Which substance in each of the following pairs has the higher molar entropy? (Assume the temperature to be \(298 \mathrm{~K}\) unless otherwise specified.) (a) \(\mathrm{CH}_{4}\) (g) or \(\mathrm{C}_{2} \mathrm{H}_{6}\) (g); (b) \(\mathrm{KCl}\) (aq) or \(\mathrm{KCl}(\mathrm{s})\); (c) \(\mathrm{Ne}(\mathrm{g})\) or \(\mathrm{Kr}(\mathrm{g})\); (d) \(\mathrm{O}_{2}(\mathrm{~g})\) at \(273 \mathrm{~K}\) and \(1.00\) atm or \(\mathrm{O}_{2}(\mathrm{~g})\) at \(450 . \mathrm{K}\) and \(1.00\) atm.

Entropy, in thermodynamics, is a measurement of the disorder or randomness within a thermodynamic system. It is a crucial concept for understanding not only physics but also chemistry, where molar entropy refers to the entropy per mole of substance.

When comparing the entropy of different substances, one must bear in mind that, generally speaking, the higher the molecular complexity and the higher the temperature, the higher the entropy will be. For example, among gases, molecules with a greater mass and a larger number of atoms tend to have higher entropy due to the increased number of ways the atoms can arrange themselves (microstates). As a rule of thumb, greater movement (like in gases or liquids) equates to greater entropy because there is more freedom for the particles to assume different configurations.

This is evident in the comparison between CH₄(g) and C₂H₆(g). Though both are gases, ethane (C₂H₆) has a higher molar entropy than methane (CH₄) because it has more atoms and therefore a more complex molecular structure, which usually comes with an increased number of possible microstates.

The branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter, is known as thermodynamics. It lays the fundamental groundwork for understanding how energy is transformed and how systems move towards entropy.

Thermodynamics is informed by several laws, one of which states that the total entropy of an isolated system can never decrease over time. It's important for students to understand that temperature plays a significant role in entropy: as temperature increases, so does the energy within the particles, which causes more movement and, consequently, a higher entropy. This is exactly why O₂(g) at 450 K will have a higher molar entropy compared to O₂(g) at 273 K.

The concept of molecular complexity relates to the number and arrangement of atoms within a molecule. As molecules become larger and more complex, the number of ways the atoms can be arranged increases dramatically, leading to a rise in entropy. This is tied to the number of microstates: a greater number of possible microstates means greater entropy.

In the context of noble gases such as neon (Ne) and krypton (Kr), the size and number of electrons directly affect molecular complexity. Krypton, being larger with more electrons, has more potential microstates compared to neon, which explains its higher molar entropy. In essence, every additional atom in a molecule or a larger atomic structure enhances the potential for disorder, aligning with why molecular complexity is a fundamental aspect of entropy.