Two carnot engine A and B use one after another. Engine A absorbs 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 engines is same than calculate value of $T_{2}$

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Solution

Given $T_{1} = 800 K$ $T_{3} = 300 K$
For first engine
$\Rightarrow$ Efficiency $(n_{1}) = 1 - \frac{T_{2}}{T_{1}} = \frac{W}{Q_{1}}$
For $2 n d$ engine
$\Rightarrow$ Efficiency $(n_{2}) = 1 - \frac{T_{3}}{T_{2}} = \frac{W}{Q_{2}}$
Since work is equal
$\Rightarrow n_{1} Q_{1} = n_{2} Q_{2}$
$\Rightarrow \frac{n_{1}}{n_{2}} = \frac{Q_{2}}{Q_{1}}$
for carnot engine $\frac{Q_{2}}{Q_{1}} = \frac{T_{3}}{T_{1}}$
So, $\frac{n_{1}}{n_{2}} = \frac{T_{3}}{T_{1}}$
$\Rightarrow \frac{1 - \frac{T_{2}}{T_{1}}}{1 - \frac{T_{3}}{T_{2}}} = \frac{T_{3}}{T_{1}}$
$\Rightarrow \frac{T_{1} - T_{2}}{T_{2} - T_{3}} = \frac{T_{1} T_{3}}{T_{2} T_{1}}$
$\Rightarrow T_{2}^{2} - 500 T_{2} - T_{3}^{2} = 0$
$\Rightarrow T_{2} = 640 K .$
Hence, the answer is $640 K .$

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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