Q$6(15$ p$.)$ The surface of a steel gear made of steel containing initially $0\mathrm{.}18$ wt$.\%$ carbon is to be gas$-$

carburized at $927C.$ Calculate the time necessary $($in hours$)$ to increase the carbon content to $0\mathrm{.}35$ wt$.\%$

at $1\mathrm{.}0mm$ below the surface. Assume that the carbon content of the surface of the gear is $1\mathrm{.}20wt.\%.$

$($D $($carbon in $-$iron$)$ at $927C=1\mathrm{.}28{10}^{-11}\frac{m^2}{s})$

Use equation: $\frac{c_x-c_0}{c_s-c_0}=1-$erf$(\frac{x}{2\sqrt[\phantom{2}]{Dt}})$