Physics307L:People/Ierides/The Balmer Series

=The Balmer Series of Hydrogen and Deuterium in Atomic Physics=

Author: Anastasia A. Ierides

Experimentalists: Anastasia A. Ierides, Alexandra S. Andrego

Institution: University of New Mexico

Address: University of New Mexico, Department of Physics and Astronomy, Albuquerque NM, 87106

Abstract
Before 1885, the year that the Balmer formula was founded by a Swiss school teacher Johann Jakob Balmer, physicists, although aware of atomic emissions, lacked the tools to predict the location of each spectral line. The Balmer equation is used in the prediction of each of the four visible emission/absorption lines of hydrogen with high precision. This had inspired the Rydberg equation, invented by a Swedish physicist Johannes Robert Rydberg. This new equation was a generalization of the Balmer formula, which in turn led to the finding of the Lyman, Paschen, and Brackett series used in predicting the absorption/emission lines of hydrogen found outside the visible spectrum.

According to the Rutherford Bohr model (devised by Neils Bohr in 1913 from the amelioration of a model created by Ernest Rutherford in 1911) of the Hydrogen atom, an electron transition that occurs between the second energy level or first excited state in the atom (corresponding to n=2) and any other higher energy level results in the Balmer lines.

The Balmer series has been helpful in astronomical and physical use for years due to the abundance of hydrogen in the universe. It has been used for several means such as spectral classification, the measure radial velocities of objects in space due to doppler shifting, and the distances to those objects.

During our experiment we found that slight discrepancy in our values could be due to some oversight when using the 'scope'. There might have gear back lash from not turning the knob all the way back before remeasuring spectra for each trial, even though we took great care in doing so. Also, the lab lasted over two days, with a week interval and during that interval another group had used the same device used for our experiment, so re-calibration for our set of data was necessary during the second day.

Introduction
The Balmer series is one of six series in which the spectral line emissions of hydrogen are designated. There are four different emission wavelengths of visible light by which the hydrogen spectrum is defined. These wavelengths can be calculated using the Balmer formula (found by Johann Balmer, 1885) written above in the "Purpose" and reflect emissions of photons by transitions of electrons between principal quantum number levels from $$n\geq 3$$ to $$ n=2\,\!$$. 

Compared to the Hydrogen atom, which contains one proton in the nucleus, the Deuterium atom, contains a proton and a neutron in its nucleus. Thus the Deuterium atom is heavier than the regular Hydrogen atom.

By observing and classifying spectra lines of the hydrogen and deuterium atoms the Balmer series can be determined. By using electrical stimulation to excite the atoms to higher energy levels we can measure the emitted photons of wavelengths equivalent to the energy of our excited electrons. Through this lab and our measurements we were able to experimentally determine Rydberg's constant, R, that is used in the Hydrogen Spectrum equation:
 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2})$$
 * $$n=3,4,5,...\,\!$$

Or more generally a modified version of the above equation:
 * $$\frac{1}{\lambda }=R(\frac{1}{m^2}-\frac{1}{n^2})$$
 * $$m=1,2,3,...\,\!$$
 * $$n=2,3,4,5,...\,\!$$
 * $$n>m\,\!$$

This process is applied to both hydrogen vapor and deuterium vapor.



Materials and Methods
Constant-Deviation Spectrometer (SER. #12610):
 * Spectrum Tube Power Supply (Model SP200)
 * 5000V
 * 10 MA

Underlined Spectrum Tubes:
 * Spectrum Tube Power Supply Model SP200 5000V
 * Spectrum Tube, Mercury Vapor S-68755-30-K
 * Spectrum Tube, Hydrogen S-68755-30-G
 * Spectrum Tube, Deuterium S-68755-30-E

