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 $T_{2} > T_{2} > T_{3} > T_{4}$ . The three engines are equally efficient if__?

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

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

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

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

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

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Three Carnot engines operate in series between a heat source at a temperature T1 and a heat sink at temperature T4 (see figure). There are two other reservoirs at temperature T2, and T3, as shown, with T2>T2>T3>T4. The three engines are equally efficient if__?

The correct option is C T2=(T21T4)1/3:T3=(T1T24)1/3Given: t1=1−T2T1=1−T2T2=1−T4T3⇒T2T1=T3T2=T4T3⇒T2=√T1T3=√T1√T2T4⇒T(1−1/4)2=T1/21T1/44⇒T3/42=T1/21T1/44⇒T2=(T21T4)1/3T3=√T2T4=√(T21T4)1/3T4=T1/31T2/34.