Figure shows an insulated cylinder of volume `V` containing monatomic gas in both the compartments. The pistone is diathermic. Initially the piston is kept fixed and the system is allowed to acquire a state of thermal equilibrium. The initial pressures and temperatures are as shown in the figure. Calculate

The heat that flows from `RHS` to `LHS`, Given `T_(2)gt T_(1)`.
Now, the pin which was keeping the piston fixed is removed. and the piston is set free to move. The piston is allowed to slide slowly, such that of mechanical equilibrium is also achieved and `T_(1)P_(2)gt P_(1)T_(2)`. Find.

Options

A. `(4)/(3)P_(1)P_(2)V((T_(2)-T_(1)))/(P_(1)T_(2)+P_(2)T_(1))`
B. `(3)/(4)P_(1)P_(2)V((T_(2)-T_(1)))/(P_(1)T_(2)+P_(2)T_(1))`
C. `(3)/(4)P_(1)P_(2)V((T_(2)+T_(1)))/(P_(1)T_(2)+P_(2)T_(1))`
D. `(3)/(4)P_(1)P_(2)V((T_(2)-T_(1)))/(P_(1)T_(2)-P_(2)T_(1))`

Answer

B

Explanation

Heat flowing from right to left
`Q=n_(1)C_(V)(T-T_(1))=(P_(1)V)/(2 R T_(1)2)(3)/(2)R{((P_(1)+P_(2))T_(1)T_(2))/(P_(1)T_(2)+P_(2)T_(1))-T_(1)}`
`=(3)/(4)P_(1)P_(2)V((T_(2)-T_(1)))/(P_(1)T_(2)+P_(2)T_(1))`