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Three Carnot engines operate in series b...

Three Carnot engines operate in series between a heat source at a temperature `T_1 = 600 K ` and heat sink at temperature `T_4 = 100K` (see figure). There are two other reservoirs at temperature `T_2 " and " T_3` as shown, with `T_1 gt T_2 gt T_3 gt T_4` The three engines are equally efficient if:

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

Text Solution

Verified by Experts

The correct Answer is:
A
`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`
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