Two Carnot engines $A$ and $B$ are operated in series. The first one, $A$, receives heat at $T_1\,(= 600\ \text{K})$ and rejects to a reservoir at temperature $T_2$. The second engine $B$ receives heat rejected by the first engine and, in turn, rejects to a heat reservoir at $T_3\,(= 400\ \text{K})$. Calculate the temperature $T_2$ if the work outputs of the two engines are equal:
Answer: (A) $500\ \text{K}$
For a Carnot engine $Q \propto T$. Let $A$ take in $Q_1$; it rejects $Q_2 = Q_1 T_2/T_1$, which $B$ takes in, and $B$ rejects $Q_3 = Q_1T_3/T_1$.
$$W_A = Q_1 - Q_2 \propto T_1 - T_2,\qquad W_B = Q_2 - Q_3 \propto T_2 - T_3$$
Equal work: $T_1 - T_2 = T_2 - T_3$, so $T_2 = \dfrac{600+400}{2} = 500\ \text{K}$.
Solution by Sreeraj P, M.Sc Physics