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Principles of Physics

Bohr Atom

by Richard M. Renneboog, M.Sc.

Fields of Study

Quantum Physics; Atomic Physics

Summary

Building on the observations of Thomson, Rutherford, Young, Einstein and others, Bohr’s model of atomic structure was a revolutionary break from classical mechanics. It yielded excellent agreement with experiment for single electron atoms, but could not be extended in any straightforward way to many-electron atoms. Bohr assumed that the angular momentum of the orbiting electron is quantized in units of h/2π where h is Planck’s constant. Bohr’s theory was an important way station on the way to the quantum mechanics of Schrödinger and Heisenberg.

Principal Terms

  • angular momentum: the rotational momentum of an object around an axis. For a particle in circular orbit it equals mass times velocity times radius of the orbit.

  • conservation of energy: a fundamental law of physics that states that the amount of energy in the universe remains constant over time. Although the energy can be transformed or transferred, it cannot be created or destroyed.

  • conservation of momentum: in physics, the principle that the total momentum in a isolated system is always constant.

  • electron: a negatively charged subatomic particle that is often bound to the positive charge of the nucleus but can also exist in a free state in an atom.

  • photoelectric effect: a phenomenon that describes the emission of electrons from matter (typically metal) upon exposure to electromagnetic radiation.

  • photon: a massless elementary particle that is the smallest possible unit, or quantum, of light and other electromagnetic radiation.

  • quantum mechanics: the branch of physics that deals with matter interactions on a subatomic scale, based on the concepts that energy is quantized, not continuous, and that elementary particles exhibit wavelike behavior.

  • wave function: a function that describes the quantum state of a system and represents the probability of finding the system in a given state at a given time.

Atomic Theory

Before the twentieth century began, it was theorized that matter consisted of atoms that contained positive and negative components. There was, however, no good workable hypothesis of the arrangement of those components within the atoms. Physicists had also observed the black lines that appeared in the transmission and absorption spectra of compounds, but the cause of the lines was unknown. In 1897, Thomson identified the electron as a subatomic particle that could be made to exist apart from its oppositely charged, and much more massive, counterpart that accounted for essentially all of the mass of an atom.

Ernest Rutherford’s gold foil experiment in 1911 demonstrated that atoms had at their core a very small, dense nucleus surrounded by what amounted to a great deal of empty space containing the few electrons. This meshed well with Thomson’s observations and formed the basis of a theory of atomic structure in which the central nucleus is surrounded by a cloud of electrons. But this alone could not provide an explanation of the relationships observed in the black lines of atomic spectra.

In 1905, Einstein published a paper describing the photoelectric effect, which had been found to be dependent on the wavelength, and therefore the frequency, of the incident light and independent of the intensity of the incident light. The hypothesis that the electrons in the atoms could accept only a photon of the correct energy in order to be emitted was the basis of quantum theory. In 1913 Bohr used the quantum theory to explain the pattern of black lines in spectra of hydrogen as corresponding to the specific energy of the photons absorbed or emitted by electrons in the atoms as they transitioned between higher and lower energy levels. The mathematics of the relationships developed into the science of quantum mechanics.

The Bohr Hydrogen Atom

The significant aspect of Bohr’s atomic theory was the manner in which it departed from the thinking of the time, which was based on classical mechanics. The assumption of the time was that the structure of the atom and the behavior of the electrons it contained must be in accord with classical, or Newtonian, physics., In order to obey the law of conservation of energy and the law of conservation of momentum in that paradigm, an orbiting body, such as an electron orbiting a nucleus, must continually radiate energy in order to maintain stability. However, because atoms clearly did not radiate energy continuously, the idea of orbiting electrons was not well received by classical physicists.

Drawing on Young’s “double slit experiment” that had demonstrated the ability of electrons to have wave characteristics, Bohr proposed the radical idea that the electrons “in orbit” about a nucleus do so either as waves or in a manner that is described by wave behavior. On the basis of the physics of objects following circular orbits, Bohr hypothesized that the angular momentum of the electrons is quantized in units of the reduced Planck’s constant, ħ = h/2π. Although he was not able to reconcile this hypothesis with classical mechanics, he was able to use his hypothesis to derive a formula that predicted the spectral lines of the hydrogen atom.

