Physics307L:People/Andrego/FormalReport

Balmer Series: Observing the Visible Spectra of Hydrogen

Author: Alexandra S. Andrego

Experimentalists: Alexandra S. Andrego & Anastasia A. Ierides

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

Albuquerque, NM 87131

aandrego@unm.edu

The visible spectrum of Hydrogen [3]

Abstract

On June 25th 1884, J.J. Balmer claimed that he could accurately determine the wavelengths of spectral lines of Hydrogen using a series he developed and named the "Balmer Series". The discovery of the quantitative relationship between wavelengths and the Rydberg constant paved the way for further advancement and experimentation in atomic spectrum theory and quantum physics. The Balmer Series lead to the ability to predict spectral lines beyond the observable for hydrogen atoms [4]. The Balmer series can be described as the designation of visible spectral lines of the emission of hydrogen atoms. The Balmer series for the Hydrogen atom contains eight visual spectral lines (32 in total) though only four are observable under the conditions of this experiment [7]. The four observable spectral lines are categorized by Ultra Violet, Violet, Blue-Green, and Red, depending on the value of the light wavelength. The use of a spectrometer allows the observation and classification of spectra lines of the hydrogen atom. By using electrical stimulation to excite the hydrogen atoms to higher energy levels, measurements of the emitted photons with wavelengths equivalent to the energy of our excited electrons can be made. Through such measurements it is possible to experimentally determine Rydberg's constant, R, that is used in the Balmer-Rydberg equation for hydrogen:

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

In this experiment our data gave us an experimental value of the Rydberg constant to be

[math]\displaystyle{ R_{experimental mean}\approx1.0973\pm 0.0025\times10^7 m^{-1}\,\! }[/math]

Introduction

--Alexandra S. Andrego 03:04, 16 November 2009 (EST)I know I need an introduction but I would like to conduct more research on the uses of the Balmer series and Deuterium before writing this up formally. I was unable to do much research due to the unfortunate and fairly inconvenient death of my 2wire, and lack of competency within the customer service department at Qwest. My apologies.

Methods

FIGURE 1: The complete set up of our Balmer Series lab. Here you can see that we have the Constant Deviation Spectrometer calibrated and pointed at our illuminated Hydrogen Bulb. From this angle it is also apparent that the use of multiple textbooks was needed to lift the Hydrogen Bulb and Spectrum Tube Power Supply so that the emitted spectra of the Hydrogen atoms could be in the visible range of the Constant Deviation Spectrometer (of which has no setting for adjusting the height of the apparatus)

FIGURE 2: The single slit aperture on our Constant Deviation Spectrometer. The width of the slit that appears in this photo was adjusted using the visible thumbscrew on the right to balance precision and clarity. Clarity of the incident light is best with a large slit width, while precision of wavelength measurements are best with a very fine slit width.

FIGURE 3: The open prism apparatus and spectrometer dial. The prism apparatus seen here had to be adjusted by hand using the visible screw tightener to lock the prism in position once calibration was complete. The spectrometer dial seen in the foreground of this picture functions through a series of gears and has to be turned by hand to acquire measurements. The gears of this dial can sometimes cause what is known as "backlash" where empty space between the gears, that is caused by a switch in direction of turns, can cause false wavelength measurements.

