Question

Predict the sign of \(\Delta S\) for each process: (a) \(\mathrm{O}_{2}(g)(1.0 \mathrm{~L}\) at \(1 \mathrm{~atm}) \longrightarrow \mathrm{O}_{2}(g)(0.10 \mathrm{~L}\) at \(10 \mathrm{~atm})\) (b) \(\mathrm{Cu}(s)\left(350^{\circ} \mathrm{C}\right.\) and \(\left.2.5 \mathrm{~atm}\right) \longrightarrow \mathrm{Cu}(s)\left(450^{\circ} \mathrm{C}\right.\) and \(\left.2.5 \mathrm{~atm}\right)\) (c) \(\mathrm{Cl}_{2}(g)\left(100^{\circ} \mathrm{C}\right.\) and \(\left.1 \mathrm{~atm}\right) \longrightarrow \mathrm{Cl}_{2}(g)\left(10^{\circ} \mathrm{C}\right.\) and \(\left.1 \mathrm{~atm}\right)\)

Short answer

(a) \(\text{ΔS} < 0\), (b) \(\text{ΔS} > 0\), (c) \(\text{ΔS} < 0\)

Step-by-step solution

Analyze Process (a)

Evaluate the change in volume and pressure for \(\text{O}_2(g)\). When a gas is compressed from \(1.0 \text{ L} \text{ at } 1 \text{ atm}\) to \(0.1 \text{ L} \text{ at } 10 \text{ atm}\), the volume decreases, which leads to a decrease in entropy \(\text{ΔS} < 0\).

Analyze Process (b)

Consider the change in temperature for solid copper \(\text{Cu}(s)\). When the temperature increases from \(350^\text{circ} \text{C} \text{ to } 450^\text{circ} \text{C}\), the entropy increases \(\text{ΔS} > 0\) since higher temperature means more thermal energy and greater disorder.

Analyze Process (c)

Evaluate the change in temperature for \(\text{Cl}_2(g)\). When the temperature decreases from \(100^\text{circ} \text{C} \text{ to } 10^\text{circ} \text{C}\), the entropy decreases \(\text{ΔS} < 0\) because lower temperature results in less thermal energy and less disorder.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships and conversions between heat and other forms of energy. It provides a comprehensive framework to analyze energy changes in chemical processes. The most relevant laws of thermodynamics for our discussion are:

• The First Law: Energy cannot be created or destroyed, only converted from one form to another.
• The Second Law: In any energy exchange, the total entropy of a system and its surroundings always increases.

The concept of entropy is vital here as it helps us understand disorder in a system. Thermodynamic principles guide us in predicting whether entropy increases or decreases during chemical reactions and phase changes.

Entropy

Entropy is a measure of disorder or randomness in a system. Higher entropy indicates greater disorder. The Second Law of Thermodynamics states that the entropy of an isolated system always increases over time. Entropy plays a key role in determining the feasibility of chemical reactions and phase transitions.
Let's review how we predict the sign of \(\text{ΔS}\) – the change in entropy:

• Compression of Gases: When a gas is compressed, its volume decreases, leading to fewer possible arrangements of molecules, thus decreasing entropy (\(\text{ΔS} < 0\)).
• Increase in Temperature: Higher temperatures increase molecular motion and disorder, leading to increased entropy (\(\text{ΔS} > 0\)).
• Decrease in Temperature: Lower temperatures reduce molecular motion and disorder, decreasing entropy (\(\text{ΔS} < 0\)).

Understanding these trends helps us predict entropy changes in different chemical processes.

Phase Changes

Phase changes occur when a substance transitions from one state of matter (solid, liquid, gas) to another. Each phase has different levels of molecular order, affecting entropy:

• Solid to Liquid: Involves melting. Since liquids have more disorder than solids, entropy increases (\(\text{ΔS} > 0\)).
• Liquid to Gas: Involves boiling. Gases have much greater disorder than liquids, leading to a significant increase in entropy (\(\text{ΔS} > 0\)).
• Solid to Gas: Involves sublimation. This direct transition leads to a very large increase in entropy (\(\text{ΔS} > 0\)).
• Gas to Liquid: Involves condensation. Since liquids are more ordered than gases, entropy decreases (\(\text{ΔS} < 0\)).
• Liquid to Solid: Involves freezing. Solids are more ordered than liquids, leading to a decrease in entropy (\(\text{ΔS} < 0\)).

By understanding these phase transitions, we can predict the entropy changes in processes involving different states of matter.

Gas Laws

Gas laws describe the behavior of gases in relation to pressure, volume, and temperature. These laws provide insights into how entropy changes under different conditions:

• Boyle's Law: At constant temperature, the pressure and volume of a gas are inversely proportional (\(P_1V_1 = P_2V_2\)). Compression of a gas (decreasing volume) increases pressure, which reduces entropy due to restricted molecular freedom.
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature (\(\frac{V_1}{T_1} = \frac{V_2}{T_2}\)). Higher temperatures increase volume, thus increasing entropy.
• Gay-Lussac's Law: At constant volume, the pressure of a gas is directly proportional to its temperature (\(\frac{P_1}{T_1} = \frac{P_2}{T_2}\)). Increased temperature raises the entropy by increasing molecular activity.

By applying these gas laws, we can predict and explain the entropy changes in various chemical processes involving gases.