# User:Alexandra S. Andrego/Notebook/Physics 307L/2009/09/28

Balmer Series Main project page
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SJK 12:33, 24 October 2009 (EDT)
12:33, 24 October 2009 (EDT)
(Same comment for Anastasia'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!

## 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 our 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 equation for hydrogen:
$\displaystyle{ \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,..\,\! }$
or more generally stated:
$\displaystyle{ \frac{1}{\lambda }=R(\frac{1}{m^2}-\frac{1}{n^2}) }$
$\displaystyle{ m=1,2,3,...\,\! }$
$\displaystyle{ n=2,3,4,5,...\,\! }$
$\displaystyle{ n\gt m\,\! }$

## Materials

Constant-Deviation Spectrometer (SER #12610)
• Spectrum Tube Power Supply (Model SP200)
• 5000V
• 10 MA
Spectrum Tubes:
• Spectrum Tube, Mercury Vapor (S-68755-30-K)
• Spectrum Tube, Hydrogen (S-68755-30-G)
• Spectrum Tube, Deuterium (S-68755-30-E)

## Safety

Some points about safety that should be considered before beginning this lab are...
• Proper handling of the mercury bulb and all glass tubing
• Refrain from toggling the power on the spectrum tube power supply to avoid damage to the mercury bulb
• Check for any damage on all cables and machinery
• Basic safety procedures used in working with electrical equipment (Such as proper grounding of the equipment)

## Set Up

To see the formal Lab guidelines that we followed visit Prof. Gold's Lab Manual
Our Set Up
Our slit width
##### To begin the experiment the following preliminary steps were taken
• Make sure that no parallax exists between the cross-hairs and the slit of the spectrometer when we focus it sharply. This is done by adjusting the spectrometer by bringing the cross-hairs into focus and sliding the ocular to a position that suits our vision.
• Bring the slit into focus by turning the large ring near the center of the viewing telescope
• Attach and position the mercury bulb into the spectrum tube power supply.
• Turn on the spectrum tube power supply and allow time(about 5 minutes) for the mercury bulb to warm up.
• Calibrate the spectrometer by using a wide slit setting to find a line of the mercury spectrum and narrow the slit until the line comes into sharp focus.
• Locate all mercury spectra lines and note the position or the value of your spectrometer dial.
• Use known values of light wavelengths to finish calibrating your system.
• Use data to correspond the wavelengths with the correct quantum numbers.
• Use our equation to solve for Rydberg's constant, R, in each case.
• Repeat this process for deuterium.

## Measurements and Data

The following table was taken from Professor Gold's Manual, pg 29 and the values were used to calibrate our spectrometer using the mercury bulb.
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 this one
Green 546.1
Yellow 1 577.0
Yellow 2 579.0
Red 690.75

The following tables are the data we took during lab using our calibrated spectroscope:

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## Calculations and Analysis

Using our raw data tables we were able to use standard function in excel to calculate our mean, standard deviation and standard error margins for each spectra line. Here you can see our final data results:

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SJK 12:29, 24 October 2009 (EDT)
12:29, 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)
The accepted value of Rydberg's constant is calculated from the following equation found on page 30 of Professor Gold's Manual:
$\displaystyle{ R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\! }$
Where $\displaystyle{ \mu\,\! }$ is the reduced mass
$\displaystyle{ R=1.0967758\times 10^7 m^{-1}\,\! }$

