Problem $6$ The solution for steady$-$state diffusion from a sphere into an infinite domain was discussed. The governing equation is:

$\frac{d}{dr}(r^2\frac{dC}{dr})=0$

The boundary conditions are:

$C=0$ when $r$

and

$-D\frac{dC}{dr}=F$ when $r=R$

The steady$-$state solution is then:

$C=\frac{F{R}^2}{dr}$

Consider now that the signaling molecule can only diffuse in a finite domain where the size of the open domain is given by $R_0,$ and the concentration at the edge of the domain is $C_0.$

$($A$)$ Replace the first boundary condition and solve for $C(r).$

$($B$)$ Compare the two solutions by sketching the concentration profiles.

$($C$)$ Is $\frac{F{R}^2}{dr}?$ Explain why $or$ why not.