Question

Which of the following Carnot's engines have maximum efficiency?
A) Working between 300 K and 0 K
B) Working between 800 K and 400 K
C) Working between 100 K and 10 K
D) Working between 300 K and 200 K

Answer 1

Hint: The efficiency of a Carnot engine depends on the temperature of the source and the sink of the engine. The source has a higher temperature and provides heat to the engine while the sink has a lower temperature and removes heat from the system.

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

Complete step by step answer
As mentioned in the hint, the efficiency of the Carnot engine depends on the temperature of the source and the sink of the Carnot engine. In the different options, given to us, we’ll determine the efficiency of Carnot engines operating at these temperatures.
For option (A), $ \eta = 1 - \dfrac{0}{{300}} = 1 $
For option (B), $ \eta = 1 - \dfrac{{400}}{{800}} = 0.5 $
For option (C), $ \eta = 1 - \dfrac{{10}}{{100}} = 0.9 $
For option (D), $ \eta = 1 - \dfrac{{200}}{{300}} = 0.33 $
Hence the Carnot engine operating between 300 K and 0 K has the highest efficiency of 1. This implies that the engine will convert all the heat energy provided to it into doing work.
Hence the correct choice is option (A).

Note
The temperature in option (A) is a theoretical one since no object can have 0 Kelvin in this universe. Also, the derived efficiency is also theoretical. In reality, there will always be energy losses in the system. So, the maximum efficiency of a Carnot engine can only reach about $ 0.7 $ practically. The Carnot engine is the most efficient engine in converting heat energy to work done in a system but no practical Carnot engine can be $ 100\% $ efficient in reality.