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Given that

volume of vessel = V

initial pressure = P

temperature is constant = T

stroke volume of pump = v

When 1st "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₁. Now, the volume of gas (V-v) at pressure P occupies the whole volume of vessel (V) at pressure P₁. So

P(V-v) = P₁V

P₁ = P[(V-v)/V]

When 2nd "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₂. Now, the volume of gas (V-v) at pressure P₁ occupies the whole volume of vessel (V) at pressure P₂. So

P₁(V-v) = P₂V

P₂ = P₁ [(V-v) / V]

P₂ = [P (V-v) / V] [(V-v) / V]

P₂ = P [(V-v) / V]²

When 3rd "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₃. Now, the volume of gas (V-v) at pressure P₂ occupies the whole volume of vessel (V) at pressure P₃. So

P₂(V-v) = P₃V

P₃ = P₂ [(V-v) / V]

P₃ = P [(V-v) / V]² [(V-v) / V]

P₃ = P [(V-v) / V]³

So final pressure is**P₃ = P [(V-v) / V]³**

Similarly, if we remove n times volume v, then pressure will be

volume of vessel = V

initial pressure = P

temperature is constant = T

stroke volume of pump = v

When 1st "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₁. Now, the volume of gas (V-v) at pressure P occupies the whole volume of vessel (V) at pressure P₁. So

P(V-v) = P₁V

P₁ = P[(V-v)/V]

When 2nd "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₂. Now, the volume of gas (V-v) at pressure P₁ occupies the whole volume of vessel (V) at pressure P₂. So

P₁(V-v) = P₂V

P₂ = P₁ [(V-v) / V]

P₂ = [P (V-v) / V] [(V-v) / V]

P₂ = P [(V-v) / V]²

When 3rd "v" volume is taken out, then the pressure decreases in the vessel.

let it becomes P₃. Now, the volume of gas (V-v) at pressure P₂ occupies the whole volume of vessel (V) at pressure P₃. So

P₂(V-v) = P₃V

P₃ = P₂ [(V-v) / V]

P₃ = P [(V-v) / V]² [(V-v) / V]

P₃ = P [(V-v) / V]³

So final pressure is

Similarly, if we remove n times volume v, then pressure will be