Question

A Carnot engine operates between temperatures of $ 600 \, \text{K} $ and $ 300 \, \text{K} $. If it absorbs $ 900 \, \text{J} $ of heat from the source, how much work does it perform?

• A. 300 J
• B. 450 J
• C. 600 J
• D. 150 J

Hint:

Carnot engine is the most efficient heat engine. Use: \[ \eta = 1 - \frac{T_C}{T_H}, \quad W = \eta Q_H \] where temperatures are in kelvin.

Answer 1

The Correct Option is A

A Carnot engine operates between two temperatures: the hot reservoir at \( T_H = 600 \, \text{K} \) and the cold reservoir at \( T_C = 300 \, \text{K} \). It absorbs \( Q_H = 900 \, \text{J} \) of heat from the hot reservoir. We need to find the work \( W \) performed by the engine.

The efficiency \( \eta \) of a Carnot engine is given by the formula: 

\(\eta = 1 - \frac{T_C}{T_H}\)

Substitute the given temperatures into the efficiency equation:

\(\eta = 1 - \frac{300}{600} = 1 - 0.5 = 0.5\)

The efficiency can also be expressed in terms of work done and heat absorbed as:

\(\eta = \frac{W}{Q_H}\)

Rearranging for work \( W \), we get:

\(W = \eta \times Q_H\)

Substitute the known values:

\(W = 0.5 \times 900 \, \text{J} = 450 \, \text{J}\)

Therefore, the work performed by the Carnot engine is 450 J, which corresponds to the second option noted earlier.

Answer 2

Efficiency of a Carnot engine is given by: \[ \eta = 1 - \frac{T_C}{T_H} \] Where: - \( T_H = 600 \, \text{K} \) - \( T_C = 300 \, \text{K} \) \[ \eta = 1 - \frac{300}{600} = 1 - 0.5 = 0.5 \] Work done \( W \) by the engine is: \[ W = \eta \cdot Q_H = 0.5 \cdot 900 = \boxed{450 \, \text{J}} \] Correction! Answer is not matching Option A, but actually: \[ \boxed{450 \, \text{J}} \Rightarrow \text{Option B} \]