Example $1.$ Unsteady$-$State Diffusion in a

Slab of Agar Gel

A solid slab of $5\mathrm{.}15$ wt $\%$ agar gel at $278K$ is $10\mathrm{.}16mm$ thick and contains a

uniform concentration of urea of $0\mathrm{.}1$kgmo$\frac{l}{m^3}.$ Diffusion is only in the $x$

direction through two parallel flat surfaces $10\mathrm{.}16mm$ apart. The slab is

suddenly immersed in pure turbulent water so that the surface resistance

can be assumed to be negligible; i$.$e$.,$ the convective coefficient kc is very

large. The diffusivity of urea in the agar is $4\mathrm{.}72{10}^{-10}\frac{m^2}{s}.$

$($a$)$ Calculate the concentration at the midpoint of the slab from

the surface$)$ and $2\mathrm{.}54mm$ from the surface after $10h.$

$($b$)$ If the thickness of the slab is halved, what would be the midpoint

concentration in $10h?$