To determine the condition for a spontaneous reaction in an isolated system, we need to analyze the thermodynamic principles involved. Let's break down the solution step by step.
### Step 1: Understanding Isolated Systems
An isolated system is one where there is no exchange of energy or matter with the surroundings. This means that the total energy of the system remains constant.
**Hint:** Remember that in an isolated system, energy and matter cannot enter or leave the system.
### Step 2: Change in Internal Energy (ΔU)
Since there is no exchange of energy, the change in internal energy (ΔU) for a spontaneous reaction in an isolated system must be zero.
**Hint:** In an isolated system, the internal energy remains constant, hence ΔU = 0.
### Step 3: Change in Entropy (ΔS)
According to the second law of thermodynamics, the total entropy of the universe tends to increase. In an isolated system, the change in entropy (ΔS) can either be greater than 0 (indicating an increase in disorder) or equal to 0 (indicating a state of equilibrium).
**Hint:** Entropy can either increase or remain constant in an isolated system, but it cannot decrease.
### Step 4: Change in Gibbs Free Energy (ΔG)
Gibbs free energy (G) is a thermodynamic potential that can predict the spontaneity of a reaction. For a reaction to be spontaneous, the change in Gibbs free energy (ΔG) must be negative (ΔG < 0). This indicates that the reaction can proceed in the forward direction.
**Hint:** A negative ΔG indicates that the reaction is spontaneous.
### Step 5: Analyzing the Options
Now, let's analyze the options provided in the question:
1. **Option 1:** ΔU = 0, ΔS = 0, ΔG = 0
2. **Option 2:** ΔU < 0, ΔS > 0, ΔG < 0
3. **Option 3:** ΔU = 0, ΔS > 0, ΔG < 0
4. **Option 4:** ΔU < 0, ΔS < 0, ΔG < 0
From our analysis:
- **Option 1** is incorrect because ΔG cannot be 0 for spontaneity.
- **Option 2** is incorrect because ΔU cannot be less than 0 in an isolated system.
- **Option 4** is incorrect because ΔU cannot be less than 0 in an isolated system.
**Conclusion:** The only option that satisfies the conditions for a spontaneous reaction in an isolated system is **Option 3: ΔU = 0, ΔS > 0, ΔG < 0**.
**Final Answer:** Option 3 is correct.
