# A Study on the Balmer Series of Hydrogen

Author: Cristhian Carrillo

Experimentalists: Cristhian Carrillo & Ginevra Cochran

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

Albuquerque, NM 87131

c28674@unm.edu

## Abstract

We present a study on the Balmer series of the Hydrogen atom. The objectives of this experiment are: (1) to study emission of light from a Hydrogen discharge source, (2) attempt to learn the empirical formulas to describe the pattern of spectral lines from Hydrogen, (3) to measure the wavelengths of the Balmer Series of visible lines from hydrogen, and (4) to learn to analyze the wavelength data to determine the Rydberg constant using the Bohr model formulation. There are a total of six different series that describe the spectral line emissions of the Hydrogen atom. Due to the constraints of this lab, we were able to observe four wavelengths from the visible spectrum of light from Hydrogen. The four observable spectral lines are categorized by Red, Blue-Green, Violet, and Ultra Violet. Electrons transitioning to different levels of quantum energy levels emit photons and as a result, we see the different wavelengths that correspond to the emissions of the photons. The use of the spectrometer allowed us to observe and classify the spectra lines of the hydrogen atom. In order to measure the energies of the excited electrons through the emitted photons with wavelengths equivalent to the energy of the electrons, the Hydrogen atoms are excited to higher energies by electrical stimulation. It is possible to use these measurements to experimentally calculate Rydberg's constant, which is used in the Rydberg equation for Hydrogen.
$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,..\,\!$

Our experimental value of the Rydberg constant was

$R_{Hydrogen,average}\approx1.103(4)\times10^7 m^{-1}\,\!$

$R_{Deuterium,average}\approx1.100(2)\times10^7 m^{-1}\,\!$

We were pretty consistent with the accepted value obtaining less than a one percent error for both hydrogen and deuterium. Our error will be discussed in the results and methods section.

## Introduction

The four visible lines of the hydrogen spectrum, described by the Balmer series and the Balmer-Rydberg equation, were first observed and characterized by Bohr using an assumption of quantized orbits in a classical physics argument [6].SJK 22:56, 21 December 2010 (EST)
22:56, 21 December 2010 (EST)
You still do not have direct quotes here. That is really bad.
Later the introduction and wide acceptance of the Balmer series allowed Neils Bohr to develop his quantum theory of atoms [6].

The Rydberg constant is named after the Swedish physicist Johannes Rydberg. Throughout the history of the 20th-century, the hydrogen atom has had a central position in it, as it is the simplest of the atoms. Hydrogen has played a key role in testing fundamental theories, and hydrogen spectroscopy is associated with successive advances in the understanding of atomic structure [9]. Thanks to advances in spectroscopy such as laser spectroscopy, the accuracy of the Rydberg constant $\R_\infty$ has been improved by several orders of magnitude in three decades. During the last decade, there has been very little progress with the improvement of the $\R_\infty$ values given by the last two adjustments of the fundamental constants in 2002 and 2006 [4]. In this experiment, we attempt to describe our procedure and analysis of the theoretical and experimental data used to deduce $\R_\infty$ and see how our analysis compares to others.

## Materials and Methods

For this experiment we used a Constant Deviation Spectrometer, a Spectrum Tube Power Supply Model SP200, 5000 Volts, 10mA and mercury and hydrogen tubes. The mercury tube was used to calibrate our constant deviation spectrometer and will be discussed in the next subtitle "Calibration of the spectrometer using mercury."
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 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.
Figure 3: 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.

### Calibration of the spectrometer using mercury

For this lab, we followed Professor Gold's Manual for the setup. First, we put the mercury spectrum tube into the spectrum tube power supply and turned it on allowing the mercury tube to warm up for a few minutes (3-5 minutes). Figure 1 shows the whole setup for the experiment. Adjusting the position of the ocular allowed us to focus the cross-hairs. We focused the slit using the large knob near the center of the apparatus. We then found a line of the mercury spectrum with the spectrometer slit wide (1/2 to 1mm). As we were trying to find lines from the spectrum we noticed that the narrower the slit was, the better it was to focus on the lines (figure 2) but we found that narrowing the slit caused loss of intensity of the light. We made sure to locate all the lines in the spectrum for mercury that we could possibly see, even though some lines were very hard to see (see figure 2). We adjusted the screw (figure 3) to the first red wavelength in Table 1 and removed the cover of of the Constant-Deviation Spectrometer's prism. We then 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 re-tightened the screw to secure the prism and hold it in place, and replaced the mercury spectrum tube with the hydrogen spectrum tube.
Table 1: Spectral wavelengths of mercury.
These accepted values of mercury were used to calibrate the constant-deviation spectrometer, obtained from Gold [2].
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

### Measuring the Balmer spectrum of hydrogen and deuterium

In this experiment, we took five measurements of the red spectral line for hydrogen and we made sure to turn the screw a quarter-turn right past the line before each successive measurement to avoid gear backlash. After this step, we then removed the hydrogen spectrum tube from the Spectrum Tube Power Supply and replaced it with the deuterium spectrum tube where we repeated 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.

## Results and Discussion

We calculated the accepted value of Rydberg's constant from the following equation found on Professor Gold's Manual, $R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\!$ where $\mu\,\!$ is the reduced mass $\mu = \frac{m_e*M}{m_e+M}\,\!$ where M is the mass of the nucleus. Our accepted value is listed in Table 2 along with all the other comparisons.
Table 2: This table shows 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%
The formula below is what we used to find our percent errors for hydrogen and deuterium:
$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$
We calculated our averages, standard deviations and standard error of the means by using Excel and Google Docs. Without these programs, it would have been more work to calculate these values with a calculator.

## Conclusions

We were very satisfied with our results and our small error we obtained for the Rydberg constant's. Based on our small percentage error, we conclude that we did the lab correctly and that we were able to see all the spectral lines for Hydrogen and Deuterium. When we first did this lab, we got our largest error with the red spectral lines. The second time around we calibrated the spectrometer by rotating and moving the prism which it is clear that our results were better and that we obtained a smaller percent error. Our error must have come from the quality of the equipment which made it hard for us to distinguish between the Rydberg constant for hydrogen and deuterium.
Perhaps using laser spectroscopy would have enhanced our data and we could have a obtained a smaller error for our calculations for the red spectral lines.

## Acknowledgements

I would like to thank my lab partner Ginny for the great help this whole semester with the labs, it was a pleasure working with her. Professor Steve Koch and Katie Richardson made it all possible with helping us with many of the set ups with the equipment and circuits. I would also like to thank Peng for giving us some ideas on how to calibrate the spectrometer and Alex Andrego for the pictures and the good example for the formatting of the formal report. I want to give a special thanks to the Fall 2010 Junior Lab Students who helped us get started with some of our labs during the semester and gave us good ideas for our reports.

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