Question

A Carnot engine used first an ideal monatomic gas and then an ideal diatomic gas. If the source and sink temperature are ${411^ \circ }C$ and ${69^ \circ }C$ respectively and the engine extracts $1000{\text{ }}J$ of heat in each cycle, then the area enclosed by $P - V$ diagram is
A. $100J$
B. $300J$
C. $500J$
D. $700J$

Answer 1

Hint: For a Carnot engine, the net efficiency is the ratio of net work done by the gas and the heat absorbed by the gas. Also, the area enclosed by the P-V diagram is equivalent to the work done by the engine. By using this concept, we can extract formulas for calculating the given problem.

Formula used:
${\eta _{carnot}} = 1 - \dfrac{{{T_L}}}{{{T_H}}} = \dfrac{W}{{{Q_1}}}$

Complete answer:
We know that the efficiency of Carnot Heat Engine is given as: -
${\eta _{carnot}} = 1 - \dfrac{{{T_2}}}{{{T_1}}}$ … (1)
where
${T_2} = $Lower Absolute Temperature = Temperature of the Sink
and, ${T_1} = $Higher Absolute Temperature = Temperature of the Source

Also, the Efficiency of Carnot Engine in terms of work done can be given as: -
${\eta _{carnot}} = \dfrac{W}{{{Q_1}}} = \dfrac{{{Q_1} - {Q_2}}}{{{Q_1}}}$ … (2)
where
$W = $Work Done in the process
${Q_1} = $Heat taken up from the Source
${Q_2} = $Heat transferred to the Sink

Temperature of the hot reservoir (source)
${T_1} = {411^ \circ }C = 684K$ (given)
$\left( {^ \circ C + 273 = K} \right)$

And Temperature of the Sink
${T_2} = {69^ \circ }C = 342K$ (given)

The heat taken up by the engine from the Reservoir ${Q_1} = 1000J$ (given). From eq. (1) and (2), we get
$\dfrac{W}{{{Q_1}}} = 1 - \dfrac{{{T_2}}}{{{T_1}}}$
$ \Rightarrow W = {Q_1}\left( {1 - \dfrac{{{T_2}}}{{{T_1}}}} \right)$ … (3)

Substituting the given values in eq. (3), we get the area enclosed by the $P - V$diagram will be equal to the work done by the engine.
$ \Rightarrow W = 1000\left( {1 - \dfrac{{342}}{{684}}} \right)$
$ \Rightarrow W = 1000\left( {\dfrac{1}{2}} \right) = 500J$

Thus, the area enclosed by the $P - V$ diagram is $500J$.

The correct option is C.

Note: In the question it is mentioned that a Carnot engine used first an ideal monatomic gas and then an ideal diatomic gas. It should be noted that the efficiency of a Carnot engine does not depend upon the nature of the gas. It implies that the nature of the gas won’t bring any change in the area enclosed by the P-V diagram.