# User:Michael R Phillips/Notebook/Physics 307L/2008/10/29

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# Balmer Series

SJK 01:05, 18 December 2008 (EST)
01:05, 18 December 2008 (EST)
Given the relative ease in acquiring data with this experiment, it's surprising that you missed many important parts of this lab. For example, the actual theoretical difference between hydrogen and deuterium which I mentioned on your summary pages. Also is the issue of gear backlash which is quite critical in this lab, and since I don't see you discussing it, I'm assuming you missed that part of the manual? I know that I was not here for one of the days, but it is discussed in the manual.

### Introduction & Safety

In this lab, we will be using a very old Constant Deviation Spectrometer to measure the Rydberg constant based on the spectra of Hydrogen and Deuterium, and also the difference between the spectral wavelengths for these two, roughly following this lab manual. As for safety, the main thing is don't get electrocuted. This means turning off and unplugging the lamp power before removing or inserting a new bulb. Other important things to note, for equipment safety rather than our own, is that we should never touch the prism because we could easily damage it with fingerprints and that we should be very careful not to break the bulbs.

SJK 00:56, 18 December 2008 (EST)
00:56, 18 December 2008 (EST)
granted there isn't a lot of equipment, but you should record what you were using.

### Calibration

Before we do any real data taking, we need to perform a simple calibration for our instrument. For this, we will use the given values of wavelengths (λ) and colors to match to our corresponding wavelengths and colors. In the end of the experiment, we will use this data to account for a kind of "calibration error" that will appear in every part of our data. Here is the data we took for initial calibration:

 Color Known Wavelength, λacc (nm) Calibration Wavelength #1, λcal1 (nm) Calibration Wavelength #2, λcal2 (nm) red 690.75 694.5 695.5 yellow1 579.0 578.2 578.5 yellow2 577.0 576.0 576.0 green 546.1 545.3 545.0 violet 435.8 435.4 435.2 deep violet 404.7 404.3 404.5

Along with the lines with given wavelengths, we found in our spectrum several low intensity lines that the lab manual indicated were from impurities in the sample. We also saw clearly the blue-green line that the manual directs us to skip.

After taking all of this calibration data, we can use the average between our two values for each color and compare these to the values that were supplied to us so that we will have a good correction for the Hydrogen and Deuterium spectral data.

Note: after taking this data and guessing as to what the method would be for calibration, we discovered that it was completely wrong! The correct way is to fix the prism rotation so that it gives us a perfect value for one of the middle wavelengths (like yellow or green). This will automatically give us good values for the rest of the experiment. This was done properly, calibrating to "yellow1" (579.0nm), before taking our actual data below.

On day two, we started from here once more, calibrating our instrument before doing anything else. This time we calibrated to green (546.1nm).

### Data

Day 1:

For Hydrogen:

 Color Accepted, λH-acc (nm) λ1 (nm) λ2 (nm) red 656.2 659.4 658.0 cyan 486.1 485.3 485.3 violet1 434.0 434.0 433.9 violet2 410.1 409.7 409.8

We were not able to conclusively locate the so-called blue line in the hydrogen spectra, and some of the sources on the web indicated that this is not a frequently used wavelength.

For Deuterium:

 Color λ1 (nm) λ2 (nm) red 659.3 657.0 cyan 485.2 485.5 violet1 433.4 433.9 violet2 410.8 *

We weren't able to get a second wavelength value for Deuterium's "violet2" because we had to allow another group to have the lights on when we arrived at that color. This may throw off our averages slightly, but shouldn't be a big deal.

One thing we noticed while getting this data, both for Hydrogen and for Deuterium, is that the intensity of the bands rose very much as a result of us just touching the power source for the tubes. We found this strange and couldn't come up with any explanation, but it seemed to be quite consistent and had nothing to do with any tilting that may occur from us touching it.

Also important to note: we measured all of this data for day 1 from the red side of the spectrum.

Day 2:

For today, we will be doing the same procedure as before but measuring from the violet side of the spectrum.

