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Experiment 10
The Rydberg Constant
10.1 Objectives
- Observe, measure, and analyze the emission spectrum of a hydrogen gas discharge tube.
- Use the data from the hydrogen emission spectrum to calculate the Rydberg constant.
10.2 Introduction
White light is made up of all the colors in the rainbow. Light that is emitted by excited gases is made of very particular colors which reveal information about the underlying structure of the atoms in the gas. For example, the light you see when you plug in a hydrogen gas discharge tube is a shade of lavender, with some pinkish tint at a higher current. If you observe the light through a spectroscope, you can identify four distinct colors in the visible light range.
10.3 Key Concepts
As always, you can find a summary on-line at Hyperphysics^1. Look for keywords: “emission, quantum”, electromagnetic spectrum, hydrogen spec-
(^1) http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html
- The Rydberg Constant
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10.4 Theory
The history of the study of the distinct colors emitted from excited gases, called spectral lines, dates back to the late 19th century, where we meet a high school mathematics teacher from Basel, Switzerland, named Johann Balmer. Balmer created an equation describing the wavelengths of the vis- ible hydrogen emission lines. However, he did not support his equation with a physical explanation. In a paper written in 1885, Balmer proposed that his equation could be used to predict the entire spectrum of hydro- gen, including the ultraviolet and infrared spectral lines. To find the nth wavelength, λn, he said to use this formula, the Balmer formula:
λn = B
n^2 (n^2 − 4)
where n is an integer greater than 2 (e.g. 3, 4, 5, etc.), and B is Balmer’s constant, a number found to make the equation fit the observed spectrum. When one solves the equation, the calculated wavelengths are very close to the four emission lines in the visible light range for a hydrogen gas discharge tube. Balmer apparently derived his equation by trial and error. Sadly, he would not live to see Niels Bohr and Johannes Rydberg prove the validity of his equation. Johannes Rydberg was a mathematics teacher like Balmer (he also taught a bit of physics). In 1890, Rydberg’s research of spectroscopy (in- spired, it is said, by the work of Dmitri Mendeleev) led to his discovery that Balmer’s equation was a specific case of a more general principle that worked for many elements, not just hydrogen. Rydberg substituted the wavenumber, 1/wavelength, for wavelength and, by applying appropriate constants, developed a variation of Balmer’s equation. The Rydberg equa- tion is given as
1 λif
= RH
n^2 f
n^2 i
Here, ni and nf are again integers. RH is the Rydberg constant. Relating it to the Balmer equation, RH = 4/B. For the hydrogen atom, nf = 2, as
(^2) http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html#c
- The Rydberg Constant
Figure 10.1: The diffraction grating.[2]
Figure 10.2: Digital spectrome- ter. Note the white USB cable and blue optical fiber.
Figure 10.3: Screenshot of Logger Pro.
10.7. Procedure
Figure 10.4: Gas discharge tube mounted in box.
10.7 Procedure
- Connect the spectrometer to the USB port of a computer. Start the data-collection program, and then choose New from the File menu.
- Connect a fiber optic cable to the threaded detector housing of the spectrometer.
- To prepare the spectrometer for measuring light emissions: In Log- ger Pro, open the Experiment menu and select Change Units Spectrometer: 1 Intensity.
- To set an appropriate sampling time for collecting emission data: In Logger Pro, open the Experiment menu and choose SetUp Sensors Spectrometer:1. In the small dialog that appears, change the Sample Time to 60 ms, change the Wavelength Smoothing to 0, and change the Samples to Average to 1.
- Turn on the hydrogen gas discharge tube. Aim the tip of the fiber optic cable at the tube.
- Observe the gas discharge tube by looking through the diffraction grating. Note how it spreads out the light into different distinct colors.
10.7. Procedure
Figure 10.5: Example of how to label the graph of the emission spectra to find the HWHM.
- Calculate the best-fit line equation for the plot, then answer Ques- tion 2.
- The Rydberg Constant
- Compare your value of the Rydberg constant from the previous question to the reference value, RH = 1. 097 × 107 m−^1. (more precise value listed in Appendix E)
- Use the RH that you calculated in Question 2 to predict the wavelength of the fifth line in the Balmer Series (n = 7). Examine your graph of the hydrogen discharge tube emissions. Does the fifth Balmer line appear as a peak in your graph? Explain why.
