Discrete Spectra and Atomic Structure
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Purpose:
To compare and contrast the emission spectra of various gases. Investigate quantitatively the emission spectrum of hydrogen and relate it to Bohr's theory of atomic structure.

Discussion:
When rarefied gases are heated they emit discrete spectra of radiation. By passing light from the gas through an interference (diffraction) grating several distinct colors are observed. Each color corresponds to a distinct wavelength. The emission spectrum from any gas is always the same. No two gases have the same emission spectrum. In a sense the characteristic emission lines of any gas act as a "fingerprint" which clearly identifies the composition of the gas.

The most easily observed emission lines in the hydrogen spectrum correspond to several lines in the Balmer series. The wavelengths of light in the Balmer series obey the following relationship:

1/l = R*[1/22 - 1/n2]        n = 3, 4, 5, ... R = 0.01097 (nm)-1

Bohr's model of the structure of atomic hydrogen is a semi-classical model. That is, some of its aspects depend on applications of classical physics. For example, Bohr assumes that the electron moves around the proton in a circle under the influence of the Coulomb force between the electron and proton. From this he determines that the total mechanical energy of the electron in orbit depends only on its distance from the proton:

E = -ke2/2r

In addition to its classical aspects, Bohr's model involves quantum innovations. First, Bohr assumed that the angular momentum of the electron was quantized:

mvr = nh/(2p),         n = 1, 2, 3, 4, 5, ...

Consequently, only certain orbits are allowed for the electron:

rn = h2/(4p2ke2m)*n2          n = 1, 2, 3, 4, 5, ...               rn = 0.053 * n2 nm

When this relationship is combined with the energy relationship, it is found that the electron orbits may only correspond to certain energies:

En = -(2p2k2e4m)/h2 * 1/n2    n = 1, 2, 3, 4, 5, ...        En = -13.6 * 1/n2    (eV)

Bohr's second quantum assumption described the nature of electron transitions between allowed energy levels. Bohr assumed that an electron could make the transition to a lower energy level if it emitted a photon in the process. The photon must carry away an amount of energy equal to the energy difference between the two levels involved in the transition.

DE = hf  or DE = hc/l

h = 6.63 * 10-34 (J*s)   c = 3 * 108 (m/s)

(The electron could also make a transition to a higher energy level. That would require the absorption of a photon carrying an amount of energy equal to the energy difference between the two levels involved in the transition. This phenomenon would be appropriate for describing hydrogen absorption spectra.)

Bohr's complete model for hydrogen predicts that emission lines correspond to the emission of a photon that occurs when an electron makes the transition from a higher energy state to a lower energy state. The photon carries off an amount of energy equal to the difference between the two states involved in the transition. The Balmer series can be understood as corresponding to the photons that are emitted when the electron makes a transition to the "n = 2" energy state from any higher (n > = 3) energy state:

DE = | E2 - En |     DE = 13.6 *[1/22 - 1/n2]   (eV)

Materials:
gas discharge tubes      Project Star spectrometers                 graph paper

Procedure:

1. Verify the Balmer relation and the prediction of Bohr's model for hydrogen emission.
• Measure the wavelength of each hydrogen emission line using the spectrometer. Observe at least 3 lines but try to observe a fourth.
• For each observed line measure the energy of each emission line using the spectrometer.
• Based on considerations of the Bohr model, guess which emission line corresponds to which initial energy level (n > =  3).  All electron transitions are to the "n = 2" state.
• Plot 1/l vs 1/n2  for your data.
• Is your graph consistent with a straight line? What is the slope? Use the slope to determine an empirical value for the Rydberg (R).
• Is your determined value of the Rydberg consistent with the value given in the introduction? Support your claim by determining the % difference.

2. Enrichment: characteristic spectra of gases
• Use the spectrometer to examine the light emitted from several different gas discharge tubes. The composition of the gas in each gas discharge tube should be labeled.
• Once you are familiar with the emission from the labeled gas discharge tubes, examine the emission from the two unlabeled tubes using your spectrometer. What can you infer about the composition of the gas in these two tubes? That is for each, which of the labeled gases has the most similar emission spectrum? Are there any differences between the emission from the unlabeled gas and the emission from the labeled gas with the most similar emission spectrum?