# 1.5: The Rydberg Formula and the Hydrogen Atomic Spectrum

- Page ID
- 210779

In an amazing demonstration of mathematical insight, in 1885 Balmer came up with a simple formula for predicting the wavelength of any of the lines in atomic hydrogen in what we now know as the Balmer series. Three years later, Rydberg generalized this so that it was possible to determine the wavelengths of any of the lines in the hydrogen emission spectrum. Rydberg suggested that all atomic spectra formed families with this pattern (he was unaware of Balmer's work). It turns out that there are families of spectra following Rydberg's pattern, notably in the alkali metals, sodium, potassium, etc., but not with the precision the hydrogen atom lines fit the Balmer formula, and low values of \(n_2\) predicted wavelengths that deviate considerably.

Rydberg's phenomenological equation is as follows:

\[ \begin{align} \widetilde{\nu} &= \dfrac{1}{ \lambda} \\[4pt] &=R_H \left( \dfrac{1}{n_1^2} -\dfrac{1}{n_2^2}\right) \label{1.5.1} \end{align} \]

where \(R_H\) is the Rydberg constant and is equal to 109,737 cm^{-1} and \(n_1\) and \(n_2\) are integers (whole numbers) with \(n_2 > n_1\).

For the Balmer lines, \(n_1 =2\) and \(n_2\) can be any whole number between 3 and infinity. The various combinations of numbers that can be substituted into this formula allow the calculation the wavelength of any of the lines in the hydrogen emission spectrum; there is close agreement between the wavelengths generated by this formula and those observed in a real spectrum.

## Other Series

The results given by Balmer and Rydberg for the spectrum in the visible region of the electromagnetic radiation start with \(n_2 = 3\), and \(n_1=2\). Is there a different series with the following formula (e.g., \(n_1=1\))?

\[\dfrac{1}{\lambda} = R_{\textrm H} \left(\dfrac{1}{1^2} - \dfrac{1}{n^2} \right ) \label{1.5.2}\]

The values for \(n_2\) and wavenumber \(\widetilde{\nu}\) for this series would be:

\(n_2\) |
2 |
3 |
4 |
5 |
... |
---|---|---|---|---|---|

\(\lambda\) (nm) |
121 | 102 | 97 | 94 | ... |

\(\widetilde{\nu}\) (cm^{-1}) |
82,2291 | 97,530 | 102,864 | 105,332 | ... |

Do you know in what region of the electromagnetic radiation these lines are? Of course, these lines are in the UV region, and they are not visible, but they are detected by instruments; these lines form a **Lyman series**. The existences of the Lyman series and Balmer's series suggest the existence of more series. For example, the series with \(n_2 = 3\) and \(n_1\) = 4, 5, 6, 7, ... is called **Pashen series**.

The above discussion presents only a phenomenological description of hydrogen emission lines and fails to provide a probe of the nature of the atom itself. Clearly a continuum model based on classical mechanics is not applicable, and as the next Section demonstrates, a simple connection between spectra and atomic structure can be formulated.

## Contributors

Michael Fowler (Beams Professor, Department of Physics, University of Virginia)

Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)