# Atomic theory 1.33 - The hydrogen spectrum

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Syllabus ref: 2.2
Hydrogen has one of the simplest emission spectra as there is only one
electron per atom that can absorb and re-emit energy.
The Hydrogen emission series
Convergence
Ionisation
The Hydrogen emission series
The electron in the ground state energy level of the hydrogen atom receives energy in the form of heat or electricity and is
promoted to a higher energy level.
It cannot remain at a higher level (excited state) for very long, and falls back to a lower level.
When the electron falls back down (relax) it must lose the energy difference between the two energy levels. This loss of
energy is performed by releasing electromagnetic energy in the form of infrared, visible light or ultraviolet radiation.
Movement of electrons between the shells is called electron transitions.
When electron transitions take place the energy emitted can be detected and its wavelength measured. This provides
information about the relative energies of the shells.
In the hydrogen atom (the simplest case with only one electron to 'jump' between shells) the energy emitted appears in
several series of lines, each series corresponding to electrons falling back to different levels. This is shown in the diagram
below.
The Lyman series corresponds to transitions between the higher shells and the lowest shell (ground state). The energy of
these transitions produces radiation in the ultra-violet region of the spectrum
The energy shells are usually given a letter 'n' to describe the specific energy level. The lowest level is n=1, the second
level is n=2 etc.
Transitions from higher shells (n&gt;2) to n=2 produce radiation in the visible region of the spectrum. It can be seen by
splitting the light using a prism or diffraction grating and projecting it onto a screen.
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Convergence
As the energy levels increase in energy they get closer together. In other words level 2 and level 3 are further apart than
level 3 and level 4. The levels converge towards a limit.
Transitions that occur in any series must also converge towards a high energy limit, as the largest transition is between
the highest energy level and the level that is characteristic of the specific series.
The highest level is sometimes refered to as the 'infinite' level, as the levels get so close together where they converge
that they are impossible to count.
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Ionisation
When dealing with the Lyman series in the ultraviolet, the transitions are down to the ground state (level 1). The
largest transition must represent a fall from the infinite level, ∞, to level 1. Viewed in reverse it can be
considered to be equal to the ionisation energy, as this is the energy required to move an electron from the
ground state to the infinite level (Note: Ionisation energy is usually expressed per mole of electrons).
M+(g) + 1e
M(g)
Consequently, the ionisation energy may be found by examining the Lyman series at the convergence limit. The
wavelength of light corresponding to the convergence limit may be converted to energy using the relationship E
= hc/l
Example: Calculate the ionisation energy of hydrogen, if the lines
of the Lymann series converge to a limit at a wavelength of 90.1
nm. h, (Planck's constant) = 6.63 x 10-34, c, speed of light = 2.99
x 108 ms-1
E = hc/l
wavelength 90.1 nm = 9.01 x 10-8 m
Therefore Energy = (6.63 x 10-34 x 2.99 x 108)/9.01 x 10-8
∴ E = 2.17 x 10-18
∴ Energy/mole = 2.17 x 10-18 x 6.02 x 1023
∴ Energy/mole = 1.308 x 106 Joules
∴ Energy = 1308 kJ mol-1
Ionisation energy of hydrogen = 1308 kJ mol-1
This method may be used to find the first ionisation energy of any element.
The wavelength (or wavenumber ) values corresponding to the convergence limits are available in data books or
in the excellent NIST physics resource .
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