Physics307L:People/Ierides/The Balmer Series Formal Lab Report

The Balmer Series of Hydrogen and Deuterium in Atomic Physics

Author: Anastasia A. Ierides

Experimentalists: Anastasia A. Ierides, Alexandra S. Andrego

University of New Mexico, Department of Physics and Astronomy, Albuquerque NM, 87131

Physics 307L Junior Lab: December 14, 2009

aierides@nm.edu

Abstract

The importance of the Balmer series lies in the prediction of absorption/emission lines of hydrogen in the visible spectrum. The Balmer series led to the ensuing discoveries of the Lyman, Paschen, and Brackett series. These series predict the spectral lines of hydrogen in the non-visible parts of the spectrum. In order to detect the Balmer Series, of first Hydrogen and then Deuterium, we used a Constant-Deviation Spectrometer (which was calibrated using a Mercury Vapor bulb), in an environment with limited light, to measure the wavelength of the emission spectra of each. The excitation of the electrons in the atoms of each gas through electrical stimulation allows a release of photons at specific wavelengths corresponding to the Balmer series (the excitation of electrons from the first ionized state, [math]\displaystyle{ n = 2 }[/math], to the second, [math]\displaystyle{ n = 3 }[/math]). Then using a generalized form of the Balmer formula, the Rydberg formula, we used our measured wavelength to compute Rydberg's constant and to verify whether or not a deviation between the Hydrogen spectra and the Deuterium spectra could be found. Our calculated value of Rydberg's constant for each Hydrogen [math]\displaystyle{ (1.0972781 \pm 0.0025) \times10^7 }[/math] per meter and Deuterium [math]\displaystyle{ (1.0983221 \pm 0.0007) \times10^7 }[/math] per meter is in close proximity with the accepted value of [math]\displaystyle{ 1.0967758 \times10^7 }[/math] per meter.

Introduction

The Balmer series a series of predicted and confirmed wavelengths of photons emitted from hydrogen spectrum belonging to the visible spectrum. It is the culmination of the excitation

of electrons from the n=2 state to the n=3,4,5, and 6 states in an atom causing a release of photons of corresponding energies [5]. These wavelengths compose the visible part of the hydrogen spectrum and are hence called the Balmer series.

Balmer Series of the Hydrogen Spectrum [9]

The prediction of these spectral lines comes from the Balmer formula, written as:

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

[math]\displaystyle{ n=3,4,5,...\,\! }[/math]

where [math]\displaystyle{ \lambda }[/math] is the wavelength to be predicted for each transition, [math]\displaystyle{ n }[/math] is the principle quantum number of the electron, and [math]\displaystyle{ R }[/math] is a constant called Rydberg's constant, derived as seen in the Appendix below [3]. This constant however is a variant one when it comes to dealing with various elements due to the reduced mass of the subatomic particles involvement in the calculation.

The theory behind this constant and the above equation is that for every element, there is some value R for which the wavelength of each photon emitted by each transition can be

predicted. The discrepancy between the "R" values of hydrogen and deuterium is ever so slight due to the mass change of deuterium from its extra neutron, which in turn may or should cause a shift in the spectral lines [7]. To test this hypothesis, a "constant-deviation" spectrometer is used within an allowed precision to experimentally attain data to support this hypothesis [3].

During our experiment we found that slight discrepancy in our values could be due to some oversight when using the 'scope'. There might have gear back lash from not turning the knob

all the way back before remeasuring spectra for each trial, even though we took great care in doing so. Also, the lab lasted over two days, with a week interval and during that interval another group had used the same device used for our experiment, so re-calibration for our set of data was necessary during the second day.

Background

Before 1885, the year that the Balmer formula was founded by a Swiss school teacher Johann Jakob Balmer, physicists, although aware of atomic emissions, lacked the tools to

predict the location of each spectral line [5]. The Balmer equation is used in the prediction of each of the four visible emission/absorption lines of hydrogen with high precision. This had inspired the Rydberg equation, invented by a Swedish physicist Johannes Robert Rydberg [6]. This new equation was a generalization of the Balmer formula, which in turn "led to the finding of the Lyman, Paschen, and Brackett series used in predicting the absorption/emission lines of hydrogen found outside the visible spectrum" [5].

According to the Rutherford Bohr model (devised by Neils Bohr in 1913 from the amelioration of a model created by Ernest Rutherford in 1911) of the Hydrogen atom, an electron

transition that occurs between the second energy level or first excited state in the atom (corresponding to n=2) and any other higher energy level results in the Balmer lines [8].

The Balmer series has been helpful in astronomical and physical use for years due to the abundance of hydrogen in the universe. It has been used for several means such as

spectral classification, the measure radial velocities of objects in space due to doppler shifting, and the distances to those objects.

The Balmer series is one of six series in which the spectral line emissions of hydrogen are designated. There are four different emission wavelengths of visible light by which the

hydrogen spectrum is defined. These wavelengths can be calculated using the Balmer formula (found by Johann Balmer, 1885) written above in the "Purpose" and reflect emissions of photons by transitions of electrons between principal quantum number levels from [math]\displaystyle{ n\geq3 }[/math] to [math]\displaystyle{ n = 2 \,\! }[/math] [5].

