$***$Measuring effective gas diffusivity.$***$ The Loschmidt cell apparatus allows measuring the effective binary diffusion coefficient of two gases in porous media. This method of measurement is used in a wide range of fields including the development of fuel cell membranes, catalysts, and carbon dioxide sequestration processes ${?}^5.$The cell has two compartments, which are packed with a porous material, and each filled with a different gas, A & B$.$ Compartments are separated by a sliding barrier. The

experiment starts by removing the sliding barrier and a gas sensor measures the local concentration of $A$ as a function of time in the upper cell. There is no significant adsorption nor reaction. The diffusion is entirely one$-$dimensional.

a$.$ Show that the following PDE and BCs apply for the mole fraction of A$.$

$\frac{del{x}_A}{\text{d}\text{e}\text{l}\text{t}}=D_{AB}\frac{de{l}^2{x}_A}{del{z}^2}$

@ $z=+-L,$del$\frac{x_A}{d}$elz$=0A^6{x}_A=\frac{1}{2}+\frac{2}{}{\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{2n+1}sin[\frac{(2n+1)}{2}\frac{z}{L}]exp[-\frac{(2n+1{)}^2{}^2}{4}\frac{t{D}_{AB}}{L^2}]=t\frac{D_{AB}}{L^2}>0\mathrm{.}2x_A$~$=\frac{1}{2}+\frac{\sqrt[\phantom{2}]{2}}{}$exp$(-\frac{{}^2}{4}\frac{t{D}_{AB}}{L^2})D_{AB}$