Materials Science and Engineering/List of Topics/Bohr Model of the Atom

In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus — similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity. This 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, referring to the Rutherford-Bohr model.

Introduced by Niels Bohr in 1913, the model's key 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, but it provided a justification for its empirical results in terms of fundamental physical constants.

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics, before moving on to the more accurate but more complex valence shell atom. A related model was originally proposed by Arthur Erich Haas in 1910, but was rejected.

History
In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, it was quite natural for Rutherford to consider a planetary model for the atom, the Rutherford model of 1911, with electrons orbiting a sun-like nucleus. However, the planetary model for the atom has a difficulty. The laws of classical mechanics, specifically the Larmor formula, predict that the electron will release electromagnetic radiation as it orbits a nucleus. Because the electron would be losing energy, it would gradually spiral inwards and collapse into the nucleus. This is a disaster, because it predicts that all matter is unstable.

Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies.

To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain motions:
 * 1) The electrons travel in orbits that have discrete Quantization (physics)|quantized momenta, and therefore quantized speeds and energies. That is, not every orbit is possible but only certain specific ones, at certain specific distances from the nucleus.
 * 2) The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another.

The significance of the Bohr model is that it states that the laws of classical mechanics do not apply to the motion of the electron about the nucleus. Bohr proposed rather that a new kind of mechanics, or quantum mechanics, describes the motion of the electrons around the nucleus. This model of electrons traveling in quantized orbits was extended into a matrix mechanics|more accurate model of electron motion about a dozen years later by Werner Heisenberg.Schrödinger equation|Another form of the same theory, modern quantum mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently and by different reasoning.

Other points are:


 * 1) When an electron makes a jump from one orbit to another, the energy difference is carried away (or supplied) by a single quantum of light (called a photon) which has an energy equal to the difference in energy between the two orbits.
 * 2) The frequency of the emitted photon is the classical orbit frequency, since photon emission correspondence principle|corresponds to classical emission of radiation. Since there are two orbits involved in emission, this is only exact when both orbits have nearly the same frequency, and this holds only when the orbits are large.

Since the frequency of a photon is proportional to its energy, rule 2 allowed Bohr to calculate the gap in energy between levels--- the level spacing is equal to Planck's constant divided by the classical orbit period. Stepping down orbit by orbit, he found that the angular momentum changed by $$h/2\pi$$ at every step.

So he proposed that the angular momentum L is quantized according to the rule
 * $$ L = n \cdot \hbar = n \cdot {h \over 2\pi} $$

where n = 1,2,3,… and is called the principal quantum number, and h is Planck's constant.

The lowest value of n is 1. This corresponds to a smallest possible radius of 0.0529 nm. This is known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton.

Bohr's condition, that the angular momentum is an integer multiple of $$\scriptstyle\hbar$$ was later reinterpreted by DeBroglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit.


 * $$n \lambda = 2 \pi r\,$$

Substituting DeBroglie's wavelength reproduces Bohr's rule. Bohr justified his rule by appealing to the correspondence principle, without providing a wave interpretation.

Source:

Wikipedia, "Bohr model"