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_(1)T_(4)^(2))^(1//3),T_(3)=(T_(1)^(2)T_(4))^(1//3)`
B. `T_(2)=(T_(1)^(2)T_(4))^(1//3),T_(3)=(T_(1)T_(4)^(2))^(1//3)`
C. `T_(2)=(T_(1)^(3)T_(4))^(1//3),T_(3)=(T_(1)T_(4)^(3))^(1//4)`
D. `T_(2)(T_(1)T_(4))^(1//2),T_(3)=(T_(1)^(2)T_(4))^(1//3)`

Answer

B

Explanation

`epsilon_(1)=1-(T_(2))/(T_(1)),epsilon_(2)=1-(T_(3))/(T_(2)),epsilon_(3)=1-(T_(4))/(T_(3))`
as `epsilon_(1)=epsilon_(2)=epsilon_(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)T_(4)^(1/2))^(1/2)rArrT_(2)=(T_(1)T_(4)^(1/2))^(1/2xx4/3)=T_(1)^(2/3)T_(4)^(1/3)rArrT_(2)=(T_(1)^(2)T_(4))^(1/3)`
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)=T_(1)^(1/3)T_(4)^(2/3)=(T_(1)T_(4)^(2))^(1/3)`