Formal Report Final Draft

The Calculation of the Rydberg Constant Based on the Balmer Series for Hydrogen

Author: Roberto Sebastian Rosales

Experimentalists: Roberto Sebastian Rosales and Matthew Cordova

Junior Lab, Department of Physics & Astronomy, University of New Mexico

Albuquerque, NM 87131

rrosales@unm.edu

Abstract

The goal of this experiment was to measure the Rydberg Constant, [math]\displaystyle{ R_H }[/math] for Hydrogen and [math]\displaystyle{ R_D }[/math] for Deuterium, by observing the Balmer Series spectral lines for the two gasses. Although we initially set out to find the Rydberg Constant for both Hydrogen and Deuterium, we switched our focus to calibrating the constant deviation spectrometer well enough to refine our calculation for Hydrogen. Our initial results are as follows: [math]\displaystyle{ R_{calc Hydrogen} = 1.0977(7)\times10^7 m^{-1}\ \ }[/math] and [math]\displaystyle{ R_{calc Deuterium} = 1.0990(2)\times10^7m^{-1}\ \ }[/math]. Upon improving our calibration techniques, we obtained a the following Hydrogen Rydberg Constant: [math]\displaystyle{ R_{calc Hydrogen}= }[/math]. The accepted value for the Rydberg Constant for Hydrogen is [math]\displaystyle{ R_{Hydrogen}= 10967758.3406 m^{-1}\ \ }[/math], and the constant for Deuterium is [math]\displaystyle{ R_{Deuterium}=10970746.1986 m^{-1} \ \ }[/math]. Our experimental calculations are surprisingly accurate considering the fact that we were relying on a very basic form of spectroscopy.

Introduction

The Balmer Series, discovered by Johann Balmer, refers to the set of spectral lines for Hydrogen that are created by the transition of electrons from high energy levels, to lower energy levels. Balmer related the wavelength of the photon emitted by these transitions to the principal quantum numbers associated with these transitions to form the Balmer formula. During this transition, an electron must emit some energy to get to the lower state, and this energy is released in the form of a photon. Because each transition is unique, meaning that the energy associated with the [math]\displaystyle{ n_2 = 6 }[/math] to [math]\displaystyle{ n_1=1 }[/math] transition is different than that of the [math]\displaystyle{ n_2 = 5 }[/math] to [math]\displaystyle{ n_2=1 }[/math] transition, the photon associated with each transition will have a unique wavelength. The Balmer Series is caused by electrons transitioning from the [math]\displaystyle{ n_1 = 3,4,5,6,... }[/math] to the [math]\displaystyle{ n_2=2 }[/math] state. Balmer derived his empirical equation by relating the wavelength of the photon emitted by this transitioning electron and the principal quantum number associated with each energy level. This equation is as follows:
[math]\displaystyle{ \lambda\ = \frac{ hn_1^2 }{ n_1^2 - n_2^2 } }[/math] ^[1]
where [math]\displaystyle{ h=3.6456*10^−7 m }[/math]. Johannes Rydberg applied Balmer's results to his own formula, and discovered that Balmer's formula was just a special case of his own. At the time, Balmer and Rydberg were unaware that "n" referred to the different quantum states. It was Niels Bohr that hypothesized that electrons can only be in certain quantum states, and the "n" corresponds to these states. From this idea, Bohr was able to predict the Rydberg Constant for an infinitely massive atomic nucleus as [math]\displaystyle{ R_\infty =\frac{m_ee^4}{(4\pi)^3\varepsilon_0^2\hbar^3c} }[/math] ^[2]. Today, we use the Rydberg formula as [math]\displaystyle{ \frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) }[/math], where the Rydberg Constant is given by [math]\displaystyle{ R_\infty = 1.0973731568525(73) \times 10^7 m^{-1} }[/math]^[3]. Of course this constant was derived by considering an infinitely massive nucleus. We were dealing with Hydrogen and Deuterium, so we had to find the accepted values for each element. These corrected constants can be obtained (and are listed in the results section) by using the following formula: [math]\displaystyle{ R_M = \frac{R_\infty}{1+m_e/M} }[/math] ^[4] where [math]\displaystyle{ M }[/math] is the mass of the atomic nucleus. In order to calculate the reduced mass versions of the Rydberg Constant for both Hydrogen and Deuterium, we observed the wavelengths at which the electrons transitioned from one energy state to another, and then used the Rydberg Equation to find [math]\displaystyle{ R_{calc Hydrogen} }[/math] and [math]\displaystyle{ R_{calc Deuterium} }[/math].

