Question

A vessel of volume $V_0$ contains an ideal gas at pressure $P_0$ and temperature T. Gas is continuously pumped out of this vessel at a constant volume -rate $dV/dt=r$ keeping the temperature constant. The pressure inside the vessel. Find (a) The pressure of the gas as a function of time, (b) the time is taken before half the origin gas is pumped out.

Answer 1

$\frac{dV}{dt}=r\Rightarrow dV=rdt$

Let the pumped out gas pressure dp

Volume of container $=V_0$ At a pump dv amount of gas has been pumped out.

$Pdv=-V_{0}df\Rightarrow P_{v}df=-V_{0}dp$

$\Rightarrow {\int }_P^P\frac{dp}{p}=-{\int }_0^1\frac{dtr}{V_0}\Rightarrow P=P{e}^{-rt/V_0}$

Half of the gas has been pump out, Pressure will be half $=\frac{1}{2}{e}^{-vt/V_0}$

$\Rightarrow \ln 2=\frac{rt}{V_0}$

$\Rightarrow t={\ln }^2\frac{{\gamma }_0}{r}$