Suppose you take a bright red dye, and release a drop (at time t = 0) in a long, thin, horizontal tube of water. You then watch the drop spread to the right and to the left. You intuitively expect that the dye will initially be highly concentrated in a small area, giving a small but bright red dot, but will gradually spread along the water tube turning it progressively a lighter shade of red.

If you actually did this experiment with a drop of dye in water, the spread of the dye would almost certainly be due principally to convection not diffusion (i.e., the dye would be carried by small water currents, not by diffusion through the water).

However, let’s pretend we don’t know that.

It turns out that the diffusion of a dye like this can be described by the equation c(x, t) =

A

√

4πDt e

−x 2

4D t , where c is the concentration of the dye, t is time, x is the distance along the tube, A is a constant, and D is called the diffusion coefficient.

a. If c has units of µM, x has units of µm and t has units of s, what are the units of D? (Hint: remember that the exponent has to be dimensionless.)

b. Show that A has units of µM µm.

c. Set D = A = 1 (in the appropriate units) and plot c(x, t)
as a surface in three dimensions. Be very careful that you plot c only for t > 0; it’s probably safest to keep t well away from 0, so plot the surface for t > 0.1, say.
We plot c only for t > 0 as we are assuming that the drop of dye is released at t = 0.

d. For each fixed t, what does the cross-section of the c surface look like? Can you use these cross-sections to show how the dye is gradually spreading out?

e. For each fixed x, what does the cross-section of the c surface look like? Can you explain the shape of these crosssections?