| 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 \[{{T}_{1}}>{{T}_{2}}>{{T}_{3}}>{{T}_{4}}.\] shown, with The three engines are equally efficient if: |
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A) \[{{T}_{2}}={{({{T}_{1}}{{T}_{4}})}^{1/2}};{{T}_{3}}={{\left( T_{1}^{2}{{T}_{4}} \right)}^{{}^{1}/{}_{3}}}\]
B) \[{{T}_{2}}={{\left( T_{1}^{2}{{T}_{4}} \right)}^{{}^{1}/{}_{3}}};{{T}_{3}}={{\left( {{T}_{1}}T_{2}^{4} \right)}^{{}^{1}/{}_{3}}}\]
C) \[{{T}_{2}}={{\left( {{T}_{1}}T_{4}^{2} \right)}^{{}^{1}/{}_{3}}};{{T}_{3}}={{\left( T_{1}^{2}{{T}_{4}} \right)}^{{}^{3}/{}_{4}}}\]
D) \[{{T}_{2}}={{\left( T_{1}^{3}{{T}_{4}} \right)}^{{}^{1}/{}_{4}}}{{T}_{3}}={{\left( {{T}_{1}}T_{4}^{3} \right)}^{{}^{1}/{}_{4}}}\]
Correct Answer: B
Solution :
\[{{n}_{1}}={{n}_{2}}={{n}_{3}}\]\[\Rightarrow \] \[1-\frac{{{T}_{2}}}{{{T}_{1}}}=1-\frac{{{T}_{3}}}{{{T}_{2}}}=1-\frac{{{T}_{4}}}{{{T}_{3}}}\]\[\Rightarrow \]\[\frac{{{T}_{2}}}{{{T}_{1}}}=\frac{{{T}_{3}}}{{{T}_{2}}}=\frac{{{T}_{4}}}{{{T}_{3}}}\]\[\Rightarrow \]\[{{T}_{2}}{{T}_{3}}={{T}_{1}}{{T}_{4}}\]and\[\frac{T_{3}^{2}}{{{T}_{2}}}={{T}_{4}}\]Solve for \[{{T}_{2}}\]and\[{{T}_{3}}.\]
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