In Smoluchowski's Theory of Coagulation we focus on an individual sphere and assume that other like particles diffuse toward it. Once they reach the sphere, they collide and form a new spherical aggregate. Once stuck together, the diffusing particles are removed from the solution. Thus, there exists a concentration gradient of particles from the free stream to the sphere. Assume the sphere has a radius of \(R_{o}\) and is surrounded by fluid containing \(n_{o}\) particles \(/ \mathrm{m}^{3}\), uniformly distributed. The particles have a diffusivity, \(D\), and are in dilute solution.

a. By simplifying the continuity equations in spherical coordinates, show that the diffusion equation reduces to:

\[\frac{\partial n}{\partial t}=D \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial n}{\partial r}\right)\]

b. What are the boundary and initial conditions for this problem? The fluid is essentially infinite in extent. (Remember we are interested in what is beyond \(r=R\) )

c. To solve the problem, we must use a trick or two. Setting:
\[y=(r-R) / R \quad \text { and } \quad C=r\left(n_{O}-n\right) / n_{O}=(y+1)\left(n_{O}-n\right) / n_{O} \text {, }\]
show that we can derive a similarity solution of the form:
\[n=n_{o}\left[1-\frac{R}{r}+\frac{R}{r} \operatorname{erf}\left(\frac{r-R}{\sqrt{4 D t}}\right)\right]\]

d. Using Fick's law, show that the flow (\# particles/time) of particles to the sphere's surface is:
\[N=4 \pi D n_{o} R\left[1+\frac{R}{\sqrt{\pi D t}}\right]\]

e. For \(t \gg R^{2} / D\) we can express the flow of particles as \(4 \pi D n_{o} R\). To determine the coagulation rate, we must account for the motion of our sphere. If the sphere is allowed to diffuse with diffusivity, \(D\), we effectively double the diffusion coefficient of all particles to \(2 D\). Finally, the number of contacts per unit volume of dispersion is just the number of contacts per particle multiplied by the number of particles per unit volume or:
\[N_{t}=8 \pi D R n_{o}^{2}\]
initially. If all the particles stick when they collide, the rate of change of the number of particles per unit volume, \(n\), is:
\[\frac{d n}{d t}=-8 \pi D R n^{2}\]
Integrate this equation (for constant \(D R\) ) and show that the characteristic coagulation time, \(T\), is given by:
\[T=\frac{1}{8 \pi D R n_{o}}\]

f. The Earth is roughly 4.5 billion years old and was formed by accretion in roughly \(500,000,000\) years. Assuming we needed to reduce the original number of planetesimals to a fraction of \(1 \times 10^{-6}\) within that time to reach our current size, what was the time constant for bombardment?