User:Ginevra Cochran/Formal Report/Rough Draft

= The Balmer series: Determining the Rydberg constant for hydrogen and deuterium =

Author: Ginevra Cochran

Experimentalists: Ginevra Cochran and Cristhian Carrillo

Junior Lab, Department of Physics & Astronomy, University of New Mexico

1919 Lomas Blvd NE

Albuquerque, NM 87131

serenity@unm.edu

Abstract
The Balmer series is the set of spectral lines produced by the transition of orbital electrons to the second energy level from above it. The Rydberg constant, the most precisely measured quantity in quantum mechanics, can be determined from the wavelengths of these spectral lines. We measured these wavelengths using a constant-deviation spectrometer and obtained a Rydberg constant for hydrogen of 1.0911685 +/- 0.0070852*107 1/m, with an error of 0.0863% from the accepted value for hydrogen. For deuterium, we obtained a Rydberg constant of 1.0991918 +/- 0.0018430*107 1/m, with an error of 0.795% from the accepted value for deuterium. From this, we concluded that our systematic error was minimized and that the Rydberg constant is dependent on the mass of the atom emitting light. 

Introduction
Johannes Rydberg, a Swedish physicist, presented the Rydberg formula in the 1880s as a relation between wavelength and differing integers.
 * $$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$

This relation was determined experimentally and predated the discovery of quantum mechanics. The Balmer series is the set of spectral lines produced by the Rydberg formula for n2=2.
 * $$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{4}\right)$$

Hydrogen has four main spectral lines, and deuterium has three. 

Calibration of the constant-deviation spectrometer using mercury
Before commencing our measurements, we adjusted the constant-deviation spectrometer (SER #12610) to suit the vision of the experimenters. We focused the cross-hairs by adjusting the position of the ocular and focused at the slit using the large knob near the center of the apparatus, eliminating parallax between the two. We elevated the spectrum tube power supply (Model SP200, 5000V, 10 mA) to align the narrow section of the our spectrum tubes with the slit which allowed light to enter the prism at the center of the constant-deviation spectrometer (see Figure 1). We slotted a mercury spectrum tube into the spectrum tube power supply. We switched on the spectrum tube power supply and allowed it to heat for 5 minutes. We opened the slit as wide as possible, focused on a red line, and narrowed the slit until the line was as sharp as possible without disappearing. We then set the constant-deviation spectrometer as far to the left of the spectrum as it would go, identified the spectral lines referred to in Table 1, and recorded the position noted on the screw drive (Figure 2). Table 1
 * {| border="2"

!Color !Accepted Wavelength (nm) !Recorded Wavelength (nm) Using a linear fit in Google Docs, we found a best-fit line which we used to adjust our later measurements. This completed the calibration of the constant-deviation spectrometer.
 * Violet (very hard to see)
 * 404.7
 * 406.9
 * Violet
 * 435.8
 * 438.9
 * Weak Blue-Green
 * skip this one
 * n/a
 * Green
 * 546.1
 * 553
 * Yellow 1
 * 577.0
 * 585.8
 * Yellow 2
 * 579.0
 * 588
 * Red
 * 690.75
 * 720
 * }
 * 579.0
 * 588
 * Red
 * 690.75
 * 720
 * }
 * }

Measurement of the Balmer spectrum of hydrogen and deuterium
We removed the mercury spectrum tube from the spectrum tube power supply and replaced it with a hydrogen spectrum tube. Moving the screw drive from left to right, we took 5 data points for each of hydrogen's 4 spectral lines (red, blue-green, and 2 violets). We then replaced the hydrogen tube with a deuterium spectrum tube. We were careful to turn the screw drive from left to right, recording 5 data points for each of deuterium's 3 spectral lines (red, blue-green, and violet).

Calculation of the Rydberg constant
We calculated our calibrated wavelengths for hydrogen and deuterium using the best-fit line we calculated from our efforts with the mercury spectrum tube. We averaged these calibrated wavelengths for each color and element and used Equation 1 to find the Rydberg constant for each average calibrated wavelength.
 * Equation 1:
 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!$$

The Rydberg constant derived from quantum mechanics has the following formulation:
 * $$R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\!$$
 * Where $$\mu\,\!$$ is the reduced mass.

We calculated our percent error for hydrogen's and deuterium's Rydberg constants using this formula and the definition of percent error:
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$



Results
Table 2
 * {| border="1"

!Element !Accepted Rydberg constant (1/m) !Calculated average Rydberg constant (1/m) !SEM of calculated constant !percent error
 * Hydrogen
 * 1.0902269*107
 * 1.0911685*107
 * 0.0070852*107
 * 0.0863%
 * Deuterium
 * 1.0905242*107
 * 1.0991918*107
 * 0.0018430*107
 * 0.795%
 * }
 * 0.795%
 * }

Our raw data, initial calibration and calculations are visible here. 

Conclusions
The values we obtained in this experiment were very close to the accepted values, as seen in Table 2. Our systematic error for this experiment was thus very low compared to our random error. The Rydberg constants for hydrogen and for deuterium are slightly different, which is rooted in the fact that the calculated Rydberg constant is dependent on the mass of the atom. 

Acknowledgments
I would like to thank my lab partner,Cristhian Carrillo, Katie Richardson, and Professor Koch for all their help in the execution of the lab, and Alex Andrego for the use of her photos documenting the experiment setup. We used Google Docs to calculate our results. 

Overall SJK Comments
23:29, 17 November 2010 (EST): This is a good start! You can see the numerous things you'll need to fix in the margins above. It will seem like a lot, and it is, but it's all do-able. The biggest things are the Introduction and lack of citations of peer-reviewed reports, and the Results, where more figures and discussion are needed. There's also the issue where I think there's an error in the analysis, which of course needs to be corrected!
 * For the follow-on week (last week of classes), I'd like you to devise a way of attacking the question of whether you really can distinguish between H and D. Think of multiple ways of asking and answering the question.  I'd also like better calibration.  And any other ideas for getting better or complementary data.  Following this, then you'll have to make the decision of whether to replace or add to these data you have already.