**Brownian Motion**

*Michael Fowler, U. *

See Applet here!

See Movie here!

In 1827 Robert Brown, a well-known botanist, was studying
sexual relations of plants, and in particular was interested in the particles
contained in grains of pollen. He began
with a plant (*Clarckia** pulchella*) in
which he found the pollen grains were filled with oblong granules about 5
microns long. He noticed that these
granules were in constant motion, and satisfied himself
that this motion was not caused by currents in the fluid or evaporation. Smaller spherical grains, which at first he
took to be oblongs end-on, but later realized weren’t, had even more vigorous
motion. He thought at first that he was
looking at the plant equivalent of sperm—they were jiggling around because they
were alive. To check this, he did the
same experiment with dead plants. There
was just as much jiggling. Perhaps all
organic matter, everything that ever was alive, still contained some mysterious
life force at this microscopic level?
Sure enough, he found the movement in tiny fragments of fossilized
wood! But then he went on to find it in
matter that never was alive—tiny particles of window glass, and even dust from
a stone that had been part of the Sphinx. The movement evidently had nothing to
do with the substance ever being alive or dead, much to Brown’s surprise. So what was causing it? Perhaps it *was* evaporation currents, or the incident light energy, or just
tiny unnoticed vibrations. But none of
these explanations was very satisfactory.

Half a century later, a new possible explanation emerged. The kinetic theory of heat developed by Maxwell, Boltzmann and others was gaining credence. If all the molecules in the fluid were indeed in vigorous motion, maybe these tiny granules were being moved around by this constant battering from all sides as the fluid molecules bounced off. But there was a problem with this explanation: didn’t it violate the second law of thermodynamics? It had been well established that energy always degrades, as friction slows movement kinetic energy goes to heat energy. This seemed to be the other way round—the molecular battering was certainly disorganized heat energy, but when the granule moved it had evidently gained kinetic energy. Since many scientists regarded the second law as an absolute truth, they were very skeptical of this explanation.

In 1888, French experimentalist Léon Gouy investigated the movement in detail, finding it to be more lively in low viscosity liquids. He established that it was unaffected by intense illumination or by strong electromagnetic fields. Despite the second law, Guoy believed—correctly—the random motion was indeed generated by thermal molecular collisions.

It’s easy to see the Brownian movement, or Brownian motion (it’s called both) by looking through a microscope at tobacco smoke in air. We have a movie here.

In 1905, Einstein published a theoretical analysis of Brownian motion. He saw it as a crucial test of the kinetic theory, even of the atomic/molecular nature of matter. Previous discussions of the phenomenon had all been qualitative—Einstein demonstrated that careful observation of Brownian motion could reveal the actual size of molecules, that is, find Avogadro’s number.

If the results of such experiments were consistent with other estimates of Avogadro’s number, based on unrelated phenomena such as gas viscosity measurements and van der Waal’s equation fit to isotherms of real gases, it would be a powerful argument for the kinetic theory. On the other hand, if there was real disagreement then the kinetic theory was in serious trouble.

Einstein’s approach was based on an analogy with osmosis:
recall that osmosis involves one substance (the solute) dissolved in another,
the solvent. The solute has larger
molecules. Now imagine a container
divided in two by a semipermeable membrane, meaning
the solvent molecules can pass through the tiny holes in it, but the solute
molecules are too big to get through.
Suppose on one side of the membrane there’s pure solvent, on the other
side solvent plus solute (assumed fairly dilute), and initially the pressure on
the two sides of the membrane is the same.
The pressure on the side with solute is generated by both solvent and
solute molecules bouncing off the membrane, so necessarily the rate at which *solvent* molecules are hitting the
membrane on this side is less than on the other side. Some percentage of the solvent molecules
hitting the membrane pass through the little holes, so what will happen is that
more will pass through from the pure solvent side, and gradually pressure will
build up on the solvent + solute side until equilibrium is reached, meaning equal
numbers of solvent molecules going each way on average.

*Einstein’s insight was
that a liquid containing a large number of tiny identical particles, such as
those observed in Brownian motion, was really no different from
a solvent containing solute molecules*. True, the Brownian particles were a lot
bigger than molecules, but they were buzzing around, and would therefore bounce
off the walls of a container, generating pressure. The formal analysis should
be the same: the kinetic theory, with equipartition of energy, predicted they
would have kinetic energy 1.5*k _{B}T*. If the
concentration of particles varied spatially, they would flow to even it
out.

Here again he used an osmosis analogy: think of a cylindrical container, with a semipermeable membrane which is like a piston, free to move. The solute concentration is initially greater to the left of the piston.

From the previous discussion, it’s clear that the solvent will flow to the left, raising the pressure there, so the piston will move to the right. The solute molecules cannot cross the piston, so the piston will move until the solute concentrations on the two sides are equal.

The rather surprising result is that if one assumes
equipartition of energy, *the pressure on
the piston from the solute on one side is the same as if those solute molecules
were freely moving in a vacuum*.

Their greatly reduced mean free path *doesn’t matter*: the pressure depends only on the concentration in
the immediate neighborhood of the piston, and the speed of the molecules. And, this is equally true if the solute
molecules are replaced by tiny but macroscopic spheres. At least, this is what Einstein asserted, and
he gave a formal proof based on an evaluation of the free energy, assuming a
dilute system (meaning interactions between the spherical granules could be
neglected).

