6.4 Bohr’s model of the hydrogen atom  (Page 2/37)

Emission spectra of the elements have complex structures; they become even more complex for elements with higher atomic numbers. The simplest spectrum, shown in [link] , belongs to the hydrogen atom. Only four lines are visible to the human eye. As you read from right to left in [link] , these lines are: red (656 nm), called the H- $\text{α}$ line; aqua (486 nm), blue (434 nm), and violet (410 nm). The lines with wavelengths shorter than 400 nm appear in the ultraviolet part of the spectrum ( [link] , far left) and are invisible to the human eye. There are infinitely many invisible spectral lines in the series for hydrogen.

An empirical formula to describe the positions (wavelengths) $\lambda$ of the hydrogen emission lines in this series was discovered in 1885 by Johann Balmer . It is known as the Balmer formula    :

The constant $R_{\text{H}}=1.09737\times {10}^7{\text{m}}^{\mathrm{-1}}$ is called the Rydberg constant for hydrogen    . In [link] , the positive integer n takes on values $n=3,4,5,6$ for the four visible lines in this series. The series of emission lines given by the Balmer formula is called the Balmer series    for hydrogen. Other emission lines of hydrogen that were discovered in the twentieth century are described by the Rydberg formula    , which summarizes all of the experimental data:

When $n_f=1,$ the series of spectral lines is called the Lyman series    . When $n_f=2,$ the series is called the Balmer series, and in this case, the Rydberg formula coincides with the Balmer formula. When $n_f=3,$ the series is called the Paschen series    . When $n_f=4,$ the series is called the Brackett series    . When $n_f=5,$ the series is called the Pfund series    . When $n_f=6,$ we have the Humphreys series    . As you may guess, there are infinitely many such spectral bands in the spectrum of hydrogen because $n_f$ can be any positive integer number.

The Rydberg formula for hydrogen gives the exact positions of the spectral lines as they are observed in a laboratory; however, at the beginning of the twentieth century, nobody could explain why it worked so well. The Rydberg formula remained unexplained until the first successful model of the hydrogen atom was proposed in 1913.