The following accepted values for the four visible wavelengths of the Balmer Series were taken from the hyperphysics website
$\displaystyle{ n=6\rightarrow n=2\,\! }$
$\displaystyle{ \lambda =410.174 nm\,\! }$
$\displaystyle{ n=5\rightarrow n=2\,\! }$
$\displaystyle{ \lambda =434.047 nm\,\! }$
$\displaystyle{ n=4\rightarrow n=2\,\! }$
$\displaystyle{ \lambda =486.133 nm\,\! }$
$\displaystyle{ n=3\rightarrow n=2\,\! }$
$\displaystyle{ \lambda =656.272 nm\,\! }$
From our data we have measured values of wavelengths to be...
$\displaystyle{ n=6\rightarrow n=2\,\! }$
$\displaystyle{ \lambda_{Hydrogen} =409.84 nm\,\! }$
$\displaystyle{ \lambda_{Deuterium} =N/A\,\! }$
$\displaystyle{ n=5\rightarrow n=2\,\! }$
$\displaystyle{ \lambda_{Hydrogen} =433.92 nm\,\! }$
$\displaystyle{ \lambda_{Deuterium} =433.3 nm\,\! }$
$\displaystyle{ n=4\rightarrow n=2\,\! }$
$\displaystyle{ \lambda_{Hydrogen} =485.96 nm\,\! }$
$\displaystyle{ \lambda_{Deuterium} =485.62 nm\,\! }$
$\displaystyle{ n=3\rightarrow n=2\,\! }$
$\displaystyle{ \lambda_{Hydrogen} =657.4 nm\,\! }$
$\displaystyle{ \lambda_{Deuterium} =655.9 nm\,\! }$
From these values are able to calculate our measured Rydberg's constant to be as follows...
$\displaystyle{ \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\! }$
$\displaystyle{ \frac{1}{\lambda }=R(\frac{n^2-4}{4n^2})\,\! }$
$\displaystyle{ R=\frac{4n^2}{\lambda(n^2-4)}\,\! }$

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

$\displaystyle{ n=5\rightarrow n=2\,\! }$
$\displaystyle{ R_{Hydrogen}=\frac{4(5)^2}{(433.92\times10^{-9} m)((5)^2-4)}\approx1.0974153\times10^7 m^{-1}\,\! }$
$\displaystyle{ R_{Deuterium}=\frac{4(5)^2}{(433.3\times10^{-9} m)((5)^2-4)}\approx1.0989856\times10^7 m^{-1}\,\! }$
$\displaystyle{ n=4\rightarrow n=2\,\! }$
$\displaystyle{ R=\frac{4(4)^2}{(485.96\times10^{-9} m)((4)^2-4)}\approx1.0984840\times10^7 m^{-1}\,\! }$
$\displaystyle{ R=\frac{4(4)^2}{(485.62\times10^{-9} m)((4)^2-4)}\approx1.0982524\times10^7 m^{-1}\,\! }$

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

The average value of our measured Rydberg's constant is...
$\displaystyle{ R_{Hydrogen,average}=\frac{(1.0979895+1.0974153+1.0984840+1.0952236)\times10^7m^{-1}}{4} }$
$\displaystyle{ =\frac{4.3891124\times10^7 m^{-1}}{4}\,\! }$
$\displaystyle{ \approx1.0972781\pm 0.0025\times10^7 m^{-1}\,\! }$
$\displaystyle{ R_{Deuterium,average}=\frac{(1.0989856+1.0982524+1.0977283)\times10^7m^{-1}}{3} }$
$\displaystyle{ =\frac{3.2949663\times10^7 m^{-1}}{4}\,\! }$
$\displaystyle{ \approx1.0983221\pm 0.0007\times10^7 m^{-1}\,\! }$
Our percent error for these calculations of the Rydberg's Constant are...
$\displaystyle{ \% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}} }$
$\displaystyle{ \% error_{Hydrogen}=\frac{1.0967758\times 10^7 m^{-1}-1.0972781\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}} }$
$\displaystyle{ \approx0.046%\,\! }$
$\displaystyle{ \% error_{Deuterium}=\frac{1.0967758\times 10^7 m^{-1}-1.0983221\times10^7 m^{-1}}{1.0967758\times 10^7 m^{-1}} }$
$\displaystyle{ \approx0.141%\,\! }$

## Notes about Our Uncertainty

-To avoid gear backlash (where we turn the measurement knob but the gears don't shift due to a small "empty space") we made sure to turn our knob at least a full turn behind our spectra line and then slowly turned the knob to our measurement.
-Due to the fact that we did this experiment over the course of two non-consecutive days, we need to take into account that we had to recalibrate the spectroscope before resuming experimentation on the second day.

## Summary

If you wish to see my informal summary of this lab follow this link

## Acknowledgments

Please note that Anastasia Ierides 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