Question

An ideal Carnot engine whose efficiency is $ 40\% $ receives heat at $ 500\,K $ . If the efficiency is to be $ 50\% $ then the temperature of the sink will be
A) 900 K
B) 600 K
C) 700 K
D) 800 K

Answer 1

Hint: The efficiency of the Carnot engine depends on the temperature of the source and the sink of the engine. The higher the difference in temperature of the source and the sink, the higher the efficiency of the engine will be:

Formula used: In this solution, we will use the following formula:
Efficiency of Carnot engine: $ \eta = 1 - \dfrac{{{T_1}}}{{{T_2}}} $ where $ {T_1} $ is the temperature of the sink and $ {T_2} $ is the temperature of the source

Complete step by step answer:
A Carnot engine that is operating between two temperatures has the maximum efficiency of converting the heat energy in the system to work done. Its efficiency depends on the temperature of the source and the sink.
We’ve been given an ideal Carnot engine whose efficiency is $ 40\% $ . And the temperature of the source is 500 K. Then we can determine the sink temperature from the formula
 $ \eta = 1 - \dfrac{{{T_1}}}{{{T_2}}} $
Substituting $ \eta = 40\% = \dfrac{{40}}{{100}} $ and $ {T_2} = 500K $ , we get
 $ \dfrac{{40}}{{100}} = 1 - \dfrac{{{T_1}}}{{500}} $
Which gives the temperature of the sink as
 $ {T_1} = 300\,K $
Now, we want the efficiency of $ 50\% $ for the same exhaust or sink temperature of $ {T_1} = 300\,K $ . Then we can write the equation for efficiency of the engine as
 $ \dfrac{{50}}{{100}} = 1 - \dfrac{{300}}{{{T_2}}} $
Solving for $ {T_2} $ , we can find the temperature of the source as
 $ {T_2} = 0.5 \times 300 = 600\,K $
Hence the temperature of the source will be $ T = 600K $ which corresponds to option (B).

Note:
We should be careful to not confuse between the two situations in which the Carnot engine is functioning. The only common factor between the two situations is the sink temperature but the efficiency of the engine in the two situations and the source temperature will be different.