A Carnot engine takes `3xx10^6cal`. of heat from a reservoir at `627^@C`, and gives it to a sink at `27^@C`. The work done by the engine is

Explanation

To solve the problem of finding the work done by a Carnot engine, we will follow these steps:

### Step 1: Convert the temperatures from Celsius to Kelvin
The temperatures given are:
- Source temperature (T1) = 627°C
- Sink temperature (T2) = 27°C

To convert these to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]

Calculating:
- \( T1 = 627 + 273.15 = 900.15 \approx 900 \, K \)
- \( T2 = 27 + 273.15 = 300.15 \approx 300 \, K \)

### Step 2: Write the formula for the efficiency of a Carnot engine
The efficiency (η) of a Carnot engine operating between two temperatures is given by:
\[ \eta = 1 - \frac{T2}{T1} \]

### Step 3: Substitute the values of T1 and T2 into the efficiency formula
Substituting the values we found:
\[ \eta = 1 - \frac{300}{900} \]

### Step 4: Calculate the efficiency
Calculating the fraction:
\[ \frac{300}{900} = \frac{1}{3} \]
Thus,
\[ \eta = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate the work done by the engine
The work done (W) by the engine can be calculated using the formula:
\[ W = Q1 \cdot \eta \]
Where \( Q1 \) is the heat absorbed from the source, which is given as \( 3 \times 10^6 \, \text{cal} \).

Substituting the values:
\[ W = 3 \times 10^6 \cdot \frac{2}{3} \]

### Step 6: Simplify to find the work done
Calculating:
\[ W = 3 \times 10^6 \cdot \frac{2}{3} = 2 \times 10^6 \, \text{cal} \]

### Final Answer
The work done by the engine is \( 2 \times 10^6 \, \text{cal} \).

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