Question

(a) In a chemical reaction, two gases combine to form a solid. What do you expect for the sign of \(\Delta S ?\) (b) How does the entropy of the system change in the processes described in Exercise \(19.12 ?\)

Short answer

In part (a), when two gases combine to form a solid, the system becomes more ordered, resulting in a decrease in entropy. Therefore, \(\Delta S < 0\). In part (b), a gas is isothermally expanded to twice its initial volume and then compressed back to its original volume. The net entropy change for this process is expected to be zero as the increase in entropy during expansion is compensated by an equal decrease during compression: \(\Delta S_{total} = 0\).

Step-by-step solution

(Part a: Entropy change when gases form a solid)

(Begin by thinking about the degrees of disorder in both the initial and final states of the system. Gases generally have a high degree of disorder, whereas solids have a more ordered arrangement of particles. When two gases combine to form a solid, the system becomes more ordered. With this knowledge, we can determine that the entropy change \(\Delta S\) should be negative, as entropy decreases when going from a more disordered state (gases) to a more ordered state (solid). So the answer for part (a) is: \(\Delta S < 0\).)

(Part b: Entropy change for processes in Exercise 19.12)

(Exercise 19.12 states that a gas is isothermally expanded to twice its initial volume and then compressed to its original volume. First, let's analyze the entropy change for each step of the process. In the isothermal expansion, the gas expands, leading to an increase in disorder, so \(\Delta S_1 > 0\). In the compression, the gas is compressed back to its original volume, which means the system has become more ordered, and thus \(\Delta S_2 < 0\). To determine the total change in the system's entropy, we have to add both changes: \(\Delta S_{total} = \Delta S_1 + \Delta S_2\). We know that the expansion and compression processes are opposite, so the net entropy change is expected to be zero since any increase in entropy during expansion should be compensated by an equal decrease during compression: \(\Delta S_{total} = 0\).)

Key concepts

Chemical Reaction

Chemical reactions involve the transformation of one or more substances into new substances. In terms of entropy, a chemical reaction can result in either an increase or decrease in the system's entropy, depending primarily on the states of the reactants and products. When we focus on a reaction where gases combine to form a solid, we observe a noticeable decrease in entropy.

• Gases possess high entropy due to their high disorder and the freedom of particles to move randomly.
• Solids have low entropy because they are more structured, with particles in fixed positions.

During such a reaction, the transition from a free-moving gaseous state to a more ordered solid state leads to a negative entropy change, indicated by \(\Delta S < 0\).
Entropy reduction is typical for reactions where disorder decreases, such as the formation of a solid from gaseous reactants.

Isothermal Process

An isothermal process is a process in which the temperature of the system remains constant. Isothermal conditions are common in many physical and chemical scenarios, including certain phases of gas expansion and compression.In the case of an isothermal gas expansion, the process involves the increase in volume of a gas at a constant temperature.

• Expansion increases the randomness and distribution of gas molecules, leading to a rise in entropy (\(\Delta S > 0\)
• More space for gas molecules results in more possible arrangements, hence more disorder.

In contrast, if the gas is isothermally compressed back to its original volume:

• The disorder decreases as gas molecules are confined to a smaller volume.
• This leads to a decrease in entropy indicated by \(\Delta S < 0\).

While individually these processes have significant entropy changes, together they balance each other out. The overall net change in entropy for a complete isothermal expansion followed by compression is typically zero \(\Delta S_{total} = 0\).
This shows how reversible isothermal processes can balance entropy changes.

Gaseous Expansion and Compression

Gaseous expansion and compression are processes where the volume occupied by a gas changes, affecting its entropy. These processes can be better understood with the principles of thermodynamics and kinetic molecular theory.**Expansion**During gaseous expansion, the gas molecules spread out to fill a larger volume.

• This results in a state of higher entropy due to the greater disorder among the molecules (\(\Delta S > 0\)).
• The molecules are more randomly distributed across a larger area.

**Compression**When a gas is compressed, its molecules are forced into a smaller volume.

• The decrease in space restricts molecular movement leading to less disorder (\(\Delta S < 0\)).
• Here, the gase's entropy decreases as the molecules occupy a denser, more organized state.

In the context of an isothermal process, such expansion followed by compression renders a unique perspective:
The net change in entropy across an expansion-compression cycle is zero (\(\Delta S_{total} = 0\)), highlighting the delicate balance achieved during reversible thermodynamic processes.