# User:Brian P. Josey/Notebook/Junior Lab/2010/11/22

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Steve Koch 03:26, 21 December 2010 (EST):Another excellent notebook. Great job this semester on maintaining an excellent primary notebook!

## Balmer Series

This week, we attempted to measure the Rydberg constant in the Balmer series lab. The Rydberg constant is a constant that is used in a formula for predicting the wavelength of a photon that is released when an excited electron drops to a lower level, or the wavelength needed to raise an electron from a lower to a more excited state. To do this, we measured the well known Balmer series, which occurs when the electrons drop from a higher excited state to the second orbital (n=2 orbital). To do this, we first calibrated our instruments with a Mercury tube, and known values for wavelength, and then measured the wavelengths of light for both hydrogen and deuterium.

## Equipment

Spectrometer
Vapor Tube Power Supply
Internal Prism
• Constant deviation spectrometer
• Spectrum Tube Power Supply Model SP200, 5000 volts, 10MA.
• Tubes containing vapor of
• mercury
• hydrogen
• deuterium

To set-up the experiment, we put a vapor tube into the power supply, and turned it on. We then aligned the constant deviation spectrometer up with tube. The spectrometer had to be calibrated first, however, to account for any discrepancies in the measurements. This is explained in the methods section below.

## Methods

Mercury
Hydrogen
Deuterium

After setting up the experiment, we proceeded to calibrate the constant deviation spectrometer. The first step in the process was to find a spectral line through the eyepiece, and focus the cross hairs on the center of the line so that both the line and the cross hairs were in focus. This is accomplished by turning the knob on the center of the arm with the eye piece. The second step was to calibrate it so that our measurements came very close with the accepted values of some known wavelengths, in this case mercury vapor. To adjust the prism, we turned the large knob marked with wavelengths to the accepted value, and then adjusted the prism itself with the knob on it so that the spectral line was focused on the cross hairs. Fortunately, being the last group in the class, the spectrometer was already very closely calibrated, as our measurement, in the table below illustrate.

We then moved on to measure the values of wavelengths of light coming from the hydrogen and deuterium vapor tubes. This was accomplished by turning the marked knob carefully while watching the lines move through the eyepiece. When the prominent lines were centered on the cross hairs of the eyepiece, we then recorded the wavelength from the marked knob. For all three vapor tubes, there was more than just the principle spectral lines, and there was even some continuity in the spectrum. The extra spectral lines could be the result of small amounts of gasses from other compounds being in the tube, while the continuous spectra is from impurities in the glass prism, which is fairly old and had to stand up to years of use. Also, because the gears that rotate the prism are not perfectly aligned in one direction, if we overshot a spectral line, we would rotate back a quarter turn and then more in more carefully on the line. This would eliminate error for the gears in our measurements.

This table summarizes both our data, and the conclusions that we inferred from the data:

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## Analysis and Results

From our measurements, we were able to measure the wavelengths of the principle spectrum lines of the mercury vapor and compare them to the accepted values provided in the lab manual. By averaging the differences between our values and the manual's, we were able to determine that there was a difference of 0.37 nm that needed to accounted for on our measurements. This value was then added to our measured wavelengths giving the "corrected measurements" on the data table. These corrected values are what we will use to determine the Rydberg constant.

Generally, the Rydberg constant, R, is used to determine the wavelength of light released from transitions of electrons in hydrogen atoms. Mathematically, this is expressed as:

$\displaystyle \frac {1} {\lambda} = \frac {1} {R} (\frac {1} {n^2} - \frac {1} {m^2})$

where:

• λ - is the wavelength of the released photon,
• R - is the Rydberg constant,
• n - is the principle quantum number of the lower energy state that the photon drops to, and
• m - is the principle quantum number of the higher energy state.

For the Balmer series, which is the only series that is in the visible portion of the light spectrum, the value of n is 2. To determine the Rydberg constant, we solve the above equation, and plug in 2 for n, this gives:

$\displaystyle R = \frac {4m^2}{\lambda (m^2-4)}$

The results from this, including the error as a result of the wavelengths is summarized in the table below:

SJK 03:25, 21 December 2010 (EST)
03:25, 21 December 2010 (EST)
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To find the constant for hydrogen and deuterium, we averaged out our results and found the standard error of mean. This gave us a Rydberg constant of 1.0962 (9) * 107 m-1 for hydrogen, and 1.0975 (6) * 107 m-1 for deuterium. To check our values against the accepted ones, we first have to calculate the Rydberg constant, R$\displaystyle \infty$ :

$\displaystyle R_\infty = \frac{m_e e^4}{8 \varepsilon_0^2 h^3 c} = 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$

where:

• me - is the rest mass of the electron,
• e - is the elementary charge,
• ε0 - is the permittivity of free space,
• h - is Planck's constant, and
• c - is the speed of light.

Then to determine the Rydberg constant for atoms with varying nuclear masses, we plug it into this formula:

$\displaystyle R_M = \frac{R_\infty}{1+m_e/M},$

where

• me - is the rest mass of the electron, and
• M - is the mass of the atomic nucleus.

This gives values of:

$\displaystyle R_{Hydrogen}= 10967758.3406 m^{-1}\ \$

$\displaystyle R_{Deuterium}=10970746.1986 m^{-1} \ \$

Comparing these values to our experimental values, we determined that the hydrogen constant was off by only, 0.052 %, and the value for the deuterium constant was off by only 0.039 %.

## Conclusion

From our data, it is clear that we were able to successfully measure the Rydberg constant for both hydrogen and deuterium by observing the Balmer series for both chemicals. While the possibility of both human error, and the error in the gears that rotate the prism could have thrown off our values, it is clear that if either had any effect, it was minimal.

## Acknowledgments and References

As always, I want to thank my lab partner, Kistin. At the same time Dr. Koch and Katie the TA also helped a significant deal. I also used a great deal of help from both Emran and Joseph for allowing us to use their notebooks as reference, and helping with the set up of the experiment.