The Bohr Model of the Atom

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On June 19, 1912, Niels Bohr wrote to his brother Harald:

"Perhaps I have found out a little about the structure of atoms."

I. Setting the Stage

The Bohr model of the atom deals specifically with the behavior of electrons in the atom. In constructing his model, Bohr was presented with several problems.

Problem #1: charged electrons moving in an orbit around the nucleus SHOULD radiate energy due to the acceleration of the electron in its orbit. The frequency of the emitted radiation should gradually change as the electron lost energy and spiraled into into the nucleus. Obviously this was not happening, because the spectral lines of a given element were sharply defined and unchanging.

Problem #2: the spectral lines did not show overtones (or harmonics). These are lines where the frequency is double, triple and so on of the fundamental frequency. The lines of the spectrum of each element were scattered about with no apparent pattern, other than the purely empirical formula of Balmer (which dates to 1885). However, no one knew what Balmer's formula meant.

J.J. Tomson's model was constructed with full knowledge of problem #1 above. What Thomson did is to extend the positive charge to the same size as the atom (radius = 10¯8 cm.) and allow the electrons to distribute themselves inside. The calculations for his "plum pudding" model, published in 1904, showed that the model did produce electron arrangements that were stable.

However, the Thomson model was conclusively destroyed by Rutherford's 1911 nucleus paper. (In the future -- 1913 and years later -- other discoveries will be made that the Thomson model fails to account for, but the Rutherford model does. Of course, Thomson, Rutherford, Bohr, etc. were not aware of these. There were even efforts in 1914 and 1915 to use the Thomson Model, but these efforts went nowhere.)

The nuclear model of Rutherford's was supported by evidence that could not be refuted. However, if electrons rotated around a nucleus, they would either rip the atom apart or self-destruct. Bohr's answer to the above problems appeared in print for the world to see in July 1913. However, as you can see from Bohr's letter to his brother, the journey to the answer started much earlier.

Bohr wrote out his ideas to date in a memo to Rutherford sometime in June/July 1912. This memo (with one missing page) still exists and if you were a qualified scholar, you could go visit it and read it!! Bohr leaves Manchester right after writing the memo and goes to Copenhagen and is married on August 1, 1912.

The critical part of Bohr's thinking was the making of two assumptions. Bohr himself described these assumptions on page 7 of his famous paper:

"(1) That the dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis.

(2) That the latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory."

Making the assumptions at all was a bit shaky, but in September 1913, at a meeting of the British Association for the Advancement of Science, Sir James Jeans remarked:

"The only justification which can be offered for the moment with regard to these hypotheses is the very important one that they work in practice."

II. Assumption #1: Electrons Move, yet are Stable

Bohr describes "stationary states" (He never uses the word the modern term "orbit.") on page 5:

"According to the above considerations, we are led to assume that these configurations will correspond to states of the system in which there is no radiation of energy states which consequently will be stationary as long as the system is not disturbed from outside."

As an electron moves in a "stationary state" it emits no radiation whatsoever. This violated a branch of science called electrodynamics (having to do with movement of charged particles and their amount of energy), but the fact is that the atom is stable and DOES NOT emit radiation in the manner predicted. It is this branch which predicts the electron will lose energy and crash into the nucleus (this is the problem #1 mentioned at the top of the file).

In this assumption is some of Bohr's daring nature. While he realized electrodynamics is useless (second part of his sentence), he proposed to use "mechanics" to describe the motion of an electron in its orbit (first part of the sentence). Mechanics deals with things like inertia, momemtum and other features of movement not involving electrical charges. He was willing to throw out well-supported scientific ideas that didn't work, but was also willing to keep other ideas that allowed him to make calculations.

The justification for Bohr deciding to assume mechanics held in the atom, but electrodynamics didn't? The results he got had two features: 1) they concurred with already known results and 2) offered an explanation for why some results were found and not others.

III. Assumption #2 - Incorporation of Planck's Constant

The "latter process" in assumption #2 is described at the end of assumption #1 -- "the passing of the systems between different stationary states"

What Bohr proposed is that the atom will emit (or gain) energy as it moves from one stationary state to another. However, the amounts of energy will not be any old amount, but only certain, fixed values. Those values will be the DIFFERENCES in energy between the stationary states.

Bohr says on p. 7:

"The second assumption is in obvious contrast to the ordinary ideas of electrodynamics but appears to be necessary in order to account for experimental facts."

The experimental facts refered to are the lines in the spectrum of hydrogen.

What Planck had discovered in 1900 was a fundamental limitation on nature. Energy is not emitted or absorbed in a continuous manner, but rather in small packets of energy called quanta. Emission and absorption occured in a DIScontinuous manner. In other words, an atom moved from one energy state to another state in steps. In the mathematical description of this process there occured a new constant of nature, discovered by Planck and named after him.

Planck's constant, symbolized by h, was involved in governing HOW MUCH energy a given quantum had. The amount of energy was directly dependent on the frequency of the radiation according to the following equation:

E = hν

This famous equation was first announced by Planck in 1900.

Bohr argued that Planck's constant should be used to help account for the stability of the atom. This reversed the technique of others who were trying to use atomic models to determine the physical significance of h. Bohr also realized that, if he was correct, his theory should produce a constant with the units of length. This constant would characterize the distance of the electron from the nucleus.

Bohr said on page 2:

The principal difference between the atom-models proposed by Thomson and Rutherford consists in the circumstance the forces acting on the electrons in the atom-model of Thomson allow of certain configurations and motions of the electrons for which the system is in a stable equilibrium; such configurations, however, apparently do not exist for the second atom-model. The nature of the difference in question will perhaps be most clearly seen by noticing that among the quantities characterizing the first atom a quantity appears -- the radius of the positive sphere -- of dimensions of a length and of the same order of magnitude as the linear extension of the atom, while such a length does not appear among the quantities characterizing the second atom, viz. the charges and masses of the electrons and the positive nucleus; nor can it be determined solely by help of the latter quantities.

. . . it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant, or as it often is called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed as this constant is of such dimensions and magnitude that it, together with the mass and charge of the particles, can determine a length of the order of magnitude required.

What Bohr wass pointing out is that Rutherford's model (with its constants of mass and charge) cannot produce a unit of length, but with the introduction of h, such a length constant could be produced. If you write h2 / me2, you get a value with the units of length and of the proper magnitude. Here are the numbers (modern values):

This length calculation yielded a value which today is named the Bohr radius. Its modern value is 5.292 x 10¯11 m and is symbolized ao.

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