A Carnot engine takes `3xx10^6` 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. `4.2xx10^(6) J`
B. `8.4xx10^(6) J`
C. `16.8 xx10^(6) J`
D. zero

Explanation

To solve the problem, we need to determine the work done by a Carnot engine that takes a certain amount of heat from a hot reservoir and releases some heat to a cold sink. Here are the steps to find the work done:

### Step 1: Identify the given values
- Heat absorbed from the hot reservoir (Q1) = \(3 \times 10^6\) cal
- Temperature of the hot reservoir (T1) = \(627^\circ C\)
- Temperature of the cold sink (T2) = \(27^\circ C\)

### Step 2: Convert temperatures to Kelvin
To use the Carnot engine formula, we need to convert the temperatures from Celsius to Kelvin:
- \(T1 = 627 + 273 = 900 \, K\)
- \(T2 = 27 + 273 = 300 \, K\)

### Step 3: Convert heat from calories to joules
Since the heat is given in calories, we need to convert it to joules. The conversion factor is \(1 \, cal = 4.184 \, J\):
- \(Q1 = 3 \times 10^6 \, cal \times 4.184 \, J/cal = 12.552 \times 10^6 \, J\)

### Step 4: Use the Carnot theorem to find Q2
According to the Carnot theorem:
\[ \frac{Q1}{T1} = \frac{Q2}{T2} \]
From this, we can express \(Q2\):
\[ Q2 = \frac{T2}{T1} \times Q1 \]
Substituting the values:
\[ Q2 = \frac{300 \, K}{900 \, K} \times 12.552 \times 10^6 \, J = \frac{1}{3} \times 12.552 \times 10^6 \, J = 4.184 \times 10^6 \, J \]

### Step 5: Calculate the work done (W)
The work done by the engine is given by:
\[ W = Q1 - Q2 \]
Substituting the values we have:
\[ W = 12.552 \times 10^6 \, J - 4.184 \times 10^6 \, J = 8.368 \times 10^6 \, J \]

### Final Answer
The work done by the engine is approximately \(8.368 \times 10^6 \, J\).