If the efficiency of a carnot engine is `eta`,then the coefficient of performance of a heat pump working between the same temperatures will be

Options

A. `1- eta`
B. `(1-eta)/eta`
C. `1/eta`
D. `1+1/eta`

Answer

To solve the problem, we need to understand the relationship between the efficiency of a Carnot engine and the coefficient of performance (COP) of a heat pump operating between the same temperatures.

### Step-by-Step Solution:

1. **Define Efficiency of a Carnot Engine**:
The efficiency (η) of a Carnot engine is defined as:
\[ \eta = \frac{W}{Q_H} \]
where:
- \( W \) is the work done by the engine.
- \( Q_H \) is the heat absorbed from the hot reservoir.

2. **Relate Work Done to Heat Released**:
In a Carnot engine, the heat released to the cold reservoir (QL) can be expressed as:
\[ Q_L = Q_H - W \]
This indicates that the work done by the engine is the difference between the heat absorbed from the hot reservoir and the heat released to the cold reservoir.

3. **Define Coefficient of Performance (COP) of a Heat Pump**:
The coefficient of performance (COP) of a heat pump is defined as:
\[ COP = \frac{Q_H}{W} \]
where:
- \( Q_H \) is the heat delivered to the hot reservoir.
- \( W \) is the work input to the heat pump.

4. **Express Work in Terms of Heat**:
From the efficiency equation, we can rearrange it to express work (W):
\[ W = Q_H - Q_L \]
Since the heat pump is operating between the same temperatures, it will absorb the same amount of heat \( Q_H \) as the Carnot engine.

5. **Relate COP to Efficiency**:
Substituting the expression for work into the COP formula gives:
\[ COP = \frac{Q_H}{Q_H - Q_L} \]
Since \( Q_L = Q_H - W \) and we know the efficiency \( \eta \) relates to these quantities, we can express \( Q_L \) in terms of \( \eta \):
\[ Q_L = Q_H(1 - \eta) \]
Therefore, we can substitute this back into the COP equation:
\[ COP = \frac{Q_H}{Q_H - Q_H(1 - \eta)} = \frac{Q_H}{Q_H \eta} = \frac{1}{\eta} \]

6. **Final Result**:
Thus, we find that the coefficient of performance (COP) of the heat pump is:
\[ COP = \frac{1}{\eta} \]

### Conclusion:
The coefficient of performance of a heat pump working between the same temperatures as a Carnot engine with efficiency \( \eta \) is given by:
\[ COP = \frac{1}{\eta} \]