The Rydberg Constant
The Rydberg Constant
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 lines of color in the visible light range. The history of the study of these 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 visible 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 hydrogen, including the ultra-violet and the infrared spectral lines. The
Balmer equation is shown below.
where m and n were integers, and h = 3654.6 10-8 cm. When one solves the equation using n
=2 and m = 3, 4, 5, or 6, 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
trail 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 (inspired, 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.
Rydberg substituted the wavenumber, 1/wavelength, for wavelength and by applying appropriate
constants he developed a variation of Balmer’s equation. The Rydberg constant bears witness to his
contribution to understanding the wave behavior of particles and helped paint a clearer picture of
emission spectra.
In 1913, Niels Bohr added to the description of the line spectra from the hydrogen discharge tube.
Bohr postulated that electrons orbited an atom in discrete energy levels. Along with Rydberg’s
work, Bohr called upon Max Planck’s investigation of black body radiation and Albert Einstein’s
determination of the energy of a photon. The combined thrust of these scientific heavyweights
resulted in “proving” Johan Balmer’s clever little formula. The Bohr equation takes the form shown
below.
Niels Bohr used this equation to show that each line in the hydrogen spectrum corresponded
to the release of energy by an electron as it passed from a higher to a lower energy level. The
energy levels are the integers in the equation, labeled ni and nf for initial and final levels, with R
representing the Rydberg constant. The term 1/λ is the wavenumber, as expressed by Rydberg in his
version of the Balmer equation.
In this experiment you will measure the emissions from a hydrogen gas discharge tube and analyze
the emission data to calculate the Rydberg constant.
6-3
