Carburization of a gear surface. We wish to carburize a gear made of $1020$ steel to a carbon content of $0\mathrm{.}4\%$ at a depth of $0\mathrm{.}5mm$ below the surface. We will place the gear in a fumace at $900C$ where the atmosphere is controlled to make the surface concentration stay at $0\mathrm{.}9\%C.$ The diffusion coefficient parameters are $D_0=2{10}^{-3}\frac{m^2}{s}$ and $E_D=142k\frac{J}{m}ol.K.$

$($a$)$ How long will the process take at $900C?$

$($b$)$ Determine a temperature at which the process would be completed in half of the time.