So we can think of the little spheres as *moving freely through space* (!), and
although their paths will actually be very different, calculations of local
pressure based on this should be correct: the pressure on the walls from the
granules is therefore given by the ideal gas law, that is,

_{}

where the average kinetic energy is
written *W*, and—if the kinetic theory
is correct—this should equal 1.5*k _{B}T*.

So how is this to be
checked experimentally? As we shall see
in a moment, the first experiment used uniformly sized tiny *spheres* in place of granules. The first
obvious thought is that if _{}is predicted to be 1.5*k _{B}T*,
perhaps one can measure the jiggling velocity of the tiny sphere a few times
and take an average. This, however, misunderstands the nature of the motion: a
molecule will bounce off the sphere around 10

A slightly less direct method of finding _{} is needed. Now, it is well-known that in an isothermal
atmosphere of an ideal gas under gravity the density falls off exponentially
with height, this is established by balancing the gravitational force on a thin
horizontal slice against the pressure difference between top and bottom. It
occurred to the French experimentalist Jean Perrin that this same argument
should apply to a “gas” of tiny uniform spheres in a fluid: their pressure is
generated by the Brownian motion. In
1908, he chose gamboge, an emulsion used for water
color, which contains bright yellow spheres of various sizes. By various ingenious tricks described in his
book he was able to separate out spheres all close to the same size. He was able to measure the size, he knew the
density—and that of the “solvent”—so he could compute the gravitational
pull. He could also measure the decrease
in density with height in isothermal equilibrium.

The calculation is as follows: for a horizontal slice of
thickness *dh*, with *n* spheres per unit volume, each of
volume *φ* and density Δ, in
a liquid of density *δ *(I’m using
Perrin’s notation here), the gravitational downward force on the slice is *ndh** φ* (Δ - * δ*), this is balanced by
the pressure difference:

_{}

This is easily integrated to give the exponential vertical density profile.

_{}

Perrin could establish by observation and measurement every
term in this equation except *W*, so
this was a way of measuring *W*,
assuming of course the validity of the kinetic theory.

Now equating *W* to
1.5*k _{B}T* gives a value for
Boltzmann’s constant, and hence via the known gas constant

Perrin repeated the experiment with a wide variety of
different substances, the experiments were very challenging, his
results for Avogadro’s number were consistently between 5×10^{23}
and 8×10^{23}. He remarked that
for the largest granules behaving like a perfect gas, *a gram molecule would weigh 200,000 tons*! The results were consistent with the other
quite different ways of finding Avogadro’s number, and these experiments
convinced even the most recalcitrant anti-atomic theory skeptics. The kinetic theory was fully established.

In 1908, Langevin gave a more direct treatment of Brownian motion. He focused on following one particle as it jiggled around. We’ll follow him in restricting the motion to one dimension—assuming the molecular collisions driving the motion are completely random, the motions in the three directions are uncorrelated, so can be treated separately and added. Finally, we’ll neglect gravity and any other external force fields.

Let’s assume, then, that we’re tracking a small spherical
object, of mass *m* and radius *a*.
It will experience a viscous drag force -6*πaηv* (Stokes’ formula).
We’ll denote the random thermal molecular collision force by *X*, which clearly averages to zero.

Assuming the equipartition of energy applies also to the kinetic energy of our sphere,

_{}

where the average is over a long time.

The equation of motion *ma*
= *F* is:

_{}

Multiplying throughout by *x*,

_{}

which can be written

_{}

Now we’ll average over a long time:

_{}

Since *X* is random,
_{} Also, _{} The operations of
averaging and taking the time derivative commute, so we can write the equation:

_{}

To solve this differential equation, write _{}

The equation becomes:

_{}

The solution is

_{}

For the actual systems examined experimentally, the exponential term dies off in far less than a microsecond, so for a particle beginning at the origin:

_{}

so

_{}_{}

Hence by doing multiple experiments and averaging,
Boltzmann’s constant *k _{B}*
can be found, and from that Avogadro’s number, as before.

Note that the constancy of _{}also appears in discussions of random molecular motion and
the random path—this is all the same thing.

** Exercises**:

1. Estimate
the decay time of the exponential term in the integrated expression for *y*(*t*) above. You’ll need to find the viscosity of water, and
estimate the sphere size as a few microns.

2. Estimate how rapidly density of yellow spheres drops with height in Perrin’s “atmosphere”.

3. Notice the average distance traveled in the last equation above depends on the kinetic energy, the size, and the viscosity. This means a tiny lead sphere would diffuse the same distance, on average, as a tiny sphere of oil of the same size. But isn’t the lead moving a lot more slowly, since it has the same average kinetic energy? Explain.

For Brown’s work, see *A
Source Book in Physics*, W. F. Magie, Harvard,
1963, page 251, where several pages from the original pamphlet are reproduced.

Albert Einstein: *Investigations
on the Theory of the Brownian Movement*,

Jean Perrin: *Brownian Motion and Molecular Reality*,

Langevin’s paper in translation: *Am*. *J*.
*Phys*. **65** (11), November 1997, 1079.

Wolfgang Pauli: *Pauli
lectures on Physics Volume 4, Statistical Mechanics*.