Question

A Carnot’s engine, whose efficiency is forty percent, takes heat from a source maintained at a temperature of 500K. It is desired to have an engine of efficiency $60\%$. Then, the intake temperature for the same exhaust must be
A) the efficiency cannot be greater than 50%
B) $1200K$
C) $750K$
D) none.

Answer 1

Hint: Let us first find out the output temperature or the sink temperature as the intimal temperature of the source is given and the efficiency of the Carnot’s engine is given. Next, this temperature will become the output temperature of the source, when the efficiency is changed. New intake temperature can be now easily calculated.
Formula used:
$\eta =1-\dfrac{{{T}_{output}}}{{{T}_{input}}}$

Complete answer:
The input temperature, efficiency of the carnot's engine is given, therefore, the output temperature or the sink temperature will be,
\[\begin{align}
  & 0.4=1-\dfrac{{{T}_{output}}}{500} \\
 & \Rightarrow {{T}_{sink}}=300K \\
\end{align}\]
Now, for this sink temperature to be equal to the output temperature of the new carnot engine,
The input temperature must be calculated as,
$\begin{align}
  & 0.6=1-\dfrac{300}{{{T}_{input}}} \\
 & \Rightarrow {{T}_{source}}=750K \\
\end{align}$

So, the correct answer is “Option C”.

Additional Information:
The carnot heat engine is a theoretical engine that operates on the carnot cycle. This basic model was developed in the year 1824. Hey, every time a thermodynamic system exists in a particular state thermodynamic cycle occurs when a system is taken through a series of different states and finally returns to the initial state back again. In this process the system may perform work on surroundings which in turn result to act as a heat engine. Heat engine acts by transferring energy from a warm region to a cool region of space and during this process it converts some of that energy into mechanical work. The cycle can also be reversed. The system may be worked upon by an external force also, in that process, can transfer thermal energy from a cooler system to a warmer one which in turn acts as a refrigerator or heat pump rather than a heat engine.

Note:
In the above question, in the second case, the efficiency of the carnot engine has changed. Hey, we were asked to find out the initial temperature so that the final temperature will be equal to the temperature of the source when initial efficiency was considered. So, the final temperature of the source must be equal to the same sing temperature even if the efficiency is changed.