6.4 Bohr’s model of the hydrogen atom  (Page 8/37)

Bohr’s model of the hydrogen atom also correctly predicts the spectra of some hydrogen-like ions. Hydrogen-like ions are atoms of elements with an atomic number Z larger than one ( $Z=1$ for hydrogen) but with all electrons removed except one. For example, an electrically neutral helium atom has an atomic number $Z=2.$ This means it has two electrons orbiting the nucleus with a charge of $q=+Ze.$ When one of the orbiting electrons is removed from the helium atom (we say, when the helium atom is singly ionized), what remains is a hydrogen-like atomic structure where the remaining electron orbits the nucleus with a charge of $q=+Ze.$ This type of situation is described by the Bohr model. Assuming that the charge of the nucleus is not $+e$ but $+Ze,$ we can repeat all steps, beginning with [link] , to obtain the results for a hydrogen-like ion:

where $a_0$ is the Bohr orbit of hydrogen, and

where $E_0$ is the ionization limit of a hydrogen atom. These equations are good approximations as long as the atomic number Z is not too large.

The Bohr model is important because it was the first model to postulate the quantization of electron orbits in atoms. Thus, it represents an early quantum theory that gave a start to developing modern quantum theory. It introduced the concept of a quantum number to describe atomic states. The limitation of the early quantum theory is that it cannot describe atoms in which the number of electrons orbiting the nucleus is larger than one. The Bohr model of hydrogen is a semi-classical model because it combines the classical concept of electron orbits with the new concept of quantization. The remarkable success of this model prompted many physicists to seek an explanation for why such a model should work at all, and to seek an understanding of the physics behind the postulates of early quantum theory. This search brought about the onset of an entirely new concept of “matter waves.”

Summary

• Positions of absorption and emission lines in the spectrum of atomic hydrogen are given by the experimental Rydberg formula. Classical physics cannot explain the spectrum of atomic hydrogen.
• The Bohr model of hydrogen was the first model of atomic structure to correctly explain the radiation spectra of atomic hydrogen. It was preceded by the Rutherford nuclear model of the atom. In Rutherford’s model, an atom consists of a positively charged point-like nucleus that contains almost the entire mass of the atom and of negative electrons that are located far away from the nucleus.
• Bohr’s model of the hydrogen atom is based on three postulates: (1) an electron moves around the nucleus in a circular orbit, (2) an electron’s angular momentum in the orbit is quantized, and (3) the change in an electron’s energy as it makes a quantum jump from one orbit to another is always accompanied by the emission or absorption of a photon. Bohr’s model is semi-classical because it combines the classical concept of electron orbit (postulate 1) with the new concept of quantization (postulates 2 and 3).
• Bohr’s model of the hydrogen atom explains the emission and absorption spectra of atomic hydrogen and hydrogen-like ions with low atomic numbers. It was the first model to introduce the concept of a quantum number to describe atomic states and to postulate quantization of electron orbits in the atom. Bohr’s model is an important step in the development of quantum mechanics, which deals with many-electron atoms.