Three Carnot engines operate in series betwee | vedclass

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(B) For Carnot engines to be equally efficient,their efficiencies must be equal: $\eta_1 = \eta_2 = \eta_3 = \eta$.Since the efficiency of a Carnot en

Vedclass · Accepted Answer

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

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(B) For Carnot engines to be equally efficient,their efficiencies must be equal: $\eta_1 = \eta_2 = \eta_3 = \eta$.Since the efficiency of a Carnot engine is given by $\eta = 1 - \frac{T_{	ext{sink}}}{T_{	ext{source}}}$,we have:$1 - \frac{T_2}{T_1} = 1 - \frac{T_3}{T_2} = 1 - \frac{T_4}{T_3} = \eta$.This implies the ratios of the temperatures are equal:$\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = k$ (where $k = 1 - \eta$).From this,we can write:$T_2 = k T_1$,$T_3 = k T_2 = k^2 T_1$,and $T_4 = k T_3 = k^3 T_1$.Thus,$k = (T_4 / T_1)^{1/3}$.Substituting $k$ back into the expressions for $T_2$ and $T_3$:$T_2 = T_1 (T_4 / T_1)^{1/3} = T_1^{2/3} T_4^{1/3} = (T_1^2 T_4)^{1/3}$.$T_3 = T_1 (T_4 / T_1)^{2/3} = T_1^{1/3} T_4^{2/3} = (T_1 T_4^2)^{1/3}$.Therefore,the correct option is $B$.

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 temperatures $T_2$ 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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