Question

Two Carnot engines ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ are operated in succession. The first one, $A$ receives heat from a source at $T_1=800\text{K}$ and rejects to a sink at $T_2\text{K}$ . The second engine $B$ receives heat rejected by the first engine and rejects it to the another sink at $T_3=300\text{K}$. If the work outputs of the two engines are equal, then the value of $T_2$ is

• A. $100\text{K}$
• B. $300\text{K}$
• C. $550\text{K}$
• D. $700\text{K}$

Answer 1

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

If both carnot engines $A\&B$ are operated in succession, then sink of engine $A$ will act as the source of engine $B$.
${\eta }_A=1-\frac{T_2}{T_1}=\frac{T_1-T_2}{T_1}$
Also for a carnot's engine,
$T\propto Q$
$\Rightarrow T_1-T_2\propto Q_1-Q_2$
From energy conservation,
$Q_1=Q_2+W$
$\Rightarrow W=Q_1-Q_2$
$\therefore {\eta }_A=\frac{T_1-T_2}{T_1}=\frac{W_A}{Q_1}....\left(1\right)$
Similarly for engine $B$, the temperature of source is $T_2$ and that of sink is $T_3$.
${\eta }_B=\frac{T_2-T_3}{T_2}=\frac{W_B}{Q_2}....\left(2\right)$
Also work output of both engine is same i.e $W_A=W_B$
Taking the equation $\left(2\right)$ and $\left(1\right)$,
$\frac{Q_1}{Q_2}=\frac{T_1}{T_2}\times \frac{T_2-T_3}{T_1-T_2}$
For a carnot's engine, $\frac{Q_1}{Q_2}=\frac{T_1}{T_2}$
Hence on equating above equations we get,
$\Rightarrow \frac{T_1}{T_2}\times \frac{T_2-T_3}{T_1-T_2}=\frac{T_1}{T_2}$
$\Rightarrow 2T_2=T_1+T_3$
$\therefore T_2=\frac{T_1+T_3}{2}=\frac{1100}{2}=550\text{K}$