Question

Ideal gases \(\mathrm{A}\) (red spheres) and \(\mathrm{B}\) (blue spheres) occupy two separate bulbs. The contents of both bulbs constitute the initial state of an isolated system. Consider the process that occurs when the stopcock is opened. (a) Sketch the final (equilibrium) state of the system. (b) What are the signs \((+,-\), or 0\()\) of \(\Delta H, \Delta S\), and \(\Delta G\) for this process? Explain. (c) How does this process illustrate the second law of thermodynamics? (d) Is the reverse process spontaneous or nonspontaneous? Explain.

Short answer

ΔH = 0, ΔS > 0, ΔG < 0; process increases entropy per the second law; reverse is nonspontaneous.

Step-by-step solution

Initial and Final State

Initially, ideal gases A (red spheres) and B (blue spheres) are in separate bulbs with a stopcock between them. Upon opening the stopcock, they mix until reaching equilibrium. In the final state, both types of spheres are uniformly distributed within the combined volume of the two bulbs. This is because gases tend to spread out to fill the available space.

Calculate ΔH (Enthalpy Change)

For ideal gases, the enthalpy change (ΔH) in an isothermal mixing process is zero because enthalpy is a function of temperature, which remains constant in this process. Thus, \(\Delta H = 0\).

Calculate ΔS (Entropy Change)

Entropy (ΔS) increases because the gases mix and their distribution across the volume becomes more random. The system moves from a state of lower entropy (before mixing) to a higher one (after mixing). Thus, \(\Delta S > 0\).

Calculate ΔG (Gibbs Free Energy Change)

The Gibbs free energy change (ΔG) is negative here because the process is spontaneous in an isolated system and is governed by \(\Delta G = \Delta H - T\Delta S\). Since \(\Delta H = 0\) and \(\Delta S > 0\), the Gibbs free energy decreases, \(\Delta G < 0\).

Second Law of Thermodynamics

This process illustrates the second law of thermodynamics, which states that the entropy of an isolated system always increases or remains constant. Here, the mixing leads to an increase in entropy, aligning with the second law.

Spontaneity of Reverse Process

The reverse process, where gases spontaneously separate into different bulbs without external work, is nonspontaneous. In isolated systems, processes that decrease entropy are not spontaneous under the second law.

Key concepts

Second Law of Thermodynamics

The second law of thermodynamics is a cornerstone of our understanding of natural processes. It states that in an isolated system, the total entropy can only stay the same or increase—never decrease. Entropy here refers to the degree of disorder or randomness in the system.
When the stopcock between the two gases is opened, the gases mix and the entropy increases since the molecules are more spread out and randomly distributed than before. - The system cannot spontaneously return to its previous state of lower entropy without external work being done. This unidirectional nature of entropy increase explains why certain processes are irreversible. The mixing of the gases perfectly demonstrates this principle, as it naturally proceeds towards greater entropy according to the second law.

Enthalpy Change

Enthalpy (\( H \)) is a measure of the total heat content of a system. It accounts for internal energy plus the product of pressure and volume. For ideal gases undergoing a mixing process at a constant temperature, such as in our scenario, the enthalpy change (\( \Delta H \) = 0) is zero.- This is because enthalpy is primarily dependent on the temperature, which does not change during the mixing of ideal gases in an isolated environment.Thus, no heat is absorbed or released during this mix, keeping \( \Delta H \) at zero. This characteristic of being constant is common in isothermal processes for ideal gases.

Entropy Change

Entropy (\( S \)) is a key concept representing disorder or randomness. When the gases in the initial exercise mix, the entropy change (\( \Delta S \)) is positive. Before the stopcock is opened, the red and blue spheres are ordered in separate bulbs. Upon mixing, their distribution becomes more uniform across the combined volume. **Understanding \( \Delta S \):**- Increases when a system becomes more disordered- Tends to be positive in spontaneous mixing processes
The rise in entropy is due to the increased number of possible microstates or configurations the gases can now adopt as they spread out, aligning with the second law of thermodynamics.

Gibbs Free Energy Change

Gibbs free energy (\( G \)) combines enthalpy, temperature, and entropy to measure the maximum usable work from a system at constant temperature and pressure. It is defined by the relation: \[ \Delta G = \Delta H - T \Delta S \]In our scenario, \( \Delta H \) is zero and \( \Delta S \) is positive, leading to a negative \( \Delta G \). - This negative value indicates that the process is spontaneous, meaning it will proceed without input from external work.**Key Points of \( \Delta G \):**- Negative \( \Delta G \): spontaneous process- Depends on signs and magnitudes of \( \Delta H \) and \( \Delta S \)Understanding \( \Delta G \) helps predict not only if a reaction will happen on its own, but also allows us to quantify how much work could possibly be extracted.