Calibration of the Spectrometer

To begin this experiment a few preliminary steps were taken to prepare for our experimentation. First we adjusted the Constant-Deviation Spectrometer (SER #12610) by bringing the cross-hairs into focus and sliding the ocular to a position that suited our vision and allowed for no parallax to exist between the cross-hairs and the slit of the spectrometer when it was focused sharply. We brought the slit into focus while looking through the eye piece by turning the large ring near the center of the viewing telescope called the spectrometer dial. We then attached and positioned the mercury bulb (Mercury Vapor Spectrum Tube (S-68755-30-K)) into the spectrum tube power supply (Model SP200) and waited approximately five minutes for the mercury bulb to warm up. Our complete set up can be seen in FIGURE 1. Calibrating the spectrometer is a very important part of the set up for this experiment. The calibration process was made easier by the use of a wide slit setting seen in FIGURE 2. After finding a line of the the mercury spectrum through the eye piece of the spectrometer and narrowing the slit until the line was focused sharply, we were able to use the known values of light wavelengths for the spectral lines of mercury (TABLE 1) to position the prism accurately [2]. The more precise our calibration was the smaller our systematic error was. The positioning of the prism was done by a series of small rotations of the prism and measurements from the spectrometer dial, seen in detail in FIGURE 3. We placed our spectrometer dial on the correct measurement for the spectral line we were observing and then attempted to correctly position our prism so that the corresponding spectral line was aligned on the cross hairs in the eyepiece of the spectrometer with a small slit width. This was best done by starting off with a large slit width and then slowly decreasing its width as we focused in on our spectral line. It is very important to understand the mechanics of a spectrometer to avoid causing unnecessary systematic error. Due to the use of gears in the spectrometer dial, "back lash" caused by the empty space between the teeth of the gears, can cause false data. It is important to always turn the dial back at least a quarter of a turn before rotating to the position at which a measurement needs to be taken. This insures that when the measurement is taken, the gears are continuously in contact and the measurements are more accurate. We repeated this process until the positioning of all components (prism, spectrometer, spectrometer dial, and slit width) yielded the correct measurements for the wavelengths of the corresponding mercury spectral line, for all visible lines. Once the calibration process was complete we were able to remove the mercury bulb and replace it with the hydrogen bulb (Hydrogen Spectrum Tube (S-68755-30-G)), and/or the deuterium bulb (Deuterium Spectrum Tube (S-68755-30-E)), and thus began our experimentation.

TABLE 1

Color	Wavelength (nm)
Deep Violet (very hard to see)	404.7
Violet	435.8
Very Weak Blue-Green	skip this one
Green	546.1
Yellow 1	577.0
Yellow 2	579.0
Red	690.75

Measuring the Spectral Lines for Hydrogen and Deuterium

After replacing the mercury bulb in the experiment with the hydrogen bulb (and again waiting the five minutes to allow the bulb to warm up and keeping in mind the "back lash" effect), we were able to easily take the measurements for the visual spectral lines of hydrogen. We began by rotating the spectrometer dial until a spectral line came into focus through the spectrometer eye piece, we made sure to do this with a wide slit width setting that made the spectral lines more easily viewable. We then narrowed our slit width and rotated our spectrometer dial once more to align the spectral line with cross hairs and to record the measurement. We repeated this process five times for each of the four visible spectral lines for Hydrogen. Anastasia read the measurements off of our spectrometer dial and I did the alignment of each spectral line. Once we felt we had taken sufficient data with the Hydrogen bulb we replaced it with a deuterium bulb and observed and recorded its spectral lines through the same method. Our data taking was spread out over the course of two non consecutive days. All of our hydrogen measurements were taken on the first day of experimentation. However for the Deuterium measurements we had to recalibrate our spectrometer on the second day of experimentation before beginning again. This recalibration could have some effects on our systematic error between our Hydrogen data and our Deuterium data.Our raw data table can be seen in our Primary Lab Notebook [6]. After collecting all of our needed measurements we used the average of all values for each spectral line to determine the best measured wavelength for each spectral color of the respective element. The standard error of mean was calculated to obtain the 68% uncertainty margins for our measurements. Once the average wavelength per spectral line was determined, we took the mean over all spectral lines of the same element and used that average to compute an experimental Rydberg constant ,R, as appears in the Balmer-Rydberg equation discussed earlier in this paper. The use of Microsoft Excel and Google Docs made the analysis of the raw data possible.

Analysis and Results

FIGURE 4: The HYDROGEN linear fit data of 1/Lambda vs the Quantum Transition Term . Here you can see the linear relationship between the inverse of the wavelength of the spectrum light and the quantum transition term that was calculated for each spectral line from the Balmer-Rydberg equation described in the formal report. The resulting slope of this graph is the experimental mean value for the Rydberg constant of the Hydrogen Balmer series.

FIGURE 5: The DEUTERIUM linear fit data of 1/Lambda vs the Quantum Transition Term. Here you can again see the linear relationship between the inverse of the wavelength of the spectrum light and the quantum transition term. Through comparision with this graph and the one seen in FIGURE 4 of the Hydrogen case, we can see very little difference between the resulting slope of the graphs and the experimental mean values for the Rydberg constants.

