# The Bohr Model

#### Learning Objective

• Explain how the Bohr model of the atom marked an improvement over earlier models, but still had limitations from its use of Maxwell's theory

#### Key Points

• The model's success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen.
• The model states that electrons in atoms move in circular orbits around a central nucleus and can only orbit stably in certain fixed circular orbits at a discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels.
• In these stable orbits, an electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic theory.

#### Terms

• unstableFor an electron orbiting the nucleus, according to classical mechanics, it would mean an orbit of decreasing radius and approaching the nucleus in a spiral trajectory.
• correspondence principleStates that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum number.
• emissionAct of releasing or giving away, energy in the case of the electron.

In atomic physics, the Bohr model depicts an atom as a small, positively charged nucleus surrounded by electrons. These electrons travel in circular orbits around the nucleus—similar in structure to the solar system, except electrostatic forces rather than gravity provide attraction.

## Development of the Bohr Model

The Bohr model was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911). Since the Bohr model is a quantum-physics-based modification of the Rutherford model, many sources combine the two: the Rutherford–Bohr model.

Although it challenged the knowledge of classical physics, the model's success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen. While the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, it also provided a justification for its empirical results in terms of fundamental physical constants.

Although revolutionary at the time, the Bohr model is a relatively primitive model of the hydrogen atom compared to the valence shell atom. As an initial hypothesis, it was derived as a first-order approximation to describe the hydrogen atom. Due to its simplicity and correct results for selected systems, the Bohr model is still commonly taught to introduce students to quantum mechanics. A related model, proposed by Arthur Erich Haas in 1910, was rejected. The quantum theory from the period between Planck's discovery of the quantum (1900) and the advent of a full-blown quantum mechanics (1925) is often referred to as the old quantum theory.

Early planetary models of the atom suffered from a flaw: they had electrons spinning in orbit around a nucleus—a charged particle in an electric field. There was no accounting for the fact that the electron would spiral into the nucleus. In terms of electron emission, this would represent a continuum of frequencies being emitted since, as the electron moved closer to the nucleus, it would move faster and would emit a different frequency than those experimentally observed. These planetary models ultimately predicted all atoms to be unstable due to the orbital decay. The Bohr theory solved this problem and correctly explained the experimentally obtained Rydberg formula for emission lines.

## Properties of Electrons under the Bohr Model

In 1913, Bohr suggested that electrons could only have certain classical motions:

1. Electrons in atoms orbit the nucleus.
2. The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits") at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, an electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic theory.
3. Electrons can only gain or lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency (ν) determined by the energy difference of the levels according to the Planck relation.

Bohr's model is significant because the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule. Although Rule 3 is not completely well defined for small orbits, Bohr could determine the energy spacing between levels using Rule 3 and come to an exactly correct quantum rule—the angular momentum L is restricted to be an integer multiple of a fixed unit:

$L=n\frac { h }{ 2\pi } =n\hbar$

where n = 1, 2, 3, ... is called the principal quantum number and ħ = h/2π. The lowest value of n is 1; this gives a smallest possible orbital radius of 0.0529 nm, known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.

## The Correspondence Principle

Like Einstein's theory of the photoelectric effect, Bohr's formula assumes that during a quantum jump, a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels. Bohr did not believe in the existence of photons.

According to the Maxwell theory, the frequency (ν) of classical radiation is equal to the rotation frequency (νrot) of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels En and En−k when k is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit n. For sufficiently large values of n (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency so that the classical orbital frequency is not ambiguous. But for small n (or large k), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.

The Bohr-Kramers-Slater theory (BKS theory) is a failed attempt to extend the Bohr model, which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.