Question

A Carnot’s engine operates with a source at 500K and sinks at 375K. The engine takes 600 Kcal of heat in one cycle, the heat rejected to sink per cycle is?
A. 250 KCal
B. 350 KCal
C. 450 KCal
D. 550 KCal

Answer 1

Hint: The net efficiency (N) of Carnot engine is given by $1 - \dfrac{{{T_{sink}}}}{{{T_{source}}}}$, where ${T_{sink}},{T_{source}}$ are the temperatures operated. Net efficiency is also defined as the ratio of work done and the heat input, $N = \dfrac{W}{{{Q_{in}}}}$. Substitute the values of obtained net efficiency, input heat to calculate the unknown work. And work is the difference of the heat rejected from the heat given, $W = {Q_{in}} - {Q_{out}}$. Substitute the values of input heat and obtained work to find the value of rejected heat.

Complete step by step answer:
We are given that a Carnot’s engine operates with a source at 500K and sinks at 375K and the engine takes 600 Kcal of heat in one cycle.
We have to calculate the heat rejected to sink per cycle.
Net efficiency of the Carnot engine is $1 - \dfrac{{{T_{sink}}}}{{{T_{source}}}}$
$
  N = 1 - \dfrac{{{T_{sink}}}}{{{T_{source}}}} \\
  {T_{sink}} = 375K,{T_{source}} = 500K \\
  N = 1 - \dfrac{{375}}{{500}} \\
  N = \dfrac{{125}}{{500}} = \dfrac{1}{4} \\
 $
Net efficiency is also defined as the ratio of work done and the heat input, $N = \dfrac{W}{{{Q_{in}}}}$
$
  N = \dfrac{W}{{{Q_{in}}}} \\
  N = \dfrac{1}{4},{Q_{in}} = 600KCal \\
  \Rightarrow \dfrac{1}{4} = \dfrac{W}{{600KCal}} \\
  \Rightarrow W = \dfrac{{600}}{4} \\
  \Rightarrow W = 150KCal \\
 $
The work done equals the net heat supplied.
$W = {Q_{in}} - {Q_{out}}$
Substitute the values of input heat supplied and the work done to calculate the heat rejected to the sink.
$
  W = {Q_{in}} - {Q_{out}} \\
  W = 150KCal \\
  {Q_{in}} = 600KCal \\
  \Rightarrow {Q_{out}} = {Q_{in}} - W \\
  \Rightarrow {Q_{out}} = 600 - 150 \\
  \Rightarrow {Q_{out}} = 450KCal \\
 $
The correct option is Option C, 450 KCal.

Note:Carnot principles are only for the cyclical devices like heat engines which states that the efficiency of a reversible heat engine is always more than the efficiency of an irreversible heat engine operating between the same two objects. A Carnot engine has to be completely reversible.