Three Carnot engines operate in series between a heat source at a… - Vidyadip

MCQ

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

A
${T_2} = {\left( {{T_1}{T_4}} \right)^{1/2}};\,{T_3} = {\left( {T_1^2{T_4}} \right)^{1/3}}$
✓
${T_2} = {\left( {T_1^2{T_4}} \right)^{1/3}};\,{T_3} = {\left( {{T_1}T_4^2} \right)^{1/3}}$
C
${T_2} = {\left( {{T_1}T_4^2} \right)^{1/3}};\,{T_3} = {\left( {T_1^2{T_4}} \right)^{1/3}}$
D
${T_2} = {\left( {T_1^3{T_4}} \right)^{1/4}};\,{T_3} = {\left( {{T_1}T_4^3} \right)^{1/4}}$

✓

Answer

Correct option: B.

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

b
$n_{1}=n_{2}=n_{3}$

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

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

$\Rightarrow \quad \mathrm{T}_{2} \mathrm{T}_{3}=\mathrm{T}_{1} \mathrm{T}_{4}$ 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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