A Carnot engine takes `12.6xx10^6` J 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. `4.2xx10^6J`
B. `8.4 xx10^(6)J`
C. `16.8xx10^(6)J`
D. Zero

Answer

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

### Step 1: Convert Temperatures to Kelvin
The temperatures given in degrees Celsius need to be converted to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]

- For the hot reservoir:
\[ T_1 = 627 + 273 = 900 \, K \]

- For the cold reservoir:
\[ T_2 = 27 + 273 = 300 \, K \]

### Step 2: Calculate the Efficiency of the Carnot Engine
The efficiency (\( \eta \)) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]

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

### Step 3: Calculate the Work Done by the Engine
The work done (\( W \)) by the engine can be calculated using the efficiency and the heat absorbed from the hot reservoir (\( Q_1 \)):
\[ W = \eta \times Q_1 \]

Given that \( Q_1 = 12.6 \times 10^6 \, J \):
\[ W = \frac{2}{3} \times 12.6 \times 10^6 \]

Calculating this:
\[ W = \frac{2 \times 12.6 \times 10^6}{3} = \frac{25.2 \times 10^6}{3} = 8.4 \times 10^6 \, J \]

### Final Answer
The work done by the engine is:
\[ W = 8.4 \times 10^6 \, J \]

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