Methods and Materials

This experiment required very few materials and the setup was quite easy. We used a constant deviation spectrometer (Adam Hilger; London, England; Serial: 12610), a Mercury gas tube (S-68755-30-K), a Hydrogen gas tube (S-68755-30-G), a Deuterium gas tube (S-68755-30-E), as well as a tube power supply (Model: SP200 5000V;10mA; Electro-Technic Products). To set up for the experiment, we simply had to place the spectrometer on the table top, point the slit opening at the tube and power supply assembly, and plug the power supply into an outlet. The general setup can be seen in Figure 1 to the right. According to Professor Gold's Lab Manual ^[5], we were supposed to adjust the ocular in order to "bring the cross-hairs into sharp focus." We found that the position of the ocular seemed to be in the right position to begin with, so we slid it back to its original position. We then adjusted the slit width by rotating the little knob right next to the slit opening. As stated in the manual, we found that not opening the slit enough resulted in a very sharp but faint spectral line. We had to find a width that allowed us to view the faint violets, but at the same time allowed for a decent resolution. We found this width to be around 0.75mm.

^{SJK 03:47, 8 December 2010 (EST)}

03:47, 8 December 2010 (EST)
It seems that throughout, your first sentences of figure captions should be converted to a title for the figure. E.g, "Figure 2 Constant deviation spectrometer."

Figure 1 - Constant Deviation Spectrometer: This is the setup that we used throughout the lab. The spectrometer can be seen at the right with the slit opening aimed at the middle of our Hydrogen tube. The tube is illuminated by the power supply (black box on the left) which is connected to a standard wall outlet. The height of the power supply and tube assembly was adjusted so that the spectrometer was viewing the middle of the tube rather than very top of the tube.

Figure 2 - Calibration Crystal/Prism: This is the prism located at the center of the spectrometer. Because the ocular portion and the slit portion of the spectrometer are fixed, the prism is what is adjusted in order to calibrate, and then rotated with the wavelength dial, which can be seen at the lower left hand corner. We loosened the screw on top of the prism and then rotated it by hand in order to calibrate the spectrometer to the Mercury spectral lines. We were also careful not to change the position of the prism.

Calibration

Initially we had to figure out how to read the dial that displayed the wavelength, and how the scale, when rotated, changed what was being viewed in the spectrometer. After some inspection of the apparatus, we realized that the crystal, shown in Figure 2, itself actually rotates as the dial is turned. So, in order to calibrate the spectrometer, we loosened the screw that holds the crystal in place, found whichever spectral line we were calibrating to in the Mercury spectrum and aligned it with the cross-hairs in the eye piece, and then tightened the screw to hold the crystal in place. Throughout this process, as well as the measuring process, we were careful to avoid gear back lash by always starting at a lower wavelength and turning the dial, which can be seen in Figure 3, towards the higher wavelength. Also, we decided to calibrate the spectrometer first to the green line for Mercury, and then take measurements for both Hydrogen and Deuterium. We then re-calibrated the spectrometer to the violet line (435.8nm) and took another set of data for both Hydrogen and Deuterium. Finally, we calibrated the spectrometer to the red line for Mercury and took another set of data. We did this hoping that we would eliminate some systematic error in our measurements. The wavelengths we used to calibrate the spectrometer are from Prof. Gold's lab manual^[4] and can be seen in Table 1

Color	Wavelength (nm)
Deep Violet	404.7
Violet	435.8
Weak Blue-Green	skip
Green	546.1
Yellow 1	577.0
Yellow 2	579.0
Red	690.75

Table1 - Known Mercury Wavlengths: These are the wavelengths for Mercury given in Professor Gold's lab manual^[4] used to calibrate the spectrometer. In a real lab setting, these calibration values would not be available, but for our purposes, we needed a reference point.

