Develop the solution of Fick's equation for Thin Films, $($for diffusion coefficient cte.$),$ starting

from the same $2$nd Fick's law:

$\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{t}}=D\frac{de{l}^{2}C}{del{x}^2}$

Using boundary conditions:

For $|x|>0,C0,$ as $,t$

For $x=0,C,$ as $,t0$

To get to:

$C(x,t)=\frac{C_i{x}^{\text{'}}}{2\sqrt[\phantom{2}]{D^{\text{'}}t}}$exp$[\frac{-x^2}{4D^{\text{'}}t}]$