For Hydrogen:

 Color Accepted, λH-acc (nm) λ1 (nm) λ2 (nm) red 656.2 662.3 663.0 cyan 486.1 487.1 487.1 violet1 434.0 434.8 434.1 violet2 410.1 411.2 410.0

After this obtaining this data, we switched to a different Hydrogen tube and measured some more data, this time starting from the red end of the spectrum.

Hydrogen 2:

 Color Accepted, λH-acc (nm) λ1 (nm) λ2 (nm) red 656.2 662.1 664.0 cyan 486.1 486.2 487.1 violet1 434.0 434.9 435.0 violet2 410.1 412.8 411.5

We could only find one working Deuterium tube, so we took one set of data measured from red and another measuring from violet but from the same tube.

For Deuterium:

 Color λ1 (nm) λ2 (nm) red 659.0 662.1 cyan 486.2 486.9 violet1 434.2 434.8 violet2 410.5 409.8

This data was from red.

Deuterium 2:

 Color λ1 (nm) λ2 (nm) red 662.0 661.0 cyan 486.7 487.1 violet1 434.5 434.9 violet2 410.7 410.0

This data was from violet.

### Analysis

The theory that underlies this experiment is straightforward: as an electron from Hydrogen (or Deuterium) gets excited from the electric current passing through the gas in a tube, it rises to a higher energy state (higher than the ground state). After a very brief amount of time, this electron returns to the ground state and emits a photon, or bundle of light energy, which has a corresponding wavelength which we perceive as color. The reason we see many colors through our spectrometer is because the electron gets excited to many different energy states (corresponding to n=3, n=4, etc) then drops to the second energy state (n=2). The reason we say it drops to the second energy state and not to the first or to ground is (1) because it's the definition of a Balmer series and (2) because we wouldn't be able to see light from any larger drops (this light would be infrared or even longer in wavelength).

Therefore, we will use all of these equations, which are a result of the above theory, to figure out our calculated Rydberg constant.

$\frac{1}{\lambda}=R(\frac{1}{2^2}-\frac{1}{n^2}), n=3,4,5,...$

which relates our measured wavelengths, λ, to the Rydberg constant, R, with a couple of constants. The only thing we need to figure out from here is which color corresponds to which quantum number, n.

The easiest way to handle all of our data is to plot λ vs n and determine a slope, a part of which will be R-dependent.

Note: Upon doing this, we decided it was ridiculous to try to do some kind of fit with a quadratic type equation like λ(n) ends up being. Instead, we just used the below relations to calculate a Rydberg constant for both Hydrogen and Deuterium using averages.

The function that we will use is:

$\lambda(n)=\frac{4n^2}{R(n^2-4)}, n=3,4,5,...$

Likewise, there is a function for n(λ)

$n(\lambda)=\sqrt{\frac{4\lambda R}{\lambda R-4}}$

which we use to decide which n corresponds to the red wavelength. From this, we found

nred = 3

ncyan = 4

nviolet1 = 5

nviolet2 = 6

for Hydrogen. We can also surmise from this that Deuterium is the same way, e.g. the quantum numbers for the same colors are the same.

There is another equation, very similar to the above one for 1/λ given in the manual,

$\frac{1}{\lambda}=R(\frac{1}{m^2}-\frac{1}{n^2}), m=1,2,3,... , n=2,3,4,... , n>m$

that is said to be more accurate (the first equation was discovered empirically while this one was a modified version to account for all wavelengths and is reinforced by Bohr's work). However, for our purposes, assuming m=2 and n=3,4,5,... is perfectly okay.

We will then compare our values of R to the actual value:

$R_{accepted}=1.0967758\times10^7m^{-1}$

using a percent error formula

$%Error=\frac{|Accepted-Measured|}{Accepted}\times100$

We will also use Standard Deviation (using Excel) and the standard error of the mean, which is just

$SEM=\frac{s}{\sqrt{N}}$

where s = Standard Deviation and N = Number of Data Points

Here is our Excel worksheet for this lab: Balmer.xlsx