Learning Objectives describe Rydberg"s theory for the hydrogen spectra. Analyze the hydrogen spectrum in regards to the energy states the electrons.

You are watching: Predict the wavelength of the fifth line in the spectrum.

In an exceptional demonstration of mathematics insight, in 1885 Balmer come up through a an easy formula because that predicting the wavelength of any type of of the lines in atom hydrogen in what we now know as the Balmer series. Three years later, Rydberg generalised this so the it was possible to recognize the wavelength of any type of of the lines in the hydrogen emission spectrum. Rydberg said that every atomic spectra formed family members with this sample (he was unaware the Balmer"s work). It transforms out the there are families of spectra adhering to Rydberg"s pattern, especially in the alkali metals, sodium, potassium, etc., but not v the precision the hydrogen atom present fit the Balmer formula, and also low worths of \(n_2\) suspect wavelengths the deviate considerably.

Rydberg"s phenomenological equation is together follows:

\< \beginalign \widetilde\nu &= \dfrac1 \lambda \\<4pt> &=R_H \left( \dfrac1n_1^2 -\dfrac1n_2^2\right) \label1.5.1 \endalign \>

where \(R_H\) is the Rydberg constant and is equal to 109,737 cm-1 and also \(n_1\) and \(n_2\) room integers (whole numbers) v \(n_2 > n_1\).

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


Other Series

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

\<\dfrac1\lambda = R_\textrm H \left(\dfrac11^2 - \dfrac1n^2 \right ) \label1.5.2\>

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

Table \(\PageIndex1\): The Lyman series of Hydrogen emissions Lines (\(n_1=1\)) \(n_2\)2345...
\(\lambda\) (nm) 121 102 97 94 ...
\(\widetilde\nu\) (cm-1) 82,2291 97,530 102,864 105,332 ...

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Do you understand in what region of the electromagnetic radiation this lines are? of course, this lines are in the UV region, and also they room not visible, yet they room detected through instruments; this lines kind a Lyman series. The existences that the Lyman series and Balmer"s series suggest the existence of an ext series. For example, the series with \(n_2 = 3\) and also \(n_1\) = 4, 5, 6, 7, ... Is dubbed Pashen series.

Mulitple series

The spectral lines room grouped into collection according to \(n_1\) values. Present are called sequentially starting from the longest wavelength/lowest frequency of the series, using Greek letters within each series. For example, the (\(n_1=1/n_2=2\)) line is dubbed "Lyman-alpha" (Ly-α), when the (\(n_1=3/n_2=7\)) line is dubbed "Paschen-delta" (Pa-δ). The very first six series have details names:

Lyman series with \(n_1 = 1\) Balmer series with \(n_1 = 2\) Paschen collection (or Bohr series) with \(n_1 = 3\) Brackett series with \(n_1 = 4\) Pfund collection with \(n_1 = 5\) Humphreys series with \(n_1 = 6\)altpredict the wavelength of the fifth line in the spectrum.