Which substance in each of the following pairs has the higher molar entropy at \(298 \mathrm{~K}\) : (a) \(\mathrm{HBr}(\mathrm{g})\) or \(\mathrm{HF}(\mathrm{g})\); (b) \(\mathrm{NH}_{3}(\mathrm{~g})\) or \(\mathrm{Ne}(\mathrm{g})\); (c) \(\mathrm{I}_{2}(\mathrm{~s})\) or \(\mathrm{I}_{2}(\mathrm{l})\); (d) \(1.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})\) at \(1.00 \mathrm{~atm}\) or \(1.0 \mathrm{~mol} \mathrm{} \operatorname{Ar}(\mathrm{g})\) at \(2.00 \mathrm{~atm}\) ?

Thermodynamics is a branch of physics that deals with heat, work, and the forms of energy involved in the transformation processes. One of its fundamental concepts is entropy, which quantifies the disorder or randomness in a system. The second law of thermodynamics states that for any spontaneous process, the total entropy of a system and its surroundings will always increase.

In the context of the exercise, we consider molar entropy, which refers specifically to the entropy of one mole of a substance. We infer from the second law that higher molar entropy implies a more disordered state of the substance, and various factors such as atomic size, molecular complexity, and phase of matter influence a substance's molar entropy.

Gas laws are a set of laws that describe the behavior of gases and how they interact with changes in temperature, volume, and pressure. Boyle's Law, Charles's Law, and Avogadro's Law are the foundational principles that help predict how a gas will behave under certain conditions.

When discussing molar entropy in gases, as in the textbook exercise, gas laws help us understand why molar entropy decreases with increasing pressure, as seen in the comparison of argon gas at two different pressures. The increase in pressure leads to a decrease in volume, which reduces the number of available microstates, thus lowering the entropy according to gas laws.

States of matter are the distinct physical forms that different phases of matter take on. The most familiar states are solid, liquid, and gas. Each state is characterized by different properties such as structure, volume, and the movement of particles. For instance, solids have a fixed shape and volume with particles vibrating in fixed positions, while liquids have a fixed volume but no fixed shape, and gases have neither a fixed volume nor shape.

In our textbook problem, when comparing the molar entropy of iodine in solid and liquid states, it is explained that substances in the liquid state have higher entropy than in the solid state. This is because the particles in the liquid state are more disordered and have a greater number of microstates available for their molecules to occupy.

Microstates are the specific, detailed configurations that a system's molecules can adopt. The concept of microstates is pivotal in statistical mechanics, which is the underpinning of thermodynamic entropy calculations. The more microstates a system has, the higher its entropy, because there's a greater number of ways in which the system's particles can be arranged.

In relation to the exercise, the molar entropy differences between various substances can be attributed to the diversity in their microstates. For instance, a polyatomic molecule like NH3 has a more intricate configuration and hence more microstates compared to a monoatomic gas like Ne. This explains why NH3 would have a higher molar entropy. Understanding microstates provides a foundational insight into why certain substances have higher entropies at a given temperature.