Question

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) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 10 \mathrm{mmHg})\)

Short answer

The order of increasing entropy (∆S) for the given processes is: (a) < (c) < (b).

Step-by-step solution

Evaluating Entropy Changes for Each Process

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.

Arranging the Processes in Order of Increasing Entropy

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)

Key concepts

Phase transitions

Phase transitions play a crucial role in understanding entropy changes in a system. During a phase transition, the physical state of a substance changes, which can result in significant changes in entropy. For example:

• When a solid turns into a liquid, the molecules become more disordered and move freely, causing an increase in entropy.
• In a transition from a liquid to a gas, as seen in process (a) with water, the molecules spread out further, leading to an even higher increase in entropy.
• Finally, transitioning from a solid directly to a gas, as in process (b) with carbon dioxide, causes the greatest increase in entropy because the substance bypasses the liquid phase entirely, allowing for maximum molecular disorder.

These transitions illustrate how entropy, a measure of disorder or randomness in a system, significantly increases when substances transition from more ordered phases like solids or liquids to less ordered ones like gases.

Pressure effects on entropy

Pressure is another important factor that influences entropy. Generally, a decrease in pressure increases entropy, and the underlying reason for this is straightforward:

At lower pressures, molecules have more space to move and spread out, which increases disorder, hence increasing entropy.

In process (b), carbon dioxide sublimates from solid to gas at a significantly reduced pressure of 10 mmHg from 1 atm, leading to a large increase in entropy.

Similarly, process (c) involves water transitioning from liquid to gas, with reducing pressure from 1 atm to 10 mmHg, contributing to a substantial entropy increase compared to process (a), which maintains constant pressure.

By understanding how decreasing pressure allows molecules to occupy a greater volume, we can comprehend how entropy surges, emphasizing the importance of pressure in thermodynamic transformations.

Entropy increase in gases

Gases inherently have higher entropy compared to solids and liquids due to their molecular motion and distribution. When substances reach the gaseous state:

• The molecules occupy the entire volume available to them, leading to high degrees of freedom and randomness.
• This is evident in processes (a) and (c), where water vaporizes into a gaseous state, increasing entropy noticeably.
• In process (b), the sublimation of solid carbon dioxide to carbon dioxide gas results in the highest entropy change, demonstrating the substantial difference in disorder when transitioning directly from a highly ordered solid to a less ordered gaseous state.

Understanding the natural tendency of gases to maximize entropy can help us appreciate the complexity and dynamics of phase changes, as gases intrinsically possess and contribute to a greater level of disorder within a thermodynamic system.