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(V+v)=PV#

#=>P_1=(PV)/(V+v)=P/(1+v/V)#

If after 2nd stroke the pressure of expanded #V+v# volume of air becomes #P_2#

then

#P_2(V+v)=P_1V#

#=>P_2=(P_1V)/(V+v)=P_1/(1+v/V)=P/(1+v/V)^2#

So after n stroke the pressure will be

#=>P_n=P/(1+v/V)^n=P(1/(1+v/V))^n#