Balmer series

Energy levels of the hydrogen atom

The Balmer series is a specific sequence of emission spectral lines in the visible electromagnetic spectrum of the hydrogen atom , the lower energy level of which is in the L-shell . It is emitted when an electron passes from a higher to the second lowest energy level . ${\displaystyle n_{1}=2}$

Other series are the Lyman , Paschen , Brackett , Pfund and Humphreys series .

spectrum

Visible part of the hydrogen spectrum. Six lines of the Balmer series are visible, as the camera's CCD sensors are also somewhat sensitive into the ultraviolet part of the spectrum.

The spectral lines of the Balmer series are named after the Swiss Johann Jakob Balmer , who recognized their mathematical law, the Balmer formula , in 1885 .

discovery

In the visible range of the hydrogen atom spectrum, four lines can be observed, the distances between which become smaller with decreasing wavelength . Starting with the longest wavelength, they are designated as Hα ( H-alpha ), Hβ, Hγ, and Hδ. Their wavelengths can be calculated using the Balmer formula: ${\displaystyle \lambda }$

${\displaystyle \lambda =A\left({\frac {n^{2}}{n^{2}-4}}\right)=A\left({\frac {n^{2}}{n^{2}-2^{2}}}\right)}$

${\displaystyle A}$is an empirical constant ( i.e. a wavelength in the ultraviolet ). Enter the whole numbers 3, 4, 5 and 6 for ( ); is the consecutive number of the shell , the principal quantum number , of the excited state concerned. ${\displaystyle A=364{,}568\,\mathrm {nm} =3645{,}68\cdot 10^{-10}\mathrm {m} }$ ${\displaystyle n}$ ${\displaystyle n\geq 2+1}$ ${\displaystyle n}$

In the ultraviolet range of the spectrum that is not visible to the human eye, further lines were discovered which are consecutively referred to as Hε, Hζ etc. and whose wavelengths can also be calculated very well for integers above 6: ${\displaystyle n}$

Lines in the hydrogen spectrum

Transition from to ${\displaystyle n}$ ${\displaystyle m}$	3 → 2	4 → 2	5 → 2	6 → 2	7 → 2	8 → 2	9 → 2	${\displaystyle \infty }$ → 2
Name of the line	Hα	Hβ	Hγ	Hδ	Hε	Hζ	Hη
Wavelength measured in nm	656.2793	486.1327	434.0466	410.1738	397.0075	388,8052	383.5387
Wavelength calculated in nm	656.278	486.132	434,045	410.1735	397.0074	388,8057	383.5397	(364.56)
colour	red	Blue green	violet	violet	violet	violet	Ultraviolet	Ultraviolet
Visibility (to the human eye)	visible						not visible

The sequence thus converges towards the wavelength as it increases from above . ${\displaystyle n}$ ${\displaystyle A}$

Generalization by Rydberg

If you put the Balmer formula according to the reciprocal of the wavelength, the wavenumber ${\displaystyle {\tilde {\nu }}}$

${\displaystyle {\tilde {\nu }}={\frac {1}{\lambda }}}$

um, the equation found by Balmer can also be expressed in the form ${\displaystyle R_{\infty }={\tfrac {4}{A}}}$

${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}}\right)}$

write in the

${\displaystyle R_{\infty }=1{,}0973731568539(55)\cdot 10^{7}\,\mathrm {m^{-1}} }$

is the Rydberg constant named after the Swedish physicist Johannes Rydberg and can be used for any natural numbers greater than 2. Only three years after Balmer's discovery therefore Rydberg Balmers generalized formula in 1888 to the named also after him Rydberg formula : ${\displaystyle n}$

${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)\quad {\text{mit}}\quad n_{1}=1,2,\dots \quad {\text{und}}\quad n_{2}\geq n_{1}+1}$

Up to this point, however, only the visible lines in the hydrogen spectrum were known, so Rydberg's equation was also a prediction of lines that could still be found. However, the discovery of the Lyman series in the ultraviolet range for by the US physicist Theodore Lyman in 1906 and the Paschen series in the infrared range for by the German physicist Friedrich Paschen in 1908 soon confirmed the correctness of Rydberg's extension . ${\displaystyle n_{1}=2}$ ${\displaystyle n_{1}=1}$ ${\displaystyle n_{1}=3}$

Ritz's combination principle

The Rydberg equation describes the hydrogen spectrum very precisely. For most other atoms, however, it does not give correct results. The Swiss mathematician Walter Ritz made progress in the description of atomic spectra in 1908 . He discovered the Ritz combination principle named after him :

Other series formulas can be formed through additive or subtractive combinations, be it of the series formulas themselves or the constants included in them.

Put simply, this means that a possible third line can be calculated from two known lines. However, not all of these calculated lines can be observed. Ritz could not explain which lines actually appear.

