In your own words, explain why (a) \(\Delta S^{\circ}\) is negative for a reaction in which the number of moles of gas decreases. (b) we take \(\Delta S^{\circ}\) to be independent of \(T\), even though entropy increases with \(T\). (c) a solid has lower entropy than its corresponding liquid.

Entropy (S) is a measure of the degree of disorder or randomness in a system. A higher entropy corresponds to a higher number of possible microstates (the arrangements of atoms or molecules in a system). When the number of moles of gas decreases in a reaction, the total number of molecules in the system also decreases, which subsequently reduces the number of possible microstates. A reduction in possible microstates leads to a decrease in entropy. Therefore, the standard entropy change (ΔS°) is negative for reactions with a decrease in moles of gas.

Although entropy increases with temperature, we often treat the standard entropy change (ΔS°) as independent of temperature when dealing with relatively small or narrow temperature ranges. This is because, in such cases, the variation in entropy with temperature is negligible for practical purposes. This simplification allows us to more easily analyze and predict the behavior of chemical reactions without having to account for temperature changes constantly. For larger temperature ranges or systems with significant temperature sensitivity, a more accurate analysis involving the temperature dependence of ΔS° might be required.

In a solid, the particles are arranged in a highly ordered, fixed structure, with a relatively low number of possible microstates. The particles are held together by strong forces, and their movement is restricted to vibrations around fixed positions. As a result, the degree of disorder or randomness is quite low in a solid. In contrast, in a liquid, the particles are more randomly arranged and have more freedom to move around. The forces binding the particles together are weaker than in a solid, allowing for a greater number of possible microstates. Therefore, the degree of disorder or randomness is higher in a liquid compared to a solid. Due to the higher number of microstates and greater disorder in a liquid, its entropy is higher than its corresponding solid.