Partitions II and IV are filled with a $0\mathrm{.}05M$ concentration of a fluorescent solute. At time $t=0,$ the partitions are opened, and the concentration is allowed to start evening out by diffusion. The diffusion coefficient is $2{10}^{-4}c\frac{m^2}{sec},$ and you may assume that the total width of the vessel containing the solution is effectively infinite $($regions I and V can be considered to be infinitely wide$).$

a$.$ Find analytical expressions for the solute concentration as a function of position and time. At what time is the concentration at $x=0$ equal to $0\mathrm{.}01M?$ Plot the concentration profile in the $x-$direction at this time.

b$.$ Numerically solve the diffusion equation, and using this numerical solution, plot the concentration profile in the $x-$direction at the time the concentration at $x=0$ equals $0\mathrm{.}01M,$ and compare with your analytical solution. Do you observe any differences between the analytical and numerical solutions? Explain why yes, or why not.

$($note: $-$erf$(x)=$erf$(-x)$