A heat engine takes up heat at 100K and liberates heat at 200K. Which of the following in an impossible amount of heat rejected by the engine if 1000J of heat is absorbed:

Options

A. 60J
B. 80J
C. 90J
D. 15J

Answer

To solve the problem step by step, we need to apply the principles of thermodynamics, specifically the efficiency of a heat engine. Here's how we can approach the question:

### Step-by-Step Solution:

1. **Identify Given Data**:
- Heat absorbed (Q1) = 1000 J
- Temperature at which heat is absorbed (T1) = 1000 K
- Temperature at which heat is rejected (T2) = 200 K

2. **Calculate Efficiency of the Heat Engine**:
The efficiency (η) of a heat engine is given by the formula:
\[ \eta = 1 - \frac{T2}{T1} \]
Substituting the values:
\[ \eta = 1 - \frac{200}{1000} = 1 - 0.2 = 0.8 \]
This means the efficiency of the engine is 80%.

3. **Calculate Work Done by the Engine**:
The work done (W) by the engine can be calculated using the formula:
\[ W = \eta \times Q1 \]
Substituting the values:
\[ W = 0.8 \times 1000 J = 800 J \]

4. **Calculate Heat Rejected (Q2)**:
The heat rejected (Q2) can be calculated using the first law of thermodynamics:
\[ Q2 = Q1 - W \]
Substituting the values:
\[ Q2 = 1000 J - 800 J = 200 J \]

5. **Determine Impossible Amount of Heat Rejected**:
According to the problem, we need to identify which amount of heat rejected is impossible. Since the minimum heat that can be rejected is equal to the work done, any amount less than the calculated heat rejected (200 J) is impossible.

The options provided in the question include values such as:
- 15 J
- 20 J
- 50 J
- 100 J

Since the minimum heat that can be rejected is 200 J, any value less than this is impossible. Therefore, 15 J is the impossible amount of heat rejected.

### Final Answer:
The impossible amount of heat rejected by the engine is **15 J**.

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