Matt's Balmer Series Notebook

Safety

^{SJK 13:54, 22 October 2010 (EDT)}

13:54, 22 October 2010 (EDT)
As with Sebastian, This too is a very good primary notebook. Very easy to follow everything. Great work! Take a look at his page for some more comments, since I put them there first: http://openwetware.org/wiki/User:Roberto_Sebastian_Rosales/Notebook/Physics_307L/2010/09/15

• The tubes are made of glass, and they should be handled with care.
• Glass tubes can get hot after even minimal use. Use caution when handling immediately after use.

Equipment

• Spectrometer - Adam Hilger; London, England; Serial: 12610
• Spectrum Tube Power Supply - Model: SP200 (5000V;10mA; Electro-Technic Products)
• Mercury Tube
• Hydrogen Tube
• Deuterium Tube

Set Up

• Place spectrometer and lamp on a sturdy, flat surface.
• Elevate the lamp to the point where the thinner part of the glass tube (while placed in the lamp) is level with the eye sight of the spectrometer. We used a stack of books.
• Fiddle with spectrometer until you obtain desirable cross hair position and general slit width.
  • Note: A thinner slit yields more accurate results, but sacrifices visibility.

Setup

Lamp

Procedure

After set up, we calibrated the spectrometer using the mercury tube. Looking up the wavelengths in Prof. Gold's lab manual, we aligned the vernier scale to a known wavelength for mercury and rotated the crystal until the desired spectral line became aligned with the cross hairs. For example, for our first set of observations we set the vernier scale to 546.1nm and carefully rotated the crystal until the green spectral line was on the cross hairs.

• The gears in the vernier scale do not perfectly mesh. For this reason, when using the scale, be sure to always approach your desired reading from the same direction when taking data. We chose to make all of our observations while moving the scale from low to higher wavelengths (from purple to red).
• A useful tip when calibrating is you don't have to align the crystal perfectly and then tighten. Once you get close, tightening the bolt attached to the crystal will slowly align the crystal for you.
  

With the spectrometer calibrated, we replaced the mercury tube with the hydrogen. After letting the lamp warm up for a moment, I attempted to locate the first purple wavelength. Now, I know the manual for this lab mentioned it would be hard to see, but neither I, Sebastian, nor Prof. Koch were able to get an accurate reading for this wavelength. For this reason, this first purple wavelength will not be included in the calculations for the Rydberg constant. Other than this first spectral line, there were no major complications in reading the spectrometer for the values of violet#2, blue-green, and red wavelengths. After recording the values for hydrogen, we replaced it with the deuterium tube. Compared to hydrogen, the spectral lines were much easier to read. There was less 'noise' visible, and the first purple wavelength, while still relatively faint, was measurable. After recording our data for deuterium, we returned to mercury to recalibrate.

• When recalibrating, we chose different spectral lines for each run to reduce systematic error.

Data

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdGVZXzNqQlB1a3lacmhNOXhHRHl4RkE |width=1000 |height=400 }}

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdHUwT3RjTDlJajZXSDg5c00tZnBTZWc |width=700 |height=200 }}

Calculations

The purpose of this lab is to calculate the Rydberg constant1
[math]\displaystyle{ R_\infty = \frac{m_e e^4}{8 \varepsilon_0^2 h^3 c} = 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1} }[/math]

With reduced mass Rydberg constant given by
[math]\displaystyle{ R_M = \frac{R_\infty}{1+m_e/M} }[/math]

The values [math]\displaystyle{ R_{hydrogen} }[/math] and [math]\displaystyle{ R_{deuterium} }[/math] were found by obtaining the average wavelength for red, blue-green, violet#2, and violet#1(deuterium only) for both hydrogen and deuterium. These average wavelengths were then used to obtain the Rydberg constant for each spectral line, and these Rydberg constants were averaged together. With the average Rydberg constant, we can obtain [math]\displaystyle{ R_{hydrogen} }[/math] and [math]\displaystyle{ R_{deuterium} }[/math].

[math]\displaystyle{ \bar\lambda=\sum_{i=1}^n\lambda_i/n }[/math]---We will have a separate [math]\displaystyle{ \bar\lambda }[/math] for each color for each element.(3 for hydrogen, 4 for deuterium)

[math]\displaystyle{ R=\frac{1}{\bar\lambda}\left(\frac{1}{2^2}-\frac{1}{m^2}\right)^{-1} }[/math]---Where m is initial energy level (i.e. for red m=3) and [math]\displaystyle{ R }[/math] is the reduced mass Rydberg constant. We will have one [math]\displaystyle{ R }[/math] for each [math]\displaystyle{ \bar\lambda }[/math](3 for hydrogen, 4 for deuterium).

[math]\displaystyle{ \bar{R}=\sum_{i=1}^n R_i/n }[/math]---Take the average [math]\displaystyle{ R }[/math] for hydrogen and deuterium separately. At this point we have [math]\displaystyle{ \bar{R}_{hydrogen} }[/math] and [math]\displaystyle{ \bar{R}_{deuterium} }[/math]. (Note:I don't know why my text is bold from here on out)

[math]\displaystyle{ R=\bar{R} +/- SEM }[/math]---Where SEM is the standard error of the mean, given by

• [math]\displaystyle{ SEM=\frac{s}{\sqrt{n}} }[/math]---Where s is the standard deviation.
  
  

With our data, the results came out to be

• [math]\displaystyle{ R_{calc Hydrogen} = (10976969 +/- 7304)m^{-1} }[/math]^{SJK 13:51, 22 October 2010 (EDT)}

  

  13:51, 22 October 2010 (EDT)
  

• [math]\displaystyle{ R_{calc Deuterium} = (10989761 +/- 1736)m^{-1} }[/math]
  

The accepted values for these constants are given in the Google doc, and are

• [math]\displaystyle{ R_{hydrogen}=10967758.3406 m^{-1} }[/math]
  
• [math]\displaystyle{ R_{deuterium}=10970746.1986 m^{-1} }[/math]
  

Our result for [math]\displaystyle{ R_{hydrogen} }[/math] is within 2 SEM. Our result for [math]\displaystyle{ R_{deuterium} }[/math] is within 11 SEM.
Our %error for [math]\displaystyle{ R_{hydrogen}=.084% }[/math] and our %error for [math]\displaystyle{ R_{deuterium}=.173% }[/math]

References

All accepted Rydberg values, physical quantities, and equations obtained from Wikipedia.
David Weiss for help with general notebook format and procedure.