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Calculating the Rydberg constant by measuring the visible spectrum lines of hydrogen and deuterium
Author/Experimentalist: Jacob R. Jaramillo

Location: University of New Mexico, Department of Physics and Astronomy, Albuquerque

E-mail: jjaramillo17@msn.com 

Abstract
Balmer's equation can be used to calculate the exact wavelength of light emitted as an election drops from one energy state to another, which is commonly referred to as the quantum number (n). In this experiment I calculated Rydberg's Constant with the use of a mercury, hydrogen and deuterium tube. Hydrogen and deuterium were selected due to their slight differences in atomic mass which results in two unique visible spectrum lines as deuterium is know as heavy hydrogen and is also a stable isotope of hydrogen. My average calculated value for the Rydberg's constant for hydrogen was $$(RHydrogen)=1.0975\times 10^7 m^-1 (3.915\times10^3)$$, which is consistent with the accepted value of $$(RHydrogen)=1.0968\times 10^7 m^-1 (6.6\times10^-12)$$. My average calculated value for the Rydberg's constant for Deuterium was $$(RDeuterium)=1.0986\times 10^7 m^-1(17.525\times10^3)$$, which is slightly off from the accepted value of $$(RDeuterium)=1.0965\times 10^7 m^-1 (6.6\times10^-12)$$. 

Introduction
Johannes R. Rydberg, was born in 1854 in Halmstad on thet west coat of Sweden. Even though Rydberg's equation is generally presented as a function of the Balmers Series, it has been proven that Rydberg's work was independent of Balmer's research. "To begin with, Rydberg discovered that it was possible to sort the spectral lines belonging to an element in a number of different series, where the lines followed each other with a regularly decreasing difference and intensity. He put numbers n the various lines in a given series, with 1 for the next line with the longest wavelength, 2 for the next line, etc. and tried to find a formula that could express the wavelengths or frequencies as a simple formula that included integers. (1) With this finding Rydberg discovered that the relationship between wavelengths and quantum numbers resembled a hyperbola. In 1890, after numerous mathematical attempts and receiving funding from the Swedish Academy of Sciences (KVA), Rydberg concluded on the following formula and published his work thus defining the Rydberg constant.

$$n=n_o-N_o/(m+u)^2$$  (2)

n is the wavenumber of the line (1/lambda) No is a constant with value of 107,921.6 while no, m and u are constants which are specific for the series.

The above formula was later re-written by Rydberg in the following, more well know method:

$$n/N_o = 1/(m_1+u_1)^2-1/(m_2+u_2)^2$$  (3)

With this great discovery, Rydberg was able to measure the Rydberg constant for Hydrogen (Rh = 109721.6) which he was later able critic to 109677 and officially release the value of 109737 as a value for a infinite nuclear mass. The constant No, is now known as the Rydberg's constant R(infinity) and is now most commonly written as:

$$R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} = 1.0973731568525(73) \times 10^7 \,\mathrm{m}^{-1}$$

The Rydberg constant according to the above formula is now represented by the value $$10973731.568527(73)m^-1$$ with an uncertainty of 6.6 x 10^-12 according to NIST. (4)

NIST uses an optical frequency comb for producing ultra precise colors of light that can trigger quantum energy jumps useful for accurately measuring the Rydberg constant. (5)



Figure 1: Optical Frequency Comb spectral display

My purpose is to analyze my data with the above formulas and compare them to the acceptable values and tolerances to find any inconsistencies with my data.

Methods and Materials
Equipment

In this experiment I used a constant deviation spectrometer which included a Pellin-Broca prism, a spectrum tube power supply (Model SP200, 5.0KV, 10mA), and three gas tubes, Mercury, Hydrogen and Deuterium.



Figure 2: Spectrometer with Pellin Broca Prism



Figure 3: Spectrum tube power supply with Deuterium tube



Figure 4: Function of the Pellin broca prism (by Jacob Jaramillo)

Set Up

I placed the Spectrum tube power supply with gas tube ~5-10mm away from the side of the Spectrometer with the adjustable slit (as show in Figure 3). I took caution by unplugging the Spectrum tube power supply, when placing the gas tube in the inside of it and when removing it as the power supply outputs 5.0KV's which is a large amount of voltage.

Step 1

One of the most critical steps of this experiment is to calibrate the Spectrometer. I preformed this task with the Mercury tube and the accepted wavelengths for the Mercury spectrum line found in the Mercury Calibration spreadsheet (Table 1). I chose to use the violet line and it's wavelength value to manually adjust the Pellin-Broca prism as this line was one of the brightest and sharpest lines. I adjusted the knob on the Spectrometer to 435.8nm and manually adjusted the Pellin Broca prism until the violet line was perfectly centered on the cross hairs of the Spectrometer eyepiece (I found it easiest to make minor adjustments to the Pellin-Broca prism while the tightening mechanism was almost completely tightened). Then I proceeded to turn the knob to each spectral line and insure that it matched as close to the accepted value as possible.

Step 2

Next I removed the Mercury tube and replaced it with the Hydrogen tube. Once the hydrogen tube was stabilized/warm (generally takes 2 minutes), I proceeded to take measurements of each spectral line, as illustrated in the Hydrogen Spectrum spreadsheet (Table 2). As noted in the Hydrogen Spectrum spreadsheet, the Hydrogen Spectrum consisted of the following lines, two violet, one teal and one red line.

Hydrogen Spectrum Data Analysis Table 2

My average calculated value for the Rydberg's constant for hydrogen was $$(RHydrogen)=1.0975\times 10^7 m^-1 (3.915\times10^3)$$, which is consistent with the accepted value of $$(RHydrogen)=1.0968\times 10^7 m^-1 (6.6\times10^-12)$$.



Figure 5: (Hydrogen Spectrum, slope of Rydberg constant)

Step 3

Same as step two; however, I used the Deuterium tube. Measurements can be found in the Deuterium Spectrum spreadsheet (Table 3). As noted in the Deuterium Spectrum spreadsheet, the Deuterium Spectrum consisted of the following lines, two violet, one teal and one red line.

Deuterium Spectrum Data Analysis Table 3

My average calculated value for the Rydberg's constant for Deuterium was $$(RDeuterium)=1.0986\times 10^7 m^-1(17.525\times10^3)$$, which is $$3.475\times 10^7 m^-1$$ off from the accepted value of $$(RDeuterium)=1.0965\times 10^7 m^-1 (6.6\times10^-12)$$.



Figure 6: (Deuterium Spectrum, slope of Rydberg constant)

Formulas

$$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2})$$ $$n=3,4,5,...\,\!$$

Calculating percent error

$$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$

Results and Discussion
As shown above my Rhydrogen value was within the accepted value while my Rdeuterium value was 3.475x10^7 off from the accepted value. Although my values were not 6.6x10^-12 within the accepted value, based on the equipment used and the fact that I didn't have an optical frequency comb, I feel that my measurements were still fairly precise.

Conclusion
As explained above, my values were either within the accepted value or extremely close. I believe the reason for my error is due to the error I found when calibrating the Spectrometer as my red spectrum was off by ~3nm. Also, although I was aware of backlash in the Spectrometer, a few times I did turn the knob the wrong way and even though I completed a full revolution before changing directions to try to reduce backlash, I feel it may have contributed.

Acknowledgments
I would like to thank my lab partner Johnny Gonzalez whom I conducted my initial data with, as well as Prof Koch who helped me with data analysis.