**Brownian
Motion**

In the phenomenon of
diffusion a concentration of some substance – a little concentration of
perfume, for example – spreads out in such a way that the radius *R* of the concentration grows like _{}, where *t* is the
elapsed time, or more simply

_{}

The
constant of proportionality is *2D*,
where *D* is the so-called *diffusion constant*.

In
a random walk back and forth along some axis, consisting of many random steps,
the expected mean square displacement X in time t has the same behavior, i.e.

_{}

(Recall that E means “expected value”, a kind of
idealized average.) In his 1905 dissertation,
Albert Einstein suggested that these two phenomena are* the same*. That is, at the
molecular level, the way diffusion works is that tiny molecules move
randomly. Since there seems no reason
why this behavior, if it exists, should not apply to all particles, even those
much bigger than single molecules, Einstein also suggested that the observable
Brownian motion should obey this law.
This is experimentally testable:
we sample the squared displacement of many Brownian particles as a
function of elapsed time t and we see if the sample mean of this quantity obeys

_{}

This we can do by plotting and looking for a
straight line relationship. If the data
lie on a line, the slope of the line determines the diffusion constant *D*.

The sample mean < > will
be a good approximation to the expected value E if we use enough particles, but we must expect that our sample
means will scatter about the (true) expected value with a standard deviation
proportional to _{}, where *N* is the
number of sampled values that go into the mean.

Einstein further suggested that the diffusion
constant *D* and the hydrodynamic
resistance *f *of the Brownian particle
must be related as

_{}

The hydrodynamic resistance of a sphere of radius *a* in a fluid of viscosity µ was known
from the work of Stokes to be _{}, so a measurement of D for the Brownian particle would
determine Boltzmann’s constant _{}, and hence Avogadro’s number. Jean Perrin won the Nobel prize in 1926 for work
done in 1907 carrying out this experimental program, and finding that
Avogadro’s number determined in this way agrees well with other determinations,
finally proving beyond doubt that molecules exist and obey statistical laws.

Procedure:

**File****ŕ****Open****ŕ****Videos****ŕ****BrownianMotion1.mov**

Play the movie and watch them
jiggle. The jiggling is Brownian motion. You may notice them going from a fairly sharp
focus to a fuzzier focus. The change in
focus is a result of movement in the z-axis (up and down). We only care about movement in the x and y
axis, so don’t be too concerned with focus change.

Each group will be assigned a
particular time interval (e.g. 5 frame steps) and will get data for as many
particles as feasible. Those assigned
large time intervals should be able to acquire data on more particles than the
groups with smaller intervals, but will get fewer data points per particle due
to the larger time jumps: you run out of
frames sooner. Make sense? A group jumping through 20 frames will move
through the clip and get to the end faster than a group jumping through 5
frames.

Let’s say you’ve been
assigned 10 frame intervals. Start off
by adjusting the frame rate. Click on
the **Clip Inspector **icon (it looks like
a piece of film with a little black triangle on the lower right edge), and
adjust the **Step Size** to 10, or
whatever you were assigned. That is the
only value you need to adjust for this lab, so click **OK**.

Now, it will probably help to
place the axes in some way that divides up the screen so you can keep track of
your points. I like putting it in the
middle and giving myself quadrants. We
only care about the difference in position so it really doesn’t matter where we
put the origin. Do whatever is convenient.

Once your axes are placed,
click on the **Track Control** icon
(looks like fireworks). The Track
Control panel should now be open. Click
on **New** and choose **Point Mass**. A box labeled **mass A** (or whatever letter you are on) will appear, click on it and
choose **Name**. Name the point mass whatever you want. Something descriptive of the particle you
will be tracking might be useful.

Choose your first
particle. Zoom in on the area containing
the particle by right clicking near the particle, choosing **Zoom**, and picking a magnification.
I find that 2.0x usually works fairly well.

Hold down Shift, which should
turn the cursor into a little box with a cross in it. Then left click as close to the center of the
particle as you can. You will
automatically be advanced to the next frame.
You will want to remove the trail left by your previous clicks by
clicking on the **Show Trails **icon
(two red rounded diamonds). Once the
trails are off, you should be able to see the particle clearly from frame to
frame.** **Keep clicking until you get
to the last frame. If at any time you
want to see the trail, click the Show Trails icon again. You will see something that is relevant in
biology: a Brownian path “explores” its
neighborhood pretty thoroughly!

