User:Ginevra Cochran/Formal Report/Final Draft

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

Author: Ginevra Cochran

Experimentalists: Ginevra Cochran and Cristhian Carrillo

Junior Laboratory, 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 (Pohl et al, [3]), 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 1.103(4) * 10⁷ 1/m, with an error of 0.6% from the accepted value for hydrogen. For deuterium, we obtained a Rydberg constant of 1.100(2) * 10⁷ 1/m, with an error of 0.2% from the accepted value for deuterium. The accepted values for the Rydberg constant of hydrogen and deuterium differ slightly, but our experiment was not precise enough to reliably detect this difference.

Introduction

Johannes Rydberg, a Swedish physicist, presented the Rydberg formula in the 1880s as a relation between wavelength and differing integers.

[math]\displaystyle{ \frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) }[/math]

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.

[math]\displaystyle{ \frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{4}\right) }[/math]

Hydrogen has four main spectral lines, and deuterium has three. In this report we will measure the Rydberg constants for hydrogen and deuterium using a Constant-Deviation Spectrometer, calibrating the spectrometer for each individual wavelength by adjusting the interior prism manually. We will also attempt to determine whether this method and equipment is capable of distinguishing between the two elements' Rydberg constants with accuracy.

Methods

Figure 1: The overall setup. The spectrum tube power supply and mercury tube are to the left, with the constant-deviation spectrometer's slit aimed at the mercury tube and the viewing scope positioned for the experimentalist. Photo credit: Alexandra Andrego, Junior Laboratory, Physics & Astronomy, University of New Mexico.

Figure 2: The inside of the constant-deviation spectrometer. The interior prism and screw drive are visible. Photo credit: Alexandra Andrego, Junior Laboratory, Physics & Astronomy, University of New Mexico.

Figure 3: The constant-deviation spectrometer slit. This width allowed the maximum definition on the spectral lines we were observing. Photo credit: Alexandra Andrego, Junior Laboratory, Physics & Astronomy, University of New Mexico.

Calibration of the constant-deviation spectrometer using mercury

Before commencing our measurements, we adjusted the Adam Hilger, London, 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 Electro-Technic Products 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 an Electro-Technic Products mercury spectrum tube (S-68755-30-K) into the Spectrum Tube Power Supply. We switched on the Spectrum Tube Power Supply and allowed it to heat for 5 minutes. We then opened the slit as wide as possible, focused on a spectral line, and narrowed the slit until the line was as sharp as possible without disappearing (see Figure 3). We adjusted the screw drive to the first red wavelength listed in Table 1, removed the cover of the Constant-Deviation Spectrometer's prism (as in Figure 2), and loosened the screw holding it in place, rotating the prism until the given line is directly in the cross-hairs of the Constant-Deviation Spectrometer's ocular. We then retightened the screw securing the prism, returned the cover to the prism, and replaced the mercury spectrum tube with an Electro-Technic Products hydrogen spectrum tube (S-68755-30-G).

Table 1: The spectral wavelengths of mercury.
We used these values to calibrate the constant-deviation spectrometer, obtained from Gold [1].

Color	Accepted Wavelength (nm)
Violet (very hard to see)	404.7
Violet	435.8
Weak Blue-Green	491.6
Green	546.1
Yellow 1	577.0
Yellow 2	579.0
Red	690.75

Measurement of the Balmer spectrum of hydrogen and deuterium

We took five measurements of the red spectral line for hydrogen, making sure to turn the screw drive a quarter-turn right past the line before each successive measurement to avoid gear backlash. We then removed the hydrogen spectrum tube from the Spectrum Tube Power Supply and replaced it with an Electro-Technic Products deuterium spectrum tube (S-68755-30-E), where we made the same series of measurements. We repeated the prism calibration of the Constant-Deviation Spectrometer with the mercury spectrum tube for one blue-green and two violet wavelengths, and recorded five measurements of each spectral line for hydrogen and deuterium, with one exception. Deuterium has only one violet spectral line, so we did not attempt to measure the second.

Calculation of the Rydberg constant

We averaged our measured wavelengths for each color and element and used Equation 1 to find the Rydberg constant for each average wavelength.

Equation 1:

[math]\displaystyle{ \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\! }[/math]

or

[math]\displaystyle{ R = \frac{\frac{1}{\lambda }}{(\frac{1}{2^2}-\frac{1}{n^2})}, n=3,4,5,6\,\! }[/math]

The accepted Rydberg constant is derived from quantum mechanics and has the following formulation:

[math]\displaystyle{ R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\! }[/math]

Where [math]\displaystyle{ \mu\,\! }[/math] is the reduced mass.

[math]\displaystyle{ \mu = \frac{m_e*M}{m_e+M}\,\! }[/math] where M is the mass of the nucleus.

We calculated our percent error for hydrogen's and deuterium's Rydberg constants using this formula and the definition of percent error:

[math]\displaystyle{ \% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}} }[/math]

All these calculations took place in Google Docs, using the SUM, ABS, STDEV, and SQRT functions, in addition to basic arithmetic.

Results and Discussion

Table 2: Our results for the Rydberg constant of hydrogen and deuterium compared to the accepted values.

Element	Accepted Rydberg constant (1/m)	Calculated average Rydberg constant (1/m)	SEM of calculated constant	percent error
Hydrogen	1.0967*107	1.103*107	0.004*107	0.6%
Deuterium	1.0971*107	1.100*107	0.002*107	0.2%

Our calculations are visible here, and our raw data here.

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  Figure 4: the hydrogen Balmer series and its best fit line

• 

  Figure 5: the deuterium Balmer series and its best fit line

Conclusions

The values we obtained in this experiment were very close to the accepted values, as seen in Table 2. The error inherent in the equipment and experiment prevented us from being able to reliably distinguish between the Rydberg constant for hydrogen and deuterium, possibly partly because their masses are so similar. Further experimentation comparing the hydrogen constant of hydrogen to that of argon, for example, could be enlightening as to the exact accuracy of our equipment.

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 Alexandra Andrego for the use of her photos documenting the experiment setup. We used Google Docs to calculate our results and MATLAB 7.8.0 (R2009a) to generate Figures 4, 5, 6, and 7.

References

[1] Gold, Michael. The University of New Mexico Dept. of Physics and Astronomy PHYSICS 307L: 'Junior Laboratory Manual Fall 2006'. http://www-hep.phys.unm.edu/~gold/phys307L/manual.pdf.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/atspect2.html

[3] Pohl, Randolf; Antognini, Aldo; Nex, François; Amaro, Fernando D.; Biraben, François; Cardoso, João M. R.; Covita, Daniel S.; Dax, Andreas et al. (2010). "The size of the proton". Nature 466 (7303): 213–216. doi:10.1038/nature09250.

[4] Coffman, Moody L. (1965). "Correction to the Rydberg Constant for Finite Nuclear Mass". American Journal of Physics 33 (10): 820–823. doi:10.1119/1.1970992.