Question

A Carnot engine working between 450K and 600K has an output of 300J per cycle. The amount of energy supplied to the engine from the source in each cycle is
a) 400J
b) 800J
c) 1600J
d) 3200J
e) None of the given option is correct

Answer 1

It is given in the question that the Carnot engine operates between 450K (the temperature of the sink ${{T}_{2}}$) and 600K ( that is the temperature of the source${{T}_{1}}$). The efficiency ($\eta$ )of the engine in terms of these two temperatures is given by, $\eta =\dfrac{{{T}_{1}}-{{T}_{2}}}{{{T}_{1}}}...(1)$. But the efficiency of the engine in general is basically the ratio of the output to the input heat supplied. Hence we can determine the input heat from the definition of efficiency of the Carnot’s engine.

Complete answer:
The Carnot’s engine is an ideal heat engine. It is an engine which does not take into consideration the factors such as friction which affects the performance of the engine.
The efficiency of any engine is given by,
$\eta =\dfrac{\text{Output (Workdone)}}{\text{Input (Heat supplied)}}$
But the heat supplied to the engine is directly proportional to its temperature. Hence we can equate equation 1 to the definition of efficiency and we get,
$\eta =\dfrac{\text{Output (Workdone)}}{\text{Input (Heat supplied)}}=\dfrac{{{T}_{1}}-{{T}_{2}}}{{{T}_{1}}}$
In the above question the temperature of the sink and the source is given as well as the output during each cycle. Hence the heat supplied from the source from the above equation we get,
$\begin{aligned} & \dfrac{\text{Output (Workdone)}}{\text{Input (Heat supplied)}}=\dfrac{{{T}_{1}}-{{T}_{2}}}{{{T}_{1}}} \\\ & \Rightarrow \dfrac{300}{\text{Input (Heat supplied)}}=\dfrac{600-450}{600} \\\ & \Rightarrow \dfrac{300}{\text{Input (Heat supplied)}}=\dfrac{150}{600} \\\ & \Rightarrow \dfrac{300}{\text{Input (Heat supplied)}}=0.25 \\\ & \Rightarrow \text{Input (Heat supplied)}=\dfrac{300}{0.25}=1200J \\\ \end{aligned}$
1200J is provided in the options.

Hence the correct answer of the above question is option e.

Note:
A Carnot’s engine has the maximum possible efficiency as it ignores all the factors that affect its efficiency. It consists of four thermodynamic changes. The first is isothermal expansion, then it undergoes adiabatic expansion further again isothermal compression and finally adiabatic compression.