Two carnot engine $A$ and $B$ use one after another. Engine $A$ absorb heat from source at $T_{1} = 800 K$ and release heat to sink of temp $T_{2} K$ . Second engine $B$ absorb heat which release by first (at $T_{2} K$ ) and release heat at another sink at temperature $T_{3} = 300 K$ . If work done by two engine is same than calculate value of $T_{2}$ .

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Solution

Let the heat released by first engine be $Q_{L}$ .
Thus, heat absorbed by first engine $Q_{H} = \frac{T_{1}}{T_{2}} Q_{L} = \frac{800}{T_{2}} Q_{L}$
So, work done by first engine $W = Q_{H} - Q_{L} = (\frac{800}{T_{2}} - 1) Q_{L}$ ......(1)
Heat absorbed by second engine is $Q_{L}$ .
Heat released by second engine $Q_{L}^{'} = \frac{T_{3}}{T_{2}} Q_{L} = \frac{300}{T_{2}} Q_{L}$
So, work done by second engine $W^{'} = Q_{L} - Q_{L}^{'} =$ $(1 - \frac{300}{T_{2}}) Q_{L}$ ...(2)
But, $W = W^{'}$
$⟹$ $\frac{800}{T_{2}} - 1 =$ $1 - \frac{300}{T_{2}}$
Or $\frac{1100}{T_{2}} = 2$
$⟹ T_{2} = 550 K$

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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