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

Answer

B

Explanation

`E_1=1-(T_2)/(T_1), E_2= 1-(T_3)/(T_2),E_3=1-(T_4)/(T_3)`
as `E_1=E_2=E_3" "(T_2)/(T_1)=(T_3)/(T_2)=(T_4)/(T_3): T_2= (T_1T_3)^(1/2)`
`T_2= (T_1T_(2)^(1/2)T_(4)^(1/2))^(1/2): " "T_(2)^(3/4)= (T_1T_(4)^(1/2))^(1/2) implies T_2= (T_1T_(4)^(1/2))^((1/2)xx(4/3))= T_(1)^(2/3)T_(4)^(1/3) implies T_2= (T_(1)^(2)T_(4))^(1/3)`
Also, `T_3=(T_2T_4)^(1/2)= (T_(1)^(2/3)T_(4)^(1/3)T_(4))^(1/2)= (T_(1)^(1/3)T_(4)^(2/3)=(T_1T_(4)^(2))^(1/3)`.