A fritionless heat engine can be 100% efficient only if its exhaust temperature is

Options

A. Equal to its input temperature
B. Less than its input temperature
C. 0 K
D. `0^@C`

Answer

To solve the question regarding the exhaust temperature of a frictionless heat engine that can be 100% efficient, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Efficiency of a Heat Engine**:
The efficiency (η) of a heat engine is defined as the ratio of the work output (W) to the heat input (Q1) from the hot reservoir. The formula for efficiency is given by:
\[ \eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1} \]
where \(Q_2\) is the heat rejected to the cold reservoir.

2. **Ideal Efficiency Condition**:
For a heat engine to be 100% efficient, all the heat from the hot reservoir must be converted into work, meaning there should be no heat rejected to the cold reservoir. This implies:
\[ Q_2 = 0 \]
Therefore, the efficiency becomes:
\[ \eta = 1 - \frac{Q_2}{Q_1} = 1 - 0 = 1 \quad \text{(or 100%)} \]

3. **Applying the Second Law of Thermodynamics**:
According to the second law of thermodynamics, it is impossible for any heat engine operating between two heat reservoirs to convert all the heat absorbed from the hot reservoir into work without rejecting some heat to the cold reservoir.

4. **Temperature Relation**:
The temperatures of the hot and cold reservoirs are denoted as \(T_1\) (hot) and \(T_2\) (cold). For a heat engine to achieve 100% efficiency, the cold reservoir must be at absolute zero, which is:
\[ T_2 = 0 \, \text{K} \]

5. **Conclusion**:
Therefore, a frictionless heat engine can only be 100% efficient if its exhaust temperature (the temperature of the cold reservoir) is:
\[ T_2 = 0 \, \text{K} \]

### Final Answer:
A frictionless heat engine can be 100% efficient only if its exhaust temperature is **0 Kelvin**.

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