An ideal gas $A$ is kept in a spherical container at a pressure of $1\mathrm{.}1$ bar. The rigid container has a diameter

of $D$ of $150cm$ and a thickness $$ of $0\mathrm{.}1cm.$ The wall of the container is permeable to A with permeability

constant $P_{AB}=9{10}^{-13}$kmo$\frac{l}{m}s\frac{.}{b}ar$ ; therefore, some $A$ is lost to air during time. The mass flux of $A$

through the wall of the container is given as$,$

$N_A=P_{AB}\frac{P_{A,1}-P_{A,2}}{},(kmo\frac{l}{m^2}(s))$

Here, $P_{A,1}$ and $P_{A,2}$ are the partial pressures of $A($bar$)$ in the container and in the air, respectively. Here, we

can assume that the partial pressure of $A$ in the air is negligibly small.

a$)$ Make component mass balance for the system at unsteady$-$state conditions.

b$)$ Obtain an expression for the variation of the pressure in the container with respect to time $P_{A,1}(t).$

c$)$ Calculate how much time it takes for the tank pressure to drop to $1$ bar from the initial value of $1\mathrm{.}1$ bar.

Gas constant $R=\frac{0\mathrm{.}08314}{b}ar.\frac{m^3}{k}$mol.$K$; Temperatures in the tank and air $=25C,$