Show that the concentration $C(\mathbf{x}, t)$ of a diffusible chemical substance in a stationary medium satisfies the partial differential equation

$$\frac{\partial C}{\partial t}=\nabla \cdot(D \nabla C)+F$$

where $D$ is the diffusivity and $F(\mathbf{x}, t)$ is the rate of supply of the chemical.

A finite amount of the chemical, $4 \pi M$, is supplied at the origin at time $t=0$, and spreads out in a spherically symmetric manner, so that $C=C(r, t)$ for $r>0, t>0$, where $r$ is the radial coordinate. The diffusivity is given by $D=k C$, for constant $k$. Show, by dimensional analysis or otherwise, that it is appropriate to seek a similarity solution in which

$$C=\frac{M^{\alpha}}{(k t)^{\beta}} f(\xi), \quad \xi=\frac{r}{(M k t)^{\gamma}} \quad \text { and } \quad \int_{0}^{\infty} \xi^{2} f(\xi) d \xi=1$$

where $\alpha, \beta, \gamma$ are constants to be determined, and derive the ordinary differential equation satisfied by $f(\xi)$.

Solve this ordinary differential equation, subject to appropriate boundary conditions, and deduce that the chemical occupies a finite spherical region of radius

$$r_{0}(t)=(75 M k t)^{1 / 5}$$

[Note: in spherical polar coordinates

$$\left.\nabla C \equiv\left(\frac{\partial C}{\partial r}, 0,0\right) \quad \text { and } \quad \nabla \cdot(V(r, t), 0,0) \equiv \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} V\right) \cdot\right]$$