Figure 3 - Wavelength Dial: This dial is located on the spectrometer adjacent to the prism/ crystal. When the dial is rotated, the prism/crystal is also rotated, therefore changing the portion of the spectrum that can be seen in the eye piece. The divisions on the dial refer to nanometers; for example, a reading of 410 would be 410nm and a reading of 66 would be 660nm.

Analysis

We used Google Docs to record and analyze our data. The raw data (columns 2 and 3) is located in Table 2 below, as well as the averages in the columns following the raw data. It should be noted that we excluded our observed wavelengths for Violet 1 for Hydrogen from our calculations. We experienced some problems when measuring the line for Violet 1 for Hydrogen. We first suspected either a bad calibration or a lack of waiting for the tube to heat up (which we now know should not have caused this problem) to be the culprit of the misreadings. We re-calibrated the spectrometer, but got the same reading of about 418nm for the Violet 1 line. We then proceeded to calibrate to a different line for Mercury, but the problem persisted. The wavelengths that we observed for the other lines seemed to be very close to the actual values. Because of this, we decided to exclude the data from our calculations.

Initial Data Analysis
For the average wavelength calculations, we took an average of all the data for one transition (i.e. all the Blue-Green wavelengths for Hydrogen) for Hydrogen and Deuterium separately using the following formula:
[math]\displaystyle{ \bar x = \frac{\sum_{i=1} x_i}{n} }[/math],
where [math]\displaystyle{ x_i }[/math] is one of the measurements and [math]\displaystyle{ n }[/math] is the total number of measurements. It is appropriate to take the average here because the observations should all be from the same parent distribution. Next, I calculated the Rydberg constant for each wavelength using the following formula:
[math]\displaystyle{ \frac{1}{\lambda} = R \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) }[/math],
where [math]\displaystyle{ n_1=2 }[/math] for the Balmer Series. And finally I calculated the average Rydberg constant for Hydrogen and Deuterium separately using the preceding formula for averages, and the Standard Error of the Mean (SEM) using the following formula:
[math]\displaystyle{ SEM\ = \frac{s}{\sqrt{n}} }[/math],
where s is the standard deviation for the sample. All of these calculations can be seen below in Table 2.

Final Data Analysis
For the numerical calculation, I first calculated the average wavelength for each transition using the same method as above. I then calculated a Rydberg constant for each transition using the Rydberg formula shown above with [math]\displaystyle{ n_1=2 }[/math] for the Balmer Series. Finally, I found the Standard Error of the Mean for the four Rydberg calculations. For the graphical calculation, I first graphed [math]\displaystyle{ \frac{1}{\lambda} }[/math] vs [math]\displaystyle{ \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) }[/math], which can be seen in Figure 4. For [math]\displaystyle{ \lambda }[/math], I used the average wavelengths calculated in the numerical section. The resulting best fit line for the data should have a slope equal to the Rydberg Constant. To find this slope, I used the LinEst function which returned the slope, as well as the uncertainty in the slope. All of these results can be found below in Table 3.

Note: All calculations were performed in Google Docs, except the formulation of the graph and the calculation of the slope via the LinEst function. The graph and linest function were used in Microsoft Excel version 14.0.5128.5000 (32-bit).

Data and Calculations

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdGVZXzNqQlB1a3lacmhNOXhHRHl4RkE |width=800 |height=750 }}

Table 2 - Raw Data Initial Attempts: This is all of our raw data, as well as the average values for our observed wavelengths and calculated Rydberg Constants.

{{#widget:Google Spreadsheet |key=0At9fxsqtusShdFhCNlRkdDkwTFA1WHdyUXpDNXl5LUE |width=800 |height=700 }}

Table 3 - Raw Data Final Attempts: This is all of our raw data, as well as the average values for our observed wavelengths and calculated Rydberg Constants for Hydrogen only.

The accepted wavelengths for the transitions in Hydrogen can be found in Table 4. Our observed wavelengths agree with these values except for the Violet 1 spectral line. Our wavelength was closer to about 418nm. This discrepancy is not attributed to anything, but I have discussed it in a previous portion of this report.

Color	Wavelength (nm)	Transition
Red	656.3	n = 3 to 2
Blue-Green	486.1	n = 4 to 2
Violet 2	434.1	n = 5 to 2
Violet 1	410.2	n = 6 to 2

Table 4 - Known Hydrogen Transitions: These are the wavelengths at which the transitions should take place for Hydrogen. These values were only used to compare our observed wavelengths.

