Physics307L F08:People/Joseph/Notebook/071024

Balmer Series

Experimentalists

Nik Joseph and Bradley Knockel

Objective

We are going to study the emission spectra for hydrogen and deuterium and from there attempt to determine their Rydberg constants. The emission spectra of both hydrogen and deuterium are generated by the transitions of each atoms' single electron from their principle quantum state to a higher state. Using Balmer's formula, [math]\displaystyle{ \frac{1}{\lambda}=R\left(\frac{1}{2^2}-\frac{1}{n^2}\right) }[/math]¹, where [math]\displaystyle{ R\, }[/math]=Rydberg constant, and measuring where these emission spectra occur, we should be able to calculate the Rydberg constant. We'll measure the spectral lines for hydrogen¹ at:

[math]\displaystyle{ n\, }[/math]	3	4	5	6
[math]\displaystyle{ \lambda\, }[/math] (nm)	656.1	486.1	434.1	410.2
Color	Red	Blue-Green	Violet	Very Violet

and we should be able to measure the Rydberg constant from there.

Equipment

Our equipment consisted of:

• Spectrum tubes filled with hydrogen, deuterium, mercury, and sodium gases
• Power supply for spectrum tubes
• "constant-deviation" spectrometer, made in-house at UNM

Setup

We began by sorting through the spectrum tubes, identifying which ones we needed. We familiarized ourselves with the spectrometer, playing the dial and focus on it; getting a general feel for the equipment. We inserted the mercury tube into the power supply and turned it on. We were looking through the eyepiece at the spectral lines, when Devon came over, removed the cover on the prism and moved the prism around. He did this so we'd have to readjust it on our own, which took a few moments. The spectrometer needed to be calibrated next. A simple way of doing this to was find a known line on the mercury spectrum, adjust the dial so you are looking at the correct wavelength, and then move the prism until the spectral line was in the cross hairs. We tried this several times, just to be sure.

To take the data was a bit cumbersome. Because the dial that adjusted wavelength was old, you have to account for lash in the gears by turning only in one direction at a time. Whenever we reversed direction we'd turn to the end of the screw, and then reverse direction.^{SJK 22:54, 6 December 2007 (CST)}

22:54, 6 December 2007 (CST)
to account for backlash, you don't have to go all the way to the end...just enough so that when you come back the other way you're fully engaged

Data

We took two sets of data: one set when we were increasing in wavelength and one set as we decreased the wavelength. We felt this would be a good idea, because it gives us a simple way of calculating error: take the difference of the two.

Hydrogen

[math]\displaystyle{ n\, }[/math]	3	4	5	6
[math]\displaystyle{ \lambda\, }[/math] (nm) decreasing	656.0	485.6	434.0	410.3
[math]\displaystyle{ \lambda\, }[/math] (nm) increasing	658.0	485.9	434.2	410.4
Color	Red	Teal	Purple	Deep Violet

Deuterium

[math]\displaystyle{ n\, }[/math]	3	4	5	6
[math]\displaystyle{ \lambda\, }[/math] (nm) decreasing	654.9	485.5	433.8	409.8
[math]\displaystyle{ \lambda\, }[/math] (nm) increasing	658.0	486.2	434.3	410.2
Color	Red	Teal	Purple	Deep Violet

Results and Analysis

Manipulating Rydberg's formula we wind up with: [math]\displaystyle{ R=\frac{1}{\frac{\lambda}{4}-\frac{\lambda}{n^2}} }[/math] which can be used to find the Rydberg constant for each wavelength. Once we have calculated those for each element we can take the mean and come up with a reasonable value. We need to take our uncertainty into account as mentioned before. Our adjusted data is:

Hydrogen

[math]\displaystyle{ n\, }[/math]	3	4	5	6
[math]\displaystyle{ \lambda\, }[/math] (nm)	657.00 [math]\displaystyle{ \pm }[/math] 2.0	485.75 [math]\displaystyle{ \pm }[/math] 0.3	434.10 [math]\displaystyle{ \pm }[/math] 0.2	410.35 [math]\displaystyle{ \pm }[/math] 0.1

Deuterium

[math]\displaystyle{ n\, }[/math]	3	4	5	6
[math]\displaystyle{ \lambda\, }[/math] (nm)	656.45 [math]\displaystyle{ \pm }[/math] 3.1	485.85 [math]\displaystyle{ \pm }[/math] 0.7	434.05 [math]\displaystyle{ \pm }[/math] 0.5	410.00 [math]\displaystyle{ \pm }[/math] 0.4

Using the formula for [math]\displaystyle{ R\, }[/math], propagating the error^{SJK 22:58, 6 December 2007 (CST)}

22:58, 6 December 2007 (CST)
So, how did you propagate the error? You would need to link to the excel file or whatever you did. Also, the uncertainties above for the averages are not correct (you are taking a mean and want to report the standard error in the mean = sigma /sqrt(N). But more than that, it's not good to average counterclockwise and clockwise, but rather, you want to take measurements going in the same direction as you calibrated the prism.

, and averaging the values we wind up with:

[math]\displaystyle{ R_{hydrogen}=1.09686(86)\times10^7 m^{-1} }[/math]

and

[math]\displaystyle{ R_{deuterium}=1.09730(142)\times10^7 m^{-1} }[/math]

Compared to the actual value of [math]\displaystyle{ R_{hydrogen}\, }[/math] which can be found here, the error comes out to [math]\displaystyle{ error\,=8.206\times10^{-5} }[/math], which is well below our measured uncertainty.^{SJK 23:01, 6 December 2007 (CST)}

23:01, 6 December 2007 (CST)
you should put the value you are comparing to here, since there are a lot of numbers on that page. Presumably, you're using the value for hydrogen...So, are you saying that you got a distinguishably different value for Deuterium versus Hydrogen? How do the error bars compare? How sure are you, and does it make sense based on the differing nuclear masses?

Conclusions

We measured Rydberg's constant very closely to the accepted value, even with our ridiculous uncertainty. Still, this lab wasn't very difficult and we learned how to use the spectral lines to analyze a material. Bradley and I discussed in great detail the reasons why or why not the Rydberg constant for deuterium would be bigger than that of hydrogen, but in the end the experiment settled the argument. In the end, data defeats rhetoric. We continued to analyze all the other samples, and we identified the "unknown" element to be neon.