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 wave 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 contains eight spectral lines though only four are observable under the conditions of this experiment. 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]

We discovered that our experimental Rydberg constant was approximately off by a small magnitude of the fifth order, and henceforth our experiment proved to be successful in determining the appropriate value for the Rydberg constant. The accepted value of the Rydberg constant is [5]:

[math]\displaystyle{ R_{accepted}=1.0973731568525\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.

Required Equipment

The equipment that was used to perform this experiment is as follows:

-Constant-Deviation Spectrometer (SER #12610)

-Spectrum Tube Power Supply (Model SP200)

• 5000V
• 10 MA

-Spectrum Tubes:

• Spectrum Tube, Mercury Vapor (S-68755-30-K)
• Spectrum Tube, Hydrogen (S-68755-30-G)
• Spectrum Tube, Deuterium (S-68755-30-E)

Methods

The Complete Set Up

The Slit Width for the Incident Light

The Open Prism Apparatus and Spectrometer Dial

Set Up

To begin the experiment a few preliminary steps were taken. First adjusting the spectrometer by bringing the cross-hairs into focus and sliding the ocular to a position that suits our vision allows for no parallax to exist between the cross-hairs and the slit of the spectrometer when it is focused sharply. To bring the slit into focus while looking through the eye piece, it is possible to turn the large ring near the center of the viewing telescope. By attaching and positioning the mercury bulb into the spectrum tube power supply the apparatus is made functional. The spectrum tube power supply requires approximately five minutes to warm up the mercury bulb. Calibration of the spectrometer is a very important part of the set up for this experiment. It is done by the use of a wide slit setting. After finding a line of the the mercury spectrum and narrowing the slit until the line is focused sharply, the known values of light wavelengths for the spectral lines of mercury can be used to position the prism, bulb, and spectrometer to achieve minimal systematic error. The positioning is done by a series of small rotations of the prism and measurements from the spectrometer dial. (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). This is repeated until the positioning of all components yields correct measurements for the wavelengths of the corresponding mercury spectra. Once the calibration process is complete the mercury bulb may be removed and replaced with the hydrogen bulb, and experimentation may commence.

The calibration technique demonstrated in this experiment used the general data from the following table in [2]

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

Experiment

After replacing the mercury bulb in the experiment with the hydrogen bulb (and waiting the five minutes to allow the bulb to warm up and keeping in mind the "back lash" effect), measurements for the visual spectral lines of hydrogen were easily taken.

Once sufficient data was obtained from the hydrogen bulb, the experiment was taken one level further, and the spectral lines of deuterium were observed through the same method as well.

Raw Data

The raw data obtained from this experiment was formatted into the table shown below.

{{#widget:Google Spreadsheet

|key=tlcu3hB5KpmJ6X9wgXnuimA |width=650 |height=305

}}

Analysis and Results

The average off all values for each spectral line was used to determine the best measured wave length for each spectral color. 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, a mean was taken over all spectral lines and used to compute an experimental Rydberg constant ,R, as appears in the Balmer-Rydberg equation discussed earlier.

The use of Microsoft Excel and Google Docs made the analysis of the raw data possible.

The spreadsheet used to perform the analysis and error propagation can be seen below.

{{#widget:Google Spreadsheet

|key=tti_lcw5NdLvkGYblz1VIxQ |width=650 |height=550

}}

From the data table above the measured experimental values of wavelengths for each spectral line observed are:

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

[math]\displaystyle{ \lambda_{Hydrogen} =409.84 nm\,\! }[/math]

[math]\displaystyle{ \lambda_{Deuterium} =N/A\,\! }[/math]

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

[math]\displaystyle{ \lambda_{Hydrogen} =433.92 nm\,\! }[/math]

[math]\displaystyle{ \lambda_{Deuterium} =433.3 nm\,\! }[/math]

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

[math]\displaystyle{ \lambda_{Hydrogen} =485.96 nm\,\! }[/math]

[math]\displaystyle{ \lambda_{Deuterium} =485.62 nm\,\! }[/math]

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

[math]\displaystyle{ \lambda_{Hydrogen} =657.4 nm\,\! }[/math]

[math]\displaystyle{ \lambda_{Deuterium} =655.9 nm\,\! }[/math]

From these values a Rydberg's constant can be 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]

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

[math]\displaystyle{ R_{Hydrogen}=\frac{4(6)^2}{(409.84\times10^{-9} m)((6)^2-4)}\approx1.0979895\times10^7 m^{-1}\,\! }[/math]

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

[math]\displaystyle{ R_{Hydrogen}=\frac{4(5)^2}{(433.92\times10^{-9} m)((5)^2-4)}\approx1.0974153\times10^7 m^{-1}\,\! }[/math]

[math]\displaystyle{ R_{Deuterium}=\frac{4(5)^2}{(433.3\times10^{-9} m)((5)^2-4)}\approx1.0989856\times10^7 m^{-1}\,\! }[/math]

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

[math]\displaystyle{ R=\frac{4(4)^2}{(485.96\times10^{-9} m)((4)^2-4)}\approx1.0984840\times10^7 m^{-1}\,\! }[/math]

[math]\displaystyle{ R=\frac{4(4)^2}{(485.62\times10^{-9} m)((4)^2-4)}\approx1.0982524\times10^7 m^{-1}\,\! }[/math]

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

[math]\displaystyle{ R=\frac{4(3)^2}{(657.4\times10^{-9} m)((3)^2-4)}\approx1.0952236\times10^7 m^{-1}\,\! }[/math]

[math]\displaystyle{ R=\frac{4(3)^2}{(655.9\times10^{-9} m)((3)^2-4)}\approx1.0977283\times10^7 m^{-1}\,\! }[/math]

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

[math]\displaystyle{ R_{Hydrogen,average}=\frac{(1.0979895+1.0974153+1.0984840+1.0952236)\times10^7m^{-1}}{4} }[/math]

[math]\displaystyle{ =\frac{4.3891124\times10^7 m^{-1}}{4}\,\! }[/math]

[math]\displaystyle{ \approx1.0972781\pm 0.0025\times10^7 m^{-1}\,\! }[/math]

[math]\displaystyle{ R_{Deuterium,average}=\frac{(1.0989856+1.0982524+1.0977283)\times10^7m^{-1}}{3} }[/math]

[math]\displaystyle{ =\frac{3.2949663\times10^7 m^{-1}}{4}\,\! }[/math]

[math]\displaystyle{ \approx1.0983221\pm 0.0007\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 take the time now to extend my greatest gratitude for Anastasia Ierides who was my lab partner for this 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 wiki lab notebook

I would also like to thank Professor Koch and Pranav for asking all the hard questions and never loosing patience with us during the long lab hours.

References

[1] 'Hydrogen Energies and Spectrum' http://hyperphysics.phy-astr.gsu.edu/Hbase/hyde.html#c4
[2] The University of New Mexico Dept. of Physics and Astronomy PHYSICS 307L: 'Junior Laboratory Manual Fall 2006' By Professor Michael Gold 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