A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volmek `V_0` in which an ideal gas is contained under the same pressure `p_0` and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas `eta` times compared to that of the other by slowly moving the piston ?

Answer

`W = p_(0)V_(0) In [(eta+1)^(2)//4eta`

Explanation

Final volume of left part `=eta xx` final volume of right-part
`V_(0) = V = eta (V_(0) - V)`
`V = ((eta-1)V_(0))/(eta+1)`
in isothermal process, work done `=p_(0)V_(0) ln ((V_(2))/(V_(1)))`
work done by left part of gas on piston is
`W_(1) = p_(0)V_(0) ln ((V_(0)+V)/(V_(0)))`
`= p_(0)_(0) ln (1+(V)/(V_(0)))`
`= p_(0)V_(0) ln (1+(eta-1)/(eta+1)) =p_(0)V_(0)ln ((2eta)/(eta+1))`
similarity, work done by right part of gas on piston is
`W_(2) = p_(0)V_(0) ln ((V_(0)+V)/(V_(0)))`
`= p_(0)V_(0) ln (1-(eta-1)/(eta+1)) = p_(0)V_(0) ln ((2)/(eta+1))`
let work done by applied force in `W_(ext)` then
`W_(1) +W_(2) +W_(ext) = 0`
`w_(ext) =- W_(1) - W_(2)`
`W_(ext) = p_(0)V_(0) ln .((eta+1)^(2))/(4eta)`