a) Rank the following in order of increasing entropy: \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) ; \mathrm{H}_{2} \mathrm{O}(\ell) ; \mathrm{H}_{2} \mathrm{O}(\mathrm{s})\) b) In general, how are the entropy of the solid, liquid, and gaseous phases of a particular compound related? Explain your reasoning.

Understanding the states of matter is a fundamental aspect of physics and chemistry. Matter can exist in several distinct forms known as 'states,' primarily solid, liquid, and gas. Solids have a fixed shape and volume with particles that are tightly packed in a regular arrangement. They exhibit strong intermolecular forces and minimal movement, residing mostly in vibrational motion around fixed positions.

Liquids have a definite volume but adapt to the shape of their container. Their particles are still close but can slide past one another, allowing liquids to flow. In liquids, intermolecular forces are weaker compared to solids, and the particles have more freedom of movement. Gases, on the other hand, have neither a definite shape nor a fixed volume. They expand to fill their container completely as the particles are far apart and move rapidly in all directions, exhibiting the weakest intermolecular forces among the three primary states.

The variation in particle arrangement and movement across these states leads to differing properties that are crucial for understanding physical phenomena and are directly related to the concept of entropy.

Thermodynamics is the branch of physical science that deals with the relations between heat, work, temperature, and energy. The laws of thermodynamics govern how energy is transferred in the form of heat and work in chemical and physical processes, predicting the directionality of these processes. One of the central concepts in thermodynamics is entropy, which is a measure of the disorder or randomness of a system.

Generally, the second law of thermodynamics states that the entropy of an isolated system never decreases over time. In other words, natural processes tend to move towards a state of maximum disorder or randomness. This concept provides a critical understanding of the behavior of molecules in different states of matter and during phase transitions. In practical terms, it explains why heat flows from a hot body to a cold one, why certain reactions are spontaneous, and it also has implications in areas like information theory and cosmology.

A phase transition is a transformation from one state of matter to another, such as from solid to liquid (melting), liquid to gas (vaporization), or solid to gas (sublimation). During a phase transition, energy is absorbed or released, and the arrangement and movement of particles change significantly.

During melting, heat is absorbed, causing the particles in a solid to vibrate more vigorously until they can break free from their fixed positions, leading to a more disordered liquid state. In vaporization, additional energy allows the particles to overcome the intermolecular forces within the liquid, dispersing into a significantly more disordered gaseous state. Similarly, sublimation involves an endothermic reaction where a solid changes directly into a gas.

The entropy of a substance increases during these transitions because the particles gain freedom of movement and the system becomes more disordered. These changes in entropy are important for understanding not only the physical properties of materials but also the energetic aspects of chemical reactions and the feasibility of processes in thermodynamics.