An ideal heat engine has an efficiency `eta`. The cofficient of performance of the engine when driven backward will be

Options

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

Explanation

To find the coefficient of performance (COP) of an ideal heat engine when it is driven backward, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Efficiency of the Heat Engine**:
The efficiency (η) of an ideal heat engine operating between two temperatures, the hot reservoir (TH) and the cold reservoir (TL), is given by:
\[ \eta = 1 - \frac{T_L}{T_H} \]
This is our Equation (1).

2. **Define the Coefficient of Performance (COP)**:
When the heat engine is driven backward (acting as a refrigerator), the COP is defined as:
\[ COP = \frac{Q_L}{W} \]
where \(Q_L\) is the heat extracted from the cold reservoir and \(W\) is the work input.

3. **Relate COP to Efficiency**:
The work done by the engine can be expressed in terms of the heat absorbed from the hot reservoir (Q_H) and the heat rejected to the cold reservoir (Q_L):
\[ W = Q_H - Q_L \]
When the engine operates backward, we can express the COP as:
\[ COP = \frac{Q_L}{Q_H - Q_L} \]

4. **Substituting for Q_H**:
From the efficiency equation, we can express \(Q_H\) in terms of \(Q_L\):
\[ Q_H = \frac{Q_L}{1 - \eta} \]
Substituting this into the COP equation gives:
\[ COP = \frac{Q_L}{\frac{Q_L}{1 - \eta} - Q_L} \]

5. **Simplifying the COP Expression**:
Simplifying the expression:
\[ COP = \frac{Q_L}{\frac{Q_L - Q_L(1 - \eta)}{1 - \eta}} = \frac{Q_L(1 - \eta)}{Q_L \eta} = \frac{1 - \eta}{\eta} \]

6. **Final Expression for COP**:
Thus, the coefficient of performance when the heat engine is driven backward is:
\[ COP = \frac{1 - \eta}{\eta} \]

### Conclusion:
The coefficient of performance of the engine when driven backward is given by:
\[ COP = \frac{1 - \eta}{\eta} \]