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

We know that $PV=nRT$

Where $n$ is the number of molecules of gas initially. Temperature $T$ is given Constant throughout the process. V is also constant as the vessel is fixed.

So $\frac{P}{n}=\text{Constant}$ ...$\left(1\right)$

Now, After $1st$ stroke, molecules of gas remained in cylinder $=n(1-\frac{1}{100})$

After $2nd$ stroke molecules of gas remain in cylinder $=n(1-\frac{1}{100}{)}^2$

After $nth$ stroke molecules of gas remain in cylender $=n(1-\frac{1}{100}{)}^n$

Let's assume the final pressure after $nth$ stroke is $P^{\prime }$, So by using equation $\left(1\right)$ we can write that $\frac{P}{n}=\frac{P^{\prime }}{n(1-\frac{1}{100}{)}^n}$

$\Rightarrow P^{\prime }=P(1-\frac{1}{100}{)}^n$