Indicate which substance in the given pairs has the higher entropy value. Explain your choices. (a) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) or \(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH}(l)\) (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) or \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)\) (c) \(2 \mathrm{H}(g)\) or \(\mathrm{H}(g)\)

Understanding the relationship between molecular complexity and entropy is crucial in chemistry. Entropy, symbolized as \( S \), is a measure of the disorder or randomness in a system. With more complexity in a molecule, such as additional atoms or bonds, the number of possible microstates increases. Microstates are the distinct ways a molecule's energy can be distributed among its atoms and bonds. When comparing \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH}(l) \) and \( \mathrm{C}_3\mathrm{H}_7\mathrm{OH}(l) \) in terms of entropy,
the principle is straightforward: the latter, being a more complex molecule, has more microstates due to its extra carbon and hydrogen atoms, resulting in a higher entropy. This is similar to having more pieces in a puzzle; the potential combinations increase, making it less predictable, thus more disordered.

Students often wonder why bigger molecules have higher entropy. It boils down to the many ways those extra atoms can vibrate, rotate, and interact compared to smaller molecules. For example, a longer chain alcohol like \( \mathrm{C}_3\mathrm{H}_7\mathrm{OH}(l) \) has more vibrational modes due to its additional bonds, leading to a higher entropy than \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH}(l) \).

Entropy is deeply influenced by the physical state of a substance. Particles in the gas phase have more freedom and can occupy much larger volumes compared to those in liquid or solid states. Consequently, gases exhibit the highest entropy. Considering the difference between \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH}(l) \) and \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH}(g) \),
we find that the gas, \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH}(g) \), has a higher entropy. Each molecule in the gaseous state can move independently of others, creating significantly more microstates. In liquids, molecules are closer together and their movements are more restricted.

It’s akin to the difference between a dance hall and a crowded elevator. In the dance hall (gas), people can move freely in all directions, which is similar to molecules in a gas having higher entropy. On the other hand, the crowded elevator (liquid) limits movement, just as the liquid state restricts molecular motion and therefore has lower entropy.

The concept of particle count affecting entropy is easy to visualize: more particles in a system mean more ways to arrange those particles, which corresponds to a higher entropy. It’s like having more guests at a party; you have more ways to organize the seating arrangements. For instance, when comparing a single hydrogen gas particle \( \mathrm{H}(g) \) to two hydrogen gas particles \( 2 \mathrm{H}(g) \),
the system with two particles has a higher entropy. This is because double the number of particles results in a greater number of possible microstates, even if each individual hydrogen atom remains fundamentally the same. This principle of particle count and entropy plays out in chemical reactions where the number of product molecules and reactant molecules can affect the total entropy of the system, influencing the reaction's spontaneity.