A carnot engine takes 6000 cal of heat from a reservoir at `627^(@)C` and gives it to a sink at `27^(@)C`. The work done by the engine is

Options

A. 2 kcal
B. 3 kcal
C. 4 kcal
D. 8 kcal

Answer

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

### Step 1: Convert temperatures from Celsius to Kelvin
The temperatures given are:
- T1 = 627°C
- 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 K (approximately 900 K)
- T2 = 27 + 273.15 = 300.15 K (approximately 300 K)

### Step 2: Identify the heat absorbed (Q1)
The heat absorbed from the hot reservoir is given as:
\[ Q1 = 6000 \text{ cal} \]

### Step 3: Use the formula for the efficiency of a Carnot engine
The efficiency (η) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T2}{T1} \]

Substituting the values:
\[ \eta = 1 - \frac{300}{900} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Calculate the work done (W)
The work done by the engine can be calculated using the formula:
\[ W = Q1 \times \eta \]

Substituting the values:
\[ W = 6000 \text{ cal} \times \frac{2}{3} \]

Calculating:
\[ W = 6000 \times \frac{2}{3} = 4000 \text{ cal} \]

### Step 5: Convert the work done to kilocalories
Since 1 kilocalorie (kcal) = 1000 calories, we convert:
\[ W = \frac{4000 \text{ cal}}{1000} = 4 \text{ kcal} \]

### Final Answer
The work done by the engine is **4 kilocalories**.

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