Physics307L:People/Cordova/Matt's Final Lab Report

Steve Koch 03:36, 19 December 2010 (EST):Considering no rough draft, this is pretty good. = Ascertainment of the Rydberg Constant Through Spectroscopy = Author: Matthew A. Cordova Experimentalists: Matthew A. Cordova & Roberto Sebastian Rosales Junior Lab, Department of Physics & Astronomy, University of New Mexico Albuquerque, NM 87123 mcordov9@unm.edu

Abstract
This lab report will investigate the Rydberg constant, a physical constant which relates to the wavelengths of photons emitted from atoms in an excited state. This will be done through spectroscopy. Specifically, we will be measuring the wavelength of photons emitted by Hydrogen and Deuterium gas observed as spectral lines isolated through diffraction. Although this can be seen as a dated method compared to modern science, the results obtained in this lab were quite acceptable. The reduced mass Rydberg constant for Hydrogen and Deuterium were found to be $$1.0977(7)*10^7m^{-1}$$ and $$1.0990(2)*10^7m^{-1}$$, respectively. The accepted values for Hydrogen and Deuterium are $$10967758.3406m^{-1}$$ and $$10970746.1986m^{-1}$$, respectively.

Introduction
As previously mentioned, atoms in an excited state emit photons when they go back down to a more stable energy level. A more accurate description, however, would be that the electrons in an atom are what reach excited levels, and the energy lost in the transaction to a lower (more stable) energy state is in the form of an emitted photon. Since the energy levels are defined as integers, there are discrete energies, i.e. wavelengths, that this photon may have. The Balmer series, first explored by Johann Balmer, deals with excited energy levels transitioning down to the n=2 state. To observe this process, my lab partner Sebastian and I will excite Hydrogen and Deuterium gas and observe the photons emitted through a constant deviation spectrometer. By sending the emitted photons through a diffracting medium, we can isolate the wavelengths and use Rydberg's equation to determine the corresponding constants for Hydrogen and Deuterium. Johannes Rydberg crafted his equation strictly from empirical data. No theory went into its conception.[1] It is expressed as  $$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$ It must be noted that this equation uses $$R_\infty$$, while we are looking for the reduced mass Rydberg constant, which is obtained from $$R_M = \frac{R_\infty}{1+m_e/M}$$Where $$e_m$$is the mass of an electron and M is the mass of the atomic nucleus. It can be seen here why using $$R_\infty$$ is a good approximation, as $$m_e$$<<M. The Rydberg constant is an important fundamental physical constant, likely due to the fact that it is one of if not the most accurately measured physical constant.[2] It is because of this accuracy that the Rydberg constant is most useful in refining quantum theories, even though no quantum theory went into its conception!

Initial Set Up
The set up for this lab, as seen by Figure 1, consists of relatively few pieces of equipment. The spectrometer should be placed on a flat, sturdy surface, while the power supply should be elevated such that the thinner section of the gas tube (while attached to the lamp) is at an even level with the eye piece of the spectrometer. Rotate the cross-hairs until they are to your liking. Lastly, set the width of slit for incoming light from the excited gas. A thinner slit yields a thinner (and therefor more accurate) spectral line, but sacrifices visibility. This concludes set up for this lab.

Calibration
With set up complete, we (Sebastian and I) proceeded to calibrate the spectrometer using a known element with known spectral lines. For this lab, we used Mercury gas. The values for the observed spectral lines can be found in Prof. Gold's lab manual. To calibrate, adjust the measuring dial (Figure 3) such that it reads a known value for an observable spectral line. For example, there is known to be a green spectral line an 546.1 nm. Set the dial such that it reads this value, and then loosen the screw which secures the refracting crystal (Figure 2) in place. Then, carefully move/rotate the crystal such that the green spectral line in centered in the cross-hairs, and re-secure the crystal. This concludes calibration.
 * Note: Be sure to always rotate the measuring dial in the same direction to avoid gear backlash. If you calibrate while rotating the dial clockwise, make all measurements in the same direction.
 * Note: When securing the crystal, make sure that the spectrometer is still calibrated. Tightening the securing screw often moves the crystal slightly.

