10.3 Bohr’s theory of the hydrogen atom  (Page 3/7)

The electron orbital energy is the sum of its kinetic and potential energy. This results in the following formula

for the orbital energies of hydrogen-like atoms    . The principal quantum number ${\displaystyle n}$ can take values ${\displaystyle n=1,2,3,...}$ Here, $E_0$ is the ground-state energy $\left(n=1\right)$ for hydrogen $\left(Z=1\right)$ and is given by

Thus, for hydrogen,

where

[link] shows an energy-level diagram for hydrogen that also illustrates how the various spectral series for hydrogen are related to transitions between energy levels.

Electron total energies are negative, since the electron is bound to the nucleus, analogous to being in a hole without enough kinetic energy to escape. As $n$ approaches infinity, the total energy becomes zero. This corresponds to a free electron with no kinetic energy, since $r_n$ gets very large for large $n$ , and the electric potential energy thus becomes zero. Thus, 13.6 eV is needed to ionize hydrogen (to go from –13.6 eV to 0, or unbound), an experimentally verified number. Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy.

Triumphs and limits of the bohr theory

Bohr did what no one had been able to do before. Not only did he explain the spectrum of hydrogen, he correctly calculated the size of the atom from basic physics. Some of his ideas are broadly applicable. Electron orbital energies are quantized in all atoms and molecules. The electrons do not spiral into the nucleus, as expected classically. These are major triumphs.

But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

Section 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.
• 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
  $\mathrm{\Delta }E=\text{hf}=E_{\text{i}}-E_{\text{f}},$
  where $\mathrm{\Delta }E$ 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 be quantized. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
  $r_n=\frac{n^2}{Z}{a}_{\text{B}}\text{(allowed orbits}n=\mathrm{1,\; 2,\; 3,\; ...),}$
  $Z$ is the atomic number of an element (the number of protons) and $a_{\text{B}}$ is defined to be the Bohr radius, which is
  $$a_{\mathrm{B}}=0.529\times {10}^{-10}\mathrm{m}$$
• Furthermore, the energies of hydrogen-like atoms are given by
  $E_n=-\frac{Z^2}{n^2}{E}_0\left(n=\mathrm{1,\; 2,\; 3\; ...}\right)\text{,}$
  where _{$E_0$} is the ground-state energy and is given by
  $$E_{\mathrm{0}}=13.6\mathrm{e}\mathrm{V}.$$
  Thus, for hydrogen,
  $E_n=-\frac{\text{13.6 eV}}{n^2}\left(n,=,\mathrm{1,\; 2,\; 3\; ...}\right)\text{.}$
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual questions