The diffusion equation for a chemical concentration $C(r, t)$ in three dimensions which depends only on the radial coordinate $r$ is

$$C_{t}=D \frac{1}{r^{2}}\left(r^{2} C_{r}\right)_{r}$$

The general similarity solution of this equation takes the form

$$C(r, t)=t^{\alpha} F(\xi), \quad \xi=\frac{r}{t^{\beta}},$$

where $\alpha$ and $\beta$ are to be determined. By direct substitution into $(*)$ and the requirement of a valid similarity solution, find one relation involving the exponents. Use the conservation of the total number of molecules to determine a second relation. Comment on the relationship between these exponents and the ones appropriate to the similarity solution of the one-dimensional diffusion equation. Show that $F$ obeys

$$D\left(F^{\prime \prime}+\frac{2}{\xi} F^{\prime}\right)+\frac{1}{2} \xi F^{\prime}+\frac{3}{2} F=0$$

and that the relevant solution describing the spreading of a delta-function initial condition is $F(\xi)=A \exp \left(-\xi^{2} / 4 D\right)$, where $A$ is a suitable normalisation that need not be found.