Three Carnot engine 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

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$T_{2} = {(T_{1} T_{4})}^{1 / 2}; T_{3} = {(T_{1}^{2} T_{4})}^{1 / 3}$

$T_{2} = {(T_{1}^{2} T_{4})}^{1 / 3}; T_{3} = {(T_{1} T_{4}^{2})}^{1 / 3}$

$T_{2} = {(T_{1}^{3} T_{4})}^{1 / 4}; T_{3} = {(T_{1} T_{4}^{3})}^{1 / 4}$

$T_{2} = {(T_{1} T_{4}^{2})}^{1 / 3}; T_{3} = {(T_{1}^{2} T_{4})}^{1 / 3}$

answer is A.

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Detailed Solution

As engines are equally efficient, we use
$η = 1 - \frac{T_{2}}{T_{1}} = 1 - \frac{T_{3}}{T_{2}} = 1 - \frac{T_{4}}{T_{3}}$
$\Rightarrow \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{2}} = \frac{T_{4}}{T_{3}}$
$\Rightarrow T_{2} = \sqrt{T_{1} T_{3}}$
and $T_{3} = \sqrt{T_{2} T_{4}}$
$\Rightarrow T_{2}^{3 / 4} = \sqrt{T_{1}^{1 / 2}} T_{4}^{1 / 4}$
$\Rightarrow T_{2} = T_{1}^{2 / 3} T_{4}^{1 / 3}$
and $T_{3} = T_{1}^{1 / 3} T_{4}^{2 / 3}$