Figure shows an insulated cylinder of volume `V` containing monatomic gas in both the compartments. The piston 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 final pressures of left and right compartment respectively are

Options

A. `(P_(1)(P_(1)+P_(2))T_(2))/(P_(1)T_(2)+P_(2)T_(1)),(P_(2)(P_(1)+P_(2))T_(1))/((P_(1)T_(2)+P_(2)T_(1)))`
B. `(P_(1)(P_(1)+P_(2))T_(1))/(P_(1)T_(2)+P_(2)T_(2)),(P_(2)(P_(1)+P_(2))T_(1))/((P_(1)T_(2)+P_(2)T_(1)))`
C. `(P_(1)(P_(1)+P_(2))T_(2))/(P_(1)T_(2)-P_(2)T_(1)),(P_(2)(P_(1)-P_(2))T_(1))/((P_(1)T_(2)+P_(2)T_(1)))`
D. `(P_(1)(P_(1)+P_(2))T_(2))/(P_(1)T_(2)+P_(2)T_(1)),(P_(2)(P_(1)-P_(2))T_(1))/((P_(1)T_(2)+P_(2)T_(1)))`

Answer

A

Explanation

Applying the first law of the thermodynamics to the cylinder.
`DeltaQ=DeltaU+DeltaW`
`DeltaQ=DeltaW=0`
`implies Delta U=0`
`n_(1)C_(V)(T-T_(1))+n_(2)(T-T_(2))=0`
`T=((P_(1)+P_(2))T_(1)T_(2))/(P_(1)T_(2)+P_(2)T_(1))`
For left compartment,
`(P_(1))/(T_(1))=(P'_(1))/(T)=P'_(1)=(P_(1)(P_(1)+P_(2))T_(2))/(P_(1)T_(2)+P_(2)T_(1))`
Similarly, `P'_(2)=(P_(2)(P_(1)+P_(2))T_(2))/((P_(1)T_(2)+P_(2)T_(1)))`