Formal Report Rough Draft

= Title =

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 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, which was actually just a special case of the Rydberg formula (discovered by Johannes Rydberg). The Rydberg formula contains the Rydberg Constant, $$R_{\infty}$$. The accuracy of the Rydberg constant is very important because the Rydberg formula has many applications, many of which have to do with astronomy. We, my lab partner Matthew Cordova and I, attempted to calculate the Rydberg Constant by measuring the wavelengths of the emissions for Hydrogen and Deuterium. This was done using a Constant Deviation Spectrometer, and a Hydrogen and Deuterium lamp. Because we were relying on a visual identification of the spectral lines, we were only able to observe the Violet 1 transition (n=6 to2), the Violet 2 transition (n=5 to2), the Blue-Green transition (n=4 to 2), and the Red transition (n=3 to 2). It should be noted that our calculated values should be compared to the reduced mass Rydberg Constant, not the Rydberg Constant for the infinitely massive nucleus. Our experimental calculated values are as follows: $$ R_{calc Hydrogen} = 1.0977(7)\times10^7 m^{-1}\ \ $$ and $$ R_{calc Deuterium} = 1.0990(2)\times10^7m^{-1}\ \ $$.

=Introduction=

As mentioned previously, a spectral series is caused by electrons transitioning from some high or excited energy state, down to a lower energy state (or by electrons transitioning from lower to higher). 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 $$n_2 = 6$$ to $$n_1=1$$ transition is different than that of the $$n_2 = 5$$ to $$n_2=1$$ transition, the photon associated with each transition will have a unique wavelength. The Balmer Series is caused by electrons transitioning from the $$ n_1 = 3,4,5,6,...$$ to the $$n_2=2$$ state. Balmer derived an empirical relation between the wavelength of the photon emitted by this transitioning electron and the principal quantum number associated with each energy level. This formula is as follows: $$\lambda\ = \frac{ hn_1^2 }{ n_1^2 - n_2^2 }$$ [1] where $$h=3.6456*10^−7 m $$. As stated above,Johannes Rydberg applied Balmer's results to his own formula, and discovered that Balmer's formula was just a special case of his own formula. Today, we use the Rydberg formula as: $$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$, where the Rydberg Constant is given by $$R_\infty = 1.0973731568525(73) \times 10^7 m^{-1}$$[2]. 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: $$R_M = \frac{R_\infty}{1+m_e/M}$$ [3] where $$M$$ is the mass of the atomic nucleus.

=Methods and Materials=

This experiment required very few materials and the setup was quite easy. We used a constant deviation spectrometer, a Mercury gas tube, a Hydrogen gas tube, a Deuterium gas tube, as well as a tube power supply. 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 [4], 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.



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


 * {| border="1"

!Color !Wavelength (nm) Table1: 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.
 * 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
 * }
 * Yellow 2
 * 579.0
 * Red
 * 690.75
 * }
 * 690.75
 * }

 =Results and Discussion=

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.

Table 2: This is all of our raw data, as well as the average values for our observed wavelengths and calculated Rydberg Constants. The accepted wavelengths for the transitions in Hydrogen can be found in Table 3. 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.


 * {| border="1"

!Color !Wavelength (nm) !Transition Table 3: These are the wavelengths at which the transitions should take place for Hydrogen. These values were only used to compare our observed wavelengths. For the average wavelength calculations, we simply 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: $$\bar x = \frac{\sum_{i=1} x_i}{n}$$, where $$x_i$$ is one of the measurements and $$n$$ 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: $$\frac{1}{\lambda} = R \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$, where $$n_1=2$$ 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: $$SEM\ = \frac{s}{\sqrt{n}}$$, where s is the standard deviation for the sample. These functions are built right in to Google Docs, so I simply used them to carry out these calculations. Our experimental results are as follows: $$ R_{calc Hydrogen} = 1.0977(7)\times10^7 m^{-1}\ \ $$ $$ R_{calc Deuterium} = 1.0990(2)\times10^7m^{-1}\ \ $$. The accepted values are as follows: $$R_{\infty}= 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$$, $$R_{Hydrogen}= 10967758.3406 m^{-1}\ \ $$ $$R_{Deuterium}=10970746.1986 m^{-1} \ \ $$.
 * 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
 * }
 * Violet 1
 * 410.2
 * n = 6 to 2
 * }
 * }

=Conclusions=

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=

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=

1. Wikipedia Johann Jakob Balmer web article http://en.wikipedia.org/wiki/Johann_Jakob_Balmer 2.National Institute of Standards Rydberg Constant for infinitely massive atomic nucleus   http://physics.nist.gov/cuu/Document/all_2002.pdf 3.Precision Measurements and Fundamental Constants link here Page 74 4.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.