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)[(2l_(1)+l_(2))/(l_(1)l_(2))]`
B. `(nRT)/(g)[(1)/(l_(1))+(1)/(l_(2))]`
C. `(RT)/(ng)[(l_(1)-3l_(2))/(l_(2)l_(1))]`
D. `(nRT)/(g)[(l_(1)-l_(2))/(l_(2)l_(1))]`

Answer

D

Explanation

For gas in upper chamber, `R_(1)V_(1) = nRT`
For gas in lower chamber, `P_(2)V_(2) = nRT`
Let the area of piston is A, `V_(1) = l_(1)A` and `V_(2)l_(2)A`
Consider equilibrium of piston


`P_(1)A = P_(1)A + mg`
`P_(2) - P_(1) = (mg)/(A), (nRT)/(V_(2)) - (nRT)/(V_(1)) = (mg)/(A)`
`nRT[(1)/(l_(2)A) - (1)/(l_(1)A)] = (mg)/(A)` , `m = (nRT)/(g)[(1)/(l_(2)) - (1)/(l_(1))]`