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

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Answer & Solution

Correct Answer

(B) \({T_2} = {\left( {T_1^2{T_4}} \right)^{1/3}};\,{T_3} = {\left( {{T_1}T_4^2} \right)^{1/3}}\)

Detailed explanation

\(n_{1}=n_{2}=n_{3}\)

\(\Rightarrow \quad 1-\frac{T_{2}}{T_{1}}=1-\frac{T_{3}}{T_{2}}=1-\frac{T_{4}}{T_{3}}\)

\(\Rightarrow \quad \frac{T_{2}}{T_{1}}=\frac{T_{3}}{T_{2}}=\frac{T_{4}}{T_{3}}\)

\(\Rightarrow \quad \mathrm{T}_{2} \mathrm{T}_{3}=\mathrm{T}_{1} \mathrm{T}_{4}\) and \(\frac{\mathrm{T}_{3}^{2}}{\mathrm{T}_{2}}=\mathrm{T}_{4}\)

Solve for \(\mathrm{T}_{2}\) and \(\mathrm{T}_{3}\)

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