User:Roberto Sebastian Rosales/Notebook/Physics 307L/2010/09/15

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 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Balmer Series Lab Data - September 15, 2010
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=Background=

The Rydberg constant is given by the following equation which can be found here:

$$R_\infty = \frac{m_e e^4}{8 \varepsilon_0^2 h^3 c} = 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$$ (which can be found here), where $$m_e, e, \varepsilon_0, h,$$ and $$c$$ are the mass of an electron, the charge of an electron, the permittivity of free space, Planck's constant, and the speed of light respectively. After reading the Wikipedia article about the Rydberg constant, I learned that when dealing with Hydrogen we must use the reduced mass version of the Rydberg constant given by: $$R_M = \frac{R_\infty}{1+m_e/M}$$ (which can be found here, where $$M$$ is the atomic mass of the nucleus. We can use the relation $$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$ (which can be found here in order to predict the Rydberg constant by measuring the wavelength of emission for known energy state transitions ($$n_1,$$ and $$n_2$$ are the quantum numbers for the electrons transitioning). For the Balmer series we will be using $$n_1 = 2$$, and the accepted values for the transitions are (and are from this Wikipedia Page:

Note
We will be comparing two calculated Rydberg constants, one for Hydrogen and one for Deuterium, to the accepted value obtained from the reduced mass Rydberg constant (not the Rydberg constant for an infinitely massive atomic nucleus).

=Equipment=
 * Spectrometer - Adam Hilger; London, England; Serial: 12610
 * Spectrum Tube Power Supply - Model: SP200 (5000V;10mA; Electro-Technic Products)
 * Mercury Tube
 * Hydrogen Tube
 * Deuterium Tube

=Procedure:= We used Prof. Gold's lab manual in order to complete this lab manual. We did however have to figure out how to calibrate the spectrometer on our own since the lab manual simply states to calibrate it to the Mercury lines.
 * First, we set up the spectrometer on the table and placed the lamp near the slit opening on top of two books.
 * After turning the lights off, we adjusted the slit width and turned the dial until we could see the lines for Mercury. We Calibrated then spectrometer to one of the lines for Mercury (see calibration section below).
 * Next, we switched out the Mercury tube with the Hydrogen tube and took measurements for the Hydrogen lines.
 * We repeated this process several times to recalibrate the spectrometer to a different Mercury line and take more measurements.
 * We also repeated this process for the Deuterium tube and took data that can be found below.

Safety

 * The only major safety hazard other than the usual electrical hazards was the temperature of the tubes after use. The tubes heated up extremely fast, so we had to wait and handle them with care when switching them out.

Setup
This lab was fairly easy to setup and took only minutes to get up and running. We simply set the spectrometer on the table top, and then positioned the Spectrum Tube Power Supply on top of two books directly in front of the slit opening. The books were used to ensure that the middle part of the tube was being viewed as opposed to the top or bottom parts of the tube. Of course the lights needed to be turned off in order to see the spectral lines through the spectrometer.

Calibration
Initially we had to figure out how to read the dial that displayed the wavelength, and how the scale, when rotated, changed what was being viewed in the spectrometer. After some inspection of the apparatus, we realized that the crystal (shown below) itself actually rotates as the dial is turned. So, in order to calibrate the spectrometer, we loosened the screw that holds the crystal in place, found whichever spectral line we were calibrating to in the Mercury spectrum and aligned it with the cross-hairs in the eye piece, and then tightened the screw to hold the crystal in place. Throughout this process, as well as the measuring process, we were careful to avoid gear back lash by always starting at a lower wavelength and turning the dial towards the higher wavelength. Also, we decided to calibrate the spectrometer first to the green line for Mercury, and then take measurements for both Hydrogen and Deuterium. We then re-calibrated the spectrometer to the violet line (435.8nm) and took another set of data for both Hydrogen and Deuterium. Finally, we calibrated the spectrometer to the red line for Mercury and took another set of data. We did this hoping that we would eliminate some systematic error in our measurements. The wavelengths we used to calibrate the spectrometer are from Prof. Gold's lab manual and are as follows: Mercury: 404.7 nm (deep violet very hard to see!!) 435.8 nm violet skip (very weak blue-green) 546.1 nm green 577.0 nm yellow 579.0 nm yellow 690.75 nm red

Issues
As mentioned in the Lab Summary, we experienced some problems when measuring the line for Violet 1 for Hydrogen. We first suspected either a bad calibration or a lack of waiting for the tube to heat up to be the culprit of the misreadings. We re-calibrated the spectrometer, but got the same reading of about 418nm for the Violet 1 line. We then proceeded to calibrate to a different line for Mercury, but the problem persisted. After obtaining reasonable data for all of the other lines for Hydrogen, my partner and I decided to move on and just record the results that we were getting. On the second day of this lab, Prof. Koch asked how we were doing and we made him aware of the issue. Upon taking his own measurement of 417.8 nm (if I recall correctly), he also realized that something had to be wrong. The three of us attempted to reposition the crystal in hopes of correcting the error, but this did not work. I believe that he also calibrated the spectrometer himself, but the same wavelength was observed. I suspect that the tube or the Hydrogen gas in the tube is the reason for the poor data, because we observed the correct wavelength of around 410 nm for the Deuterium tube. We wanted to test a different Hydrogen tube, but the other tube had a note attached that read "Broken," so we did not test it. So, with the suggestion from my lab partner and Prof. Koch, I decided not to include the data for Violet 1 in our $$ R_{calc Hydrogen}$$ calculation. A Rydberg constant for the Violet 1 average wavelength is available in the Google Spreadsheet shown in my "Data and Calculations" section, but it is not a part of any other calculations.

