User:Tyler Wynkoop/Tyler's Page/Balmer

Balmer Series Lab Summary
Background

The Balmer series is a set of spectral lines that characterizes hydrogen. When an element in gaseous form is heated, it emits light. More importantly, this light is not continuous throughout the spectrum, but rather several specific wavelengths, or spectral lines. These wavelengths are characteristic of the element that is heated. For hydrogen, these wavelengths are called the Balmer series. Hydrogen has four spectral lines in the visible spectrum: two shades of violet, light blue, and a vibrant red.

These emissions of light correspond to electrons dropping from excited states to ground states. When the electrons fall, they each release a photon corresponding the energy lost from transitioning states. Each spectral line indicates an electron from a specific excited state to the ground state, where the excited state is different for each spectral line.

Johann Balmer produced an equation in 1885 that determined the wavelength of an emitted photon as a function of the number of the excited state, where each excited state was assigned an integer in sequential order, and the ground state is n=2. Balmer's original equation was

$$\lambda\ = B\left(\frac{m^2}{m^2 - n^2}\right) = B\left(\frac{m^2}{m^2 - 2^2}\right)$$

where λ is the wavelength of the emitted photon, B is Balmer's contant, n=2 for the ground state, and "m" is an excited state. However, three years later, the physicist Johannes Rydberg developed, from Balmer's equation, a new equation that accounted for all transitions within hydrogen, including those from one excited state to another. Rydberg's equation is:

$$\frac{1}{\lambda} = R_\mathrm{H}\left(\frac{1}{2^2} - \frac{1}{n^2}\right) \quad \mathrm{for~} n=3,4,5,...$$

where R_H is the Rydberg constant. This formula can then be adapted to describe other atoms using element-specific constants. In this lab, our goal was to measure the Rydberg constant for Hydrogen.

Set Up

In this lab, we started with a lamp with variable bulbs, each with a different element in them. We then used a prism to separate the different wavelengths, and view the resulting spectrum through an eyepiece. A dial moves the eyepiece to view specific spectral lines and displays the corresponding wavelength. To begin, we calibrated the eyepiece with a known spectral line: 435.8 nm with a mercury bulb. We then inserted a hydrogen bulb and a deuterium and looked at the brightest lines, measuring their wavelength. Knowing the wavelength, we then calculated the Rydberg constant.

Data and Results I took the data, and Dan recorded, as usual. The data taken is represented here:

The accepted value for the Rydberg constant for hydrogen is $$R_\mathrm{H} = 1.0973731 /times 10^7 \frac{1}{m} $$. Our data yielded $$R_\mathrm{H} = 1093(5)\times10^{4} \frac{1}{m}$$ and for deuterium $$R_\mathrm{H} = 1097(1)\times10^{4} \frac{1}{m}$$. This resulted in errors significantly smaller than a single percent for both deuterium and hydrogen.

Error

Our equipment was sensitive only to a single decimal (or four significant figures), and the results that we got were extremely consistent and accurate. There was some "cheating" that happened however when we viewed the hydrogen. Our hydrogen seemed to have been contaminated, resulting in an overpowering yellow spectral line that shouldn't have been present, as well as displaying nearly the entire spectrum. This made identifying the lines that were supposed to be present a challenge. We had to look up known data in order know where we should approximately find the spectral lines. They were indeed present, but the contamination of the sample made several more lines which we threw out as bad data. The lab asked us to reason whether or not comparing Hydrogen and Deuterium would yield any results. Since our hydrogen sample was grossly contaminated, I am unable to answer.