Summarizing Bohr’s Theory of the Hydrogen Atom

Bohr’s Theory of the Hydrogen Atom Summary

• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula

  \(\cfrac{1}{\lambda }=R\left(\cfrac{1}{{n}_{\text{f}}^{2}}-\cfrac{1}{{n}_{\text{i}}^{2}}\right),\)

  where \(\lambda \) is the wavelength of the emitted EM radiation and \(R\) is the Rydberg constant, which has the value

  \(R=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-1}\text{.}\)

• The constants \({n}_{i}\) and \({n}_{f}\) are positive integers, and \({n}_{i}\) must be greater than \({n}_{f}\).
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by

  \(\Delta E=\text{hf}={E}_{\text{i}}-{E}_{\text{f}},\)

  where \(\Delta E\) is the change in energy between the initial and final orbits and \(\text{hf}\) is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by

  \(L={m}_{e}{\text{vr}}_{n}=n\cfrac{h}{2\pi }(n=1, 2, 3 \dots ),\)

  where \(L\) is the angular momentum, \({r}_{n}\) is the radius of the \(n\text{th}\) orbit, and \(h\) is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by

  \({r}_{n}=\cfrac{{n}^{2}}{Z}{a}_{\text{B}}\text{(allowed orbits}\phantom{\rule{0.25em}{0ex}}n=1, 2, 3, ...),\)

  \(Z\) is the atomic number of an element (the number of electrons is has when neutral) and \({a}_{\text{B}}\) is defined to be the Bohr radius, which is

  \({a}_{\text{B}}=\cfrac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}\)

• Furthermore, the energies of hydrogen-like atoms are given by

  \({E}_{n}=-\cfrac{{Z}^{2}}{{n}^{2}}{E}_{0}(n=1, 2, 3 ...)\text{,}\)

  where _{\({E}_{0}\)} is the ground-state energy and is given by

  \({E}_{0}=\cfrac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}=\text{13.6 eV.}\)

  Thus, for hydrogen,

  \({E}_{n}=-\cfrac{\text{13.6 eV}}{{n}^{2}}(n,=,1, 2, 3 ...)\text{.}\)

• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Glossary

hydrogen spectrum wavelengths

the wavelengths of visible light from hydrogen; can be calculated by \(\cfrac{1}{\lambda }=R\left(\cfrac{1}{{n}_{\text{f}}^{2}}-\cfrac{1}{{n}_{\text{i}}^{2}}\right)\)

Rydberg constant

a physical constant related to the atomic spectra with an established value of \(1.097×{\text{10}}^{\text{7}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{\text{−1}}\)

double-slit interference

an experiment in which waves or particles from a single source impinge upon two slits so that the resulting interference pattern may be observed

energy-level diagram

a diagram used to analyze the energy level of electrons in the orbits of an atom

Bohr radius

the mean radius of the orbit of an electron around the nucleus of a hydrogen atom in its ground state

hydrogen-like atom

any atom with only a single electron

energies of hydrogen-like atoms

Bohr formula for energies of electron states in hydrogen-like atoms: \({E}_{n}=-\cfrac{{Z}^{2}}{{n}^{2}}{E}_{0}(n=\text{1, 2, 3,}\phantom{\rule{0.25em}{0ex}}\dots )\)