Consider a bar of material of length, $L,$ with uniform initial solute concentra$-$

tion, $C_o.$ Suppose additional solute is absorbed at the left and right surfaces

and distributed through the bar by diffusion. The left and right surfaces are

maintained at concentrations $C_L$ and $C_R,$ respectively $(C_L{C}_R).$ Assume

the diffusivity, $D,$ is constant. Provide an analytical expression for the solute

concentration at any location in the bar for long times $($times long enough for

diffusion to extend along the entire length, $L).$

Suggestion: To use one of the standard $1-$dimensional solutions we have

considered, you will need to use a dimensionless concentration that is zero on

the boundaries. You can get a dimensionless concentration to be zero on the

boundaries by subtracting the steady$-$state concentration profile from the time$-$

dependent concentration and dividing the result by a similar scaling factor.