# Gas station without pumps

## 2014 January 16

### Fourth day of freshman design seminar

Today we continued the design exercise I started last week, designing a photospectrometer.

1. What is Beer-Lambert Law?
2. What is absorbance?
3. What is Bragg’s Law?
4. What is a molar extinction coefficient? (also known as molar absorption coefficient or molar absorptivity)

The lack of answers lead me into a small lecture on each of the points, so that they could see the connections to photospectrometry:

• Absorbance is a measure of how much light is absorbed (or diffused) by a sample.  It is expressed as $A_{\lambda} = - \log \frac{I_{out}}{I_{in}}$.  The λ is the wavelength of light at which the intensity is measured.  I pointed out (towards the end of the mini-lecture) that whether the logarithm is base 10 or base e depends on who is doing it—chemists tend to use base 10, physicists almost always base e, and biologists vary depending on who taught them about absorbance.
• Beer-Lambert Law is an expression for computing absorbance: $A_{\lambda} = \epsilon_{\lambda} l c$, where $\epsilon_{\lambda}$ is the molar extinction coefficient (a property of the substance) in $cm^{-1} M^{-1}$, $l$ is the length of the light path (in cm), and $c$ is the concentration of the absorbing substance (in M).  I pointed out that concentration of a substance in solution could be computed from a known extinction coefficient and a measurement of the absorbance at the wavelength of the extinction coefficient.  (This is the main use of absorbance in biomolecular labs.)  I also pointed out that extinction coefficients had the same problem of absorbance of coming in base e or base 10 scaling.
I also passed around a couple of disposable cuvettes for students to look at, talking about how they had a well-calibrated 1cm path inside (fixing length).  One thing I goofed on in class—I said that the cuvettes were made out of acrylic, but when I got home I checked the boxes and found that I had polystyrene Brand cuvettes, 100 macro and 100 semi-micro size.  The polystyrene cuvettes are a bit cheaper than the acrylic ones, but not quite as transparent in the UV (they go down to about 295nm instead of 270nm for acrylic and 220nm for the UV semimicro cuvettes) [http://www.brandtech.com/cuvettegraph.pdf].  I also measured the outside size of the cuvettes (which Brand does not document): the 2.5ml “macro” size is 1.24cm each way, and the 1.5ml “semi-micro” size is 1.22cm each way at the top, and 0.99cm by 1.20cm at the windows.
• Bragg’s Law expresses the amount that light with wavelength $\lambda$ and input angle $\theta_{i}$ from the normal is diffracted from a grating with line spacing d:
$\sin(\theta_{i}) + \sin(\theta_{m}) = \frac{m\lambda}{d}$.  I had students compute the diffraction angle for the first spot (m=1) for the green laser (520 nm) and a diffraction grating with 1000nm ruling.
I also passed around a \$7 spectroscope that I had bought for homeschool physics and had students look at the fluorescent lights with it.

After the mini-lecture, we started the design exercise.  Since they were new to design thinking and needed scaffolding, I did this as a whole class exercise:

• First I reviewed what we knew of the inputs (a chemical sample in water in a cuvette) and the output (a plot of absorbance versus wavelength for about 300nm to 700nm, though polystyrene seems to be transparent for near infrared  (at least to 1000nm).
• Then I started asking them for components, explaining the concept of a block diagram (though I did not provide the connections between the blocks yet). They started with
• a device to spread out the light according to wavelength,
• (with a little prompting) a slit to get only one wavelength.

I pointed out that the spreading out the light spreader and slit could be thought of as a single unit (a monochromator) and that it had to be adjustable to get different wavelengths. They then added a sample holder for the cuvette.  It took a lot of prompting to get them to add a measuring device for the light, but they got there.  A little more prompting got them to add a computer interface for recording the spectrum.

• We also talked about omitting the monochromator slit and using an array of sensors (like a cellphone camera) to record the whole spectrum at once.

At that point we were running short on time, so I had them go to the boards in groups of 3 to try to flesh out the design with more detail.  After five minutes, we really were out of time and I had another meeting  to go to, so I assigned them their first written homework—a fleshed out design for the spectrophotometer.  They can do it individually or in groups of up to three people, and it is due in one week.

I’m very curious what they come up with. I’m hoping that the students will add a lot of details to the design (partly by looking at examples on the web, partly by thinking about what the needs are for each component and how the components have to work together).  I’m afraid that the students are still in  in the habit of regurgitating what the teacher has told them, rather than finding things on their own or thinking things through without leading questions, so I’m trying not to get my hopes up too high.