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Figures shows the emission spectrum of hydrogen. Only four emission lines, three in the blue and one in the red line of the spectrum, are present.
The emission spectrum of atomic hydrogen: The spectral positions of emission lines are characteristic for hydrogen atoms. (credit: “Merikanto”/Wikimedia Commons)
Figures shows the emission spectrum of iron. Numerous overlapping emission lines are present in the visible part of the spectrum.
The emission spectrum of atomic iron: The spectral positions of emission lines are characteristic for iron atoms.

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- 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) λ of the hydrogen emission lines in this series was discovered in 1885 by Johann Balmer . It is known as the Balmer formula    :

1 λ = R H ( 1 2 2 1 n 2 ) .

The constant R H = 1.09737 × 10 7 m −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:

1 λ = R H ( 1 n f 2 1 n i 2 ) , where n i = n f + 1 , n f + 2 , n f + 3 , . . .

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.

Limits of the balmer series

Calculate the longest and the shortest wavelengths in the Balmer series.

Strategy

We can use either the Balmer formula or the Rydberg formula. The longest wavelength is obtained when 1 / n i is largest, which is when n i = n f + 1 = 3 , because n f = 2 for the Balmer series. The smallest wavelength is obtained when 1 / n i is smallest, which is 1 / n i 0 when n i .

Solution

The long-wave limit:

1 λ = R H ( 1 2 2 1 3 2 ) = ( 1.09737 × 10 7 ) 1 m ( 1 4 1 9 ) λ = 656.3 nm

The short-wave limit:

1 λ = R H ( 1 2 2 0 ) = ( 1.09737 × 10 7 ) 1 m ( 1 4 ) λ = 364.6 nm

Significance

Note that there are infinitely many spectral lines lying between these two limits.

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Check Your Understanding What are the limits of the Lyman series? Can you see these spectral lines?

121.5 nm and 91.1 nm; no, these spectral bands are in the ultraviolet

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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