If the work done by a Carnot engine working between two temperatures 600K & 300K is used as the work input in Carnot refrigerator, working between 200K & 400K, find the heat (in J) removed from the lower temperature by refrigerator? The heat supplied to engine in 500J

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Understand the Carnot Engine
The Carnot engine operates between two temperatures, \( T_1 = 600 \, K \) (hot reservoir) and \( T_2 = 300 \, K \) (cold reservoir). The heat supplied to the engine is given as \( Q_H = 500 \, J \).

### Step 2: Calculate the Efficiency of the Carnot Engine
The efficiency \( \eta \) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
Substituting the values:
\[ \eta = 1 - \frac{300}{600} = 1 - 0.5 = 0.5 \]

### Step 3: Calculate the Work Done by the Carnot Engine
The work done \( W \) by the engine can be calculated using the efficiency:
\[ W = \eta \cdot Q_H \]
Substituting the values:
\[ W = 0.5 \cdot 500 \, J = 250 \, J \]

### Step 4: Understand the Carnot Refrigerator
The Carnot refrigerator operates between two temperatures, \( T_C = 200 \, K \) (cold reservoir) and \( T_H = 400 \, K \). The work done on the refrigerator is the work output from the Carnot engine, which we calculated as \( W = 250 \, J \).

### Step 5: Calculate the Coefficient of Performance (COP) of the Carnot Refrigerator
The coefficient of performance \( COP \) of a Carnot refrigerator is given by:
\[ COP = \frac{T_C}{T_H - T_C} \]
Substituting the values:
\[ COP = \frac{200}{400 - 200} = \frac{200}{200} = 1 \]

### Step 6: Calculate the Heat Removed by the Refrigerator
The heat removed \( Q_C \) from the cold reservoir can be calculated using the formula:
\[ Q_C = COP \cdot W \]
Substituting the values:
\[ Q_C = 1 \cdot 250 \, J = 250 \, J \]

### Final Answer
The heat removed from the lower temperature by the refrigerator is \( 250 \, J \).

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