Question

The temperature of the sink of a Carnot's engine is 300 K and the efficiency of the engine is 0.25. If the temperature of the source of the engine is increased by 100 K, the efficiency of the engine increases by

• A. 0.50
• B. 0.25
• C. 0.15
• D. 0.40

Hint:

For Carnot engines, remember that the efficiency depends on the temperatures of the sink and the source, and the efficiency increases as the temperature of the source increases.

Answer 1

The Correct Option is C

The efficiency of a Carnot engine is given by: \[ \eta = 1 - \frac{T_{\text{sink}}}{T_{\text{source}}} \] where \( \eta \) is the efficiency, \( T_{\text{sink}} \) is the temperature of the sink, and \( T_{\text{source}} \) is the temperature of the source. Given: - \( T_{\text{sink}} = 300 \, \text{K} \) - Initial efficiency \( \eta = 0.25 \) - \( T_{\text{source}} \) can be calculated as: \[ 0.25 = 1 - \frac{300}{T_{\text{source}}} \] Solving for \( T_{\text{source}} \): \[ T_{\text{source}} = \frac{300}{0.75} = 400 \, \text{K} \] When the temperature of the source is increased by 100 K: \[ T_{\text{new source}} = 400 + 100 = 500 \, \text{K} \] Now, calculate the new efficiency \( \eta_{\text{new}} \): \[ \eta_{\text{new}} = 1 - \frac{300}{500} = 1 - 0.6 = 0.4 \] The increase in efficiency is: \[ \Delta \eta = 0.4 - 0.25 = 0.15 \] Thus, the correct answer is option (3), 0.15.

Answer 2

Step 1: Recall efficiency formula of Carnot engine
The efficiency \( \eta \) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_C}{T_H} \]
where \( T_C \) is the temperature of the sink and \( T_H \) is the temperature of the source.

Step 2: Given data
- \( T_C = 300 \, K \)
- Initial efficiency \( \eta_1 = 0.25 \)

Step 3: Find initial source temperature \( T_H \)
Using efficiency formula:
\[ 0.25 = 1 - \frac{300}{T_H} \implies \frac{300}{T_H} = 0.75 \implies T_H = \frac{300}{0.75} = 400 \, K \]

Step 4: New source temperature after increase
\[ T_H' = 400 + 100 = 500 \, K \]

Step 5: Calculate new efficiency \( \eta_2 \)
\[ \eta_2 = 1 - \frac{300}{500} = 1 - 0.6 = 0.4 \]

Step 6: Calculate increase in efficiency
\[ \Delta \eta = \eta_2 - \eta_1 = 0.4 - 0.25 = 0.15 \]

Step 7: Conclusion
The efficiency of the engine increases by 0.15.