A Quantum Mechanical Model of the Atom

The history of physics from ancient times to the modern day, focusing on light and matter. Danish physicist Niels Bohr developed his theory of the atom in 1913. Bohr's theory had electrons orbit in different ‘shells' like planets in different orbits around the Sun. The further an electron is from the centre, the easier it is to remove. Spectral lines are cause by electrons moving in and out of different shells.

Last updated on 5th August 2017 by Dr Helen Klus

1. Bohr's model of the atom

Danish physicist Niels Bohr developed his theory of the atom in 1913, two years after the first Solvay Conference on Physics[1]. Bohr's theory of the atom solved the problems with New Zealand physicist Ernest Rutherford's atomic theory. It also explained, among other things, fluorescence, the photoelectric effect, and spectral lines, as well as the periodic nature of elements.

1.1 Principle quantum number n

Bohr realised that electrons can only be found in specific places, and suggested that electrons orbit the nucleus of atoms in circles, with quantised orbital angular momentums, at quantised orbital distances. These are known as shells.

Each shell has an 'n' value. This is known as the principle quantum number where shell n=1 is the closest shell to the nucleus of the atom, n=2 is the second closest, and so on.

Bohr's model of the atom, showing a small positive nucleus, electrons orbit in levels, or ‘shells'. Levels have lower energies the closer they are to the nucleus, when an electron moves into a lower energy shell, it emits the excess energy in the form of a photon.

Bohr's model of the atom. The red dot is the nucleus, known at the time to contain protons, and the green dots represent two possible locations for an electron. E = hν where E = energy, h = Planck's constant, and ν = frequency. Image credit: Brighterorange/CC-SA.

Each shell is associated with a quantised energy and can only contain a finite number of electrons. It takes less energy for an electron to remain in the closest shell to the nucleus, and more energy to remain further away.

No matter how little energy electrons have, they cannot move closer to the nucleus than the lowest possible shell. This is why negatively charged electrons are not pulled into the positively charged nucleus.

Bohr showed that electrons do not move gradually from one shell to the next. Instead, they seem to disappear from one shell and instantly appear in another.

It takes energy for an electron to move outwards, and it releases energy when it moves inwards. Energy is taken and given in the form of photons, and so it is quantised, which means that only certain energies are allowed.

This can result in atomic emission, fluorescence, and charge-exchange. It also explains the nature of the photoelectric effect, spectral lines, and the periodic nature of the elements.

1.2 Energy levels and spectral lines

Absorption spectra occur when atoms absorb photons, and their electrons gain enough energy to move into an orbit further from the nucleus. Following the Planck relation, this energy corresponds to a specific frequency of light, and therefore a specific wavelength is missing from the spectrum.

Emission spectra occur when electrons drop to a lower energy level and emit photons, again this light has an energy determined by the Planck relation, and so the photons always have a specific wavelength.

Diagram showing how emission spectra are produced by electrons when they move into lower-energy atomic shells.

Emission lines from the Balmer series, where electrons move to the second closest shell to the nucleus and emit energy in the form of photons. Image credit: Helen Klus/CC-NC-SA.

The energy differences between the shells of hydrogen-like atoms are given by the Rydberg formula[2]:

= RZ2 (

where λvac is the wavelength of the light emitted in a vacuum, R is the Rydberg constant, equal to about 1.09×107 m-1, Z is the atomic number, equal to the number of protons in the nucleus, and n1 and n2 are the shell numbers, where n1 corresponds to the closest shell to the nucleus.

Diagram showing the different energy levels in a hydrogen atom, and the lines produced when electrons move between them.

Depiction showing the different ways that electrons can move within a hydrogen atom, each possibility produces a different spectral line. Image credit: Szdori/OrangeDog/CC-A.

The Stark effect occurs because an electric field pulls negatively charged electrons in the opposite direction to the positively charged nucleus. This can lower the energy of electrons in otherwise identical states, causing spectral lines to split.

1.3 Electronic structure and the periodic table

Over the next few years, Rutherford showed that the nucleus of a hydrogen atom is composed of a single positively charged particle, the proton, and that the nuclei of other atoms also contain protons[3][4].

Atoms have the same number of protons as they do electrons. Positive ions have fewer electrons than protons, and negative ions have more.

