# User:Anastasia A. Ierides/Notebook/Physics 307L/2009/09/28

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SJK 12:33, 24 October 2009 (EDT)
12:33, 24 October 2009 (EDT)
(Same comment for Alex's notebook)This is an excellent primary notebook! Everything appears to be here that would be needed to make it very easy to repeat your measurements. Great use of photos and spreadsheets, and also latex equations. Great work!

## Balmer Series Lab

Partner: Alex Andrego

### Equipment

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

### Safety

Before we begin some points of safety must be noted:

1. First and foremost your safety comes first and then the equipments'
2. Be careful when handling the Mercury Vapor Bulb (in case you need to) and all glass tubing
3. Check the cords, cables, and machinery in use for any damage or possible electrocution points on fuses of machinery by making sure the power cords' protective grounding conductor must be connected to ground
4. Make sure the areas containing and around the experiment are clear of obstacles

### Purpose

The purpose of this lab is to observe and classify spectra lines of the hydrogen and deuterium atoms. 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 should also be able to experimentally determine Rydberg's constant, R, that is used in the Hydrogen Spectrum equation:

${\displaystyle {\frac {1}{\lambda }}=R({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}})}$
${\displaystyle n=3,4,5,...\,\!}$

Or more generally a modified version of the above equation:

${\displaystyle {\frac {1}{\lambda }}=R({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}})}$
${\displaystyle m=1,2,3,...\,\!}$
${\displaystyle n=2,3,4,5,...\,\!}$
${\displaystyle n>m\,\!}$

This process will be applied to both hydrogen vapor and deuterium vapor.

### Brief Description of Balmer Series

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 ${\displaystyle n\geq 3}$ to ${\displaystyle n=2\,\!}$. [1]

### Brief Description of Deuterium

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.

### Setup & Procedure

For this lab we are using Gold's Lab Manual Gold's Physics 307L Manual

Our Set Up
Our slit width
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

### Data, Tables, & Analysis

Table taken from page 29 of this link [2]

The open prism apparatus and measuring gear
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

key=tlcu3hB5KpmJ6X9wgXnuimA width=650 height=300

}}

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: {{#widget:Google Spreadsheet

key=tHKTX_HPEkRAqhsbZKtjGrg width=650 height=705

}}

SJK 12:28, 24 October 2009 (EDT)
12:28, 24 October 2009 (EDT)
It looks like in the "combined results" part of the spreadsheet you are averaging the hydrogen and deuterium values? I don't think you actually use those values further, but I wanted to point out that they shouldn't be averaged together, because they should have two different true means (due to mass difference)

### Calculations

According to page 30 of Gold's manual [3], the present-day accepted value Rydberg's constant is calculated as:

${\displaystyle R={\frac {\mu e^{4}}{8\epsilon _{0}^{2}ch^{3}}}\,\!}$

where ${\displaystyle \mu \,\!}$ is the reduced mass

${\displaystyle R=1.0967758\times 10^{7}m^{-1}\,\!}$

According to [4], the accepted values for the four visible wavelengths of hydrogen in the Balmer series are:

${\displaystyle n=6\rightarrow n=2\,\!}$
${\displaystyle \lambda =410.174nm\,\!}$
${\displaystyle n=5\rightarrow n=2\,\!}$
${\displaystyle \lambda =434.047nm\,\!}$
${\displaystyle n=4\rightarrow n=2\,\!}$
${\displaystyle \lambda =486.133nm\,\!}$
${\displaystyle n=3\rightarrow n=2\,\!}$
${\displaystyle \lambda =656.272nm\,\!}$

From our measured values for the wavelengths, we have:

${\displaystyle n=6\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=409.84nm\,\!}$
${\displaystyle \lambda _{Deuterium}=N/A\,\!}$
${\displaystyle n=5\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=433.92nm\,\!}$
${\displaystyle \lambda _{Deuterium}=433.3nm\,\!}$
${\displaystyle n=4\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=485.96nm\,\!}$
${\displaystyle \lambda _{Deuterium}=485.62nm\,\!}$
${\displaystyle n=3\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=657.4nm\,\!}$
${\displaystyle \lambda _{Deuterium}=655.9nm\,\!}$

From these values we can calculate our measured Rydberg's constant:

${\displaystyle {\frac {1}{\lambda }}=R({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}}),n=3,4,5,6\,\!}$
${\displaystyle {\frac {1}{\lambda }}=R({\frac {n^{2}-4}{4n^{2}}})\,\!}$
${\displaystyle R={\frac {4n^{2}}{\lambda (n^{2}-4)}}\,\!}$

