm{O}_{2}(\mathrm{g})\) at \(0^{\circ} \mathrm{C}\) has higher entropy than at \(-50^{\circ} \mathrm{C}\).

Step 5: Analyze Situation (c)

Gaseous iodine, \(\mathrm{I}_{2}(\mathrm{g})\), is more disordered compared to solid iodine, \(\mathrm{I}_{2}(\mathrm{s})\), at the same temperature, resulting in higher entropy for the gas phase.

Step 6: Analyze Situation (d)

At lower pressure, gas molecules have more space and can spread out more, leading to greater disorder. Hence, \(\mathrm{O}_{2}(\mathrm{g})\) at 0.01 bar has higher entropy than at 1 bar.

Key Concepts

Molecular Disorder

Entropy is often described as a measure of the disorder within a molecular system. The more ways that the particles can be arranged, the higher the entropy. Imagine perfectly ordered crystals, which have lower entropy, because their molecular positions are fixed and predictable. In contrast, a system with impurities, like silicon with traces of other elements, has greater entropy.

• Impurities disrupt the regular lattice structure of pure substances.
• The presence of different atoms increases the possible configurations of molecules.

As a result, molecular disorder in systems with more diverse components will have increased entropy. This principle is perfectly demonstrated in situation (a). Pure silicon is more ordered than silicon containing boron or phosphorus, leading to less entropy.

Temperature Effect on Entropy

Temperature plays a significant role in determining the entropy of a system. The principle is straightforward: higher temperatures equate to higher entropy.

• As temperature increases, molecules have more kinetic energy.
• Molecules move faster and can access more states or configurations.

Consider \(O_{2}(g) \) at different temperatures, as in scenario (b). At 0°C, the molecules have more energy compared to at -50°C, allowing them greater freedom to move and occupy diverse microstates. Generally, higher energy levels lead to higher entropy because the system's disorder or randomness increases.

Phase Changes

Phase transitions between solid, liquid, and gas greatly affect a substance's entropy. This is because different phases have varying degrees of molecular freedom.

• Solids have tightly packed molecules that vibrate in place, resulting in low entropy.
• Liquids allow molecules to move around, increasing entropy.
• Gases have molecules that are free to move randomly in all directions, maximizing entropy.

In scenario (c), solid iodine (\(I_{2}(s)\)) has lower entropy compared to gaseous iodine (\(I_{2}(g)\)). This happens because gas molecules are significantly more dispersed and disordered than in solids, making gas the phase with the highest entropy.

Pressure Effect on Entropy

Entropy is also influenced by pressure, especially in gaseous substances. Pressure changes affect the volume that molecules can occupy.

• High pressure results in gas molecules being closer together, thus reducing entropy.
• Low pressure allows molecules to spread out, increasing entropy.

For gases like \(O_{2}(g) \), as in situation (d), lower pressure at 0.01 bar means more space for the molecules to disperse, thereby increasing disorder and entropy. Conversely, at 1 bar, the molecules are more confined, resulting in lower entropy due to reduced molecular freedom.