Abstract

In this experiment we used a spectroscope to measure the spectrum lines of hydrogen, sodium and mercury.  The values for sodium and mercury where used to calibrate the spectroscope and thus improve the accuracy of the Hydrogen measurements.  From the hydrogen spectra we found the Rhyberg constant to be 1.0590E7 +/- 6.06E4.
 

Introduction

Using a manual diffraction grating spectroscope, the position of the spectrum lines for heated gases of hydrogen, mercury and sodium were measured.  From that we obtained the values for the wavelengths of those lines.  The measured wavelengths for mercury and sodium where compared to the expected values and used to refine the values for certain parameters of the scope.  From this we were thus able to better evaluate the measurements for the hydrogen spectrum.  Using these measurements and adjusting for the presence of air we were are to compute the value of the Rhyberg constant to be 1.0590E7 +/- 6.06E4 (See Analysis for note on error).
 

Experiment

The experiment was essentially handled in three phases but each phase was almost identical except for the differance lights being looked at and some differances in which data one chose to record for each segment.  For this experiment we used a spectroscope (Fig 1) which included a highly accurate scale by which to measure the angular position of the eye piece.  The light from each source was projected through a narrow slit down the telescope portion of the spectroscope and onto a fine diffraction grating (rated to be 6000 lines per cm, calculated value in Results/Analsys).  The diffraction grating occupied the central pivot position on the stand and the eyepiece was moved around it, thus allowing one to find the angular locations of the first and second order diffrracrion maxima with respect to both sides of the central maxima.  During part of the initial setup process a light was retroreflected off the grating so as to ensure that it was properly aligned and to establish a zero position on the scale (102 41' +/- 1')

Results/Analysis

The first data collected was that for the Na lamp for which the positions of the double yellow lines were recorded for both first and second order for both the right and left of zero as follows :

We know that l = d/p*sin(q), where l is the wavelength, d is the width of slits in the grating, p is the order of the line beign measured and q is the angular displacement of that line.

Upon examing the data (taking an average of points on left and right) we see that :

Where the yellow is the data and the pink is the expected (note the pink is not straight but its variance is too small to be seen on graph as is the expected error on data)

Something is obviously wrong here.  Upon consideration I decided to include two sources of systematic error, first that the grating wasn't really aligned straight and secondly that the grating width was other than what the manufacturer said it was.  Doing a least squares fit to bring the data into line with expectation we find a ~ -2.1 degree displacement in the grating position and that the grating has more like 5580 lines per cm.  Which gives data that looks like :

Now if we continue on to the mercury spectrum.  For the mercury spectrum I only measured the first and second spectrum on the left side, and will use the assumption that the value for d as given above is correct, however using the fact that we can not average one side with the other means that there is a potential for another systematic coming from the possiblity of being off center with the zero point.  By fitting the mercury data to the expectation and requiring that the 1st order and 2nd order data give similar values for the wavelength we get values for the displacement of the screen and the systematic error in the zero point to be respectively -1.96 and +3.0.  I'll be the first to admit that 3.0 is unjustifiably large and the only conclusion I can make is that the apparatus was disturbed at some point during the experiment.

Even with these corrections though the data isn't very good, in that it is skewed high and I have no reasonable explanation why that should be so.  The pink and yellow are the collected data.  The blue line below them is the expectation.

Well this isn't so unreasonable since we are within one error tolerance here barely.

Considering the Hydrogen Spectrum and using values calculated above as calibration data we can then calculate the Rydberg constant from 1/l = R * (1/2^2-1/n^2) for n =3,4,5 ... , where R is the Rydberg constant and l is the wavelength.  This can of course be found by finding the slope of the plotted data.

The slope of this line being R was found to be 1.0590E7 +/- 6.06E4, which is close though not as close as one might like to the value of 1.0967E7 which is expected.  I suspect that the small error on my figure was merely an accident caused by the small amount of data being used.  The data was not adjusted for variations in atmosphere as that correction was an order of magnitude smaller than the already posited error on the wavelength.

From the equation R_M = R_Infinity (1+m_e/M)^-1, where R_M is the Ryhberg constant for an atom of nuclear mass M, and m_e is the mass of the electron.  We can see that this gives us a value of R_Infinity = 1.0597E7 +/- 6.07E4.
 

Conclusion

This experiment's ultimately goal was to measure the value of the Ryhberg constant for Hydrogen and in the process it was neccesary to calibrate the equipment by making runs against the known spectral lines of mercury and sodium.  Ultimately the result obtained was relatively close to the expected value however, I feel that the error estimate obtained from the graph is almost certainly way to low do to lack of data.  When doing experiments of this nature it is definately useful to have some knowledge of the expected results in advance so as to provide a check for the values one does get.