Interpretation by Bohr's atomic model

The formulas that had been found purely empirically up to this point could be understood for the first time using Bohr's atomic model . According to this, the spectral lines can be traced back to the transition of electrons to another energy level. Using Bohr's model, the general equation for these transitions is:

${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)\quad {\text{mit}}\quad n_{2}\geq n_{1}+1}$

The first term in brackets ,, is the so-called basic term , the second ,, is called the running term. If you stick to the basic term and vary in the running term, the following series, named after their discoverers, result. With the exception of Hα (red), Hβ (blue-green), Hγ, Hδ, Hε and Hζ (all violet) they are in the ultraviolet and infrared range of the frequency spectrum. ${\displaystyle {\tfrac {1}{n_{1}^{2}}}}$ ${\displaystyle {\tfrac {1}{n_{2}^{2}}}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$

Surname	n 1	n 2	formula	Spectral range / color
Lyman series	1	2, 3, 4, ...	${\displaystyle {\tilde {\nu }}=R_{\infty }\left(1-{\frac {1}{n_{2}^{2}}}\right)}$	Vacuum UV (121 nm → 91 nm)
Balmer series	2	3, 4, 5, ...	${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{2^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$	red, blue-green, 4 × violet, then transition to near UV → 365 nm
Paschen series	3	4, 5, 6, ...	${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{3^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$	IR-A (1875 nm → 820 nm)
Brackett series	4th	5, 6, 7, ...	${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{4^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$	IR-B (4050 nm → 1460 nm)
Pound series	5	6, 7, 8, ...	${\displaystyle {\tilde {\nu }}=R_{\infty }\left({\frac {1}{5^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$	IR-B (7457 nm → 2280 nm)

In the Bohr model of the atom, in contrast to the Balmer formula, the constant is not a purely empirical quantity. Rather, the value can be traced back directly to the natural constants used in the calculation . Also the restriction to integer values ​​for and as well as the condition ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$

${\displaystyle n_{2}\geq n_{1}+1}$

follow from this model. The variables or are then the main quantum numbers for that basic or excited state to which the electron falls back, or the higher-energy, further excited state from which it decays, i.e. H. As with the Balmer series, a transition between electrons is generally also possible between two excited states. ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{2}}$

The top right shows the energy level diagram of the hydrogen atom and visualizes the above equations ( in the figure will take the name and instead of the designation used ${\displaystyle n_{1}}$ ${\displaystyle m}$ ${\displaystyle n_{2}}$ ${\displaystyle n}$ ) on the left vertical axis is removed. On the right vertical axis, the associated excitation energy, measured from the ground state, is given in eV . The distance between the energy levels is true to scale. In the horizontal direction, the first transitions are shown as examples for each series. The associated principal quantum numbers of the state are given above. The distance between the lines, i.e. H. in the horizontal direction, is not to scale, but chosen to be the same size for reasons of clarity. The figure shows that all lines in a series end at the same energy level. The Hα line of the Balmer series is thus a transition from  = 3 to  = 2. ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$

On the far right in the series, the respective series limit is shown with dots, i.e. H.

${\displaystyle n_{2}\longrightarrow \infty .}$

The electron is then no longer bound to the atomic nucleus, the atom is ionized . For the Lyman series, Bohr's equation gives an energy of 13.6 eV. This value also agrees well with the experimentally determined value for the ionization energy of the hydrogen atom in the ground state.

The question of which of the lines that are possible according to Ritz's combination principle actually occur is clarified by the selection rules . These result from quantum mechanical calculations.

history

The discoverer Balmer examined the light emanating from gas discharges in hydrogen because he suspected that there is a causal connection between the light emission and the structure of the atoms . The emitted light, spectrally broken down with a grating , shows the four discrete lines in the visible range ( line spectrum ). In 1884 Balmer found the law of education (see above) with the constant . ${\displaystyle A=3645{,}6\cdot 10^{-10}\mathrm {m} }$

He considered his discovery to be a special case of an as yet unknown, more general equation that could also be valid for other elements. This assumption is confirmed by later studies of spectra of atoms or ions with only one electron in the outermost shell. For Balmer, however, the physical meaning of remained unclear . ${\displaystyle n}$

See also

• Moseley's law
• Pickering series

literature

• Johann Jakob Balmer: Note on the spectral lines of hydrogen. In: Wiedemann's Annalen der Physik und Chemie 25 (1885), pp. 80-87, also to be found as: Annalen der Physik , Volume 261, Issue 5, 1885, pp. 80-87 (accessed on November 1, 2010)

Individual evidence

1. ↑ Source: Helmut Vogel: Gerthsen Physik. Springer-Verlag: Berlin Heidelberg, 18th edition 1995, p. 623