A carnot engine takes 900 Kcal of heat from a reservoir at `723^(@)C ` and exhausts it to a sink at `30^(0)C` the work done by the engine is

Options

A. `2.73xx10^(6) cal`
B. `3.73xx10^(6) cal`
C. `6.27xx10^(5) cal`
D. `3.73xx10^(5) cal`

Explanation

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

### Step 1: Convert temperatures from Celsius to Kelvin
The temperatures given are:
- Reservoir temperature \( T_1 = 723^\circ C \)
- Sink temperature \( T_2 = 30^\circ C \)

To convert these temperatures to Kelvin:
\[ T_1 = 723 + 273 = 996 \, K \]
\[ T_2 = 30 + 273 = 303 \, K \]

### Step 2: Convert heat absorbed from kilocalories to calories
The heat absorbed by the engine from the reservoir is given as:
\[ Q_1 = 900 \, \text{kcal} \]
To convert kilocalories to calories:
\[ Q_1 = 900 \times 1000 = 9 \times 10^5 \, \text{calories} \]

### Step 3: Use the Carnot efficiency formula to find \( Q_2 \)
The efficiency \( \eta \) of a Carnot engine can be expressed as:
\[ \eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_2}{T_1} \]
From this, we can find \( Q_2 \):
\[ \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \]
Thus,
\[ Q_2 = Q_1 \cdot \frac{T_2}{T_1} \]
Substituting the values:
\[ Q_2 = 9 \times 10^5 \cdot \frac{303}{996} \]
Calculating \( Q_2 \):
\[ Q_2 = 9 \times 10^5 \cdot 0.304 = 2.736 \times 10^5 \, \text{calories} \]

### Step 4: Calculate the work done by the engine
The work done \( W \) by the engine is given by:
\[ W = Q_1 - Q_2 \]
Substituting the values we found:
\[ W = 9 \times 10^5 - 2.736 \times 10^5 \]
Calculating \( W \):
\[ W = (9 - 2.736) \times 10^5 = 6.264 \times 10^5 \, \text{calories} \]

### Final Answer
The work done by the engine is:
\[ W \approx 6.27 \times 10^5 \, \text{calories} \]

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