Predict which member of each of the following pairs has the greater standard entropy at \(25^{\circ} \mathrm{C} :(\mathbf{a}) \mathrm{C}_{6} \mathrm{H}_{6}(l)\) or \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\) (b) \(\mathrm{CO}(g)\) or \(\mathrm{CO}_{2}(g),(\mathbf{c}) 1 \mathrm{mol} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) or 2 \(\mathrm{mol} \mathrm{NO}_{2}(\mathrm{g})\) (d) \(\mathrm{HCl}(g)\) or \(\mathrm{HCl}(a q) .\) Use Appendix \(\mathrm{C}\) to find the standard entropy of each substance.

Entropy, often denoted by the symbol S, is a fundamental concept in thermodynamics that measures the degree of disorder or randomness in a system. When we talk about molecular complexity, we refer to the number of atoms and the complexity of the interactions between them within a molecule. Generally, the more complex a molecule is, the higher its entropy.

For instance, comparing carbon monoxide (CO) with carbon dioxide (CO₂), CO₂ has a greater number of atoms which leads to more possible ways the energy can be distributed throughout the molecule. As a result, CO₂ has a higher standard entropy than CO. Molecular complexity significantly influences a system's entropy, as a larger, more complex molecule will have more vibrational modes, allowing for more microstates that a system can exist in.

The concept of entropy of states of matter is crucial to understanding the behavior of substances under different conditions. Entropy varies with the physical state of a material: solids, liquids, and gases each have distinct entropy values.

In general, gases have the highest standard entropy because their molecules are far apart and move with considerable freedom. Liquids follow since they have some structure but less than solids. Finally, solids possess the lowest entropy due to their relatively fixed and orderly arrangement of atoms. To illustrate, benzene (C₆H₆) in gaseous form has greater standard entropy than in its liquid form, because the gas molecules can spread out and move independently, creating a state of higher disorder.

The branch of physics called thermodynamics deals with heat and temperature and their relation to energy and work. It lays out the fundamental principles governing the transfer and transformation of energy in systems. Its laws help us understand phenomena such as why heat flows from hot to cold objects, how engines produce work, and why certain reactions are spontaneous.

At the heart of thermodynamics is the concept of entropy, which can help explain physical processes and predict the direction of spontaneous change. For example, the second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. This explains why substances spontaneously transition from solid to liquid to gas at increasing temperatures, since these changes typically lead to an increase in entropy.

The concept of molecular disorder is intertwined with entropy. It refers to the degree to which molecules within a system are organized. An ordered state, such as a crystal lattice of a solid, has low entropy because the molecules are arranged in a predictable pattern. Conversely, in a disordered state, like a gas, molecules are distributed randomly and move freely, contributing to high entropy.

Molecular disorder becomes particularly apparent when comparing different states of matter or the same substance under different conditions. Consider hydrochloric acid (HCl) in gas versus aqueous form. When HCl is dissolved in water, it interacts with water molecules, dramatically increasing the disorder of the system compared to its gaseous state, and hence also increasing its standard entropy.