Bohrs Theory of the Hydrogen Atom

The nuclear theory of the atom assumed that the negatively charged electron was in orbit about a more massive nucleus. However, Maxwell s theory of electromagnetism requires that when charged matter changes its direction, it must emit radiation as it accelerates. But electrons in atoms don t emit radiation as they orbit the nucleus, as far as scientists could tell. 

Bohr reasoned that perhaps energy was not the only quantity that could be quantized. If a particle were traveling in a circular orbit about a nucleus, suppose its angular momentum were quantized  

Bohr made certain assumptions, statements that were not to be justified but assumed as true, and from these statements he derived certain mathematical expressions about the electron in the hydrogen atom. His assumptions were  

In the hydrogen atom, the electron moves in a circular orbit about the nucleus. Mechanically, the centripetal force that curves the path of the electron is balanced by the coulombic force of attraction between the oppositely charged particles (the negatively charged electron and the positively charged proton in the nucleus). 

Only certain orbits are allowed, each orbit having a quantized value of its angular momentum. 

Einstein s work paved the way for the solution of yet another nineteenth-century mystery in physics the emission spectra of atoms. 

The colored Images are called spectral lines, p) The line emissbn spectrum of hydrogen atoms. 

In 1913, not too long after Planck s and Einstein s discoveries, a theoretical explanation of the mission spectrum of the hydrogen atom was presented by the Danish physicist Niels Bohr. Bohr s treatment is very complex and is no longer considered to be correct in all its details. Thus, we will concentrate only on his important assumptions and final results, which do account for the spectral lines. 

When a high voltage is applied between the fori , some of the sodium ions in the pickle are converted to sodium atoms in an excited state. These atoms emit the characteristic yellow light as they relax to the ground state. 

When Bohr first tackled this problem, physicists already knew that the atom contains electrons and protons. They thought of an atom as an entity in which electrons whirled around the nucleus in circular orbits at high velocities. This was an appealing model because it resembled the motions of the planets around the sun. In the hydrogen atom, it was believed that the electrostatic attraction between the positive solar proton and the negative planetary electron pulls the electron inward and that this force is balanced exactly by the outward acceleration due to the circular motion of the electron. 

Each canponent color is focused at a definite position, according to its wavelength, and forms a colored image of the slit on the photographic plate. The colored irtuxges are called spectral lines, (b) The line emission spectrum of hydrogen atoms. 

Every element has a unique emission spectrum. The characteristic lines in atomic spectra can be used in chemical analysis to identify unknown atoms, mnch as fingerprints are used to identify people. When the lines of the emission spectrum of a known element exactly match the lines of the emission spectrum of an imknown sample, die identity of the sample is established. Although the utility of this procediue was recognized some time ago in chemical analysis, the origin of these lines was nnknown rmtil early in the twentieth century. 

Color emitted by hydrogen atoms in a discharge tube. The color observed results from the combination of the colors emitted in the visible spectrum. 

In addition to explaining the photoelectric effect. Planck s quantum theory and Einstein s ideas made it possible for scientists to unravel another nineteenth-century mystery in physics atomic line spectra. 

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To make an informed guess for your first value of ot, you may wish to reread the section on the Bohr theory of the hydrogen atom and the Schroedinger wave functions for the hydrogen atom in a good physical or general chemistry book (see Bibliography). 

This section started with the discovery of Soddy and Fajans on radioactive decay around 1910 and the relationship of radioactive decay to the periodic table. At this point in the history, we understand the periodic table and we understand the role of isotopes in the periodic table. We have not yet understood the structure of the modern Table, i.e. first row two elements, second row eight elements, etc. That understanding can be based on Bohr theory of the hydrogen atom originally developed in 1911 and is summarized in Bohr s famous article in Zeitschrift fur Physik (Bohr 1922). 

It was the apparent violation of this requirement that led to the postulate of quantization in the Bohr theory of the hydrogen atom. However, we are concerned here with the classical result in which the charge does radiate. Our objective is to describe the emitted radiation some distance r from the emitter. 

A more detailed account of the Bohr theory of the hydrogen atom is given in Chap. 2 and Apps. II and III. 

While h is quite small in the macroscopic world, it is not at all insignificant when the particle under consideration is of subatomic scale. Let us use an actual example to illustrate this point. Suppose the Ax of an electron is 10-14 m, or 0.01 pm. Then, with eq. (1.2.1), we get Apx = 5.27 x 10-21 kg m s-1. This uncertainty in momentum would be quite small in the macroscopic world. However, for subatomic particles such as an electron, with mass of 9.11 x 10-31 kg, such an uncertainty would not be negligible at all. Hence, on the basis of the Uncertainty Principle, we can no longer say that an electron is precisely located at this point with an exactly known velocity. It should be stressed that the uncertainties we are discussing here have nothing to do with the imperfection of the measuring instruments. Rather, they are inherent indeterminacies. If we recall the Bohr theory of the hydrogen atom, we find that both the radius of the orbit and the velocity of the electron can be precisely calculated. Hence the Bohr results violate the Uncertainty Principle. 

This is an instance of a fundamental result in quantum mechanics, that any measured component of orbital angular momentum is restricted to integral multiples of h. The Bohr theory of the hydrogen atom, to be discussed in the next chapter, can be derived from this asssumption alone. 

Rydberg constant /rid-berg/ A constant that occurs in formula for the frequencies of spectral lines in atomic spectra. For the hydrogen atom it has the value 1.0968 x 10 m"i. The value of the Rydberg constant can be calculated from the BOHR THEORY of the hydrogen atom and from quantum mechanics. These calculations showed that ... 

The simplest model to employ for the estimation of the energy needed to free the electron uses the Bohr theory of the hydrogen atom. To recall this model, a single electron is attracted to a... 

Quantum mechanics is basically statistical in nature. Knowing the state, we cannot predict the result of a position measurement with certainty we can only predict the probabilities of various possible results. The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture. 

The main purpose of this book is the discussion of bonding in several important classes of molecules. Before starting this discussion, we shall review briefly the pertinent details of atomic structure. Since in our opinion the modern theories of atomic structure began with the ideas of Niels Bohr, we start with the Bohr theory of the hydrogen atom. 

How is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle (In fact, it was this inconsistency, along with the theory s limited application to non-hydrogen-like systems, that limited Bohr s theory.)... 

This is the Rydberg constant, R, from the hydrogen atom spectrum. Quantum mechanics therefore predicts the experimentally determined hydrogen atom spectrum. At this point, quantum mechanics predicts everything that Bohr s theory did and more, and so supersedes the Bohr theory of the hydrogen atom. 

The values in Eq. (17.2-29) differ from the assumed values h, 2h, 3h,..in the Bohr theory of the hydrogen atom. The Bohr theory also did not provide for states of zero angular momentum or for different values of corresponding to a given value of L. ... 

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