=Data and Calculations= All of the raw data can be found in the following spreadsheet, as well as the calculations for our experimental Rydberg constant for both Hydrogen and Deuterium. Be sure to scroll all the way to the right for our Rydberg constant calculations.

'''Note: Because of a problem or error in the measurements taken for Violet1 for Hydrogen, we decided not to include this in our calculation for the average Rydberg constant (only for Hydrogen). My partner Mathew Cordova and Prof Koch both agreed that we should leave that out of our calculation. Again, although I did calculate an average wavelength and Rydberg constant for Violet1, it is not included in the average Rydberg constant calculation or the Standard error of the mean calculation.''' For the average wavelength calculations, we simply took an average of all the data for one transition (i.e. all the Blue-Green wavelengths for Hydrogen) for Hydrogen and Deuterium separately. $$\bar x = \frac{\sum_{i=1} x_i}{n}$$, where $$x_i$$ is one of the measurements and $$n$$ is the total number of measurements. I was unsure about whether taking the average of the wavelengths was allowed or a good idea, but in class we discussed that if the observations are normally distributed then taking the averages in order to calculate the Rydberg constant is allowed. Also, the average Rydberg constant for both Hydrogen and Deuterium are calculated in this same manner. Next, I calculated the Rydberg constant for each wavelength using the following formula: $$\frac{1}{\lambda} = R \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$, where $$n_1=2$$ for the Balmer Series. And finally I calculated the average Rydberg constant for Hydrogen and Deuterium separately using the preceding formula for averages, and the Standard Error of the Mean (SEM) using the following formula: $$SEM\ = \frac{s}{\sqrt{n}}$$, where s is the standard deviation for the sample (the formula is taken from here). These functions are built right in to Google Docs, so I simply used them to carry out these calculations. Our value for the Rydberg constant is reported as $$ R = \bar R +/- SEM$$, or the mean value for R +/- the standard error of the mean.

Experimental Values:
From our data, we calculated the following values for the Rydberg constant: $$ R_{calc Hydrogen} = (10976969 +/- 7304)m^{-1}\ \ $$ $$ R_{calc Deuterium} = (10989761 +/- 1736)m^{-1}\ \ $$

Accepted Values:
$$R_{\infty}= 1.097\;373\;156\;852\;5\;(73) \times 10^7 \ \mathrm{m}^{-1}$$ $$R_{Hydrogen}= 10967758.3406 m^{-1}\ \ $$ $$R_{Deuterium}=10970746.1986 m^{-1} \ \ $$
 * The accepted value for $$R_{\infty}$$ can be found here, and the accepted values for Hydrogen and Deuterium were obtained by carrying out the calculation for the reduced mass version of the Rydberg constant. The values used for the mass of the electron, proton, and neutron can be found at the following Wikipedia pages: Electron Mass, Proton Mass, and Neutron Mass.

Comparison
I think that our data and calculations were fairly accurate for this lab considering the spectrometer that we were using. Using the following equation we calculated the percent error: $$ % Error = \frac{|x_{calculated} - x_{actual}|}{x_{actual}}*100$$.
 * We were within 2 SEM's for the $$R_{calc Hydrogen}$$ calculation compared to the accepted value for Hydrogen.
 * We were withing 11 SEM's for the $$ R_{calc Deuterium}$$ calculation compared toe the accepted value for Deuterium.
 * Our %error for $$R_{calc hydrogen}=0.084 %$$ and our %error for $$R_{deuterium}=.173 %$$

Error
The error in this lab would most likely come from errors in the calibration or the spectrometer, misreading the value on the dial, and the width of the slit opening. The calibration was kind of hard because a slight movement of the crystal would result in a drastic change in position of the line being viewed. Also, if the screw on the crystal is tightened too much, it can skew the position of the dial. Reading the dial is also difficult because there are not fine divisions. Because of this, we had to round every wavelength to the tenth of a nanometer. As mentioned in the lab manual (which can be found in the "Procedure section"), we had to find a decent slit width that would allow for an intense enough viewing, but also for a fine resolution. Opening the slit too much would result in a very thick line, which would obviously increase our error when we tried to put the cross hairs on the center of the line, while too narrow of a slit width would decrease the intensity of the lower wavelengths to a point at which that are almost not viewable. All of the preceding topics would result in systematic error. Overall I feel like we recorded decent data considering the instruments we were using.

=References=

I looked at David Weiss' and Tom Mahony's lab notebooks for some help on calculating the SEM for our Rydberg constants. I referenced Wikipedia multiple times and the links can be found near the information that was obtained from Wikipedia.


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