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: $$\Delta \eta = \eta_2 - \eta_1 = 25$$ 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).