# Question #f7837

Mar 21, 2017

By applying Boyle's law of gas we can easily determine the pressure of air in the closed vessel of volume $V$ after n stroke of the piston of the vacuum pump connected with the vessel,if the volume of air exhausted in each stroke is $v$

Given that the original pressure of V volume of air in the vessel is P. In the first stroke it is expanded to a volume $V + v$,where v represents the inner volume of the barrel of the piston.If the expanded air acquires pressure ${P}_{1}$ during first stroke,then by Boyle's law we have

${P}_{1} \left(V + v\right) = P V$

$\implies {P}_{1} = \frac{P V}{V + v} = \frac{P}{1 + \frac{v}{V}}$

If after 2nd stroke the pressure of expanded $V + v$ volume of air becomes ${P}_{2}$

then

${P}_{2} \left(V + v\right) = {P}_{1} V$

$\implies {P}_{2} = \frac{{P}_{1} V}{V + v} = {P}_{1} / \left(1 + \frac{v}{V}\right) = \frac{P}{1 + \frac{v}{V}} ^ 2$

So after n stroke the pressure will be

$\implies {P}_{n} = \frac{P}{1 + \frac{v}{V}} ^ n = P {\left(\frac{1}{1 + \frac{v}{V}}\right)}^{n}$