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An ideal gas is enclosed in a container ...

An ideal gas is enclosed in a container of volume V at a pressure P. It is being pumped out of the container by using a pump with stroke volume V. What is final pressure in container after n-stroke of the pump? (assume temperature remains same)

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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 for 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 stroke, 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 2nd stroke,
`P_(1)V=P_(2)(V+V)`,
i.e., `P_(2)=P_(1)[V/(V+V)]`
Substituting the value of `P_(1)` from equation (1) in the above `P_(2)=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)`.
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