Show a vertical cylindrical vessel separated in two parts by a frictionless piston free to move along the length of vessel. The length of the cylinder is 90 cm and the piston divides the cylinder in the ratio of 5:4. Each of the two parts of the vessel contains 0.1 mole of an ideal gas. The temperature of the gas is 300K in each part. Calculate the mass of the piston.

Explanation

`Let l_(1) and l_(2)` be the upper part and the lower part of the cylinder respectively. Clearly, `l_(1)=50cm` and `l_(2)=40cm`. Let the pressures in the upper and lower parts be `p_(1)` and `p_(2)` respectively. Let the area of creoss section of the cylinder be `A`. The temperature in both parts is `T=300K`.
Consider the equilbrium of the piston . the forces acting on the piston are
(a) its weight mg
(b) `p_(1)` A downward, by the upper part of the gas
and (c) `p_(2)` A upward, by the lower part of gas .
Thus , `p_(2) A =p_(1)A +mg` ....(i)
Using `pV=nRt` for the upper and the lower parts
`p_(1)l_(1)A=nRT` ...(ii)
and `p_(2)l_(2)A=nRT` (iii)
Putting `p_(1) A and p_(2)A` from (ii)and(iii) into(i),
`(nRT)/(l_(2))=(nRT)/(l_(1))+mg`
Thus, `m=(nRT)/(g)[(1)/(l_(2))-(1)/(l_(1))]`
`=((0.1 mol)(8.3JK^(-1) mol^(-1))(300K))/(9.8ms^(-2)[(1)/(0.4m)-(1)/(0.5m)]`
`=12.7kg.`