Physics307L F09:People/Gonzalez/Balmer Series Summary

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Balmer Series Lab Summary

History

In 1885 Johann Balmer discovered (or created) an formula for the computing the wavelengths from visible spectral lines of hydrogen:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda =\frac{hm^{2}}{m^{2}-n^{2}}}

In later years it was discovered that Balmer's formula was a special case in the Rydberg formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{1}{\lambda }=R_{h}\left(\frac{1}{n^{2}_{1}}-\frac{1}{n^{2}_{2}} \right)}

However, it would not be until 1913 that the Rydberg constant R could be explained, when the Bohr model was created.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R=\frac{\mu e^{4}}{8\epsilon ^{2}_{0}h^{3}c}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mu}

is the reduced mass of the atom.

Summary of procedures

A picture of the prism on the spectrometer and the measuring knob.

Measuring the spectral lines of mercury was the first step in order to properly calibrate the spectrometer to the prism inside. Once the prism was at a proper angle and good measurements were taken and compared to the given mercury lamp numbers then we were able to measure the hydrogen spectral lines followed by the spectral lines of deuterium. The data collecting simply involved looking into the spectrometer until a line of a particular color could be seen, the color was recorded and the line was aligned with the cross-hairs on the spectrometer, once the line was properly aligned the number on the knurled ring was recorded as well, this number would be the wavelength of the given line in nanometers. Data was collected, but unfortunately, only two complete measurements were taken from the hydrogen lamp. Below are the computed Rydberg constants for both hydrogen and deuterium.


Purpose of the experiment

The Balmer series shows us that the frequency of the line spectrum is related to an integer (n), which is the quantum number, this hints that light emits energy in fixed amounts, which eventually shows that light is quantized. The photon's energy is directly related to its frequency and therefore can be measured through its wave number. The higher the quantum number the more energy that the photon has and the shorter the wavelength will become.

Data Analysis

This is a good example of what was seen through the spectrometer.
SJK 12:08, 21 October 2009 (EDT)
12:08, 21 October 2009 (EDT)
As I mentioned on your primary notebook page, this analysis information belongs on your primary notebook page...with a more brief summary on this page. I am glad to see the work, though!

The following is the R value measured for hydrogen, its average, standard error, and the standard error of the mean.
{{#widget:Google Spreadsheet |key=t4DHDBtz_LCUrvENKiFGhJQ |width=500 |height=300 }} Since the Violet 1 numbers were so far off due to measuring the wrong line i made two sets of calculations, one including Violet 1, the other excluding it.
The numbers given show us that the average value for R is R=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.0807300X10^{7}m^{-1}} , while the R value without Violet 1 measurements are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.0975266X10^{7}m^{-1}} , the R value without the Violet 1 values are far closer to the accepted value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R=1.0967758X10^{7}m^{-1}}

SJK 12:15, 21 October 2009 (EDT)
12:15, 21 October 2009 (EDT)
Good job reporting the value with uncertainty and attempting the concise notation. You actually should put the uncertainty directly after the value (not after the exponent), though. See for example on this page. Thus, you should write
1.09753(56) x 10^7 1/m
(note different rounding, too)

And actually you did it wrong for the including violet numbers. Those values are sort of nonsense (as you point out) ... but if you did want to report it correctly, you put your uncertainty on the wrong digits. It should be 1.080(17).

The R value with Standard error of the mean is R=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.08073X10^{7}(17)m^{-1}}
Without the Violet 1 numbers R=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.09752X10^{7}(56)m^{-1}}


Therefore my best guess and ranges are:
Best guess: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.0975266X10^{7}m^{-1}}
range 1 (R+SEM)=10980900Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m^{-1}}
range 2 (R-SEM)=10969700Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m^{-1}}
This still falls out of range for the accepted value. more measurements should be taken.

SJK 12:21, 21 October 2009 (EDT)
12:21, 21 October 2009 (EDT)
I like how you use the SEM and are comparing your reasonable range (68% confidence interval) to the accepted value. Some things to push you further: (a) where did you get the accepted value? Did you compute it? Where did you get the values that went into your compuation? (wikipedia?) (b) Yes, the accepted value is outside of your range...but by how much? 2 SEMs? 10 SEMs? It's actually less than 2 SEM, and thus you could be consistent. (c) graphical representation of the data with error bars would really help see how close you are to the accepted value

The deuterium R values are as follows:
{{#widget:Google Spreadsheet |key=t3tkHrF7rdrKpM1ya3JTwjw |width=500 |height=200 }}

As before the R value with the range is computed the same way. Giving us the following:
R=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.0998X10^{7}(6)m^{-1}}

Best Guess: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R=10992433}
Range 1: (R+SEM)=10998544
Range 2: (R_SEM)=10986322

Reasons For Error

Several things could've gone wrong and did go wrong in this experiment. More measurements could've reduced error, as well as being more careful in measuring the line spectrum. The large measuring knob has a lot of gear back lash, which was noticed a few times by myself and my lab partner Jacob. Also we found that reading the dial proved difficult for us, and at one time we realized that we may have been reading it wrong.SJK 12:22, 21 October 2009 (EDT)
12:22, 21 October 2009 (EDT)
Were you careful to turn in the same direction to account for backlash?

What I've learned

This lab helped me to better understand the importance of the Balmer series and how it was one among the many steps taken into the creation of quantum mechanics. I now recognize the similarities and importance between the line spectrum and their relation to the emitted photons and the gas necessary to emit them. Other than the lab I am better understanding how to use google docs and more important, beginning to have a greater appreciation and understanding for error and data analysis involving error. SJK 12:23, 21 October 2009 (EDT)
12:23, 21 October 2009 (EDT)
Yes, I agree, you're definitely learning a lot about error analysis...good use of SEM here and comparison to accepted value. For next labs, keep learning!

Acknowledgments

The picture above was not taken by me, but by Anatastasia and Alex.(Steve Koch 11:58, 21 October 2009 (EDT): Good job crediting ... would be much better to credit in the figure caption so that a reader would be sure to notice the credit.)
The history on the Balmer series was found in both lab manual and on Wikipedia's webpage about Johann Balmer
And a link to a website that had a hydrogen line spectrum as was seen through the spectrometer. Hydrogen

Other Links

A link to my notebook for this lab. Balmer Series notebook
Jacob's page. User:Jacob_R_Jaramillo