The One-Electron Atom, the Rydberg Constant and the Balmer Series

In 1885, Balmer showed that the lines observed in the visible spectrum of hydrogen is described by the relationship

1/λ = R(1/22 – 1/n22)

where n2 is an integer value greater than 2, and R is the Rydberg constant having the value 1.097x107m-1. By substituting a second integer value in place of 2, the Balmer series relationship could be extended to describe other series of lines that could be identified in the spectrum of hydrogen. These are the Lyman, Paschen, Brackett and Pfund series. Their relationships are predicted by the more general form of the Balmer series equation, which is

1/λ = R(1/n12 – 1/n22).

It was then realized that the above formula would be obeyed by the emission spectrum of neutral hydrogen, singly ionized helium, doubly ionized lithium and so on, which are all single electron levels.

The quantum nature of the transitions involved is strongly suggested by the fact that n1 and n2 are integers. The precision with which wavelengths could be determined from spectra produces a very accurate value of the Rydberg constant. Calculating the energy of a particular level as

E = -(Z2/n2)(e2/4Πε0)2(m/2ħ2).

Bohr found that the difference in energy between two specific energy levels can be described by the equation

E2 – E1 = -(mZ2/2 ħ2) (e2/4Πε0)2 (1/n12 – 1/n22),

which is essentially the same equation as that describing the Balmer series and yields an expression for the value of the Rydberg constant as

R =(mZ2/2 ħ3c) (e2/4Πε0)2.

Here for simplicity we have assumed an infinitely heavy nucleus and that the atom has z-1 units of electrical charge.

The success of this prediction, in the excellent agreement found between the observed and calculated values of the Rydberg constant, were very strong arguments in favor of Bohr’s postulate that the electron in an atom has quantized angular momentum.

Sample Problem

Calculate the “mass” of the electron given the following values:

c = 2.99 X 108 m.sec-1

h = 6.625 X 10-34 J.sec

e = 1.60 X 10-19 C

R = 1.097 x 107m-1

Answer:

First ensure that all values are in the appropriate units. The value of c must be converted to m.sec-1, as 2.99 X 108 m.sec-1. Rearrange the equation

R =(mZ2/2 ħ3c) (e2/4Πε0)2

to obtain

m = R (2 ħ3c/Z2)(e2/4Πε0)-2

Substitute the given values, and carry out the calculation to obtain

9.1 x 10-31kg

The Legacy of the Bohr Atom

In practical terms, the Bohr atom has value in the present day as learning tool for understanding the basic principles of quantum mechanics. That, however, is the foundation of the standard model of the physical universe and our understanding of nuclear physics. Harvesting solar energy efficiently depends on the development of materials that exhibit the photoelectric effect as efficiently as possible, maximizing the absorption each photons that cause the release of electrons. Spectral analysis enables the identification of elements and compounds both on Earth and in distant stars. At the quantum level, atomic structure controls the movement of electrons in the transistors of computer chips, which is expected to be maximized by the development of computers that function on the quantum level using light transmission rather than just the current nanometer scale that depends on the physical movement of electrons. Understanding the Bohr atom and its role in the development of the theoretical basis of these and future technologies is the Bohr atom’s essential legacy.

Citation Types

Type
Format
MLA 9th
Renneboog, Richard M. "Bohr Atom." Principles of Physics, edited by Donald R. Franceschetti, Salem Press, 2016. Salem Online, online.salempress.com/articleDetails.do?articleName=POP_0020.
APA 7th
Renneboog, R. M. (2016). Bohr Atom. In D. R. Franceschetti (Ed.), Principles of Physics. Salem Press. online.salempress.com.
CMOS 17th
Renneboog, Richard M. "Bohr Atom." Edited by Donald R. Franceschetti. Principles of Physics. Hackensack: Salem Press, 2016. Accessed September 18, 2025. online.salempress.com.