A colleague of yours is walking down the floor leading to your office, which is located mid$-$way along a very long $(100m)$ hallway. He is carrying a vial containing $10g$ of a noxious smelling gas. He trips $20m$ before reaching your $($open$!)$ office door, and the vial breaks.

The gas rapidly mixes vertically and horizontally within the hallway, which is $2m$ wide and $3m$ high. The human nose will detect the gas at concentrations greater than $10\frac{g}{l}.$ Assume an isotropic diffusion, $D=0\mathrm{.}05\frac{m^2}{s}.$

$($a$)$ What governing equation describes the evolution of gas concentration in the hall?

$($b$)$ At what time after the spill do you smell the gas?

$($c$)$ When does the smell, as perceived by humans, disappear from the hallway?

$C(x,t)=\frac{M}{A_{yz}*\sqrt[\phantom{2}]{4Dt}}$exp$(-\frac{x^2}{4Dt})$