The cylinder shown in the figure has conducting walls and temperature of the surrounding is T, the piston is initially in equilibrium, the cylinder contains n moles of a gas, Now the piston is displaced slowly by an external agent to make the volume double of the initial. Find work done by external agent in term of n, R, T

Explanation

`1^(st)` Method:
Work done by external agent is positive, because `F_(ext)` and displacement are in the same direction. Applying equilibrium condition when pressure of the gas is `P`
`PA +F_(ext) =P_(atm)A`
`F_(ext) = P_(atm)A - PA`
`W_(ext) =int_(0)^(d) F_(ext) dx`
`= int_(0)^(d) P_(atm) Adx - int_(0)^(d) PA dx = P_(atm) A int_(0)^(d) dx - int_(V)^(2V) (nRT)/(V) dV`
`= P_(atm) Ad - nRT In2`
`= P_(atm). V_(0) - nRT In2 = nRT (1-In2)`
`2^(nd)` Method
Applying work energy theorem on the piston
As `W_(all) = DeltaK.E`
`DeltaK.E = 0` (given)
`W_(gas) +W_(atm) +W_(ext) = 0`
`nRT In (V_(f))/(V_(i)) - nRT +W_(ext) = 0`
`W_(ext) = nRT (1- In2)`