Physics307L:People/Carrillo/FormalRoughDraft

= 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 measrue 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
 * $$R_{Hydrogen,average}\approx1.099\pm 0.008\times10^7 m^{-1}\,\!$$




 * 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]. 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_infinity 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_infinity 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_infinity 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."



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.  Adusting 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 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).  As we were turning the screw that rotates the prism, we made sure to record the position on the dial which corresponds to the mercury lines (see figure 3).  To adjust our measurements, we used a linear fit in Google Docs and this completed the calibration part of constant-deviation spectrometer.

Measuring the Balmer spectrum of hydrogen

 * After we finished calibrating, we removed the mercury tube from the power supply and slotted the hydrogen tube into the power supply and let it warm up for about 3-5 minutes before we started taking data. We started from the very far left of the spectrum and moved the screw from left to right making sure that we did not move it to the left because this would give us a systematic error. We then measured five wavelengths for each of the four colors in the hydrogen spectrum.  We repeated the same procedure for deuterium.



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
 * $$R=1.0967758\times 10^7 m^{-1}\,\!$$


 * The following accepted values for the four visible wavelengths of the Balmer Series were taken from the hyperphysics website
 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda =410.174 nm\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda =434.047 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda =486.133 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda =656.272 nm\,\!$$


 * Using the results from the data set 2 I calculated the values for the wavelengths. Please follow this link [[Image:Balmer_calculations.xlsx|excel spreadsheet]] to see the standard deviation and standard error of the mean for our data. The values below are what I calculated in the excel spread sheet.
 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =417.88 nm\,\!$$
 * $$\lambda_{Deuterium} =N/A\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =433.66 nm\,\!$$
 * $$\lambda_{Deuterium} =433.7 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =483.44 nm\,\!$$
 * $$\lambda_{Deuterium} =483.17 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =644.19 nm\,\!$$
 * $$\lambda_{Deuterium} =642.07 nm\,\!$$


 * Using these values,I was able to calculate our measured Rydberg constant.
 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!$$
 * $$R=\frac{4n^2}{\lambda(n^2-4)}\,\!$$


 * $$n=6\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}\approx1.0768641\times10^7 m^{-1}\,\!$$


 * $$n=5\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}\approx1.0980733\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}\approx1.0979720\times10^7 m^{-1}\,\!$$


 * $$n=4\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}\approx1.1032048\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}\approx1.1038212\times10^7 m^{-1}\,\!$$


 * $$n=3\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}\approx1.1176826\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}\approx1.1213730\times10^7 m^{-1}\,\!$$


 * Below are the average values of the Rydberg constant for Hydrogen and Deuterium.
 * $$R_{Hydrogen,average}\approx1.0989562\pm 0.008\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium,average}\approx1.1077221\pm 0.007\times10^7 m^{-1}\,\!$$


 * Below are the calculated percent errors.
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$
 * $$\% error_{Hydrogen}\approx0.20%\,\!$$
 * $$\% error_{Deuterium}\approx.998%\,\!$$

Conclusions

 * We very happy with our results and our small error we obtained for the Rydberg constant's. Based on our small percentage error, I conclude that Ginny and I did the lab correctly and that we were able to see all the spectral lines for Hydrogen and Deuterium although we had the most error with the red spectral lines for both deuterium and hydrogen.  I am still not sure why we had the most error with our data for the red spectral lines because the red spectral line was one of the easiest colors to see and so the trouble with seeing the line was not the problem.  It may be the case that we did not calibrate the spectrometer properly.  We did not calibrate more than once because we were able to take all of our measurements all in one day, therefore, we did not have to worry about the error due to different calibrations.


 * 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.

General SJK Comment
Steve Koch 06:58, 8 December 2010 (EST): Please see Ginny's page for ideas for extra data: http://openwetware.org/wiki/User:Ginevra_Cochran/Formal_Report/Rough_Draft