Question

A gas is enclosed in a vessel of volume $V$ at a pressure $P$. It is being pumped out of the vessel by means of a piston pump with a stroke volume $\frac{V}{100}$. What is the final pressure in the vessel after $n$ strokes of the pump? Assume no change in temperature during pumping out of gas.

Answer 1

Given initial pressure $=P$ and volume $=V$

Let the initial temperature be $T$

So, $PV=nRT$

Stroke volume of tyhe pump $=\frac{V}{100}$

Assuming an isothermal process.

After pumping out gas of volume $\frac{V}{100}$ out of $V,$ the no. of moles left in the container $=n\left(\frac{V-\frac{V}{100}}{V}\right)=\frac{99n}{100}$

As volume $V$ of the container is constant,

$\Rightarrow P_1=P\times \frac{99}{100}$

Again when gas is pumped out pressure will become $P_2=P_1\times \frac{99}{100}=P{\left(\frac{99}{100}\right)}^2$

After $n$ pumping pressure inside container $=P{\left(\frac{99}{100}\right)}^n$

Hence, the answer is $P{\left(\frac{99}{100}\right)}^n.$