Problem 2 - Chemistry and Chemical Reactivity

Question

Which substance has the higher entropy? (a) a sample of pure silicon (to be used in a computer chip) or a piece of silicon containing a trace of another element 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

Boron/phosphorus silicon, \( \mathrm{O}_{2} \) at \(0^{\circ} \mathrm{C}\), \( \mathrm{I}_{2} \) gas, \( \mathrm{O}_{2} \) at \(0.01 \text{ bar}\) have higher entropy.

Step 1: Understanding Entropy and Impurities

Entropy is a measure of disorder or randomness in a system. In part (a), a sample of silicon containing impurities (like boron or phosphorus) will have higher entropy compared to pure silicon. This is because the presence of different elements introduces more ways to arrange atoms, increasing disorder.

Step 2: Effect of Temperature on Entropy

Part (b) involves comparing entropy at two different temperatures. Higher temperature generally leads to higher entropy because molecules have more kinetic energy and more available microstates. Therefore, \( \mathrm{O}_{2} \) at \( 0^{\circ} \mathrm{C} \) has higher entropy compared to \( -50^{\circ} \mathrm{C} \).

Step 3: Phase Changes and Entropy

In part (c), consider the phases of iodine. \( \mathrm{I}_{2} \) as a gas has higher entropy than \( \mathrm{I}_{2} \) as a solid because gases have significantly more ways for molecules to move and distribute energy than solids.

Step 4: Effect of Pressure on Entropy in Gases

For part (d), consider that lower pressure allows gas molecules to occupy a larger volume, increasing the number of accessible microstates. Thus, \( \mathrm{O}_{2} \) at \( 0.01 \text{ bar} \) pressure has higher entropy than at \(1 \text{ bar} \) pressure.

Key Concepts

Impurities in solids

When discussing impurities in solids and their effect on entropy, imagine how the order and arrangement of atoms are impacted when different elements are present. In a pure solid, like silicon used for computer chips, atoms are arranged in a very structured, regular pattern. Think of it as an orderly row of people standing in line. However, introducing a trace element, such as boron or phosphorus, disrupts this order.

Here's why this leads to higher entropy:

• The presence of different atoms creates more possible ways to arrange the particles.
• This increase in possible arrangements translates to more disorder.
• Imagine mixing a small quantity of a different colored bead into a line of uniform beads; the orderliness decreases.

Therefore, a silicon sample with impurities will typically have higher entropy than one that is completely pure, because of these many additional possible configurations.

Temperature effects on gases

Temperature profoundly affects the entropy of gases due to the increased motion of their molecules. At higher temperatures, gases have greater kinetic energy, which means their molecules move more vigorously.

Consider the following effects of increasing temperature on gases:

• Molecules have more energy and can move in more directions, increasing the possible "microstates" or configurations.
• At low temperatures, like \(-50^{\circ} \mathrm{C}\), molecules are sluggish and have fewer movement options, leading to lower entropy.
• Conversely, at higher temperatures, such as \(0^{\circ} \mathrm{C}\), molecules can spread out more and have increased disorder, resulting in higher entropy.

In essence, the greater the temperature, the higher the entropy, because temperature amplifies molecular motion and the disorder of the system.

Phase changes

Phase changes play a significant role in determining entropy levels. The key here is understanding the difference in molecular dynamics between different phases of a substance — particularly solute (solid) and vapor (gas).

• In solids like \(\mathrm{I}_{2}(\mathrm{s})\), molecules are tightly packed and have limited movement, contributing to low entropy.
• In gases, such as \(\mathrm{I}_{2}(\mathrm{g})\), molecules are free to move throughout the available space in a disordered manner.
• The transition from solid to gas involves a massive increase in the number of ways molecules can arrange themselves, thus significantly increasing entropy.

So, when considering substances at the same temperature, like iodine, the gaseous form will have much higher entropy compared to the solid because of the freedom and randomness in molecular motion.

Pressure effects on gases

The pressure of a gas environment can dramatically affect its entropy by altering the volume that gas molecules can occupy.

Here's how pressure impacts gas entropy:

• At lower pressures, gas molecules are allowed to expand and spread over a larger volume.
• This expansion increases the possible arrangements and microstates, leading to higher entropy.
• Conversely, at higher pressures, the gas is compressed to a smaller volume, decreasing the possible configurations and thus lowering entropy.

For instance, \(\mathrm{O}_{2}(\mathrm{g})\) at \(0.01 \text{ bar}\) experiences higher entropy compared to \(1 \text{ bar}\) pressure because of the increased space and fewer restrictions on molecular movement. Lowering the pressure is akin to reducing crowding, which increases freedom and the number of possibilities within the system.