Our results from this experiment can be seen in TABLE 2 below. The transition column refers to the structure of the Balmer series and the Balmer-Rydberg equation discussed earlier in this paper. All error margins included in the data table are the results of the standard error of mean calculation performed by excel. The excel spreadsheet with these calculations can be refered to in our Primary Lab Notebook under the Analysis section [6].

{{#widget:Google Spreadsheet

|key=t_1tTZFJkNFGN2kIa2cLLfg |width=832 |height=150

}}

From the values listed in TABLE 2, a Rydberg's constant was calculated from the Balmer-Rydberg equation as follows:

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

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

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

Where the relationship between [math]\displaystyle{ \frac{1}{\lambda }\,\! }[/math] and the term [math]\displaystyle{ \frac{n^2-4}{4n^2}\,\! }[/math] were graphed using excel and the resulting slope for both the Hydrogen and Deuterium cases was the average experimental value of the Rydberg Constant (see FIGURE 4 for the Hydrogen Linear Fitting and FIGURE 5 for the Deuterium Linear Fitting). Excel was used for the creation of theses graphs and for the linear fit seen on both figures. For any and all excel spreadsheets please refer to our Primary Lab Notebook [6].

The average values for the measured Rydberg's constants are:

[math]\displaystyle{ R_{Hydrogen,average}\approx1.0973\pm 0.0025\times10^7 m^{-1}\,\! }[/math]

[math]\displaystyle{ R_{Deuterium,average}\approx1.0983\pm 0.0007\times10^7 m^{-1}\,\! }[/math]

We discovered that our experimental Rydberg constant for the Hydrogen Spectrum was approximately off by a small magnitude of the fifth order, and henceforth our experiment proved to be successful in estimating an appropriate value for the Rydberg constant for the Hydrogen Spectrum Balmer Series. However,our experimental mean value for the Deuterium Balmer Series was not as successful. Our uncertainty in our experimental Rydberg constant for the Deuterium spectrum does not put the accepted value in our attained range. This goes to show that our experiement set-up on day two of our data trials (Deuterium data was only taken on the second day of trials and had a different calibration), falls short in statistically minimizing both our random and systematic error. The accepted value of the Rydberg constant, for which we have based all comparisons, is [5]:

[math]\displaystyle{ R_{accepted}=1.0973731568525\times10^7 m^{-1}\,\! }[/math]

Discussion

In order to fully comprehend the results above, it is beneficial to compare the experimental results with the accepted values.

The accepted value of Rydberg's constant is calculated from the following equation found on page 30 of Professor Gold's Manual.

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

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

[math]\displaystyle{ R=1.0967758\times 10^7 m^{-1}\,\! }[/math]

The following accepted values for the four visible wavelengths of the Balmer Series were taken from the hyperphysics website

[math]\displaystyle{ n=6\rightarrow n=2\,\! }[/math]

[math]\displaystyle{ \lambda =410.174 nm\,\! }[/math]

[math]\displaystyle{ n=5\rightarrow n=2\,\! }[/math]

[math]\displaystyle{ \lambda =434.047 nm\,\! }[/math]

[math]\displaystyle{ n=4\rightarrow n=2\,\! }[/math]

[math]\displaystyle{ \lambda =486.133 nm\,\! }[/math]

[math]\displaystyle{ n=3\rightarrow n=2\,\! }[/math]

[math]\displaystyle{ \lambda =656.272 nm\,\! }[/math]

Through comparison it is evident that the measured values attained through this experiment were successfully close to the accepted values listed above. It is evident from results such as these that the systematic error was successfully minimized for this experiment. When comparing the results from the hydrogen bulb and the deuterium bulb, it can be noticed that both results have significantly different true means for each wavelength and the Rydberg constant. According to the theory behind the Balmer series, and Rydberg constant, this is due to the difference in mass that exists between the two atoms. This should therefore be true for all atoms of varying mass.

Uncertainty and Error

Though this experiment proves to have minimized major sources of systematic error, there were still some areas of which could have been improved on.

