For each of the following pairs of substances, which substance has the greater value of \(S^{\circ}\) ? a. \(\mathrm{C}_{\text {graphite }}(s)\) or \(\mathrm{C}_{\text {diamond }}(s)\) b. \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) or \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)\) c. \(\mathrm{CO}_{2}(s)\) or \(\mathrm{CO}_{2}(g)\)

Standard molar entropy, denoted as \(S^{\circ}\), is a way to quantify the degree of disorder or randomness associated with one mole of a substance at a standard reference state. The units for standard molar entropy are typically joules per mole kelvin (J/(mol·K)). Greater entropy values suggest a higher level of disorder and a greater dispersal of energy within the system.

In understanding \(S^{\circ}\), it's important to note that the entropy of a substance is intrinsically linked to the freedom of its particles. As a general rule, gases have higher entropy than liquids, which in turn have higher entropy than solids. This is because particles in a gas have the most freedom to move and occupy space compared to particles in liquids and solids. Consequently, predicting which substance has greater \(S^{\circ}\) often involves considering its physical state under standard conditions as well as the complexity of its molecular structure.

Carbon allotropes such as graphite and diamond exhibit different structures and, thus, different standard molar entropies. Graphite consists of layers of carbon atoms arranged in a hexagonal lattice, where each layer is weakly bonded to the next. This structure allows for more randomness in the arrangement of atoms and easier movement within the layers.

On the other hand, diamond features a rigid, tetrahedral lattice where each carbon atom is strongly bonded to four others, forming a very stable and highly ordered structure. The high order and strong bonding in diamond minimize the disorder and possible microstates, leading to a lower entropy in comparison to graphite. In summary, between graphite and diamond, graphite has the higher standard molar entropy due to its more disordered structure.

The phases or states of matter—solid, liquid, and gas—are differentiated by the degree of order and strength of interaction among their constituent particles. In solids, particles are closely packed in a fixed volume and shape, exhibiting vibrations but minimal movement. Liquids have more freedom, allowing particles to slide past one another while generally maintaining a fixed volume. Gases possess the most freedom, with particles that can move independently and completely fill their container.

Increasing Entropy Across Phases

Order decreases significantly from solids to liquids to gases, which also correlates with increasing entropy. The sequence of phases—solid to liquid to gas—is thus a sequence of increasing entropy. When a substance transitions from a solid to a liquid to a gas, the particles become less ordered and more widely distributed, leading to higher randomness, more possible microstates, and ultimately, higher entropy.

Phase transitions, such as melting or vaporizing, involve significant changes in the properties of a substance, including its entropy. Consider the transition from a solid to a liquid (melting) or from a liquid to a gas (boiling).

Melting and Boiling

During melting, the solid structure breaks down, allowing particles to move past one another more freely in the liquid state. The increased motion and disorder cause a rise in entropy. Boiling presents an even more drastic change—particles in a liquid are confined to a certain volume, but upon vaporization, they disperse throughout the available space, greatly increasing the substance's entropy.

Entropy changes are also reversible. For example, when a gas condenses into a liquid, or a liquid freezes into a solid, entropy decreases because the particles are becoming more ordered and less dispersed. Thus, a substance's phase at a particular temperature and pressure can significantly influence its standard molar entropy.