Question

A Carnot engine operates between \( 500 \, \text{K} \) and \( 300 \, \text{K} \). If it absorbs \( 1000 \, \text{J} \) of heat from the source, the amount of work done by the engine is:

• A. \( 400 \, \text{J} \)
• B. \( 600 \, \text{J} \)
• C. \( 200 \, \text{J} \)
• D. \( 500 \, \text{J} \)

Hint:

For a Carnot engine, use the efficiency formula \( \eta = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \) to find the work done as \( W = \eta \times Q_{\text{in}} \).

Answer 1

The Correct Option is A

- The efficiency of a Carnot engine is given by: \[ \eta = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \] - Here, \( T_{\text{hot}} = 500 \, \text{K} \), \( T_{\text{cold}} = 300 \, \text{K} \): \[ \eta = 1 - \frac{300}{500} = 1 - \frac{3}{5} = \frac{2}{5} = 0.4 \] - Efficiency is also the ratio of work done to heat absorbed: \[ \eta = \frac{W}{Q_{\text{in}}} \] - Given \( Q_{\text{in}} = 1000 \, \text{J} \): \[ 0.4 = \frac{W}{1000} \implies W = 0.4 \times 1000 = 400 \, \text{J} \]