Question

Predict the sign of \(\Delta S\) for each process: (a) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)(350 \mathrm{~K}\) and 500 torr \() \longrightarrow\) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)(350 \mathrm{~K}\) and 250 torr \()\) (b) \(\mathrm{N}_{2}(g)(298 \mathrm{~K}\) and \(1 \mathrm{~atm}) \longrightarrow \mathrm{N}_{2}(a q)(298 \mathrm{~K}\) and \(1 \mathrm{~atm})\) (c) \(\mathrm{O}_{2}(a q)(303 \mathrm{~K}\) and \(1 \mathrm{~atm}) \longrightarrow \mathrm{O}_{2}(g)(303 \mathrm{~K}\) and \(1 \mathrm{~atm})\)

Short answer

(a) Positive, (b) Negative, (c) Positive.

Step-by-step solution

Process (a) Entropy Change

Analyze the process where \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)\) (ethanol vapor) at 350 K and 500 torr goes to ethanol vapor at 350 K and 250 torr. Since the temperature remains constant and the pressure decreases, the gas will expand. Expansion of a gas increases randomness or disorder, hence the entropy \(\Delta S\) will be positive.

Process (b) Entropy Change

Consider the process where \(\mathrm{N}_{2}(g)\) (nitrogen gas) at 298 K and 1 atm transitions to \(\mathrm{N}_{2}(a q)\) (aqueous nitrogen) at the same temperature and pressure. Dissolving a gas into a solvent generally decreases the entropy because the gas molecules are more constrained in the liquid phase than in the gas phase. Therefore, \(\Delta S\) will be negative for this process.

Process (c) Entropy Change

Examine the process where \(\mathrm{O}_{2}(a q)\) (aqueous oxygen) at 303 K and 1 atm goes to \(\mathrm{O}_{2}(g)\) (oxygen gas) at the same temperature and pressure. When a gas is released from an aqueous solution, the molecules go from a more ordered state to a less ordered state. Thus, the entropy \(\Delta S\) will be positive.

Key concepts

Entropy

Entropy is a measure of the randomness or disorder of a system. It is a fundamental concept in thermodynamics used to predict the spontaneity of chemical processes. When predicting the sign of \(\text{ΔS}\) for a chemical reaction, remember:

• An increase in the number of gas molecules usually increases entropy.
• Phase transitions from solid to liquid to gas increase entropy.
• Dissolving a solid or liquid into solution usually increases entropy.
• Dissolving a gas into a liquid usually decreases entropy.

For our examples: In process (a), the expansion of ethanol vapor increases entropy because gas molecules occupy a larger volume, increasing randomness. Hence, \(\text{ΔS} > 0\).
In process (b), dissolving nitrogen gas in water decreases entropy as the gas molecules become more ordered in the liquid phase. Therefore, \(\text{ΔS} < 0\).
In process (c), releasing oxygen gas from an aqueous solution increases entropy, as the gas phase is more disordered compared to the aqueous phase. Thus, \(\text{ΔS} > 0\).

Thermodynamics

Thermodynamics deals with energy changes and the equilibrium of systems. The second law of thermodynamics states that for any spontaneous process, the total entropy of a system and its surroundings always increases. This principle helps us determine whether a process will occur naturally:

• Spontaneous processes generally increase the entropy of the universe.
• Processes that increase the system's entropy are often favorable.

For example, in process (a), the gas expansion is spontaneous and increases system entropy. In contrast, process (b) is less favorable because dissolving nitrogen gas into water decreases system entropy. Process (c) is also spontaneous, as releasing gas molecules into the air increases the system's entropy.

Phase Transitions

Phase transitions involve changes between different states of matter (solid, liquid, gas). They are accompanied by changes in entropy:

• Solid to liquid and liquid to gas transitions increase entropy.
• Gas to liquid and liquid to solid transitions decrease entropy.

When analyzing entropy changes, examine the phase transition type:
In process (a), there is no phase transition (ethanol remains in gas form), but the expansion still increases entropy.
Process (b) features a phase-like transition from gas to liquid, reducing system entropy.
Process (c) involves a phase transition from aqueous (dissolved) form to gas, significantly increasing entropy as molecules move to a more disordered state.

Gas Expansion

Gas expansion occurs when gas molecules spread out to occupy a larger volume. This process always increases entropy due to the increase in disorder:

• Lowering pressure allows gas to expand, increasing entropy.
• At constant temperature, expanding gas means molecules have more accessible microstates.

In our specific exercise:
In process (a), decreasing pressure causes ethanol vapor to expand, resulting in greater randomness and higher entropy. This is a classic example of how gas expansion positively affects entropy values.

Dissolution of Gases

Dissolution involves dissolving a gas into a liquid, which generally decreases system entropy:

• Gas molecules free in the air are more disordered compared to when dissolved in a liquid.
• Dissolution confines gas molecules to a smaller volume, increasing order.

For instance, in process (b), \(\text{N}_2(g) \to \text{N}_2(aq)\), entropy decreases, indicating the gas molecules become more ordered in the aqueous state.
Conversely, in process (c), the dissolution process is reversed. When \(\text{O}_2(aq) \to \text{O}_2(g)\), entropy increases as the gas is released from the liquid, resulting in less order and greater randomness.