Hydrogen gas is used in industry to manufacture a sheet material. At the end of the process hydrogen remains in the solution in the material. A large sheet of material $40mm$ thick contains $H_2$ having a uniform concentration of $3\mathrm{.}1kg-$molm. The sheet is exposed to a fluid streams that causes the concentration of the dissolved $H_2$ to be reduced to zero at both surfaces.

This condition is maintained constant thereafter. The diffusivity of hydrogen is $D_A=8{10}^{-7}\frac{m^2}{s}.$

$($a$)$ Determine the concentration of dissolved $H_2$ after $190$ seconds $($practically this is the concentration at the center of the sheet$)$

$($b$)($Safety$)$ To ensure safe marketing of the material, the hydrogen concentration must be reduced to $0\mathrm{.}002kg-mo\frac{l}{m^3},$ what time will it take to reduce the concentration to this value?

Data:

$C_{A0}=3\mathrm{.}1kg-\frac{\text{m}\text{o}\text{l}}{m^3}$;$C_{A(infinite)}=0.$