Arrange the entropy changes of the following processes, all at \(25^{\circ} \mathrm{C},\) in the expected order of increasing \(\Delta S,\) and explain your reasoning: (a) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 1 \mathrm{atm})\) (b) \(\mathrm{CO}_{2}(\mathrm{s}, 1 \mathrm{atm}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g}, 10 \mathrm{mm} \mathrm{Hg})\) (c) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 10 \mathrm{mmHg})\)

Firstly, examine each process and determine the phase change happening and the pressure change. For process (a), water is changing from a liquid state to a gaseous state at constant atmospheric pressure. This is an entropy increasing process as the system changes from a more ordered state (liquid) to a less ordered state (gas).For process (b), carbon dioxide is changing from a solid state to a gaseous state and the pressure is decreasing from 1 atm to 10 mmHg. Changing from solid to gas means a significant increase in entropy. Additionally, decrease in pressure also increases entropy. For process (c), water is changing from a liquid state to a gaseous state and the pressure decreases from 1 atm to 10 mmHg. Just like process (b), this process significantly increases entropy due to the phase change and the decrease in pressure.

Now arrange the processes according to the expected increase in entropy. Since process (a) only involves a change in phase at constant pressure, it would have the least increase in entropy. Process (c) involves the same phase change as process (a) but also includes a decrease in pressure, hence it would have a larger increase in entropy compared to process (a). Lastly, process (b) has the largest increase in entropy. This is not only because it involves a change from solid to gas (which is a more significant change compared to liquid to gas), but also because of the decrease in pressure like process (c). Therefore, the order of increasing ∆S would be: (a) < (c) < (b)