Compared to the Hydrogen atom, which contains one proton in the nucleus, the Deuterium atom, contains a proton and a neutron in its nucleus. Thus the Deuterium atom is heavier

than the regular Hydrogen atom. By observing and classifying spectra lines of the hydrogen and deuterium atoms the Balmer series can be determined [7]. By using electrical stimulation to excite the atoms to higher energy levels we can measure the emitted photons of wavelengths equivalent to the energy of our excited electrons.

Materials and Methods

Apparatus: Constant-Deviation Spectrometer

Figure 1 A Pellin-Broca constant-deviation prism [10] has a wavelength dependent index of refraction which makes it act as a spectrometer [3].

A constant-deviation spectrometer (SER. #12610) in use during this lab consists of a Pellin-Broca constant-deviation prism whose incident and exit angles are fixed with an appropriate rotation of the prism. The only requirement in this setting consists of keeping a the incident light ray and the axis of the analyzing telescope perpendicular [3].

Figure 2 The actual prism inside the constant-deviation spectrometer separating the spectral lines. This prism is adjusted in order to calibrate the spectrometer [11]

Calibration of the Spectrometer

Figure 3. Constant-Deviation Spectrometer and Power Supply [11]. Here the spectrometer is set up in front of the bulb to begin calibration. Books were necessary to be placed underneath the power supply in order to achieve the best resolution possible through the spectrometer.

In order to calibrate our spectrometer we used a mercury vapor tube (S-68755-30-K). To start, the mercury tube was placed inside the spectrum tube power supply (Model SP200) and warmed for about five to ten minutes. Meanwhile, the spectrometer was adjusted such that the cross-hairs were brought into focus by adjusting the ocular to suit our vision. The slit was then brought into focus using the large ring near the center of the viewing telescope, turning the screw only in one direction to ensure that "gear backlash" can be avoided. This setup can be seen in Figure 3.

Figure 4. Slit Width. This is the slit width from the spectrometer. It was adjusted using the large knob to the right of the slit. In order to find the various spectral lines, a wide slit was used, whereas the more focused thin slit was used to focus in on the wavelength for each spectral line.

Using the spectrometer with a wide slit as seen in Figure 4, a line of the mercury spectrum could be distinguished and then, with a narrowed slit, the line was brought into focus until it was quite narrow and sharp. Noting the orientation and value of the spectrometer dial, as seen in Figure 5, the corresponding wavelengths of light were applied as given in Table 1 [3]:

Figure 5 The open prism apparatus and measuring gear are visible here. The spectrometer dial was used to orient the correct wavelength to the corresponding spectral line.

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

The calibration of the spectrometer was then completed and the compilation of data was commenced.

Measurement of the Hydrogen and Deuterium Spectrum

This is our raw data: {{#widget:Google Spreadsheet |key=tlcu3hB5KpmJ6X9wgXnuimA |width=650 |height=300 }}

Using our raw data tables we used the functions in Excel for mean and standard deviation to find the standard error margins. From that we formulated a total mean: {{#widget:Google Spreadsheet |key=tHKTX_HPEkRAqhsbZKtjGrg |width=650 |height=705 }}

Results and Discussion

^{SJK 00:39, 30 November 2009 (EST)}

00:39, 30 November 2009 (EST)
Notice that besides the photos of the apparatus above, you do not have any other figures. Whereas in almost any experimental research report, there will be several figures to cleanly and clearly display the results. So, definitely you're going to need to create graphs from the data, as I mentioned above. Also, this results section right now looks like a string of calculations, which would be fairly confusing to the reader. Plus, there is very little in the way of discussion: What does it mean? What are you trying to ascertain, how are statistical and systematic errors looking, etc. I think in this section in particular, you'd benefit quite a bit by reading some published, peer-reviewed physics or astronomy papers that you're interested in. You'll see how figures and discussion work. Just as I mentioned above, you'll want to number each figure and have an excellent caption for the reader. Furthermore, your results and discussion text will refer to every figure and comment on them.

^{SJK 18:02, 29 November 2009 (EST)}

18:02, 29 November 2009 (EST)
Unless you have other ideas, I think for your "extra data" session, you should focus on the issue of whether you can discern the difference between H and D with your apparatus (I think the answer is "no"). You can approach the issue statistically (how small could you get the uncertainty with your instrument, and how does this compare with the difference between H and D. This will make your results section longer, and allow you to have more discussion. Also, you can expand your introduction to talk about difference between H and D.

Conclusions

^{SJK 00:42, 30 November 2009 (EST)}

00:42, 30 November 2009 (EST)
I think you are on the right track for conclusions, but more can be included: "In conclusion, we were able to precisely measure the Rydberg constant for both hydrogen and deuterium. The accepted value for the Rydberg constant for Hydrogen was statistically significantly different from our measurement (___ SEMs discrepancy), indicating some systematic error. Our measurements for H and D were statistically signficantly different, although theoretically their difference should be too small for us to discern with this instrument. Possible reasons for these results are ... and can be investigated further by ...

