entropy of a system decreases

Options

A. When heat is supplied to a system at constant temperature
B. When heat is taken out from the system at constant temperature
C. At equilibrium
D. In any spontaneous process

Answer

To solve the problem of determining when the entropy of a system decreases, we can analyze the options provided step by step.

### Step 1: Understanding Entropy
Entropy (S) is a measure of the disorder or randomness of a system. It can be defined mathematically as:
\[ dS = \frac{dQ_{\text{rev}}}{T} \]
where \( dQ_{\text{rev}} \) is the reversible heat transfer and \( T \) is the absolute temperature.

### Step 2: Analyzing Option A
The first option states that when heat is supplied to a system at constant temperature, the entropy of the system decreases.

1. **Heat Supplied**: If heat is supplied to the system, \( dQ \) is positive.
2. **Effect on Entropy**: Since \( dQ > 0 \) and \( T > 0 \), it follows that:
\[ dS = \frac{dQ}{T} > 0 \]
This indicates that the entropy of the system increases, not decreases.

**Conclusion for Option A**: This option is incorrect.

### Step 3: Analyzing Option B
The second option states that when heat is taken out from the system at constant temperature, the entropy of the system decreases.

1. **Heat Removed**: If heat is taken out, \( dQ \) is negative.
2. **Effect on Entropy**: Since \( dQ < 0 \) and \( T > 0 \), we have:
\[ dS = \frac{dQ}{T} < 0 \]
This indicates that the entropy of the system decreases.

**Conclusion for Option B**: This option is correct.

### Step 4: Analyzing Option C
The third option discusses the equilibrium state of a system.

1. **Equilibrium**: In equilibrium, the system and surroundings have the same entropy.
2. **Effect on Entropy**: If the system is in equilibrium, there is no net change in entropy. Thus, we cannot say that the entropy of the system decreases or increases; it remains constant.

**Conclusion for Option C**: This option is incorrect.

### Step 5: Analyzing Option D
The fourth option relates to spontaneous processes.

1. **Spontaneous Process**: A spontaneous process is characterized by an increase in the total entropy of the system and surroundings.
2. **Effect on Entropy**: For a process to be spontaneous, \( dS_{\text{total}} > 0 \). This implies that the entropy of the system cannot decrease in a spontaneous process.

**Conclusion for Option D**: This option is incorrect.

### Final Conclusion
Among the given options, only **Option B** is correct: "When heat is taken out from the system at constant temperature, the entropy is going to decrease."

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