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Three Carnot engines operate in series between a heat source at a temperature $T_{1}$ and a heat sink at temperature $T_{4}$ (see figure). There are two other reservoirs at temperature $T_{2}$ and $T_{3}$ , as shown, with $T_{1}>T_{2}>T_{3}>T_{4}$ . The three engines are equally efficient if:

Option 1)
Option 2)
Option 3)
Option 4)

Answers (1)

best_answer

Efficiency of a carnot cycle -

$\eta =\frac{W}{Q_{1}}=1-\frac{T_{2}}{T_{1}}$

$T_{1}\, and\, T_{2}$ are in kelvin

- wherein

$T_{1}=$ Source temperature

$T_{2}=$ Sink Temperature

$\left ( T_{1} > T_{2}\right )$

$\varepsilon _{1}=1-\frac{T_{2}}{T_{1}}\\\\\varepsilon _{2}=1-\frac{T_{3}}{T_{2}}\\\\\varepsilon _{3}=1-\frac{T_{4}}{T_{3}}\\\\\varepsilon _{1}=\varepsilon _{2}=\varepsilon _{3}$

$\Rightarrow 1-\frac{T_{2}}{T_{1}}= 1- \frac{T_{3}}{T_{2}}=1-\frac{T_{4}}{T_{3}}$

$\frac{T_{2}}{T_{1}}=\frac{T_{3}}{T_{2}}=\frac{T_{4}}{T_{3}} \\\\\Rightarrow \frac{T_{2}}{T_{1}}=\frac{T_{3}}{T_{2}}\\\\\Rightarrow T_{2}=\; \sqrt{T_{1}T_{3}}--------(1)\\\\SIMILARLY\\\\ \: \: T_{3}=\sqrt{T_{2}T_{4}}-----(2)PUT \: IN \: (1)\\\\T_{2}=\sqrt{T_{1}T_{3}}\\\\T_{2=\sqrt{T_{1}\sqrt{T_{2}T_{4}}}}\\\\T_{2}=T_{1}^{\frac{2}{3}}T_{4}^{\frac{1}{3}}$

Option 1)

Option 2)
Option 3)

Option 4)

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