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

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
Calibration
 * We followed Professor Gold's Manual for the setup.
 * 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



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


 * The general equation:


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


 * We calculated the accepted value of Rydberg's constant from the following equation found on Professor Gold's Manual:
 * $$R=\frac{\mu e^4}{8\epsilon _0^2ch^3}\,\!$$
 * Where $$\mu\,\!$$ is the reduced mass
 * $$R=1.0967758\times 10^7 m^{-1}\,\!$$


 * The following accepted values for the four visible wavelengths of the Balmer Series were taken from the hyperphysics website
 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda =410.174 nm\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda =434.047 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda =486.133 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda =656.272 nm\,\!$$


 * Using the results from the data set 2 I calculated the values for the wavelengths. Please follow this link [[Image:Balmer_calculations.xlsx|excel spreadsheet]] 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.


 * $$n=6\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =417.88 nm\,\!$$
 * $$\lambda_{Deuterium} =N/A\,\!$$
 * $$n=5\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =433.66 nm\,\!$$
 * $$\lambda_{Deuterium} =433.7 nm\,\!$$
 * $$n=4\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =483.44 nm\,\!$$
 * $$\lambda_{Deuterium} =483.17 nm\,\!$$
 * $$n=3\rightarrow n=2\,\!$$
 * $$\lambda_{Hydrogen} =644.19 nm\,\!$$
 * $$\lambda_{Deuterium} =642.07 nm\,\!$$


 * Using these values,I was able to calculate our measured Rydberg's constant.
 * $$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!$$
 * $$R=\frac{4n^2}{\lambda(n^2-4)}\,\!$$


 * $$n=6\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}=\frac{4(6)^2}{(417.88\times10^{-9} m)((6)^2-4)}\approx1.0768641\times10^7 m^{-1}\,\!$$


 * $$n=5\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}=\frac{4(5)^2}{(433.66\times10^{-9} m)((5)^2-4)}\approx1.0980733\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}=\frac{4(5)^2}{(433.7\times10^{-9} m)((5)^2-4)}\approx1.0979720\times10^7 m^{-1}\,\!$$


 * $$n=4\rightarrow n=2\,\!$$
 * $$Ra_{Hydrogen}=\frac{4(4)^2}{(483.44\times10^{-9} m)((4)^2-4)}\approx1.1032048\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}=\frac{4(4)^2}{(483.17\times10^{-9} m)((4)^2-4)}\approx1.1038212\times10^7 m^{-1}\,\!$$


 * $$n=3\rightarrow n=2\,\!$$
 * $$R_{Hydrogen}=\frac{4(3)^2}{(644.19\times10^{-9} m)((3)^2-4)}\approx1.1176826\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium}=\frac{4(3)^2}{(642.07\times10^{-9} m)((3)^2-4)}\approx1.1213730\times10^7 m^{-1}\,\!$$


 * Below are the average values of the Rydberg constant for Hydrogen and Deuterium.
 * $$R_{Hydrogen,average}\approx1.0989562\pm 0.008\times10^7 m^{-1}\,\!$$
 * $$R_{Deuterium,average}\approx1.1077221\pm 0.007\times10^7 m^{-1}\,\!$$


 * Below are the calculated percent errors.
 * $$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$$
 * $$\% error_{Hydrogen}\approx0.20%\,\!$$
 * $$\% error_{Deuterium}\approx.998%\,\!$$

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.