Question

One mole of an ideal gas 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 isothermally the gas volume under the piston $\eta$ times by slowly raising the piston? Neglect friction.

Answer 1

Let the pressure of the gas under the piston be $P$ and $A$ be the cross-section area of the piston.

$\therefore$ $F+PA=P_{o}A$

$⟹F=(P_o-P)A$

Also $V_1=V$ and $V_2=\eta V$ (given)

$\therefore$ $W={\int }_{V_1}^{V_2}Fdx$

OR $W={\int }_{V_1}^{V_2}(P_o-P)Adx=$ ${\int }_{V_1}^{V_2}(P_o-P)dV$

Using $PV=nRT$ $⟹P=\frac{nRT}{V}$

$⟹$ $W={\int }_{V_1}^{V_2}{P}_{o}dV-$$nRT{\int }_{V_1}^{V_2}\frac{dV}{V}$

OR $W=P_o(V_2-V_1)-nRT$ $ln\frac{V_2}{V_1}$

OR $W=P_{o}V(\eta -1)-nRT$ $ln\left(\eta \right)$

$⟹$ $W=RT\left[\right(\eta -1)-n$ $ln\left(\eta \right)]$ $(\because P_{o}V=RT)$