Question

Two Carnot engines $A$ and $B$ are operated in series. The first engine, $A$, receives heat at $800\text{K}$and rejects to a reservoir at temperature $T$. The second engine $B$ receives the heat rejected by the first engine and in turn rejects to a heat reservoir at $300\text{K}$. Find the value of $T$, when the work outputs of the engines are equal.

• A. $0\text{K}$
• B. $1100\text{K}$
• C. $550\text{K}$
• D. $300\text{K}$

Answer 1

The correct option is C $550\text{K}$


Work done by engine $A$,

$W_A=Q_1-Q_2$

Work done by engine $B$,

$W_B=Q_2-Q_3$

As given, $W_A=W_B$

$Q_1-Q_2=Q_2-Q_3$

Dividing on both side by $Q_1$,

$1-\frac{Q_2}{Q_1}=\frac{Q_2}{Q_1}-\frac{Q_3}{Q_1}$

$\Rightarrow 1-\frac{Q_2}{Q_1}=\frac{Q_2}{Q_1}(1-\frac{Q_3}{Q_2})$

$\Rightarrow 1-\frac{T}{T_1}=\frac{T}{T_1}(1-\frac{T_3}{T})$

$\Rightarrow \frac{T_1}{T}-1=1-\frac{T_3}{T}$

$\Rightarrow \frac{T_1}{T}+\frac{T_3}{T}=2$

​​​​​​​$\Rightarrow \frac{T_1+T_3}{T}=2$

​​​​​​​$\Rightarrow T=\frac{T_1+T_3}{2}=\frac{800+300}{2}$

$\Rightarrow T=550\text{K}$

​​​​​​​​​​​​​​Hence, option $\left(C\right)$ is the correct answer.