Our experimental results for our initial data from Table 2 are as follows:

[math]\displaystyle{  R_{calc Hydrogen} = 1.0977(7)\times10^7 m^{-1}\ \  }[/math]
[math]\displaystyle{  R_{calc Deuterium} = 1.0990(2)\times10^7m^{-1}\ \  }[/math].

Our experimental results for our final data for Hydrogen only from Table 3 are as follows:

 [math]\displaystyle{  R_{numerical Hydrogen} = 1.097(1)\times10^7 m^{-1}\ \  }[/math]
 [math]\displaystyle{  R_{graphical Hydrogen} = 1.107(5)\times10^7 m^{-1}\ \  }[/math]

The accepted values are as follows:
[math]\displaystyle{ R_{\infty}= 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1} }[/math], [math]\displaystyle{ R_{Hydrogen}= 10967758.3406 m^{-1}\ \ }[/math]
[math]\displaystyle{ R_{Deuterium}=10970746.1986 m^{-1} \ \ }[/math].

Conclusions

^{SJK 03:51, 8 December 2010 (EST)}

03:51, 8 December 2010 (EST)
I like your conclusions. Maybe investigating whether you can be consistent with both hydrogen and deuterium would be a good thing to explore tomorrow. Also would be good to address the question of whether you can distinguish the difference between H and D spectra using your instrument.

To conclude this report, I would like to reflect back on the experiment that was performed as well as compare our results to the accepted values. My lab partner and I set out to measure one of the most precisely known physical constants, the Rydberg Constant. We accomplished this by observing the spectral lines for both Hydrogen and Deuterium, and then calculating the Rydberg constant based on the known quantum transitions for each wavelength. Considering the fact that we were using a constant deviation spectrometer (which is acknowledged to be an antiquated instrument in the lab manual), our calculations do not differ from the actual values by a whole lot. Our calculated Rydberg Constant for Hydrogen was within 2 SEM's (0.084% error) from the accepted value, and our calculated Rydberg Constant for Deuterium was within 11 SEM's (o.173% error) from the accepted value. We can assume that there was a significant source of systematic error in our Deuterium measurements. The source of this error is unknown to me because we were taking data for both Hydrogen and Deuterium simultaneously. One would expect a similar error for Hydrogen, but this is not the case. Again, the error seems to be expected since we were using old instruments and techniques. In conclusion, one can see why the Rydberg constants is one of the most precisely known physical constants. We were able to calculate it up to four or five decimals using the aforementioned method.

Acknowledgments

^{SJK 03:52, 8 December 2010 (EST)}

03:52, 8 December 2010 (EST)
Good acknowledgements

I would like to thank my lab partner Matthew Cordova for assisting with the data taking for this lab. Also, I would like to thank Professor Koch for helping us with trying to figure out what the problem was for the Violet 1 measurements. Although they did not personally help me, I need to also thank David Weiss and Tom Mahony for making their lab notebooks available on OpenWetWare. By looking at their notebooks, I was able to do the SEM calculations.

References

^{SJK 03:53, 8 December 2010 (EST)}

03:53, 8 December 2010 (EST)
As noted above and in class, you'll need to directly cite a couple original research reports. These may be most easy in the "introduction" section.

1. Wikipedia Johann Jakob Balmer web article http://en.wikipedia.org/wiki/Johann_Jakob_Balmer
2.
3.National Institute of Standards Rydberg Constant for infinitely massive atomic nucleus http://physics.nist.gov/cuu/Document/all_2002.pdf
4.Precision Measurements and Fundamental Constants link here Page 74
5.Gold, Michael Physics 307L: Junior Lab Manual (2006) link here

General SJK Comments

04:35, 8 December 2010 (EST): For follow-on data, one thing you can explore is the best way to calibrate the data. Is it really the best thing to calibrate to a bunch of different lines and then average together values obtained from those lines? Can you think of better ways? Also, you can plot your data as 1/lambda versus the (1/n^2-1/m^2) factor. This should look linear and if so, you can do a linear fit to find the rydberg constant. Finally, you should explore the question of whether you could distinguish between hydrogen and deuterium, given the instrument you have.