Efficiency of a heat engine working between a given source and sink is 0.5 . Coefficient of performance of the refrigerator working between the same source and the sink will be

Options

A. 1
B. `0.5`
C. `1.5`
D. 2

Answer

To solve the problem, we need to find the Coefficient of Performance (COP) of a refrigerator working between the same source and sink as a heat engine with a given efficiency. Let's break it down step by step.

### Step 1: Understand the Given Information
We are given that the efficiency (η) of a heat engine working between a source (hot reservoir) and a sink (cold reservoir) is 0.5.

### Step 2: Use the Efficiency Formula
The efficiency of a heat engine is defined as:
\[ \eta = 1 - \frac{T_L}{T_H} \]
where \(T_H\) is the temperature of the hot reservoir and \(T_L\) is the temperature of the cold reservoir.

### Step 3: Set Up the Equation
From the efficiency given:
\[ 0.5 = 1 - \frac{T_L}{T_H} \]
Rearranging this gives:
\[ \frac{T_L}{T_H} = 1 - 0.5 = 0.5 \]

### Step 4: Express the Temperature Ratio
From the ratio \(\frac{T_L}{T_H} = 0.5\), we can express it as:
\[ T_L = 0.5 T_H \]

### Step 5: Find the Coefficient of Performance (COP) of the Refrigerator
The COP of a refrigerator is defined as:
\[ COP = \frac{T_L}{T_H - T_L} \]

### Step 6: Substitute the Values
Substituting \(T_L = 0.5 T_H\) into the COP formula:
\[ COP = \frac{0.5 T_H}{T_H - 0.5 T_H} \]
This simplifies to:
\[ COP = \frac{0.5 T_H}{0.5 T_H} = 1 \]

### Step 7: Conclusion
Thus, the Coefficient of Performance (COP) of the refrigerator is:
\[ COP = 1 \]

### Final Answer
The correct option is **1**.

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