**BIEN 435 Laboratory**

**Measurement of the
Diffusion Coefficient of a Dye in Water**

*Introduction:*
Diffusion coefficients are difficult to obtain directly. An alternative to direct measurement is to
obtain measurements of the distribution of a diffusing species and apply a
known mathematical analysis to back out the desired quantity.

*Overview: * You will need some way to measure
concentration of a substance, which will be an optical technique where the
attenuation of the light passed through a substance depends on the
concentration of the species. You will
then look at diffusion in a test tube, which will represent one-dimensional
diffusion. A critical part of this
experiment is the calibration of the optical device. Two different dyes will be used, and you will
test the results to determine whether the diffusion coefficients of the two
dyes are different.

*Null Hypothesis:*
The calculated diffusion coefficient of the dye is independent of the
time at which the measurements are taken..

*Calibration: *The optical device sends laser light
through the sample. The sample
attenuates the laser light by an amount that depends on both the path length of
the light and the concentration of the dye.
A simple relation for this is Beer’s law:

where I is the intenstity
at the detector, I_{0} is the intensity of the incident light, α is an
attenuation coefficient that depends on concentration, and *l* is the path length of light through the medium.

The figure to the right shows the
setup for the test tube. The detector is
a photoresistor.
Its resistance changes according to the amount of light striking
it. You need to determine the
relationship between its resistance (output) and concentration (input)

1.
Fill
the test tube with water.

2.
To
determine the volume of water, weigh the test tube, then
weigh the test tube when filled with water.
The difference is the weight of the water. The volume can be obtained by dividing by
density.

3.
With
just the water in the test tube, take a reading of the resistance on the multimeter.

4.
Add
1 drop of the dye to the water and mix thoroughly. Take another reading of the voltage.

5.
Continue
this process, adding the dye a drop at a time until you have reached 5 drops.

6.
From
this data set determine the calibration of concentration as a function of
resistance. (Is it linear?)

*Experimental Procedure: * Carefully place as
drop of dye on the cap of the test tube.
Obtain measurements of voltage at 5 distances from the cap, 0.5 cm, 1.0
cm, 1.5 cm, 2 cm and 2.5 cm. Take these
readings after 2.5 minutes and after 5 minutes.
Repeat these measurements so that you have 5 total sets of data.

*Data analysis:*

Eq. 2

Examine the equation for 1-dimensional diffusion as a
function of time and space Eq. 2. You have all the data to compare this to
theory, except the Diffusion coefficient.
Use a least squares method to determine the Diffusion coefficient.

If you take the logarithm of this equation, it becomes:

_{}

Assuming that all data were taken at the same time, you can
plot _{} as a function of _{}and perform a linear least squares fit. The slope of this will be _{}, and the y-intercept will be _{}. From the slope and
knowledge of the time you can determine D.
That is, if

_{}, then _{}.

Next you will want to know if this method gives you
different values for *D* at the two
different times. Calculate the values of
D for all 5 sets of data and for both times.
Now perform a Student’s T test to test the null hypothesis.