A gas is enclosed in a vessel of volume `V` at a pressure `P`. It is being pumped out of the vessel by mean of a piston-pump with a stroke volume. `v`. What is the final pressure in the vessel after `'n'` strokes of the pump ? Assume temperature reamains constant.

Explanation

According to ideal gas equation `PV = nRT`, at constant temperature for a given mass, `PV = P^(1)V^(1)`
Now as stroke volume is `v` during 1st stroke fpr constant mass (say m) volume changes from `V` to `(V + v)` and so if pressure changes from `P` to `P_(1)`, the above equation yields
`PV = P_(1)(V + v)`, i.e., `P_(1) = P[(V)/(V + v)]` .......(1)
After the first storke, the gas left in the vessel has again volume `V` but at pressure `P_(1) ("with mass" m_(1)ltm)`. Now the second stroke will take place from these initial conditions and if `P_(2)` is the pressure of the gas in the cylinder at the end of `2`nd stroke,
`P_(1)V = P_(2)(V + v)`, i.e., `P_(2) = P_(1) = P[(V)/(V + v)]`

Substituting the value of `P_(1)` from equation `(1)` in the above `P_(2) = P_(1) = P[(V)/(V + v)]^(2)`
Repeating the same for `n` strokes, the pressure of the gas in the vessel after `n^(th)` stroke will be.
`P_(n) = P[(V)/(V + v)]^(n) = P[(1)/(1 + (v // V))]^(n)`