# Physics307L:People/Rivera/Notebook/Formal Report

# Deducing the Rydberg Constant by Measuring the Balmer Series Spectral Lines of Hydrogen and Deuterium

## Abstract

We investigated the Rydberg Constant and Balmer series using a "Constant Deviation Spectrometer". We looked at the 3rd, 4th, 5th and 6th spectral lines for both Hydrogen and Deuterium and took measurements on their wavelengths. The Balmer series or Balmer lines in atomic physics, is the designation of one of a set of six different named series describing the spectral line emissions of the hydrogen atom.(1) Then using the Balmer Formula we were able to determine the Rydberg Constant for **hydrogen [math]R_h=1.093(+/-.005)*10^7m^-1[/math]** and **deterium [math]R_d=1.095(+/-.004)*10^7m^-1[/math]**. Giving us a **0.32%** error in relation to the accepted value for hydrogen and a **0.21%** error in relation to the accepted value for deuterium. The accepted value for the Rydberg constant in bot Hydrogen and Deuterium is [math]R=1.097(+/-.004)*10^7m^-1[/math] in fact according to a paper in Physical Review Letters **(4)** the difference in their values shows up in the millionths position in their decimal which is out of the range of my significant figures.

## Introduction

The Authors investigated the Balmer Series to obtain the Rydberg constant for hydrogen and deuterium. The series is investigated by looking at the 3rd, 4th, 5th and 6th electron transitions which characterize the the Balmer series. These transitions are in the visible range and can bee seen with the spectrometer.**(1)**

Our investigation was aimed at uncovering the Rydberg Constant for Hydrogen and Deuterium. The Rydberg constant comes from the ratio [math]\frac{4}{B}=R_\mathrm{H}[/math] in the Balmer Equation. Where n is the principle quantum number of the electron, and [math]\lambda[/math] is the wavelength of the light emitted in the spectral line. Given by the Balmer formula**(1)**.

The theory is that there is a constant that will allow us to predict the [math]\lambda[/math] of the spectral lines of an element. This constant is referred to as the Rydberg constant.**(3)** We looked at Hydrogen and Deuterium to see if we could find a difference between the two elements. Given the difference in the two elements atomic mass we would expect to find a small difference in their Rydberg constants. The question is whether or not our apparatus has the precision for us to be able to determine that difference.

## Methods and Materials

We used a constant deviation spectrometer to read the wavelength of the spectral lines for hydrogen and deuterium (pictured at right). The light box holds the samples and allows us to excite them by passing a current though the gas to excite the atoms and causing them to radiate photons then pass through the slit to the prism where the photons separate into their respective energies creating the spectrum seen through the eye piece. The device can be found in the Junior Physics Lab at the UNM PandA Building

### Calibration

We started by focusing the eye piece so that the cross hairs and the slit are both in sharp focus with no parallax between them. Next we used the mercury tube to find the lines of mercury and to calibrate the device. We used the following lines to calibrate our device:

Violet | Green | Yellow | Yellow | Red |

435.8 nm | 546.1 nm | 577 nm | 579 nm | 690.75 nm |

To calibrate we found a line and turned the prism and the dial until the line and the dial showed the correct wavelength for the spectral line. When we were finished we took several readings on the mercury to make sure that the readings were reasonable across the dial. While taking data we made sure to only turn the dial counter clockwise as suggested in the manual so that we could remove any discrepancies brought about in the dial's mechanics.

### The Hydrogen Spectrum

We first used the hydrogen sample and began taking readings on the spectral lines.We observed the 3rd(Red), 4th(Blue), 5th(Violet 2) and 6th(Violet 1) spectral lines of hydrogen. We took 10 wavelength ([math]\lambda[/math]) readings for each line. We made sure we were turning the dial counterclockwise to keep the dial mechanics from interfering with our data. Next we did the same for deuterium making sure to move the dial counterclockwise for each reading.

### Sodium D Spectral Lines

We looked at the sodium D spectral lines**(5)** and tried to discern between the [math]\lambda=589.0 nm[/math] line and the [math]\lambda=589.6 nm[/math]. We looked at these lines to test the precision of the spectrometer to determine if we could reasonably expect a difference in the Rydberg constant for deuterium and hydrogen.

