One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a t

Question

One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a temperature T. The space over the piston opens into the atmosphere. What work has to be performed in order to increase the gas volume isothermally under the piston $\eta$ times by slowly raising the piston? The friction of the piston against the cylinder walls is negligibly small.

Accepted Answer

$dw = F.dx = \left( P_{0} - P ight)A.dx = \left( P_{0} - P ight).dV $$${	herefore w = \int_{V}^{\eta V}{\left( P_{o} - \frac{nRT}{V} ight)dV} = P_{0}(\eta V - V) - RT.\ln\frac{V\eta}{V} }{P_{0}V(\eta - 1) - RT.\ln\eta = RT\left( \eta - 1 - \ln\eta ight)}$$

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