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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 vesse] has again volume V but at pressure `P_(1) ` (with mass `m_(1) lt ` m). 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) = [ (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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