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 of the gas being taken out equals 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.

Options

A. `2P_()e^(-rt//V_(0))`
B. `3P_()e^-rt//V_(0))`
C. `-P_()e^(-rt//V_(0))`
D. `P_(0)e^(-rt//V_(0))`

Answer

D

Explanation

The temperature of the gas is constant, therefore
`rArr P(dV)/(dt)+V(dp)/(dt+V(dP)/(dt)=0`
`rArr V(dP)/(dt)=-rP`
`rArr int_(P_(0))^(P)(dP)/(P)=-(1)/(V_(0))int_(0)^(1)rdt`
`rArr In (P)/(P_(0))=(-rt)/(V_(0))`
`rArr P=P_(0)e^(-rt//V_(0)`