Question

Carnot engine has an efficiency of \(\frac{1}{7}\). When the temperature of the sink is reduced by \(60\) K, its efficiency is doubled. The temperature of the source is

• A. \(47^{\circ }\text{C}\)
• B. \(147^{\circ }\text{C}\)
• C. \(43^{\circ }\text{C}\)
• D. \(143^{\circ }\text{C}\)

Hint:

Use \(\eta=1-\dfrac{T_2}{T_1}\) for both situations and subtract.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The efficiency of a Carnot engine is \(\eta=1-\dfrac{T_2}{T_1}\).

Step 2: Case 1
\(\dfrac{1}{7}=1-\dfrac{T_2}{T_1}\), so \(\dfrac{T_2}{T_1}=\dfrac{6}{7}\).

Step 3: Case 2
The sink is lowered by 60 K and the efficiency is \(\dfrac{2}{7}\): \(\dfrac{T_2-60}{T_1}=\dfrac{5}{7}\).

Step 4: Calculation
Subtract the two relations: \(\dfrac{60}{T_1}=\dfrac{1}{7}\), so \(T_1=420\) K.

Step 5: Result
\(T_1=420-273=147^{\circ}\text{C}\).

Final Answer:
The temperature of the source is 147 degrees Celsius. \[ \boxed{\text{(B) }147^{\circ}\text{C}} \]