User:Arianna Pregenzer-Wenzler/Notebook/Junior Lab/Balmer Series Summary

Balmer Series Summary

 * My Course Page

Brief summary look up Balmer Series in lab manual linked below, to see what was basically required for this lab.
 * lab manual]
 * notebook entry for Balmer Series lab

Analysis

Off the top of my head (with details doubled checked using my physics 330 book, "Quantum Physics...", Eisberg and Resnick) the physics in this lab is as follows.
 * Review basic atomic physics

When we the mercury bulb into the lamp and pluged it in we added energy (in the form of electrity) to the mecury atom causing large numbers of electrons to leave their groundstate (lowest energy state, n=1) and enter an excited state. Once in an excited state the electrons want to return to lower energy and do so be releasing their energy in the form of photons. These energy emmisions are quantized (only specific values) of the form E= n*h*neu; where neu is the frequency, h is Planck's constant, and n is the principle quantum number. What we see as visible light is actually EM radiation in the spectrum corresponding to frequencies aproximately between 7.5 to 4.3 *10^14 Hz (or wavelengths between 700nm(red) and 400nm(violet). When the "white" light from from the mecury (or hydrogen, or deuterium) bulb passes through the prism in the spectroscope it is seperated into its different componets (bands of color) according to their diferent engeries charactized by different frequencies because differet frequencies have different indexes of refraction inside this medium (the prism).  By using the spectroscope we can identify the values of the energy being emmited in the visible range by the atom in question by reading off the wavelength associated with a particular band of color.


 * In this lab we looked at the Balmer series for hydrogen and deuterium and used our measured values of the wavelength of their spectrums to determine the Rydberg constant for hydrogen. The Balmer series looks at photons emmited with wavelengths in and close to the visible range by electrons starting at an excited state n>2 (ie:3,4,5,....) and ending at n=2.


 * The accepted value for the Rydberg constant for hydrogen is R=1.0967758*10^7 /m. It is calculated from the following formula, and since we are working with the Balmer series this means n_1 = 2
 * $$\frac{1}{\lambda_{\mathrm{vac}}} = R_{\mathrm{H}} \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$

[wiki page I copied this formula from]

The Rydberg constant for deuterium can be calculated using the Rydber constant (different than the Rydberg constant for hydrogen) and mulitiplying it by the correction for reduced mass which takes into account electron mass and allows you to account for the addition of a neutron in the neucleus. [Rydberg constant, lower formulas came from here] I would expect the Rydberg constant for deuteterium to be closer to the Rydberg constant (larger then the R. constant for Hydrogen)
 * $$R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} = 1.0973731568525(73) \times 10^7 \,\mathrm{m}^{-1}$$
 * $$R_M = \frac{R_\infty}{1+m_e/M} \ $$
 * where,
 * $$R_M \ $$ is the Rydberg constant for a certain atom with one electron with the rest mass $$m_e \ $$
 * $$M \ $$ is the mass of the atomic nucleus.


 * Using these formulas my Rydberg constant for deuterium I calculate the R. constant for deuterium to be aprox. 1.0970743*10^7/m where I just used 2X mass proton for M. My assumption was correct in that my calculation for R. deuterium gives it as larger than R. hydrogen.

The fact that the Rydberg constant for deuterium is larger than than the Rydberg constant for hydrogen might be useful in regards to the data analysis on our Rydberg constants for these two elements.


 * Esentilally we analysed our data by taking all our experimentaly measured values for the wavelength of a given band of color (6 measurments/band of color)and computing a Rydberg constant for each of them. This means 6*4,or 24 Rydberg constants for hydrogen and 18 Rydberg constants for dueterium (one less band for deuterium, because conditions when we started measuring (probably the lights were on) made it such that we couldn't really see the deep purple band).  We then took the six Rydberg constants that came from each band and computed its SEM (Standard error of the mean) which was the standard deviation of our Rydberg constants divided by the square root of the number of trials, where the number of trials was 6.  This was a creation of my lab partner Daniel who was trying to write MATLAB code that would allow us to plot the Rydberg constants calculated from our measurments of the wavelenghts of their individual spectral bands for each element with an error bar corresponding to the SEM.  Daniel's code eventually worked and we got a plot of our Rydberg constants as point around the accepted value of the Rydberg constant for hydrogen.  On the purple end of the hydrogen/deuterium spectrum we were quite close to the accepted value of the R. constant, but our values of R. for cyan, and even more so for red were way off.  I am atributing this to error in the calibration of our spectrometer (see my comments in what I learned).  Looking at our experimental data in this manner was interesting because it really gave us a clear picture of where our collected data fell relative to the accepted value.  Just looking at our measured values for wavelength, our average values of our Rydberg constant, and the value of our % error that we stuck in our code I would never have guessed that our data was so inaccurate.