For this lab we are using Gold's Lab Manual Gold's Physics 307L Manual
 * 1) First we adjusted the spectrometer, bringing the cross-hairs into focus by sliding the ocular to suit our vision
 * 2) Then we brought the slit into focus, turning the large ring near the center of the viewing telescope making sure to turn the screw in only one direction to avoid error due to "gear back lash"
 * 3) Next, we attached the mercury bulb and turned on the mercury tube power supply to let it warm up
 * 4) Using the spectrometer with a wide slit, we found a line of the mercury spectrum and then narrowed the slit until the line became narrow and sharp
 * 5) Then we located as many mercury spectra as we could and noted the orientation and value of our spectrometer dial
 * 6) While using the wavelengths of light given in Gold's Manual (page 29), we finished calibrating the system
 * 7) To solve for Rydberg's constant we correlated our data to the appropriate quantum numbers and used the equation given (page 30)
 * 8) Finally, we repeated this process for deuterium as well.
 * Note: The first two steps are to make certain that no parallax exists between the cross-hairs and the slit when in sharp focus

Table taken from page 29 of this link


 * {| border="1"

!Color !Wavelength (nm)
 * Deep Violet (very hard to see)
 * 404.7
 * Violet
 * 435.8
 * Very 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
 * }



This is our raw data:

Using our raw data tables we used the functions in Excel for mean and standard deviation to find the standard error margins. From that we formulated a total mean:

Results and Discussion
From our measured values for the wavelengths, we have:
 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =409.84 nm\,\!$$
 * $$\lambda_{Deuterium} =N/A\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =433.92 nm\,\!$$
 * $$\lambda_{Deuterium} =433.3 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =485.96 nm\,\!$$
 * $$\lambda_{Deuterium} =485.62 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =657.4 nm\,\!$$
 * $$\lambda_{Deuterium} =655.9 nm\,\!$$

And according to, the accepted values for the four visible wavelengths of hydrogen in the Balmer series are:
 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda =410.174 nm\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda =434.047 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda =486.133 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda =656.272 nm\,\!$$

From these values we can calculate our measured Rydberg's constant:
 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!$$
 * $$\frac{1}{\lambda }=R(\frac{n^2-4}{4n^2})\,\!$$
 * $$R=\frac{4n^2}{\lambda(n^2-4)}\,\!$$


 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =409.84 nm\,\!$$
 * $$R_{Hydrogen}=\frac{4(6)^2}{(409.84\times10^{-9} m)((6)^2-4)}\approx1.0979895\times10^7 m^{-1}\,\!$$


 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =433.92 nm\,\!$$
 * $$R_{Hydrogen}=\frac{4(5)^2}{(433.92\times10^{-9} m)((5)^2-4)}\approx1.0974153\times10^7 m^{-1}\,\!$$
 * $$\lambda_{Deuterium} =433.3 nm\,\!$$
 * $$R_{Deuterium}=\frac{4(5)^2}{(433.3\times10^{-9} m)((5)^2-4)}\approx1.0989856\times10^7 m^{-1}\,\!$$


 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =485.96 nm\,\!$$
 * $$R=\frac{4(4)^2}{(485.96\times10^{-9} m)((4)^2-4)}\approx1.0984840\times10^7 m^{-1}\,\!$$
 * $$\lambda_{Deuterium} =485.62 nm\,\!$$
 * $$R=\frac{4(4)^2}{(485.62\times10^{-9} m)((4)^2-4)}\approx1.0982524\times10^7 m^{-1}\,\!$$


 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =657.4 nm\,\!$$
 * $$R=\frac{4(3)^2}{(657.4\times10^{-9} m)((3)^2-4)}\approx1.0952236\times10^7 m^{-1}\,\!$$
 * $$\lambda_{Deuterium} =655.9 nm\,\!$$
 * $$R=\frac{4(3)^2}{(655.9\times10^{-9} m)((3)^2-4)}\approx1.0977283\times10^7 m^{-1}\,\!$$

