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)gtT_(2)gtT_(3)gtT_(4)`. The three engines are equally efficient if :

Options

A. `T_2=(T_1T_4^2)^(1//3),T_3=(T_1^2T_4)^(1//3)`
B. `T_2=(T_1T_4)^(1//2),T_3=(T_1^2T_4)^(1//3)`
C. `T_2=(T_1^3T_4)^(1//4),T_3=(T_1T_4^3)^(1//4)`
D. `T_2=(T_1^2T_4)^(1//3),T_3=(T_1T_4^2)^(1//3)`

Answer

D

Explanation

`eta_1=1-T_2/T_1, eta_2=1-T_3/T_2, eta_3=1-T_4/T_3`
Given `eta_1 = eta_2 =eta_3`
`T_2/T_1=T_3/T_2=T_4/T_3`
`T_(2)^(2)=T_1T_3 " "...(i)`
`T_3^2 =T_2T_4" "...(ii)`
From (i) and (ii)
` T_3 = (T_1T_(4)^(2))^(1//3), T_2 = (T_(1)^(2)T_4)^(1//3)`