4.4. More about diffusion[[Student version, December 8, 2002]] 115
fluid.
Moreover, what works for a pollen grain holds equally for the individual molecules in a fluid.
They too will wander from their positions at any initial instant. We don’t need to see individual
molecules to confirm this experimentally. Simply release a large numberNof ink molecules at
one point, for example, with a micropipette. Each begins an independent random walk through
the surrounding water. We can come back at timetand examine the solution optically using a
photometer. The solution’scolorgives the number densityc(r)ofink molecules, which in turn
allows us to calculate the mean-square displacement〈r(t)^2 〉asN−^1
∫
d^3 rr^2 c(r). By watching the
ink spread, we not only can verify that diffusion obeys the law Idea 4.6, but also can find the
value of the diffusion constantD.For small molecules, in water, at room temperature, one finds
D≈ 10 −^9 m^2 s−^1 .Amore useful form of this number, and one worth memorizing, isD≈ 1 μm^2 /ms.
Example Pretend that the interior of a bacterium could be adequately modeled as a sphere
of water of radius 1μm,and a cell of your body as a similar sphere of radius 10μm.
About how long does it take for a sudden supply of sugar molecules at, say, the
center of the bacterium to spread uniformly throughout the cell? What about for a
cell in your body?
Solution: Rearranging Equation 4.6 slightly and substitutingD=1μm^2 /ms,we
find that the time is around (1μm)^2 /(6D)≈ 0. 2 msfor the bacterium. It takes a
hundred times longer for sugar to spread through the bigger cell.The estimage just made points out an engineering design problem that larger, more complex
cells need to address: Although diffusion is very rapid on the micrometer scale, it very quickly
becomes inadequate as a means of transporting material on long scales. As an extreme example,
youhavesome single cells, the neurons that stretch from your spinal cord to your toes, that are
about a meter long! If the specialized proteins needed at the terminus of these nerves had to arrive
there from the cell body by diffusion, you’d be in trouble. Indeed, many animal and plant cells
(not just neurons) have developed an infrastructure of “plumbing,” “highways,” and “trucks,” all
to carry out such transport (see Section 2.2.4). But on the subcellular, 1μmlevel, diffusion is fast,
automatic, and free. And indeed, bacteria don’t have all that transport infrastructure; they don’t
need it.
4.4.2 Diffusion obeys a simple equation
Although the motion of a colloidal particle is totally unpredictable, Section 4.1.2 showed that a
certainaverageproperty of many random walks obeys a simple law (Equation 4.5a on page 104).
But the mean-square displacement is just one of many properties of the full probability distribution
P(x, t)ofparticle displacements after a given timethas passed. Can we find any simple rule
governing thefulldistribution?
Wecould try to use the binomial-coefficient method to answer this question (see the Example
on page 101). Instead, however, this section will derive an approximation, valid when there are
very many steps between each observation. (Section 4.6.5′on page 134 explores the validity of
this approximation.) The approximation is simpler and more flexible than the binomial coefficient
approach, and will lead us to some important intuition about dissipation in general.
It’s possible experimentally to observe the initial position of a colloidal particle, watch it wan-