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 $550\text{K}$

$\Rightarrow$ For a Carnot's engine the formula for efficiency is,
$\eta =1-\frac{Q_2}{Q_1}=1-\frac{T_2}{T_1}$
$\Rightarrow \frac{Q_1-Q_2}{Q_1}=\frac{T_1-T_2}{T_1}...\left(i\right)$
The work output $\left(W\right)$ of a Carnot's engine is,
$W=Q_1-Q_2...\left(ii\right)$
From Eq.$\left(i\right),\left(ii\right)$ it is clear that $(Q_1-Q_2)\propto (T_1-T_2)$
$\Rightarrow W\propto (T_1-T_2)...\left(iii\right)$
For engine $A$,
Temperature of source $T_1=800\text{K}$
Temperature of sink $T_2=T\text{K}$
$\Rightarrow$For engine $B$, sink of engine $A$ is acting as a source reservoir. Hence for engine $B$,
Temperature of source, $T_1^{\prime }=T\text{K}$
Temperature of sink, $T_2^{\prime }=300\text{K}$


Since workoutputs of both engines $A$ and $B$ are equal,
$W_A=W_B$
$\frac{W_A}{W_B}=1...\left(iv\right)$
Now for engines from Eq.$\left(iii\right)$, we can write
$\Rightarrow W_A\propto (T_1-T_2)$
Or, $W_A\propto (800-T)$
$W_B\propto (T_1^{\prime }-T_2^{\prime })$
Or, $W_B\propto (T-300)$
Substituting in Eq.$\left(iv\right)$
$\Rightarrow \frac{(800-T)}{(T-300)}=1$
Or, $2T=1100$
$\therefore T=550\text{K}$