User:John Callow/Notebook/Junior Lab 307/2009/09/16

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Balmer Series
Link to my summary

The procedure I followed may be found in Dr. Gould's manual section 4 on the Balmar series.

Equipment List
Spectrometer

Spectrum tube power supply Model SP200 5000volts 10MA Electro Technic Products Chicago ILL. 606400

mercury vapor tube, hydrogen tube, deuterium tube

Flash light (recommended to read measurements being taken.)

Safety
For experimenters

Careful when setting up the different tubes, the power supply is capable of a pretty nasty shock so it's recommended to set up with everything unplugged and if not that at least make sure it's off.

The lab is done in the dark so keep the area clean.

For the equipment

Handle the prism on the spectrometer by the frosted edges to avoid dirtying it with fingerprints.

Make sure the wires and such on the power supply are ok and nothing looks broken before setup.

Handle the tubes being careful not to drop them. Also try to hold then by the bottom or top parts to avoid getting fingerprints on them.

Setup
With the tube power supply off and preferably unplugged place the mercury tube in the power supply. Then align the spectrometer's open slit up with the a tube in the power supply. The tube power supply I used was not very tall so I placed some books under it to help raise it up as seen in the picture. Once everything is set up plug in and turn on the power supply while turning off the lights.

Now looking through the eyepiece adjust it so the cross hairs are in focus. After that adjust the prism holding it by the frosted edges in the spectrometer holding it by the frosted edges so that you are able to clearly see the light from the tube when looking in the eyepiece. To calibrate set the dial to the wavelength of the expected value for the green line (546.1nm). Now try to rotate the prism by hand and get the green line as close to the middle of the cross hairs as possible. The view should appear similar to the picture on the left though likely a bit sharper in person as the picture was taken by my phone. Finish up by tightening the prism in place to prevent it from rotating while using the dial. Now it is ready for taking measurements for calibration. As a note, you don't have to calibrate to green or really any line as long as all the lines are visible and measurable. The reason for doing this is so that it is easier to tell if a number is in the range it should be without finding a formula that adjusts to the scale.

Data and Calculations
All measurements done turning dial left to right. All data in nanometers unless otherwise specified.

There were some issues measuring the red line. The mercury tube I used had several red lines in the area and so it was unclear which was the correct one to measure. With the mean of every other line being so close to the accepted value and because I don't have access to any other tubes to test I feel it is best to throw out the data for red. If however it was clear at which red line the measurement should have taken place I would keep the data.

Also because if excluding red the values are so close to the expected I am going to make the assumption due to lack of time that 1 on the spectrometer scale is 1nm. If I had more time (and a bit more experience with maple or excel) a linear line of best fit would be a good function to convert 1 on the spectrometer scale to nanometers. Just looking at the data though it looks almost like the function would be about Nanometers = (1+1/200)*(spectrometer reading)-B where B is much less than one so the difference is negligible.

To find the Rydberg constant in these google excel sheets I used $$\frac{1}{\lambda} = R \left(\frac{1}{2^2}-\frac{1}{m^2}\right)$$ from the manual and found the values of m from this article on the Balmer series in the overview section.

So my best guess of the Rydberg constant is the averages of the above or

R = (1.099 +- .00025)*10^7 meters^-1. From the manual the accepted value is about 1.097*10^7 meters^-1 so I came pretty close. My actual error from measuring is probably more like +-1nm for each measurement as I did not adjust to the scale of the spectrometer with a best fitting line which seems like that would have gotten me even closer. Even so the estimated error is only

(1.099-1.097)*100/1.099 = .182%.

I'm unsure whether or not yellow should be included which is why there is a question mark next to it. There was a continuous area of lines from red to blue that had a somewhat well defined yellow strip in the middle. On the right I have a picture taken with my phone of what I was seeing with a circle around the yellow. My phone isn't very good and as such the picture didn't come out very clear but the lines are visible. They were much clearer though looking through the spectrometer myself.

According to in this article on the Balmer series I don't think that these lines should not have been showing up. There is no value of m to satisfy the formula if n = 2 at this wavelength for hydrogen. There is a chance that the tube was miss labeled but both hydrogen tubes and the deuterium tubes I used showed this pattern. My best guess would be that there was some contaminants whether in the tube or from other light sources in the room(such as computer screens or light leaking through the door). Also I should have been able to measure another violet line. When running the experiment I thought there might be one but it was barely visible and so unmeasurable. Widening the slit and or moving the tube closer to the slit didn't seem to help much.

So my best estimate is (1.097+-.00025)*10^7 m^-1.

Now the manual asks me to calculate the differences between the expected R and H_alpha line of deuterium and hydrogen. From the manual $$R = \frac{u e^4}{8 \varepsilon_0^2 h^3 c}$$ where u is the reduced mass. I didn't know where to find the reduced mass, but from this article on reduced mass it can be found using the formula $$u = \frac{m_e*m_n}{m_e+m_n}$$ using 9.1094*10^-31 kg for the mass of the electron, 1.6726*10^-27 kg as the mass for the proton, and 1.6749*10^-27 kg for the mass of each neutron in deuterium we find

u = 9.10775*10^-31 kg.

Now plugging this value into the formula for R and using e = 1.602176*10^-19 coulombs epsilon 0 = 8.854 A^2*s^4/kg h = 6.626*10^-34 J*s c = 3*10^8 m/s we get

Rydberg constant for deuterium = 1.0965*10^7 m^-1 Now plugging this back into the $$\frac{1}{\lambda} = R \left(\frac{1}{2^2}-\frac{1}{m^2}\right)$$ with m = 3 we find that $$H_a$$ = 657nm. So the difference is around 1 nanometer with the Rydberg constant difference being around .0002. I've done so much rounding and the values are so close together that I'm unable to give anything other than ballpark values for this. But looking at my measurements using the spectrometer there seemed to be about a 1nm difference, and 1nm differences seemed to be something the spectrometer used could measure. As for the reason why this happens, deuterium has a lower reduced mass than hydrogen and from the formula it is clear how this changes R.


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