Question

One reversible heat engine operates between $900\text{K}$ and $T_2\text{K}$, and another reversible heat engine operates between $T_2\text{K}$ and $400\text{K}$. If both the engine have the same efficiency, then the temperature $T_2$ must be equal to

• A. ${600}^{\circ }\text{C}$
• B. $300\text{K}$
• C. $600\text{K}$
• D. $650\text{K}$

Answer 1

The correct option is C $600\text{K}$

Given that,
For engine $1$,
$T_1=900\text{K}$
For engine $2$,
$T_2^{\prime }=400\text{K}$

Efficiency of engine $1$ = Efficiency of engine $2$
${\eta }_1={\eta }_2$
$\because$ Both are reversible heat engine.
$\therefore {\eta }_1=1-\frac{T_2}{T_1}=1-\frac{T_2}{900}$

${\eta }_2=1-\frac{{T^{\prime }}_2}{T_2}=1-\frac{400}{T_2}$
$\because {\eta }_1={\eta }_2$
$\Rightarrow 1-\frac{T_2}{900}=1-\frac{400}{T_2}$
$\Rightarrow T_2=\sqrt{900\times 400}=30\times 20=600\text{K}$

Why this question? Tips: If two heat engines have same efficiency, then Intermediate temperature = $\sqrt{\text{Higher temperature}\times \text{Lower temperature}}$