Question

Two Carnot engines $A$ and $B$ are operated in series. The first one $A$ receives heat from a source at temperature $800\text{K}$ and rejects heat to a reservoir at temperature $T\text{K}$. The second engine $B$ receives the heat rejected by the first engine and in turn rejects to a heat reservoir at temperature $300\text{K}$ Find the value of temperature $T$ if the efficiencies of two engines are equal.
(Take $\sqrt{6}=2.45$ for calculation)

• A. $550\text{K}$
• B. $300\text{K}$
• C. $25\times {10}^4\text{K}$
• D. $490\text{K}$

Answer 1

The correct option is D $490\text{K}$

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, the efficiencies of two engines are equal.
$\Rightarrow {\eta }_A={\eta }_B$
$\Rightarrow 1-\frac{T_2}{T_1}=1-\frac{T_2^{\prime }}{T_1^{\prime }}$
$\Rightarrow \frac{T_2}{T_1}=\frac{T_2^{\prime }}{T_1^{\prime }}$
$\Rightarrow \frac{T}{800}=\frac{300}{T}$
$\Rightarrow T^2=24\times {10}^4$
$\Rightarrow T=100\times \sqrt{24}=2\sqrt{6}\times 100$
substituting $\sqrt{6}=2.45$,
$\therefore T=4.90\times 100=490\text{K}$