Now, pull out the **Data Table** and **Graph**. Recall, that it’s
over on the right hand side. Put the
cursor over the spotted white space next to the scroll bar, when it turns into
a double headed arrow, left click, hold and drag out the data table.

RIGHT CLICK and choose **Copy Data**, then paste the data into an
Excel spreadsheet. Remember, you don’t
need to highlight the headers (t, x, y), they come along for the ride with the
data. Once you are in Excel, label the
units in the header. Be aware that the x
and y values are in pixels.

Repeat this process for as
many particles as you can. Be sure to
start a new TRACK (with a new name, like mass B for the 2^{nd}
track) for each new particle. When you paste data into Excel for a new
particle, be sure it is in its own region in the spreadsheet, separated from
the tables for other particles.

__Analysis__

Make sure your data is well
laid out and labeled, something like columns A, B, and C in Fig. 1 is
recommended.

Figure 1: Layout of Brownian
motion data.

First you need to start two
new columns. These columns will contain
the amount the particle moved from frame to frame. The data you pasted in tell you where the
particle was at time 0, 0.415 s, 0.83 s and so forth (the times will be larger
or smaller depending on what frame number you use). This does not tell you how far the particle
traveled in the 0.415 s between your recordings of its position. In order to obtain that information you need
to make a column where you subtract the first data point from the second data
point. So if you found the particle was
at x=0 pixels when t=0 s, and at x=0.368 pixels when t=0.415 s, your first
entry in your new column would be 0.368 pixels.
You don’t need to do this by hand though; Excel will do the work for
you. See the Calculations section of the
Excel handout for directions on how to make Excel do the math. Repeat the process for the Y direction also.

Now, you need to square the
values from your new columns, which are D and E if Fig. 1 and make two new
columns. As you did when getting the
difference, make Excel do the math.
Repeat the process for the Y direction also.

Once you have two columns
containing the square of the difference in position between frames, take the
average of these numbers. In the empty
cell at the bottom of the column of squares, which are columns F and E in Fig.
1, you would type =Average(F4:F27), and this will give you the mean of the
squares. Similarly for the y direction. (Of course the cells you use will undoubtedly
be different from these, so use the appropriate range for your table.)

Repeat this for all your data
points until you have means of the squared displacements in each direction for
all the particles.

Once you have the mean
squares, make a column of the time interval you were using in seconds, and
place the means of squares from the x and y direction next to the time
interval, as in Fig. 2.

In the cell below the mean
squares of x and y, calculate the mean of each column. Then take the standard deviation of each
column. The formula for standard
deviation is STDEV, so you will need something like =STDEV(BE15:BE23) in the
next available cell (see Fig. 2).

Fig. 2: Layout of Brownian Motion Data

We determined the mean square
displacement over and over, for many particles:
our best estimate is the sample mean of all these determinations, found
at the bottom of Fig. 2. How good is
it? We know roughly how the individual
determinations scatter around the mean:
that is the observed sample standard deviation, also computed at the bottom
of Fig. 2. Since we used 9 samples to
get the sample mean, the standard deviation of this sample mean should be that
sample standard deviation divided by the square root of 9, or 3, which is about
1. That is, we seem to have determined
the mean x squared for this time interval to be about 8.3, plus or minus 1.

Since we have determined _{} for only one value of *t*, we can’t know if these quantities are
proportional. For that, we need to see
what the other groups found! To finish,
we will have all groups report their values.
Together we will make a plot, and if all goes well, determine the slope *2D*.

From this measurement it is
possible to extract Avogadro’s number, using Einstein’s relation (1.4) between *D* and the hydrodynamic resistance *f*.
It implies

_{}

Here R is the gas constant, T is the absolute
temperature (about 297 K when the video was made), _{}is viscosity of water (about
0.00091 Pa-s at this temperature) and *a*, the radius of the little polystyrene spheres we are watching, is
0.93 microns. Finally, the digitized
microscope images have a scale 4.165 pixels per micron. Get a class value for Avogadro’s number.