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The Bohr Model of the Atom
Modern Physics
The Bohr model pictures electrons in fixed orbits around the nucleus, each with a definite energy. It explains why hydrogen emits light only at certain wavelengths.
What the Bohr model says
In 1913 Niels Bohr proposed that electrons move in circular orbits around the nucleus, but only certain orbits are allowed. Each allowed orbit has a definite energy, and an electron in one of them does not radiate energy, even though classical physics said an orbiting charge should. The lowest energy orbit, n = 1, is the ground state. Higher orbits have higher, less negative, energies, and the radius grows as n squared, with the ground-state radius, the Bohr radius, equal to about 5.29 x 10^-11 m. Bohr's key rule was that the electron's angular momentum is quantised in multiples of h/2pi.
Energy levels and spectra
An electron can jump between levels by absorbing or emitting a photon whose energy exactly equals the difference between the two levels. A jump to a higher level absorbs a photon, and a jump to a lower level emits one. This is why hydrogen gives a line spectrum rather than a continuous rainbow: only certain photon energies are possible, so only certain wavelengths appear. The series of lines ending at n = 1 is the Lyman series in the ultraviolet, the series ending at n = 2 is the Balmer series, part of which is visible, and the series ending at n = 3 is the Paschen series in the infrared. Each element has its own spectrum, like a fingerprint.
Limits of the model
The Bohr model was a breakthrough, but it is only a stepping stone. It predicts hydrogen's spectrum accurately, yet it cannot explain the spectra of larger atoms, the intensities of spectral lines, or the fine splitting of lines seen in a magnetic field. The full quantum mechanical model replaces definite orbits with orbitals, regions where the electron is likely to be found, and it explains the whole periodic table. Even so, the Bohr picture remains a useful introduction, and the energy-level idea it introduced is still central to lasers, LEDs and atomic clocks, where transitions between levels produce light of a precise frequency.
Worked example: photon from a hydrogen transition
An electron in hydrogen falls from n = 2 to n = 1. Energy levels: E_n = -13.6 / n^2 eV. E1 = -13.6 / 1 = -13.6 eV. E2 = -13.6 / 4 = -3.4 eV. Photon energy: E = E2 - E1 = -3.4 - (-13.6) = 10.2 eV. Convert: 10.2 x 1.60 x 10^-19 = 1.63 x 10^-18 J. Wavelength: lambda = h c / E = (6.63 x 10^-34)(3.0 x 10^8)/(1.63 x 10^-18) = 1.22 x 10^-7 m = 122 nm. This is the Lyman-alpha line, in the ultraviolet.
Common mistakes
- Saying electrons orbit like planets in a definite path: the Bohr orbits are a useful model, but in full quantum mechanics electrons occupy probability clouds, not sharp paths.
- Forgetting the minus sign: bound energy levels are negative, with zero at infinity, so a transition down releases a positive photon energy.
- Using the 13.6 eV formula for other atoms: it applies to hydrogen; other atoms have different level structures.
Practice questions
- What is the energy of the n = 3 level in hydrogen?
Show answer
E3 = -13.6/9 = -1.51 eV.
- What photon energy is emitted in a transition from n = 3 to n = 2?
Show answer
E = E3 - E2 = -1.51 - (-3.4) = 1.89 eV.
- What is the ionisation energy of hydrogen from its ground state?
Show answer
13.6 eV, the energy needed to take the electron from -13.6 eV to zero.
Frequently asked questions
Does the Bohr model work for all atoms?
It works well for hydrogen, but it fails for atoms with more than one electron and for fine details, which need full quantum mechanics.
Why does the orbiting electron not spiral into the nucleus?
Because angular momentum is quantised, only certain orbits are allowed, and the electron does not radiate while it stays in one of them.