Two Carnot engines ‘A’ and ‘B’ are operated in succession. The first one, A receives heat from a source at $T_{1} =$ 800K and rejects to a sink at $T_{1}$ K . The second engine B receives heat rejected by the first engine and rejects to another sink at $T_{3} =$ 300K . If the work outputs of two engines are equal, then the value of $T_{2}$ is

100 K

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300 K

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550 K

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700 K

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Solution

The correct option is C 550 K
Let, $Q_{1}$ =heat absorbed in first engine, $Q_{2}$ = heat released in first engine=heat absorbed in second engine, $Q_{3}$ =heat released in second engine,
$T_{1}$ =source temperature of first engine, $T_{2}$ = sink temperature of first engine=source temperature of second engine, $T_{3}$ =sink temperature of second engine
For the 2 Carnot Engines, $\frac{Q_{2}}{Q_{1}} = \frac{T_{2}}{T_{1}}$
$\frac{Q_{3}}{Q_{2}} = \frac{T_{3}}{T_{2}}$
The condition given is, $W_{1}$ = $W_{2}$
$\Rightarrow Q_{2} - Q_{1} = Q_{3} - Q_{2} \Rightarrow 2 Q_{2} = Q_{1} + Q_{3}$ $\Rightarrow 2 T_{2} = T_{1} + T_{3} \Rightarrow 2 T_{2} = 300 + 800 = 1100$
Hence required temperature is 550K

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Two Carnot engines A and B are operated in succession. The first one, A receives heat from a source at $T_{1}$ = 800 K and rejects to sink at $T_{2}$ K. The second engine B receives heat rejected by the first engine and rejects to another sink at $T_{3}$ = 300 K. If the work outputs of two engines are equal, then the value of $T_{2}$ is