Q. 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
A) $100\ K$
B) $300\ K$
C) $550\ K$
D) $700\ K$
Explanation
(c) $\eta_{A} = \frac{T_{1} - T_{2}}{T_{1}} = \frac{W_{A}}{Q_{1}} \Rightarrow \eta_{B} = \frac{T_{2} - T_{3}}{T_{2}} = \frac{W_{B}}{Q_{2}}$
$\therefore\frac{Q_{1}}{Q_{2}} = \frac{T_{1}}{T_{2}} \times \frac{T_{2} - T_{3}}{T_{1} - T_{2}} = \frac{T_{1}}{T_{2}}\ \therefore W_{A} = W_{B}$
$\therefore T_{2} = \frac{T_{1} + T_{3}}{2} = \frac{800 + 300}{2} = 550K$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.