According to our data, although the Deuterium spectral lines varied from the Hydrogen lines in wavelength, as seen by the percentage error in the Rydberg Constant of each, the variance is slight. The largest value by which the wavelengths varied was designated in the red wavelength measurement as seen in our data tables. But also according to our data, the wavelength measurements of each color seemed to be shifted from the Hydrogen in the Deuterium spectrum.

Acknowledgements

Please note that Alexandra S. Andrego was my lab partner for this lab. I thank her for her collaboration in the acquisition of data and helpful analysis.

Thanks to Professor Gold for using his lab manual [3] as a guide for the calibration process as well as a guide for data collection

Also thanks to the many authors of Wikipedia for the useful open-access articles discussing the Balmer series [5], Rydberg's constant [6], and Deuterium [7].

Finally, I extend my gratitude to Professor Steven Koch as well as his teaching assistant, Pranav Rathi, for their extensive feedback as well as guidance throughout this lab.

References

[1] "Hydrogen energies and spectrum." Test Page for Apache Installation. N.p., n.d. Web. 13 Dec. 2009. http://hyperphysics.phy-astr.gsu.edu/Hbase/hyde.html#c4

[2] "Hydrogen spectrum." Test Page for Apache Installation. N.p., n.d. Web. 13 Dec. 2009. http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/hydspec.html

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

[4] "User:Anastasia A. Ierides/Notebook/Physics 307L/2009/09/28 - OpenWetWare." Main Page - OpenWetWare. N.p., n.d. Web. 13 Dec. 2009. http://openwetware.org/wiki/User:Anastasia_A._Ierides/Notebook/Physics_307L/2009/09/28

[5] "Balmer series - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 14 Dec. 2009. http://en.wikipedia.org/wiki/Balmer_series

[6] "Rydberg constant - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 14 Dec. 2009. http://en.wikipedia.org/wiki/Rydberg_constant

[7] "Deuterium - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 14 Dec. 2009. http://en.wikipedia.org/wiki/Deuterium

[8] "Bohr model - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 14 Dec. 2009. http://en.wikipedia.org/wiki/Bohr_model

[9] "Balmer series." Department of Mathematics, Texas A&M University. N.p., n.d. Web. 14 Dec. 2009. http://www.math.tamu.edu/~comech/balmer/balmer.html

[10] "Pellin-Broca prism - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 14 Dec. 2009. http://en.wikipedia.org/wiki/Pellin-Broca_prism

[11] "Physics307L F07:People/Rivera/Notebook/Formal Report - OpenWetWare." Main Page - OpenWetWare. N.p., n.d. Web. 14 Dec. 2009. http://openwetware.org/wiki/Physics307L_F07:People/Rivera/Notebook/Formal_Report

Appendix

Derivation of the Rydberg Equation [5]

We can start from the equation of total energy of an electron in the nth energy state derived from the Bohr model:

[math]\displaystyle{ E_\mathrm{total} = - \frac{m_e e^4}{8 \epsilon_0^2 h^2}. \frac{1}{n^2} \ }[/math]

The change in energy of an electron transitioning from one energy state with a value [math]\displaystyle{ n }[/math] to another is:

[math]\displaystyle{ \Delta E = \frac{ m_e e^4}{8 \epsilon_0^2 h^2} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \ }[/math]

Using [math]\displaystyle{ \frac{1}{ \lambda} = \frac {E}{hc} \rightarrow \Delta{E} = hc \Delta \frac{1}{\lambda}\,\! }[/math] to change the units to wavelength, we get

[math]\displaystyle{ \Delta \left( \frac{1}{ \lambda}\right) = \frac{ m_e e^4}{8 \epsilon_0^2 h^3 c} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \ }[/math]

where

[math]\displaystyle{ h \ }[/math] is Planck's constant,

[math]\displaystyle{ m_e \ }[/math] is the rest mass of the electron,

[math]\displaystyle{ e \ }[/math] is the elementary charge,

[math]\displaystyle{ c \ }[/math] is the speed of light in vacuum, and

[math]\displaystyle{ \epsilon_0 \ }[/math] is the permittivity of free space.

And the Rydberg constant for Hydrogen is found as:

[math]\displaystyle{ R_H=\frac{m_e e^4}{8 \epsilon_0^2 h^2}\,\! }[/math]

List of Used Constants

[math]\displaystyle{ \mu\,\! }[/math] is the reduced mass of an atom

[math]\displaystyle{ e=1.602\times10^{-19} C\,\! }[/math]

[math]\displaystyle{ \epsilon_0=8.854\times10^{-12} F\cdot m^{-1}\,\! }[/math]

[math]\displaystyle{ c=2.998\times10^8 m\cdot s^{-1}\,\! }[/math]

[math]\displaystyle{ h=6.626\times10^{-34}J\cdot s\,\! }[/math]

Rydberg's constant for hydrogen is calculated to be approximately:

[math]\displaystyle{ R\simeq1.0967758\times10^7m^{-1}\,\! }[/math]