A vessel of volume `V` is evacuated by means of a piston air pump. One piston stroke captures the volume `Delta V`. How many strokes are needed to reduce the pressure in the vessel `eta` times ? The process is assumed to be isothermal, and the gas ideal.

Answer

`n = (In eta)/(In(1+DeltaV//V))`

Explanation

`PV = constant`
`PV = P_(1) (V +DeltaV)`
`P_(1) = (PV)/(V +DeltaV) ..(1)`
`P_(1)V = P_(2)(V +DeltaV)`
`P_(2) = (P_(1)V)/(V +DeltaV) ..(2)`
from (1) and (2)
`P_(2) = P [(V)/(V +DeltaV)]^(2)`
similarly `P_(n) = P [(V)/(V + DeltaV)]^(n)`
according to problem
`(P_(n))/(P) = (1)/(eta)`
`(1)/(eta) = ((V)/(V + DeltaV))^(n)`
`- ln(eta) = nln ((V)/(V +DeltaV))`
`(ln eta)/(n = ln(1+(DeltaV)/(V)))`