${\displaystyle n=6\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=409.84nm\,\!}$
${\displaystyle R_{Hydrogen}={\frac {4(6)^{2}}{(409.84\times 10^{-9}m)((6)^{2}-4)}}\approx 1.0979895\times 10^{7}m^{-1}\,\!}$

${\displaystyle n=5\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=433.92nm\,\!}$
${\displaystyle R_{Hydrogen}={\frac {4(5)^{2}}{(433.92\times 10^{-9}m)((5)^{2}-4)}}\approx 1.0974153\times 10^{7}m^{-1}\,\!}$
${\displaystyle \lambda _{Deuterium}=433.3nm\,\!}$
${\displaystyle R_{Deuterium}={\frac {4(5)^{2}}{(433.3\times 10^{-9}m)((5)^{2}-4)}}\approx 1.0989856\times 10^{7}m^{-1}\,\!}$
${\displaystyle n=4\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=485.96nm\,\!}$
${\displaystyle R={\frac {4(4)^{2}}{(485.96\times 10^{-9}m)((4)^{2}-4)}}\approx 1.0984840\times 10^{7}m^{-1}\,\!}$
${\displaystyle \lambda _{Deuterium}=485.62nm\,\!}$
${\displaystyle R={\frac {4(4)^{2}}{(485.62\times 10^{-9}m)((4)^{2}-4)}}\approx 1.0982524\times 10^{7}m^{-1}\,\!}$

${\displaystyle n=3\rightarrow n=2\,\!}$
${\displaystyle \lambda _{Hydrogen}=657.4nm\,\!}$
${\displaystyle R={\frac {4(3)^{2}}{(657.4\times 10^{-9}m)((3)^{2}-4)}}\approx 1.0952236\times 10^{7}m^{-1}\,\!}$
${\displaystyle \lambda _{Deuterium}=655.9nm\,\!}$
${\displaystyle R={\frac {4(3)^{2}}{(655.9\times 10^{-9}m)((3)^{2}-4)}}\approx 1.0977283\times 10^{7}m^{-1}\,\!}$

The average value of our measured Rydberg's constant can be calculated as:

${\displaystyle R_{average}={\frac {1}{n}}\sum {R_{i}}}$

where ${\displaystyle n\,\!}$ is the total number of ${\displaystyle R_{i}\,\!}$ So,

${\displaystyle R_{Hydrogen,average}={\frac {(1.0979895+1.0974153+1.0984840+1.0952236)\times 10^{7}m^{-1}}{4}}}$
${\displaystyle ={\frac {4.3891124\times 10^{7}m^{-1}}{4}}\,\!}$
${\displaystyle \approx 1.0972781\pm 0.0025\times 10^{7}m^{-1}\,\!}$
${\displaystyle R_{Deuterium,average}={\frac {(1.0989856+1.0982524+1.0977283)\times 10^{7}m^{-1}}{3}}}$
${\displaystyle ={\frac {3.2949663\times 10^{7}m^{-1}}{4}}\,\!}$
${\displaystyle \approx 1.0983221\pm 0.0007\times 10^{7}m^{-1}\,\!}$

The error for our measured value relative to the accepted value is then given by:

${\displaystyle \%error={\frac {R_{accepted}-R_{measured}}{R_{accepted}}}}$
${\displaystyle \%error_{Hydrogen}={\frac {1.0967758\times 10^{7}m^{-1}-1.0972781\times 10^{7}m^{-1}}{1.0967758\times 10^{7}m^{-1}}}}$
${\displaystyle \approx 0.046\%\,\!}$
${\displaystyle \%error_{Deuterium}={\frac {1.0967758\times 10^{7}m^{-1}-1.0983221\times 10^{7}m^{-1}}{1.0967758\times 10^{7}m^{-1}}}}$
${\displaystyle \approx 0.141\%\,\!}$
• The ${\displaystyle \pm 0.0025\,\!}$ and ${\displaystyle \pm 0.0007\,\!}$ come from the SEM of the values that we used to calculate the mean.

• The lab lasted over two days, with a week interval. During that interval another group had used the same device that we were using for ours, so re-calibration for our set of data was necessary during the second day.
• As for using the 'scope, we took care to avoid gear back lash by turning the knob all the way back before remeasuring spectra for each trial.

### Balmer Series Lab Summary

This is the link to my Balmer Series Lab Summary: Balmer Series Lab Summary

### Acknowledgments

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