A gas is enclosed in a vessel of volume V at a pressure P. It is being pumped out of the vessel by means 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 remains constant.

A gas is enclosed in a vessel of volume V at a pressure P. It is being

1.	A gas is enclosed in a vessel of volume V at a pressure P. It is being pumped out of the vessel by means 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 remains constant.
Answer» Solution :According to ideal gas equation PV = `mu` RT, at CONSTANT temperature for a given MASS, `PV=P^(1)V^(1)` Now as STROKE volume is VM 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) 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)` fromequation (1) in the above `P_(2)=P[V/(V+v)]^(2)` Repeatingthe same for n stroks, 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)`.