# 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

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

• 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
$\displaystyle \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,..\,\!$
The general equation:
$\displaystyle \frac{1}{\lambda }=R(\frac{1}{m^2}-\frac{1}{n^2})$
$\displaystyle m=1,2,3,...\,\!$
$\displaystyle n=2,3,4,5,...\,\!$
$\displaystyle n>m\,\!$
We calculated the accepted value of Rydberg's constant from the following equation found on 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
$\displaystyle n=6\rightarrow n=2\,\!$
$\displaystyle \lambda =410.174 nm\,\!$
$\displaystyle n=5\rightarrow n=2\,\!$
$\displaystyle \lambda =434.047 nm\,\!$
$\displaystyle n=4\rightarrow n=2\,\!$
$\displaystyle \lambda =486.133 nm\,\!$
$\displaystyle n=3\rightarrow n=2\,\!$
$\displaystyle \lambda =656.272 nm\,\!$
• 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.
$\displaystyle n=6\rightarrow n=2\,\!$
$\displaystyle \lambda_{Hydrogen} =417.88 nm\,\!$
$\displaystyle \lambda_{Deuterium} =N/A\,\!$
$\displaystyle n=5\rightarrow n=2\,\!$
$\displaystyle \lambda_{Hydrogen} =433.66 nm\,\!$
$\displaystyle \lambda_{Deuterium} =433.7 nm\,\!$
$\displaystyle n=4\rightarrow n=2\,\!$
$\displaystyle \lambda_{Hydrogen} =483.44 nm\,\!$
$\displaystyle \lambda_{Deuterium} =483.17 nm\,\!$
$\displaystyle n=3\rightarrow n=2\,\!$
$\displaystyle \lambda_{Hydrogen} =644.19 nm\,\!$
$\displaystyle \lambda_{Deuterium} =642.07 nm\,\!$
• Using these values,I was able to calculate our measured Rydberg's constant.
$\displaystyle \frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,6\,\!$
$\displaystyle R=\frac{4n^2}{\lambda(n^2-4)}\,\!$

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

$\displaystyle n=5\rightarrow n=2\,\!$
$\displaystyle R_{Hydrogen}=\frac{4(5)^2}{(433.66\times10^{-9} m)((5)^2-4)}\approx1.0980733\times10^7 m^{-1}\,\!$
$\displaystyle R_{Deuterium}=\frac{4(5)^2}{(433.7\times10^{-9} m)((5)^2-4)}\approx1.0979720\times10^7 m^{-1}\,\!$
$\displaystyle n=4\rightarrow n=2\,\!$
$\displaystyle Ra_{Hydrogen}=\frac{4(4)^2}{(483.44\times10^{-9} m)((4)^2-4)}\approx1.1032048\times10^7 m^{-1}\,\!$
$\displaystyle R_{Deuterium}=\frac{4(4)^2}{(483.17\times10^{-9} m)((4)^2-4)}\approx1.1038212\times10^7 m^{-1}\,\!$

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

• Below are the average values of the Rydberg constant for Hydrogen and Deuterium.
$\displaystyle R_{Hydrogen,average}\approx1.0989562\pm 0.008\times10^7 m^{-1}\,\!$
$\displaystyle R_{Deuterium,average}\approx1.1077221\pm 0.007\times10^7 m^{-1}\,\!$
• Below are the calculated percent errors.
$\displaystyle \% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$
$\displaystyle \% error_{Hydrogen}\approx0.20%\,\!$
$\displaystyle \% error_{Deuterium}\approx.998%\,\!$

## 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.