Question

A heat engine is supplied with 250 kJ/s of heat at a constant fixed temperature of ${{227}^{\circ }}C$; the heat is rejected at ${{27}^{\circ }}C$, the cycle is reversible, then what amount of heat is rejected?
a) 24 kJ/s
b) 223 kJ/s
c) 150 kJ/s
D) None of the above

Answer 1

Hint: We have the temperature of sink as:${{T}_{1}}={{270}^{\circ }}C$ and temperature of source as: ${{T}_{2}}={{27}^{\circ }}C$. Also, heat supplied is: ${{Q}_{1}}=250\text{ }kJ{{s}^{-1}}$. So, by using the formula of efficiency of the heat engine, find the heat rejected: ${{Q}_{2}}$.

Formula used:
$\eta =1-\dfrac{{{T}_{1}}}{{{T}_{2}}}=1-\dfrac{{{Q}_{1}}}{{{Q}_{2}}}$, where $\eta $ is the efficiency of heat engine, ${{T}_{1}}$ is the initial temperature, ${{T}_{2}}$ is the final temperature, ${{Q}_{1}}$ is the heat supplied and ${{Q}_{2}}$ is the heat rejected.

Complete step by step answer:
We have:
$\begin{align}
  & {{T}_{1}}={{227}^{\circ }}C \\
 & {{T}_{2}}={{27}^{\circ }}C \\
 & {{Q}_{1}}=250\text{ }kJ{{s}^{-1}} \\
\end{align}$
Now, convert all the temperatures from degree Celsius to degree Kelvin.
We have:
$\begin{align}
  & {{T}_{1}}=227+273 \\
 & =500K
\end{align}$
$\begin{align}
  & {{T}_{2}}=27+273 \\
 & =300K
\end{align}$
Now, by using the formula for efficiency of heat engine, we have:
$\begin{align}
  & \eta =1-\dfrac{{{T}_{2}}}{{{T}_{1}}} \\
 & =1-\dfrac{500}{300} \\
 & =\dfrac{300-500}{300} \\
 & =-\dfrac{200}{300} \\
 & =-\dfrac{2}{3}......(1)
\end{align}$
Also, we know that, the efficiency of heat engine is equal to:
$\eta =1-\dfrac{{{Q}_{1}}}{{{Q}_{2}}}$
Now, by substituting the value of efficiency of the heat engine from equation (1), find the value of heat rejected.
We get:
 $\begin{align}
  & \Rightarrow -\dfrac{2}{3}=1-\dfrac{250}{{{Q}_{2}}} \\
 & \Rightarrow \dfrac{250}{{{Q}_{2}}}=1+\dfrac{2}{3} \\
 & \Rightarrow \dfrac{250}{{{Q}_{2}}}=\dfrac{5}{3} \\
 & \Rightarrow {{Q}_{2}}=\dfrac{250\times 3}{5} \\
 & \Rightarrow {{Q}_{2}}=150kJ{{s}^{-1}} \\
\end{align}$
So, the amount of heat rejected by the heat engine is $150kJ{{s}^{-1}}$ .

So, the correct answer is “Option C”.

Additional Information:
A heat engine is a device that converts heat to work. It takes heat from a reservoir then does some work like moving a piston, lifting weight etc and finally discharges some heat energy into the sink.

Note:
Remember to convert both the given temperatures in Kelvin scale first. The temperature can be converted by adding 273 to each given value of temperature.