Answer
$t=2.3\times10^{6}s$ (about 27 days)
Work Step by Step
First of all, let's apply Fick's law of diffusion $m=\frac{(DA\Delta C)t}{L}$ to find the time.
$m=\frac{(DA\Delta C)t}{L}=>t=\frac{mL}{DA\Delta C}=\frac{mL}{DA(C_{2}-C_{1})}$
$t=\frac{mL}{DA(C_{2}-0)}=\frac{mL}{DAC_{2}}-(1)$
Let's apply the ideal gas law $PV=nRT$ to find the value of $C_{2}$
$PV=nRT=\frac{m}{M}RT=>P=(\frac{m}{V})\frac{RT}{M}=\frac{\rho RT}{M}$
$\rho=\frac{PM}{RT}=C_{2}$ ; Let's plug known values into this equation.
$C_{2}=\frac{(2400\space Pa)(0.018\space kg/mol)}{(8.31\space J/mol\space K)(293\space K)}=1.77\times10^{-2}kg/m^{3}-(2)$
Substituting values into equation (1) gives,
$$t=\frac{(2\times10^{-3}kg)(0.15\space m)}{(2.4\times10^{-5}m^{2}/s)(3\times10^{-4}m^{2})(1.77\times10^{-2}kg/m^{3})}$$
$t=2.3\times10^{6}s$ (about 27 days)
