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Difference between Internal Energy and Enthalpy

Three Carnot ...

Question

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

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

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

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

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

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Solution

The correct option is C $T_{2} = (T_{1}^{2} T_{4})^{1 / 3}; T_{3} = (T_{1} T_{4}^{2})^{1 / 3}$
Given:
$T_{1} > T_{2} > T_{3} > T_{4}$
$η_{1} = η_{2} = η_{3}$

According to question, $η_{1} = η_{2} = η_{3}$

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

[ $∵$ Three engines are equally efficient]

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

$\Rightarrow T_{2} = \sqrt{T_{1} T_{3}}$ ... $(i)$

$T_{3} = \sqrt{T_{2} T_{4}}$ ... $(i i)$

From $(i)$ and $(i i)$

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

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

Hence, option $(B)$ is correct.

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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