User:Cristhian Carrillo/Notebook/Physics 307L/2010/10/13

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Balmer Series

  • Note that Ginny was my lab partner for this lab.

Purpose

  • Observe the Balmer Series of Hydrogen and Deuterium.
  • The purpose of this lab is to study the Balmer Series in the Hydrogen spectrum.
  • Determine the Rydberg constant for Hydrogen.
  • Compare Hydrogen with Deuterium.
  • Learning how to calibrate an optical spectrometer using the known mercury spectrum.

Equipment

  • Constant Deviation Spectrometer
  • Spectrum Tube Power Supply Model SP200, 5000 volts, 10MA.
  • Mercury and Hydrogen tubes

Safety

  • Be careful with the equipment.
  • Check to make sure that the electrical wires have no electrocution points.
  • Be careful with the glass tubes.
  • One very important thing is to turn the screw that rotates the prism in one direction only to avoid gear back lash

Setup

Our Set Up
Our slit width

Calibration

  • Turn on the mercury tube and let it warm up for a few minutes
  • Find a line of the mercury spectrum with the spectrometer slit wide (1/2 to 1mm).
  • The narrower the slit, the better
  • Keep in mind that narrowing the slit causes loss of intensity of the light.
  • Locate all mercury lines that you can.
  • When you are turning the screw which rotates the prism, note the positionis on the dial which correspond to the mercury lines.

Steps to begin the experiment

  • Must bring the slit into focus by turning the large ring near the center of the viewing microscope.
  • Attach and position the mercury bulb into the spectrum tube power supply.
  • After turning on the power supply, allow the mercury bulb to warm up for about five minutes.
  • When calibrating the spectrometer, use a wide slit setting to find a line of the mercury spectrum and narrow the slit until the line comes into a sharp focus.
  • You must locate all mercury spectra lines and note the position or the value of your spectrometer dial.
  • Use the known values of the light wavelengths to finish calibrating the system.
  • Use the data to find the correct quantum numbers corresponding to the wavelengths.
  • Use the equation to solve for Rydberg's constant R in each case.
  • Repeat this process for deuterium.

Calculations and Analysis

The open prism apparatus and measuring gear

{{#widget:Google Spreadsheet |key=0AqzpS6URre5adHk5SGNuTjFwYUwzaHNKRV9nRHhfVVE |width=800 |height=900 }}

  • After we calibrated with mercury we measured the wavelengths in hydrogen and deuterium by reading the dial
  • For my calculations I used the second data set for hydrogen and deuterium since I thought that we took more than enough measurements.

Equations Used for Calculations

  • For Hydrogen
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,..\,\!}
The general equation:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{1}{\lambda }=R(\frac{1}{m^2}-\frac{1}{n^2})}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m=1,2,3,...\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=2,3,4,5,...\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n>m\,\!}
We calculated the accepted value of Rydberg's constant from the following equation found on Professor Gold's Manual:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\!}
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mu\,\!} is the reduced mass
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=6\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda =410.174 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=5\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda =434.047 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=4\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda =486.133 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=3\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda =656.272 nm\,\!}
  • Using the results from the data set 2 I calculated the values for the wavelengths. Please follow this link File:Balmer calculations.xlsx to see the standard deviation and standard error of the mean for our data. The values below are what I calculated in the excel spread sheet.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=6\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Hydrogen} =417.88 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Deuterium} =N/A\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=5\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Hydrogen} =433.66 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Deuterium} =433.7 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=4\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Hydrogen} =483.44 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Deuterium} =483.17 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=3\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Hydrogen} =644.19 nm\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_{Deuterium} =642.07 nm\,\!}
  • Using these values,I was able to calculate our measured Rydberg's constant.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R=\frac{4n^2}{\lambda(n^2-4)}\,\!}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=6\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Hydrogen}=\frac{4(6)^2}{(417.88\times10^{-9} m)((6)^2-4)}\approx1.0768641\times10^7 m^{-1}\,\!}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=5\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Hydrogen}=\frac{4(5)^2}{(433.66\times10^{-9} m)((5)^2-4)}\approx1.0980733\times10^7 m^{-1}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Deuterium}=\frac{4(5)^2}{(433.7\times10^{-9} m)((5)^2-4)}\approx1.0979720\times10^7 m^{-1}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=4\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Ra_{Hydrogen}=\frac{4(4)^2}{(483.44\times10^{-9} m)((4)^2-4)}\approx1.1032048\times10^7 m^{-1}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Deuterium}=\frac{4(4)^2}{(483.17\times10^{-9} m)((4)^2-4)}\approx1.1038212\times10^7 m^{-1}\,\!}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n=3\rightarrow n=2\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Hydrogen}=\frac{4(3)^2}{(644.19\times10^{-9} m)((3)^2-4)}\approx1.1176826\times10^7 m^{-1}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Deuterium}=\frac{4(3)^2}{(642.07\times10^{-9} m)((3)^2-4)}\approx1.1213730\times10^7 m^{-1}\,\!}


  • Below are the average values of the Rydberg constant for Hydrogen and Deuterium.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Hydrogen,average}\approx1.0989562\pm 0.008\times10^7 m^{-1}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R_{Deuterium,average}\approx1.1077221\pm 0.007\times10^7 m^{-1}\,\!}
  • Below are the calculated percent errors.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \% error_{Hydrogen}\approx0.20%\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \% error_{Deuterium}\approx.998%\,\!}

Error

  • Reasons for our error could be because...
  • We might have calibrated the spectroscope wrong
  • Some line spectra were hard to see
  • Gear backlash: We might have turned the dial in both directions

Acknowledgements

  • I would like to thank my lab partner Ginny for the great help.
  • Thanks to Peng for giving us some ideas as to how to calibrate the spectrometer.
  • I would like to thank Professor Koch and Katie for the help.
  • Alex Andrego for the pictures.