This experiment was conducted over the course of two non-consecutive days, where it is known that the spectrometer was used by other members of the facility between data taking. This caused a source of systematic error in that the spectrometer had to be recalibrated for a second day's worth of experimentation. Precise and consistent calibration is very important for this lab because it allows better comparison of results and more accurate measurements.

Our error percentiles for the calculations made in this experiment for the Rydberg's Constant can be calculated as:

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

[math]\displaystyle{ \% error_{Hydrogen}=\frac{1.0967758\times 10^7 m^{-1}-1.0972781\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}} }[/math]

[math]\displaystyle{ \approx0.046%\,\! }[/math]

[math]\displaystyle{ \% error_{Deuterium}=\frac{1.0967758\times 10^7 m^{-1}-1.0983221\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}} }[/math]

[math]\displaystyle{ \approx0.141%\,\! }[/math]

Acknowledgments

I would like to extend my greatest gratitude for Anastasia Ierides who was my partner for the Balmer Series Lab. Her enthusiasm and work ethic made this experiment one of my favorites.

Google Docs and Microsoft Excel were used to format and post our raw data and error analysis to our primary lab notebook and this formal report

I would also like extend my utmost appreciation to Professor Koch and Pranav for monitoring and providing guidance through this entire process.

References

[1] 'Hydrogen Energies and Spectrum'. http://hyperphysics.phy-astr.gsu.edu/Hbase/hyde.html#c4
[2] 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
[3] Merikanto, 'File:Emission spectrum-H.png'. Wikepedia, May 2006 http://en.wikipedia.org/wiki/File:Emission_spectrum-H.png
[4] Banet, Leo, 'Evolution of the Balmer Series'. Am. J. Phys. 34, 496 (1966), DOI:10.1119/1.1973077 http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000034000006000496000001&idtype=cvips&gifs=yes&ref=no
[5] Mohr, Peter J., Taylor, Barry N., and David B. Newell, 'CODATA recommended values of the fundamental physical constants: 2006'. Rev. Mod. Phys. 80, 633 (2008), DOI:10.1103/RevModPhys.80.633 http://physics.nist.gov/cuu/Constants/codata.pdf
[6] User:Alexandra S. Andrego/Notebook/Physics 307L/2009/09/28, 'Balmer Series'. http://www.openwetware.org/wiki/User:Alexandra_S._Andrego/Notebook/Physics_307L/2009/09/28
[7] Wood, R. W.,'An Extension of the Balmer Series of Hydrogen and Spectroscopic Phenomena of Very Long Vacuum Tubes'. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 97, No. 687 (Aug. 3, 1920), pp. 455-470 http://www.jstor.org/stable/93837
[8] Rigden, John S., James O'Connell, and Reviewer, Am. J.,'Hydrogen: The Essential Element'. American Journal of Physics Phys. 71, 189 (2003), DOI:10.1119/1.1522705 http://scitation.aip.org.libproxy.unm.edu/getpdf/servlet/GetPDFServlet?filetype=pdf&id=AJPIAS000071000002000189000001&idtype=cvips&prog=normal
[9] Hänsch, T. W., M. H. Nayfeh, S. A. Lee, S. M. Curry, and I. S. Shahin,'Precision Measurement of the Rydberg Constant by Laser Saturation Spectroscopy of the Balmer α Line in Hydrogen and Deuterium'. Department of Physics, Stanford University, Stanford, California 94305. The American Physical Society, Vol 32, 1336-1338. 1974. DOI:10.1103/PhysRevLett.32.1336 http://link.aps.org/doi/10.1103/PhysRevLett.32.1336
[10] Andreae, T., W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, and T. W. Hänsch,'Precision Measurement of the Rydberg Constant by Laser Saturation Spectroscopy of the Balmer α Line in Hydrogen and Deuterium'. Max-Planck-Institut für Quantenoptik, W-8046 Garching, Germany. The American Physical Society, Vol 69, 1923-1926. 1992. DOI:10.1103/PhysRevLett.69.1923 http://link.aps.org/doi/10.1103/PhysRevLett.69.1923
[11] Laws, Edward. 'Mathematical Methods for Oceanographers: An Introduction'. John Wiley and Sons Inc. NY, New York. Ed 1, x. 1997. ISBN: 0-471-16221-3.