Data Recording
With the spectrometer calibrated, the data was ready to be recorded. For this lab, the spectral lines of Hydrogen and Deuterium gas are to be examined. There are four observable wavelengths for both Hydrogen and Deuterium for the n=6,5,4,3 transitions. However, for Hydrogen the first spectral line (n=6) was impossible to identify, so it shall be excluded from the data analysis. Other than this, there were no complications in recording the data. Multiple trials were completed, each calibrated to a different Mercury wavelength in an attempt to reduce error in the equipment.

Data
Data recorded and analyzed through GoogleDocs.

Calculations
The purpose of this lab is to calculate the Rydberg constant[4] $$R_\infty = \frac{m_e e^4}{8 \varepsilon_0^2 h^3 c} = 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$$ With reduced mass Rydberg constant given by $$R_M = \frac{R_\infty}{1+m_e/M}$$ The values $$R_{hydrogen}$$ and $$R_{deuterium}$$ were found by obtaining the average wavelength for red, blue-green, violet#2, and violet#1(deuterium only) for both hydrogen and deuterium. These average wavelengths were then used to obtain the Rydberg constant for each spectral line, and these Rydberg constants were averaged together. With the average Rydberg constant, we can obtain $$R_{hydrogen}$$ and $$R_{deuterium}$$.

$$\bar\lambda=\sum_{i=1}^n\lambda_i/n$$---We will have a separate $$\bar\lambda$$ for each color for each element.(3 for hydrogen, 4 for deuterium) $$R=\frac{1}{\bar\lambda}\left(\frac{1}{2^2}-\frac{1}{m^2}\right)^{-1}$$---Where m is initial energy level (i.e. for red m=3) and $$R$$ is the reduced mass Rydberg constant. We will have one $$R$$ for each $$\bar\lambda$$(3 for hydrogen, 4 for deuterium). $$\bar{R}=\sum_{i=1}^n R_i/n$$---Take the average $$R$$ for hydrogen and deuterium separately. At this point we have $$\bar{R}_{hydrogen}$$ and $$\bar{R}_{deuterium}$$. (Note:I don't know why my text is bold from here on out) $$R=\bar{R} +/- SEM$$---Where SEM is the standard error of the mean, given by With our data, the results came out to be The accepted values for these constants are given in the Google doc, and are
 * $$SEM=\frac{s}{\sqrt{n}}$$---Where s is the standard deviation.
 * $$ R_{calc Hydrogen} = 1.0977(7)*10^7m^{-1}$$
 * $$ R_{calc Deuterium} = 1.0990(2)*10^7m^{-1}$$
 * $$R_{hydrogen}=10967758.3406 m^{-1}$$
 * $$R_{deuterium}=10970746.1986 m^{-1}$$

Conclusion
To conclude, I will say that even though this lab was done using dated methods, the results are quite accurate. There is some systematic error, as seen by the fact that the results are multiple SEM's away from the accepted values. This is due to human error in taking the observations (the human eye is only so accurate) and the condition of the equipment (the gas tubes are old, and the spectrometer seems to be an antique). Finally, I propose the question can one differentiate Hydrogen from Deuterium gas using this method? In short, no. When you look at the deviation in our calculated values, the range in acceptable values are dangerously close. With these uncertainties, it would be impossible to confidently distinguish Hydrogen versus Deuterium gas.

Acknowledgments
Special thanks goes to my lab partner Sebastian Rosales. Also to Professor Koch for help with all aspects for this lab from set up to data analysis. Acknowledgment must be given to any and all persons who provide their lab notebooks on OpenWetWare. The knowledge gained from viewing these is immeasurable.