Module 7.3 Energy levels
Navigation
- Home page of the learning unit
- Building block 1: The Dalton atomic model
- Module 2: The order of the elements
- Unit 3: Electrical phenomena
- Unit 4: Excursus - Radioactivity
- Module 5: The Rutherford scattering experiment
- Module 6: In learning stations to the atomic nucleus
- Module 8: Ionisation energies
Module 7.3 Energy levels
However, we can not only identify these levels by their energy, but also assign whole numbers to them, because in 1885 the mathematician Johann Jakob Balmer developed a formula that was able to show the mathematical relationship between the wavelengths of the visible hydrogen spectrum. The lines could no longer only be observed, but also calculated. The equation known today as the Balmer formula can be represented as follows:
\(\lambda=A\left(\frac{n^2}{n^2-2^2}\right)\)
with A = 364.56 nm and n = 3,4,5,...
Task
Check whether the Balmer formula provides the riwavelengths for the emission spectrum of hydrogen by calculating λ for n = 3,4,5,6.
Also calculate the wavelength λ for n = 7,8,9.
Balmer's formula also predicted that hydrogen must emit further lines in the ultraviolet light range that cannot be recognised with the naked eye. These are the wavelengths you calculated for n = 7,8,9. The discovery of these lines a short time later showed that Balmer's formula was correct.
In 1888, the physicist Johannes Rydberg published a generalisation of Balmer's formula:
\(\frac{1}{\lambda}=R_\infty\cdot\left(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2}\right)\)
with \(R_\infty\)= 1.097 - 10⁷ \(\frac{1}{m}\)and n₁, n₂ = 1,2,3,4... and n₁ < n₂
for n₁= 2, the wavelengths that can also be calculated using the Balmer formula are obtained. However, if you set n₁= 1 or 3 and vary n₂, this results in wavelengths that are in the ultraviolet and infrared range of light and are therefore invisible to the human eye. In 1906, the physicist Theodore Lyman was able to prove the wavelengths in the ultraviolet range for n₁= 1 and in 1908, the physicist Friedrich Paschen was able to prove the wavelengths in the infrared range for n₁= 3. This confirmed the predictive power and accuracy of the Rydberg formula.
Question
In Unit 7.1, we established that the reason why atoms of an element always emit exactly one specific line spectrum must lie in the structure of the atoms themselves.
Now, with the help of Einstein's relationship between wavelength and energy and the formulae of Balmer and Rydberg, we have established that atoms can have different energy levels and that the transitions between these levels are the cause of the specific line spectra. However, the question remains unanswered:
- What do all atoms of an element have in common?
And from this follows the question:
- Which "component" of the atom absorbs and releases the energy so that line spectra are created?
Resolution of the questions
Our previous model of the structure of an atom includes the atomic nucleus, which is made up of protons and neutrons, is very small compared to the size of the whole atom and makes up the majority of the atomic mass, and the atomic shell, in which the electrons are located.
All atoms of an element have the same number of protons in the nucleus and the same number of electrons in the shell. This is what all atoms of an element have in common.
Which atomic particles are able to absorb energy when excited?
The protons are firmly bound in the atomic nucleus and can only be released from it by bombardment with other particles (see: Building blocks of the nucleus 1).
We do not yet know the distribution of electrons in the atomic shell, but they are not as tightly bound there as the protons in the nucleus and can therefore be easily excited by the supply of energy and "jump back and forth" between energy levels.
The integer n1 and n2 therefore represent the energy levels between which the electrons in the atomic shell transition. With the help of Balmer and Rydberg's formulae, we can visualise the structure of the electron shell. The electrons can be located at certain energy levels to which we can assign whole numbers.