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A vessel of volume `V_(0)` contains an ideal gas at pressure `p_(0)`and temperature T. Gas is continuously pumped out of this vessel at a constant volume-rate `dV/dt=r` keeping the temperature constant, The pressure inside the vessel. Find (a) the pressure of the gas as a function of time,(b) the time taken before half the original gas is pumped out. |
Answer» We have, ` dV/dt = r ` ` rArr dV = rdt ` Let the pressure pumped out gas = dp ` ` Volume of container = `V_0` At a pump dV amount of gas has been pumped out ` PdV = -(V_0) dP ` ` rArr Prdt = - (V_0) dP` ` rArr dp/P = (- rdt / V_0) ` On integration , we get ` P = (e^ (-rt/V_0))` ` Half of the gas been pumped out , pressure will be half ` i.e. 1 = (1/2 e ^ (-rt/ V_0))` ` rArr ` 1 n 2 = rt/ V_0 ` ` rArr t = In 2 xx (V_0/r )` |
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