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

Options

A. `0^(@)C `
B. `0K`
C. equal to its input temerpature
D. half of its input temperature

Answer

To determine the exhaust temperature at which a frictionless heat engine can be 100% efficient, we can follow these steps:

### Step 1: Understand the Concept of Efficiency
The efficiency of a heat engine is defined as the ratio of the work output to the heat input. For an ideal engine, the maximum efficiency is given by the Carnot efficiency.

### Step 2: Recall the Carnot Efficiency Formula
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
where:
- \(T_1\) is the temperature of the heat source (input temperature),
- \(T_2\) is the temperature of the heat sink (exhaust temperature).

### Step 3: Set the Efficiency to 100%
For the engine to be 100% efficient, we set:
\[ \eta = 1 \]
Substituting into the efficiency formula, we have:
\[ 1 = 1 - \frac{T_2}{T_1} \]

### Step 4: Solve for \(T_2\)
Rearranging the equation gives:
\[ 1 - 1 = -\frac{T_2}{T_1} \]
This simplifies to:
\[ 0 = -\frac{T_2}{T_1} \]
Thus, we can conclude:
\[ \frac{T_2}{T_1} = 0 \]
This implies that:
\[ T_2 = 0 \text{ K} \]

### Step 5: Conclusion
The exhaust temperature \(T_2\) must be 0 Kelvin for the heat engine to achieve 100% efficiency.

### Final Answer
The exhaust temperature must be **0 Kelvin**.

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