# Lab 06: Balmer Series

Steve Koch 04:01, 21 December 2010 (EST):Missing standard error of the mean and comparison of discrepancy with this SEM.

### Summary

The Balmer Series is of one of a set of six different series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula which was discovered by Johann Balmer in 1885. The visible spectrum of light from hydrogen displays four wavelengths, 410.2 nm, 434.1 nm, 486.1 nm, and 656.3 nm. These correspond to electron transitioning from n = 6,5,4,3 to n = 2 respectively, where n is the principal quantum number of the electron. These are the first four wavelengths in the Balmer Series.

In our lab we used a Constant Deviation Spectrometer to observe and measure the spectral lines of mercury, hydrogen, and deuterium. Deuterium is an isotope of hydrogen that has a proton and a neutron in its nucleus. We measured the wavelengths of light emitted from a mercury tube and compared them against known values of the spectral lines for mercury to get a correction for systematic error caused by the instrument. Then we measured the wavelengths of the first four lines of the Balmer series for hydrogen, and the first three lines of the Balmer series for deuterium. We were unable to locate the fourth line for deuterium. We used our corrected wavelength measurements to determine the Rydberg constant.

Part of the Deuterium Emission Spectrum as seen through the Spectrometer.
Courtesy of Emran Qassem.

### Analysis

The Rydberg constant can also be found with the following equation when the wavelengths and corresponding transitions are known:

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

where n=3, 4, 5, 6 in the Balmer Series.

Solving for R gives:

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

Calculating R for each wavelength for the first three terms in the Balmer series for hydrogen and deuterium, and then averaging the results for hydrogen and deuterium separately gives:

RHydrogen = (RHalpha + RHbeta + RHgamma) / 3 = (10953407.40062 + 11017462.67835 + 11001027.49597) / 3 = 10990630.85831m − 1
RDeuterium = (RDalpha + RDbeta + RDgamma) / 3 = (10970091.26506 + 10989086.46351 + 11014513.82487) / 3 = 10991230.51781m − 1

The known values for RHydrogen and RDeuterium are found by using the following equation:

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

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

These values are:

$R_{Hydrogen}= 10967758.3406 m^{-1}\ \$
$R_{Deuterium}=10970746.1986 m^{-1} \ \$

The calculated percent errors are given by:

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

These values are:

${Hydrogen}\approx0.21%\,\!$
${Deuterium}\approx0.19%\,\!$

### Conclusion

We were able to gather some good data since we have a low percent error. I think the change factor I used was off and it didn't compensated for the systematic error from the spectrometer very well. I would like to generate a new one with more than just the readings and known values for mercury. In the future I think this lab should include a few more elements so that the differences in the Rydberg constant for small nuclei become apparent. That was very interesting to me when I discussed it with Katie, the TA.