As far as the difference between hydrogen and deuterium; looking at theis spectrums there really wasn't a noticeable difference even though we didn't measure a deep purple band for deuterium. I think we didn't see the deep purple band initially and we didn't really think about why we should see one (because the only difference between hydrogen and deuterium is a neutron so they really shouldn't be that different) so we did't really look for it. Unfortunately we could not make our code, in the form that we wrote it, mark the Rydberg constants for hydrogen and deuterium differntly. I would have suspected from my calculation of the R. constant for deuterium, that our measured R. constants for deuterium would have had larger values than the corresponding hydrogen R. constants, but looking at our data I think the opposite was actually true, (either that or they are mixed, more analysis than I have time for at the moment would be necessary to deterimine this for sure).

Here is our MATLAB code and plot, plublished into word, scroll to the end to find the plot. [MATLAB code]

What I Learned


 * I just learned to be even more careful about saving my work when editing these pages, since I just finished twenty minutes of writing, signed my name, hit save and got told that I wasn't allowed to contribute (apparentely it loged me out just as I hit save), so here it goes again....
 * I got comfortable using a spectrometer. By finding a band of color then using the slit to narrow it down to a fine line and then centering the crosshairs on its center I feel like I got more and more precise measurments of the values of the given wavelengths.  After a bit of practice I also got to where I could read the value of the wavelength off the dial correctly.


 * The most satisfying this I learned was how to linearly plot my data so that the final value I was trying to find experimentially ( in this case the Rydberg constant, R) was given as the slope of my measurments. This was accomplished by taking the equation for R and using the averages of our measured values to give use an equation in the form of Constant=R*Constant.  The equation for R is [1/wavelength]=R*[(1/2^2)-(1/n^2)] where the 2 is the lowere limit of n given by the fact that we are working with the Balmer series, n = 3,4,5,... depending on the wavelength, and the wavelength is just the average value of the six measurments we made of that particular band using the spectrometer.  The satisfying part of this was seeing it mathmatically, translating this knowledge into workable code is still a bit beyond me.


 * What I would have done differently is made sure I actually understood how we were supposed to calibrate the spectrometer before continuing on to the collection of data that first day in lab. I am actually still wondering if the data we collected that first day on the mercury spectrum could have been used to calibrate the spectrometer.  I had assumed that we used our insturment to measure the mercury spectrum and then turned that data into a linear plot that we could compare to a similar plot of the accepted measurments of the mercury spectrum, maybe by looking at the difference in the slopes.  Then we would use that difference to adjust the rest of our collected data.  The catch was that we couldn't figure out how to plot the mercury data we collected, we had a bunch of values for wavelength, but what did we plot them against?  Could we have ploted our expermental wavelengths agains the accepted wavelenghts?
 * Given how we calibrated our spectrometer on day 2, by taking a known value of the wavelength of a band of color and moving the prism until that band lined up with the crosshairs, I still would have done things slightly differently if I were to do it over again. We took the know value for the lighter purple and lined it up with the crosshairs, then checked to make sure our spectrometer was reading the values of the other bands in the mercury spectrum at close to the accepted values.  If I were to do it again, I would have used a band closer to the center of the mercury spectrum, since I noticed that our values were most off the accepted values at the opposite end of the spectrum (red) from where we did our most accurate calibration.  I'm also wondering if since the crosshairs were slightly off center (to the right I believe) as we checked the remaing bands of the spectrum off the accepted values as we move away from the purple band we did our calibration on, if we shoud have taken this into account when trying to center in on the bands (rather than putting the crosshairs right in the center of the bands should we have put them on the right edge, or would this have just introduced more error).
 * I still feel like I'm working in the dark on data anaylsis. I really don't have either the computer knowledge or the mathmatical knowledge.  I can (and need to) get a book for the math, but for the computer anaylsis I feel pretty daunted. Here's another idea for the first day of lab, you could do a hands on lecture on data analysis by giving everyone some data and showing them some of the basics of manipulating it in Excel, and at the same time telling them why they would want to do so.*Arianna Pregenzer-Wenzler 01:31, 30 September 2008 (EDT):*Arianna Pregenzer-Wenzler 01:06, 29 September 2008 (EDT):