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 sink at ${T_2}K.$. 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 .... $K$
Diffcult
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(c) ${\eta _A} = \frac{{{T_1} - {T_2}}}{{{T_1}}} = \frac{{{W_A}}}{{{Q_1}}}$
==>${\eta _B} = \frac{{{T_2} - {T_3}}}{{{T_2}}} = \frac{{{W_B}}}{{{Q_2}}}$
$\therefore$ $\frac{{{Q_1}}}{{{Q_2}}} = \frac{{{T_1}}}{{{T_2}}} \times \frac{{{T_2} - {T_3}}}{{{T_1} - {T_2}}} = \frac{{{T_1}}}{{{T_2}}}$
$\therefore$ ${W_A} = {W_B}$
$\therefore$ ${T_2} = \frac{{{T_1} + {T_3}}}{2} = \frac{{800 + 300}}{2} = 550K$
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