A carnot engine takes in 3000 kcal 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. `4.2 xx 10^(6)` J
B. `8.4 xx 10^(6)` J
C. `16.8 xx 10^(6)` J
D. zero

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the temperatures from Celsius to Kelvin
The temperatures need to be converted to Kelvin because the efficiency formula for a Carnot engine requires absolute temperatures.

- **T1 (Temperature of the reservoir)**:
\[ T1 = 627°C + 273 = 900 \, K \]

- **T2 (Temperature of the sink)**:
\[ T2 = 27°C + 273 = 300 \, K \]

### Step 2: Calculate the efficiency of the Carnot engine
The efficiency (η) of a Carnot engine is given by the formula:
\[ η = 1 - \frac{T2}{T1} \]
Substituting the values we found:
\[ η = 1 - \frac{300}{900} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 3: Calculate the work done by the engine
The work done (W) by the engine can be calculated using the formula:
\[ W = η \times Q1 \]
Where \(Q1\) is the heat absorbed from the reservoir. Given \(Q1 = 3000 \, kcal\):
\[ W = \frac{2}{3} \times 3000 \, kcal = 2000 \, kcal \]

### Step 4: Convert the work done from kcal to joules
To convert kcal to joules, we use the conversion factor \(1 \, kcal = 4184 \, J\):
\[ W = 2000 \, kcal \times 4184 \, J/kcal = 8,368,000 \, J = 8.368 \times 10^6 \, J \]

### Final Answer
The work done by the engine is approximately:
\[ W \approx 8.36 \times 10^6 \, J \]

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