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: $Q_1=250kJ{s}^{-1}$. So, by using the formula of efficiency of the heat engine, find the heat rejected: $Q_2$.

Formula used:
$\eta =1-{\displaystyle \frac{T_1}{T_2}}=1-{\displaystyle \frac{Q_1}{Q_2}}$, 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:
$\begin{array}{rl}& T_1={227}^{\circ }C\\ & T_2={27}^{\circ }C\\ & Q_1=250kJ{s}^{-1}\end{array}$
Now, convert all the temperatures from degree Celsius to degree Kelvin.
We have:
$\begin{array}{rl}& T_1=227+273\\ & =500K\end{array}$
$\begin{array}{rl}& T_2=27+273\\ & =300K\end{array}$
Now, by using the formula for efficiency of heat engine, we have:
$\begin{array}{rl}& \eta =1-{\displaystyle \frac{T_2}{T_1}}\\ & =1-{\displaystyle \frac{500}{300}}\\ & ={\displaystyle \frac{300-500}{300}}\\ & =-{\displaystyle \frac{200}{300}}\\ & =-{\displaystyle \frac{2}{3}}......(1)\end{array}$
Also, we know that, the efficiency of heat engine is equal to:
$\eta =1-{\displaystyle \frac{Q_1}{Q_2}}$
Now, by substituting the value of efficiency of the heat engine from equation (1), find the value of heat rejected.
We get:
 $\begin{array}{rl}& \Rightarrow -{\displaystyle \frac{2}{3}}=1-{\displaystyle \frac{250}{Q_2}}\\ & \Rightarrow {\displaystyle \frac{250}{Q_2}}=1+{\displaystyle \frac{2}{3}}\\ & \Rightarrow {\displaystyle \frac{250}{Q_2}}={\displaystyle \frac{5}{3}}\\ & \Rightarrow Q_2={\displaystyle \frac{250\times 3}{5}}\\ & \Rightarrow Q_2=150kJ{s}^{-1}\end{array}$
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.