American physicist Robert Millikan had determined the electric charge of the electron in 1909, which allowed him to determine the atomic weight of each element using the mass to charge ratio[5].

In 1919, American chemist Irving Langmuir showed that atoms can share electrons, and this is the basis for all molecular chemistry[6].

Bohr and British chemist Charles Bury were eventually able to show why elements possess properties that occur periodically when they are ordered by atomic weight[7][8]. This is related to how electrons arrange themselves within atoms.

Bohr and Bury showed that as the elements go up in atomic weight, they also gain another electron. Generally, the first shell of an atom must be filled first, the second shell is then filled, and so on.

The number of electrons allowed in each shell follows two simple rules:

  • Firstly, the furthest shell to be filled can never contain more than eight electrons.

  • Secondly, the maximum number of electrons in every other shell can be calculated using the formula Maximum number = 2n2.

Here n refers to the shell number, the closest being n=1. This shows that the first shell can contain no more than 2 electrons, the second 8, the third 18, the fourth 32, and so on.

Diagram showing the arrangement of electrons in the first 36 elements on the periodic table.

The arrangement of electrons in the first 36 elements (click here to view all on one page). Image credit: modified by Helen Klus, original images by Greg Robson/CC-SA.

Modern period tables are split into groups and periods. The period refers to the shell number furthest from the nucleus to contain an electron. The group refers to the number of electrons in this shell. The exception is helium (He); helium only has two electrons but is in group VIII because it has a complete outer shell.

The elements in between groups II and III mostly have two electrons in their outer shell, with the exception of Lawrencium (La), which has three, and ten elements, including copper (Cu), gold (Au), silver (Ag), and platinum (Pt), that only have one.

This is because an atom cannot have 9 electrons in its outer shell, and so once its third shell, which is capable of containing 18 electrons, gains 8, the 9th must reside in the fourth shell, while the third is filled. This pattern is repeated while the fourth shell gains 32 electrons, and so on.

Depiction of a modern periodic table.

A modern periodic table. Image credit: Armtuk/CC-SA.

The periodic behaviour of elements is due to the fact that atoms are least reactive when they have a full outer shell. They are most reactive when they are closest to gaining a full shell, i.e. when they have 1 or 7 electrons in their outer shell.

Atoms with the same number of electrons in their outer shell are generally more reactive the larger their period.

This explains the behaviour of ions described by British natural philosopher Michael Faraday in 1834[9]; sodium chloride (NaCl) readily forms into negative chlorine ions (Cl-) and positive sodium ions (Na+) because sodium has a single electron in its outer shell, and so it is more stable without it. Chlorine, on the other hand, has 7 electrons in its outer shell; it would be more stable with an extra electron.

The sodium atom loses an electron and becomes a positive ion, and the chlorine atom gains an electron becoming a negative ion.

2. References

  1. Bohr, N., 1913, 'On the constitution of atoms and molecules', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26, pp.1-25.

  2. Rydberg, J. R., 1890, 'On the structure of the line-spectra of the chemical elements', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 29, pp.331-337.

  3. Rutherford, E., 1919, 'Collision of alpha Particles with Light Atoms IV. An Anomalous Effect in Nitrogen', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37, pp.581-587.

  4. Soddy, F., 1920, 'Name for the Positive Nucleus', Nature, 106, pp.502-503.

  5. Millikan, R. A., 1909, 'A new modification of the cloud method of measuring the elementary electrical charge, and the most probable value of that charge', Physical Review, 29, pp.560-561.

  6. Langmuir, I., 1919, 'The Arrangement of Electrons in Atoms and Molecules', Journal of the American Chemical Society, 41, pp.868-934.

  7. Bury, C. R., 1921, 'Langmuir's theory of the arrangement of electrons in atoms and molecules', Journal of the American Chemical Society, 43, pp.1602-1609.

  8. Bohr, N., 1923, 'On the application of quantum theory to the atomic structure', Zeitschrift für Physik, 13, pp.117-165.

  9. Faraday, M., 1834, 'Experimental Researches in Electricity — Seventh series', Philosophical Transactions of the Royal Society of London, 124, pp.77-122.

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5. History of Quantum Spin

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7. History of Nuclear Physics

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