Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature `T_0`, while Box contains one mole of helium at temperature `(7/3)T_0`. The boxes are then put into thermal contact with each other, and heat flows between them until the gasses reach a common final temperature (ignore the heat capacity of boxes). Then, the final temperature of the gasses, `T_f` in terms of `T_0` is

Options

A. `T_f = (3)/(7) T_0`
B. `T_f = (7)/(3)T_0`
C. `T_f = (3)/(2) T_0`
D. `T_f = (5)/(2) T_0`

Answer

C

Explanation

Internal energy of a system remains constant or charge in internal energy of the system is zero.
The system of two rigid boxes is shown in figure.
`Delta U_A = I xx (5 R)/(2) (T_f - T_0)`
`Delta U_B = 1 xx (3 R)/(2) (T_f - (7)/(3) T_0)`
Now, `DeltaU_A + Delta U_B = 0`
`(5 R)/(2) (T_f - T_0) + (3 R)/(2) (T_f - (7 T_0)/(3)) = 0`
`5 T_f - 5 T_0 + 3 T_f - 7 T_0 = 0`
`rArr 8 T_f = 12 T_0`
`rArr T_f = (12)/(8) T_0 = (3)/(2) T_0`.