A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above the piston is `l_(1)`, and that below the piston is `l_(2)`, such that `l_(1)gtl_(2)`. Each part of the cylinder contains n moles of an ideal gas at equal temeprature T. If the pistion is stationary, its mass, m, will be given by : (R is universal gas constant and g is the acceleration due to gravitey)

Options

A. `(RT)/g [(2J_1 + J_2)/(J_1 J_2)]`
B. `(nRT)/g[1/J_2 + 1/J_1]`
C. `(RT)/(ng)[(J_1- 3J_2)/(J_2 J_1)]`
D. `(nRT)/g[(J_1 - J_2)/(J_2J_1)]`

Answer

D

Explanation

For gas in upper chamber, `P_1V_1=nRT`
For gas in lower chamber, `P_2V_2=nRT`
Let the area of piston is A, `V_1=l_1A " and " V_2=l_2A`
consider equilibrium of piston
`P_2A=P_1A+mg`
`P_2-P-1=(mg)/(A) , " "(nRT)/(V_2)-(nRT)/(V_1)=(mg)/(A)`
`nRT[(1)/(l_2A)-(1)/(l_1A)]= (mg)/(A), m=(nRT)/(g)[(1)/(l_2)-(1)/(l_1)]`.