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2015-04-02T12:19:04+05:30

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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
P_n=P [\frac{V-v}{V} ]^n
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