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 = (36)^(1//3) xx 100 K " " T_3 = 6^(1//3) xx 100 K`
B. `T_2 = 360 K , T_3 = 60K`
C. `T_2 = (36)^(1//4) xx 100 K , T_3 = 6^(1//4) xx 100 K`
D. None of these

Answer

A

Explanation

`epsi_1 = 1 - T_2/T_1 , epsi_2 = 1 - (T_3/T_2 , epsi_3 = 1 - T_4/T_3`
as ` epsi_1 = epsi_2 = epsi_3`
` T_2/T_1 = T_3/T_2 = T_4/T_3 , T_2 = (T_1 T_3)^(1/2)`
` T_2 (T_1 T_2^(1/2) T_4^(1/2))^(1/2)`
`T_2^(3/4) = (T_1^2 T_4^(1/2) )^(1/2) rArr T_2 = (T_1 T_4^(1/2) )^(1/2 xx 4/3) = T_1^(3/2) T_4^(1/3) `
`rArr T_2 = (T_1^2 T_4)^(1/3) = (600 xx 600 xx 100)^(1//3) = (36)^(1//3) xx 100`
Also ` T_3 = (T_2 T_4)^(1/2) = (T_1^(2/3) T_4^(1/3) T_4)^(1/2) = (T_1^(2/3) T_4^(4/3) )^(1/2)`
` rArr T_3 = T_1^(1//3) T_4^(2//3) " " T_3 = (T_1T_4^2)^(1//3) = (600 xx 100 xx 100 )^(1/3) = 6^(1//3) xx 100`