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The cylinder shown in the figure has con...

The cylinder shown in the figure has conducting walls and temperature of the surrounding is T, the pistonj 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

Text Solution

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1st Method: Work done by external agent is positive, because Fext and displacement are in the same direction . Since walls are conducting therefore temperature ramains constant.
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 1n2`
`= P_("atm"). V_(0) - nRT 1n2 = nRT (1 - 1n)`
2nd Method : Applying work erergy theorem on the piston
`Delta k = 0`
`W_("all") = Delta k`
`W_("gas") + W_("atm") + W_("ext") = 0`
`nRT 1n (V_(1))/(V_(1)) - nRT + W_("ext") = 0`
`W_("ext") = nRT (1 - 1n2)`
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