The average value of our measured Rydberg's constant can be calculated as:
 * $$R_{average}=\frac{1}{n}\sum {R_i}$$

where $$n\,\!$$ is the total number of $$R_i\,\!$$ So,
 * $$R_{Hydrogen,average}=\frac{(1.0979895+1.0974153+1.0984840+1.0952236)\times10^7m^{-1}}{4}$$
 * $$=\frac{4.3891124\times10^7 m^{-1}}{4}\,\!$$
 * $$\approx1.0972781\pm 0.0025\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium,average}=\frac{(1.0989856+1.0982524+1.0977283)\times10^7m^{-1}}{3}$$
 * $$=\frac{3.2949663\times10^7 m^{-1}}{4}\,\!$$
 * $$\approx1.0983221\pm 0.0007\times10^7 m^{-1}\,\!$$

The error for our measured value relative to the accepted value is then given by:
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$
 * $$\% error_{Hydrogen}=\frac{1.0967758\times 10^7 m^{-1}-1.0972781\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}}$$
 * $$\approx0.046%\,\!$$
 * $$\% error_{Deuterium}=\frac{1.0967758\times 10^7 m^{-1}-1.0983221\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}}$$
 * $$\approx0.141%\,\!$$


 * The $$\pm 0.0025\,\!$$ and $$\pm 0.0007\,\!$$ come from the SEM of the values that we used to calculate the mean.



Conclusions
According to our data, although the Deuterium spectral lines varied from the Hydrogen lines in wavelength, as seen by the percentage error in the Rydberg Constant of each, the variance is slight. The largest value by which the wavelengths varied was designated in the red wavelength measurement as seen in our data tables. But also according to our data, the wavelength measurements of each color seemed to be shifted from the Hydrogen in the Deuterium spectrum. 

Acknowledgements
Please note that Alexandra S. Andrego was my lab partner for this lab. Her version of this lab can be found here
 * Prof. Gold's Lab Manual served as a loose guideline for our lab procedure and our calibration wave lengths
 * We used Google Docs to format and post our raw data and error analysis to our wiki notebook
 * Our accepted values for the Balmer Series came from hyperphysics.com
 * Wikipedia had a great article on the Balmer Series and we used it to confirm our results and understanding for this lab
 * Wikepedia 2 is an article on Deuterium

Derivation of the Rydberg Equation
We can start from the equation of total energy of an electron in the nth energy state derived from the Bohr model:


 * $$ E_\mathrm{total} = - \frac{m_e e^4}{8 \epsilon_0^2 h^2}. \frac{1}{n^2} \ $$

The change in energy of an electron transitioning from one energy state with a value $$n$$ to another is:


 * $$ \Delta E = \frac{ m_e e^4}{8 \epsilon_0^2 h^2} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \ $$

Using $$\frac{1}{ \lambda} = \frac {E}{hc} \rightarrow \Delta{E} = hc \Delta \frac{1}{\lambda}\,\! $$ to change the units to wavelength, we get
 * $$ \Delta \left( \frac{1}{ \lambda}\right) = \frac{ m_e e^4}{8 \epsilon_0^2 h^3 c} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \ $$

where
 * $$h \ $$ is Planck's constant,
 * $$m_e \ $$ is the rest mass of the electron,
 * $$e \ $$ is the elementary charge,
 * $$c \ $$ is the speed of light in vacuum, and
 * $$\epsilon_0 \ $$ is the permittivity of free space.

And the Rydberg constant for Hydrogen is found as:
 * $$R_H=\frac{m_e e^4}{8 \epsilon_0^2 h^2}\,\!$$

List of Used Constants

 * $$\mu\,\!$$ is the reduced mass of an atom
 * $$e=1.602\times10^{-19} C\,\!$$
 * $$\epsilon_0=8.854\times10^{-12} F\cdot m^{-1}\,\!$$
 * $$c=2.998\times10^8 m\cdot s^{-1}\,\!$$
 * $$h=6.626\times10^{-34}J\cdot s\,\!$$
 * Rydberg's constant for hydrogen is calculated to be approximately:
 * $$R\simeq1.0967758\times10^7m^{-1}\,\!$$