ThermodynamicsHard
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.
Options
A.$RT\left( \eta - 1 - \ln\eta \right)$
B.$RT\left( 1 - \eta + \ln\eta \right)$
C.$RT\ln\eta$
D.$- RT\ln\eta$
Solution
$dw = F.dx = \left( P_{0} - P \right)A.dx = \left( P_{0} - P \right).dV $$${\therefore w = \int_{V}^{\eta V}{\left( P_{o} - \frac{nRT}{V} \right)dV} = P_{0}(\eta V - V) - RT.\ln\frac{V\eta}{V} }{P_{0}V(\eta - 1) - RT.\ln\eta = RT\left( \eta - 1 - \ln\eta \right)}$$
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