This problem addresses the process by which an inhomogeneous alloy becomes homogenized. The

initial concentration as a function of position $x$ is given by the following sinusoidal distribution $($just like

we discussed in lecture$)$:

$C\frac{?}{b}=ar(C)+{}_{0}sin\frac{x}{l}$

The concentration evolves with time as:

$C\frac{?}{b}=ar(C)+{}_{0}sin(\frac{x}{l})exp\frac{-t}{}$

You can notice that at long times $($for large values of $t),$ the concentration becomes $\frac{?}{\text{b}\text{a}\text{r}}(C).$ It is uniform

throughout, namely the concentration has homogenized.

The time constant $$ tells you the rate at which this homogenization takes place.

Use Fick's second law to find $.$ Your answer can be in terms of $D_B,$ and $l.$