Spreading burst

Your Turn 4d on page 121 claimed that, in one-dimensional diffusion, an observer sitting at a fixed point sees a transient pulse of concentration pass by. Make this statement more precise, as follows: Write the explicit solution of the diffusion equation for release of a million particles from a point source in three dimensions. Then show that the concentration measured by an observer at fixed distance r from the initial release point peaks at a certain time.

a. Find that time, in terms of r and D.

b. Show that the value of concentration at that time is a constant times r−3, and evaluate the constant numerically