## Results and Discussions

The excel sheet used to analyze this data can be found at File:BalmerSeries.xls

### Hydrogen

We manipulated the Balmer Formula to give us [math]R=\frac{1}{\frac{\lambda}{4}-\frac{\lambda}{n^2}}[/math] and used it to find the Rydberg Constant from our data. We first got the mean for each for each set. Then from this we calculated our std deviation and standard error. We then calculated the Rydberg constant for each set of data using the mean for each quantum number.

Next we took our calculations for the Rydberg constant and got their mean and standard deviation. We used these and looked at the accepted value for **R** to find our experimental difference and relative error with relation to the experimental value.

We also plotted the wave number([math]\frac{1}{\lambda}[/math]) versus the quantum number to show their linear relationship.

### Deuterium

We manipulated the Balmer Formula to give us [math]R=\frac{1}{\frac{\lambda}{4}-\frac{\lambda}{n^2}}[/math] and used it to find the Rydberg Constant from our data. We first got the mean for each for each set. Then from this we calculated our std deviation and standard error. We then calculated the Rydberg constant for each set of data using the mean for each quantum number.

Next we took our calculations for the Rydberg constant and got their mean and standard deviation. We used these and looked at the accepted value for **R** to find our experimental difference and relative error with relation to the experimental value. The accepted value for the Rydberg constant for Deuterium is 1.097m^-1 when rounded to the thousandths.

We also plotted the wave number([math]\frac{1}{\lambda}[/math]) versus the quantum number to show their linear relationship.

### Sodium D Spectral Lines

Looking at the two sodium lines [math]\lambda=589.0 nm[/math] and [math]\lambda=589.6 nm[/math]. We determined that we couldn't reasonably tell the difference between the two lines. At times we felt that we could discern the two line but couldn't be sure if it wasn't just an optical illusion within the device and we couldn't discern their wavelengths on the dial.

## Conclusions

We found the Rydberg constant for hydrogen to be **[math]R_h=1.093(+/-.005)*10^7m^-1[/math]** giving us a **0.32%** error in relation to the accepted value of **[math]1.097*10^7m^-1[/math]**.

We also found the Rydberg constant for deuterium to be **[math]R_d=1.095(+/-.004)*10^7m^-1[/math]** giving us a **0.21%** error in relation to the accepted value of **[math]1.097*10^7m^-1[/math]**.

Looking at the significant figures in our number and our observations of the Na lines I wouldn't expect to see a difference in our numbers for the Rydberg constant for Hydrogen and Deuterium. If we had a more precise device I would expect to see better numbers with less error and I would also expect to see a difference between the two Rydberg constants of hydrogen and deuterium(heavy hydrogen), given that fact that their masses are different and would give us different excitation levels.

Future measurements should be taken with a more precise apparatus. Possibly using an eyepiece with higher magnification and a dial that has more precision. Possibly even using a computer operated detector that could more precisely determine the wavelengths of the spectral lines allowing us to resolve the differences between hydrogen and deuterium.

## Acknowledgments

Thanks to Brian Ritter who assisted me with this experiment.

Also thanks to Dr. Koch who allowed us time to better understand our experiments.

## References

1) "Balmer series." Wikipedia, The Free Encyclopedia. 27 Oct 2007, 23:47 UTC. Wikimedia Foundation, Inc. 4 Nov 2007 <http://en.wikipedia.org/w/index.php?title=Balmer_series&oldid=167525817>.

2) Physics 307L: Lab Book, 14 July 2006, Professor Micheal Gold, <http://www-hep.phys.unm.edu/%7Egold/phys307L/manual.pdf>

3) "Rydberg constant." Wikipedia, The Free Encyclopedia. 1 Nov 2007, 19:59 UTC. Wikimedia Foundation, Inc. 4 Nov 2007 <http://en.wikipedia.org/w/index.php?title=Rydberg_constant&oldid=168571795>.

4)Absolute Frequency Measurement of the $2S-8S/D$ Transitions in Hydrogen and Deuterium: New Determination of the Rydberg Constant,de Beauvoir, B. and Nez, F. and Julien, L. and Cagnac, B. and Biraben, F. and Touahri, D. and Hilico, L. and Acef, O. and Clairon, A. and Zondy, J. J.,Phys. Rev. Lett.,Volume 78,number 3,pages 440--443,January 1997,doi 10.1103/PhysRevLett.78.440,publisher American Physical Society.

5)"Sodium." Wikipedia, The Free Encyclopedia. 8 Dec 2007, 22:04 UTC. Wikimedia Foundation, Inc. 9 Dec 2007 <http://en.wikipedia.org/w/index.php?title=Sodium&oldid=176644612>.