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 establish the relationship between the efficiency of a Carnot engine and the coefficient of performance (COP) of a heat pump operating between the same two temperatures.

### Step-by-Step Solution:

1. **Understanding the Efficiency of a Carnot Engine**:
The efficiency (η) of a Carnot engine operating between two temperatures \( T_H \) (hot reservoir) and \( T_C \) (cold reservoir) is given by the formula:
\[ \eta = 1 - \frac{T_C}{T_H} \]
where \( T_H \) is the absolute temperature of the hot reservoir and \( T_C \) is the absolute temperature of the cold reservoir.

2. **Understanding the Coefficient of Performance of a Heat Pump**:
The coefficient of performance (COP) of a heat pump is defined as the ratio of the heat delivered to the hot reservoir (\( Q_H \)) to the work input (\( W \)):
\[ \text{COP} = \frac{Q_H}{W} \]
For a Carnot heat pump, the COP can also be expressed in terms of the temperatures:
\[ \text{COP} = \frac{T_H}{T_H - T_C} \]

3. **Relating COP to Efficiency**:
From the efficiency formula, we can rearrange it to express \( T_H - T_C \):
\[ T_H - T_C = T_H(1 - \eta) \]
Substituting this into the COP formula gives:
\[ \text{COP} = \frac{T_H}{T_H - T_C} = \frac{T_H}{T_H(1 - \eta)} = \frac{1}{1 - \eta} \]

4. **Final Expression**:
Therefore, we conclude that the coefficient of performance of the heat pump is related to the efficiency of the Carnot engine by the following formula:
\[ \text{COP} = \frac{1}{\eta} \]

### Conclusion:
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:
\[ \text{COP} = \frac{1}{\eta} \]