NEET-XII-Physics
24: Kinetic Theory of Gases
- #37A vessel of volume V0 contains an ideal gas at pressure p0 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.Ans : `` \begin{array}{l}\text{Let}\mathit{\text{P}}\text{be the pressure and}\mathit{\text{n}}\text{be the number of moles of gas inside the}\,\mathrm{\,vessel\; at\; any\; given\; time\,}t.\text{}\\ \text{}\\ \text{Suppose a small amount of gas of}\mathit{\text{dn}}\text{moles is pumped out and the decrease in pressure is}\mathit{\text{dP.}}\\ \\ \text{Applying equation of state to the gas inside the vessel, we get}\\ (P-dP){V}_{o}=(n-dn)RT\\ \Rightarrow P{V}_{o}-dP{V}_{o}=nRT-dnRT\\ \text{But}P{V}_{o}=nRT\\ \Rightarrow {V}_{o}dP=dnRT...\left(1\right)\\ \text{The pressure of the gas taken out is equal to the inner pressure}\text{.}\\ \text{Applying equation of state, we get}\\ (P-dP)dV\text{=}dnRT\\ \Rightarrow PdV=dnRT...\left(2\right)\\ \text{From eq.}\left(1\right)\text{and eq.}\left(2\right)\text{, we get}\\ {V}_{o}dP=PdV\\ \Rightarrow \frac{dP}{P}=\frac{dV}{{V}_{o}}\\ \frac{dV}{dt}=r\\ \Rightarrow dV=rdt\\ \\ \Rightarrow dV=-rdt...\left(3\right)\left[\,\mathrm{\,Since\,}\,\mathrm{\,pressures\,}\,\mathrm{\,decreases\,},\,\mathrm{\,rate\,}\,\mathrm{\,is\,}\,\mathrm{\,negative\,}\right]\\ \text{Now,}\\ \frac{dP}{P}=\frac{-rdt}{{V}_{o}}\left[\,\mathrm{\,From\,}\,\mathrm{\,eq\,}.\left(3\right)\right]\\ \left(a\right)\\ {\text{Integrating the equation P = P}}_{0}\text{to P = P and time t = 0 to t = t, we get}\\ \underset{{P}_{o}}{\overset{P}{\int }}=\underset{0}{\overset{t}{\int }}\\ \Rightarrow \,\mathrm{\,ln\,}P-\,\mathrm{\,ln\,}{P}_{o}=-\frac{rt}{{V}_{o}}\\ \Rightarrow \,\mathrm{\,ln\,}\left(\frac{P}{{P}_{o}}\right)=-\frac{rt}{{V}_{o}}\\ \Rightarrow P={P}_{o}{e}^{\frac{-rt}{{V}_{o}}}\end{array}``
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`` \begin{array}{l}\left(b\right)\\ \text{P =}\frac{{P}_{o}}{2}\\ \frac{{P}_{o}}{2}={P}_{o}{e}^{\frac{-rt}{{V}_{o}}}\\ \Rightarrow {e}^{\frac{rt}{{V}_{o}}}=2\\ \Rightarrow \frac{rt}{{V}_{o}}=\,\mathrm{\,ln\,}2\\ \Rightarrow t=\frac{{V}_{o}\,\mathrm{\,ln\,}2}{r}\end{array}``
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