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

The cylinder shown in the figure has conducting walls and temperature

1.	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
Answer» `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)`

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