Question

The infinite number of the flask is connected to one another as shown. The volumes and pressure in each flask vary as shown above. The stopcocks are initially closed. Then find the common pressure in atm when all the stopcocks are opened if (P = 30 atm).

Answer 1

Solution:-

Let $n_T$ be the total no. of moles.

$\therefore n_T=n_1+n_2+.........\infty .....\left(1\right)$

From ideal gas equation,

$PV=nRT$

$\Rightarrow n=\frac{PV}{RT}$

From ${eq}^n\left(1\right)$, we have

$\frac{P_T{V}_T}{RT}=\frac{P_1{V}_1}{RT}+\frac{P_2{V}_2}{RT}+........\infty$

$\frac{P_T{V}_T}{RT}=\frac{1}{RT}(P_1{V}_1+P_2{V}_2+..........\infty )$

$\Rightarrow P_T{V}_T=P_1{V}_1+P_2{V}_2+..............\infty$

Now according to the question-

$P_T{V}_T=2P.V+P.\frac{V}{2}+\frac{P}{2}.\frac{V}{4}+...........\infty$

$\Rightarrow P_T{V}_T=PV(2+\frac{1}{2}+\frac{1}{8}+.........\infty )$

As the above series is in G.P..

$\therefore P_T{V}_T=PV\left(\frac{2}{1-\frac{1}{4}}\right)$

$\Rightarrow P_T{V}_T=\frac{8}{3}PV.....\left(2\right)$

Now

$V_T=V_1+V_2+............\infty$

$V_T=V+\frac{V}{2}+..........\infty$

$\Rightarrow V_T=V(1+\frac{1}{2}+\frac{1}{4}+..............\infty )$

$\Rightarrow V_T=V\left(\frac{1}{1-\frac{1}{2}}\right)=2V$

Substituting the value of $V_T$ in ${eq}^n\left(2\right)$, we have

$P_T.2V=\frac{8}{3}PV$

$\Rightarrow P_T=\frac{4}{3}P$

Given that $P=30atm$

$\therefore P_T=\frac{4}{3}\times 30=40atm$

Hrence the common pressure in atm when all the stopcocks are opened is $40$.