Question #41989

An ideal monatomic gas of molar mass $M$ undergoes a cyclic process. During the high-pressure isothermal expansion phase at temperature $T_0$, the volume increases from $V_0$ to $2V_0$ at a constant volumetric rate $\frac{dV}{dt} = C$. Simultaneously, a microscopic valve of area $A$ is opened, allowing gas to effuse into a vacuum. The valve is closed during the rest of the cycle. If the effusion rate is given by $\frac{dn}{dt} = -\frac{P A}{\sqrt{2\pi M R T}}$, what is the ratio of the work done by the gas during this expansion phase to the work that would have been done if the valve remained closed? (Let $\beta = \frac{A}{C} \sqrt{\frac{R T_0}{2\pi M}}$)