The Bohr Atom

The relation between the speed v of the electron in. a circular orbit about the nucleus and the radius r of the orbit can be derived by use of Newton s laws of motion. A geometrical construction shows that the acceleration of the electron toward the center of the orbit is v2/r, and hence the force required to produce this acceleration is mv2/r. This force is the force of attraction Ze2/r2 of the electron and the nucleus hence we write the equation 

It may be noted that this equation corresponds to the virial theorem (Sec. 1-4). The term on the left is twice the kinetic energy, and that on the right is the potential energy with changed sign. 

The angular momentum for the electron in its orbit is mrv. Bohr s postulate for quantizing the circular orbits is represented by the equation 

In this equation n is the quantum number for the hydrogen atom, assumed to have the values 1 (for the normal state of the atom), 2 (for the first excited state), 3, 4, 5, and so on. 

These two equations are easily solved. It is found that the radius of the circular Bohr orbit for quantum number n is equal to W/4xaZnt. This can be written as n ao/Zy in which a0 has the value 0.530 A. The speed of the electron in its orbit is found to be v = 2irZe2/nh. For the normal hydrogen atom, with Z = 1 and n = 1, this speed is 2.18 X 108 cm/sec, about 0.7 percent that of the speed of light. 

As the structure of atoms was being deduced, experiments suggested that the nucleus of the atom was positively charged, whereas electrons around the outside have a negative charge. You probably learned this model of the atom early in your time as a student, perhaps even in grade school, and so you are likely to accept it 

This picture of the atom allows us to begin answering some of the questions about light bulbs that we posed in the insight section. Recall that we said that all lamps are based on the principle that an excited substance can emit light as it loses 

The concept of an exdted state is general in the ener levels of molecules, though we will only encounter its application to electronic ener levels. 

Line spectrum of hydrogen. Each line corresponds to the wavelength of the energy emitted when the electron of a hydrogen atom, which has absorbed energy, falls back to a lower principal energy level. 

The Bohr model of the hydrogen atom described the electron revolving in certain allowed circular orbits around the nucleus. 

When an excited electron returns to the ground state, energy is emitted as a photon is released. The color (wavelength) of the light is determined by the difference in energy between the two states (excited and ground). 

The intensity of the dots shows that the electron spends more time closer to the nucleus. 

14Svante Arrhenius, bom near Uppsala, Sweden, 1859. Ph.D. University of Stockholm. Nobel Prize in chemistry 1903. Professor Stockholm. Died Stockholm 1927. 

A way out of this dilemma was suggested by Bohr15 in 1913 [9, 10]. He retained the classical picture of electrons orbiting the nucleus in accord with Newton s laws, but subject to the constraint that the angular momentum of an electron must be an integral multiple of h/2n  

Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from (1) the Plank equation Eq. 4.3, AE = hv and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. 

15Niels Bohr, bom Copenhagen, 1885. Ph.D. University of Copenhagen. Professor, University of Copenhagen. Nobel Prize in physics 1922. Founder of the Copenhagen school interpretation of quantum theory. Died Copenhagen, 1962. 

Equation 4.13 expresses the total (kinetic plus potential) energy of the electron of a hydrogenlike atom in terms of four fundamental quantities of our universe electron charge, electron mass, the permittivity of empty space, and Planck s constant. From Eq. 4.13 the energy change involved in emission or absorption of light by a hydrogenlike atom is simply 

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Whereas zirconium was discovered in 1789 and titanium in 1790, it was not until 1923 that hafnium was positively identified. The Bohr atomic theory was the basis for postulating that element 72 should be tetravalent rather than a trivalent member of the rare-earth series. Moseley s technique of identification was used by means of the x-ray spectra of several 2ircon concentrates and lines at the positions and with the relative intensities postulated by Bohr were found (1). Hafnium was named after Hafma, the Latin name for Copenhagen where the discovery was made. 

To understand the origins of dispersion forces, let us consider two Bohr atoms, each of which consists of an electron orbiting around a nucleus comprised of a proton, having a radius ao, often referred to as the first Bohr radius . It is obvious that a Bohr atom has no permanent dipole moment. However, the Bohr atom can be considered to have an instantaneous dipole moment given by... 

Whether the Bohr atomic model or the quantum mechanical model is introduced to students, it is inevitable that they have to learn, among other things, that (i) the atomic nucleus is surrounded by electrons and (ii) most of an atom is empty space. Students understanding of the visual representation of the above two statements was explored by Harrison and Treagust (1996). In the study, 48 Grade 8-10... 

The Bohr atom went a long way toward explaining the nature of atoms, but there were problems. Although scientists could calculate the emission spectrum of hydrogen using the Bohr model, the model could not account for the spectra of heavier atoms. The biggest problem with the Bohr atom, however, lay in its lack of a... 

