A carnot engine has the same efficiency
(i) between `100K and 500K` and
(ii) between `TK`and `900K`. Caculate the temperature `T` of the sink.

Explanation

To solve the problem, we need to use the formula for the efficiency of a Carnot engine, which is given by:

\[ \eta = 1 - \frac{T_2}{T_1} \]

where \( T_1 \) is the temperature of the hot reservoir (source) and \( T_2 \) is the temperature of the cold reservoir (sink).

### Step 1: Calculate the efficiency of the first Carnot engine
The first Carnot engine operates between temperatures \( T_1 = 500 \, K \) and \( T_2 = 100 \, K \).

Using the efficiency formula:

\[ \eta_1 = 1 - \frac{T_2}{T_1} = 1 - \frac{100}{500} \]

Calculating this gives:

\[ \eta_1 = 1 - 0.2 = 0.8 \]

### Step 2: Set up the efficiency equation for the second Carnot engine
The second Carnot engine operates between temperatures \( T_1 = 900 \, K \) and \( T_2 = T \).

Since both engines have the same efficiency, we can write:

\[ \eta_2 = 1 - \frac{T}{900} \]

### Step 3: Equate the efficiencies
Since \( \eta_1 = \eta_2 \):

\[ 0.8 = 1 - \frac{T}{900} \]

### Step 4: Solve for \( T \)
Rearranging the equation gives:

\[ \frac{T}{900} = 1 - 0.8 \]

This simplifies to:

\[ \frac{T}{900} = 0.2 \]

Multiplying both sides by 900 gives:

\[ T = 900 \times 0.2 = 180 \, K \]

### Conclusion
The temperature \( T \) of the sink for the second Carnot engine is \( 180 \, K \).

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