Question

If the ratio of the absolute temperature of the sink and source of a Carnot engine is changed from 2:3 to 3:4, the efficiency of the engine changes by:

• A. \( 25\% \)
• B. \( 40\% \)
• C. \( 50\% \)
• D. \( 15\% \)

Hint:

Carnot efficiency depends on the temperature difference between the hot and cold reservoirs. A smaller temperature difference reduces efficiency.

Answer 1

The Correct Option is A

To determine how the efficiency of a Carnot engine changes when the ratio of the absolute temperatures of the sink and source changes, we follow these steps: Given: 
- Initial temperature ratio: \( \frac{T_{\text{sink}}}{T_{\text{source}}} = \frac{2}{3} \) 
- New temperature ratio: \( \frac{T_{\text{sink}}}{T_{\text{source}}} = \frac{3}{4} \) 

Step 1: Calculate the Initial Efficiency The efficiency \( \eta \) of a Carnot engine is given by: \[ \eta = 1 - \frac{T_{\text{sink}}}{T_{\text{source}}} \] For the initial ratio: \[ \eta_1 = 1 - \frac{2}{3} = \frac{1}{3} \approx 33.33\% \] 

Step 2: Calculate the New Efficiency For the new ratio: \[ \eta_2 = 1 - \frac{3}{4} = \frac{1}{4} = 25\% \] 

Step 3: Determine the Change in Efficiency
The change in efficiency \( \Delta \eta \) is: \[ \Delta \eta = \eta_2 - \eta_1 = 25% - 33.33% = -8.33\% \] However, we are interested in the absolute change relative to the initial efficiency: \[ \text{Percentage Change} = \left| \frac{\Delta \eta}{\eta_1} \right| \times 100\% = \left| \frac{-8.33\%}{33.33\%} \right| \times 100\% \approx 25\% \] 

Final Answer: \[ \boxed{25\%} \] This corresponds to option (1).