The shells play a leading role in the structure of the Periodic Table. This graphic representation is borrowed from the Bohr atomic model. Historically, the shells were assigned letters, nowadays... 

M Astatine is isolated in tiny amounts from reactor materials. The Bohr atomic model shows the tightly packed electron shell. One can formally see" the instability. It was the last of the 92 naturally occurring elements to be found. 

Niels Bohr, "On the Constitution of Atoms and Molecules, Pt.I, Binding of Electrons by Positive Nuclei," PhiLMag. 26 (1913) 125 Pt. II, "Systems Containing Only a Single Nucleus," Phil.Mag. 26 (1913) 476502 "Pt. Ill, Systems Containing Several Nuclei," Phil.Mag. 26 (1913) 857875. See John Heilbron and Thomas Kuhn, "The Genesis of the Bohr Atom," HSPS 1 (1969) 211290. 

Ibid., 122, 130, 138 Bohr, "On the Constitution of Atoms and Molecules." And John Heilbron and Thomas Kuhn, "The Genesis of the Bohr Atom," 245250. 

Bohr s hydrogen atom model of 1913 had provided inspiration to a few physicists, like Kossel, who were interested in chemical problems but to very few chemists concerned with the explanation of valence. First of all, the Bohr atom had a dynamic character that was not consistent with the static and stable characteristics of ordinary molecules. Second, Bohr s approach, as amended by Kossel, could not even account for the fundamental tetrahedral structure of organic molecules because it was based on a planar atomic model. Nor could it account for "homopolar" or covalent bonds, because the radii of the Bohr orbits were calculated on the basis of a Coulombic force model. Although Bohr discussed H2, HC1, H20, and CH4, physicists and physical chemists mainly took up the problem of H2, which seemed most amenable to further treatment. 11... 

Explain how the Bohr atomic model differs from the Rutherford atomic model, and explain the observations and inferences that led Bohr to propose his model. 

Electrons Changing Energy Level in the Bohr Atom... 

In the Bohr atom, as it is commonly now depicted, electrons -which have a mass just 0.00055 times that of the proton, but an equal and opposite electric charge - orbit around a nucleus of protons and neutrons, packed together with an awesome density. If matter were uniformly as dense as the nucleus rather than containing so much empty space, a thimbleful would weigh about a billion tonnes. ... 

Which is all very well, but the Bohr atom is wrong. The picture of a dense nucleus surrounded by electrons is accurate enough, but they do not follow nice elliptical orbits like those of the planets. Venus and Mars follow Newton s laws, but electrons are governed by the... 

Born, M. and Lande, A. (1918). [The absolute calculation of crystal properties with the aid of the Bohr atom model]. Sitsungsber. Preuss. Akad. Wissen. Berlin 45, 1048-68 (in German). 

Fio. 2-3.—At the left is represented the circular orbit of the Bohr atom. At the right is shown the very eccentric orbit (line orbit), with no angular momentum, that corresponds somewhat more closely to the description of the hydrogen atom in its normal state given by quantum mechanics. 

The first attempts to interpret Werner s views on an electronic basis were made in 1923 by Nevil Vincent Sidgwick (1873—1952) and Thomas Martin Lowry (1874—1936).103 Sidgwick s initial concern was to explain Werner s coordination number in terms of the sizes of the sub-groups of electrons in the Bohr atom.104 He soon developed the attempt to systematize coordination numbers into his concept of the effective atomic number (EAN).105 He considered ligands to be Lewis bases which donated electrons (usually one pair per ligand) to the metal ion, which thus behaves as a Lewis acid. Ions tend to add electrons by this process until the EAN (the sum of the electrons on the metal ion plus the electrons donated by the ligand) of the next noble gas is achieved. Today the EAN rule is of little theoretical importance. Although a number of elements obey it, there are many important stable exceptions. Nevertheless, it is extremely useful as a predictive rule in one area of coordination chemistry, that of metal carbonyls and nitrosyls. 

The angular momentum or an electron moving in an orbit of the type described by Bohr is ail axial vector L = r x p, formed from the radial distance r between electron and nucleus and the linear momentum p of the electron relative lo a fixed nucleus. Figure 2 shows the customary method used to illustrate the axial vector L in terms of the orbital morion of any object, of which the electron of the Bohr atom is only one example. Although Bohr s planetary model needed only circular orbits lo explain the spectral lines observed in the spectrum of a hydrogen atom, subsequent... 

While the Bohr atom is of no help in understanding the splittings of the Balmer lines, using it we can calculate the field at which a state is ionized by an electric field. Consider a H atom with its nucleus at the origin in the presence of an electric field in the z direction. The potential experienced by an electron moving along the z axis is given by... 

When Eq. 2.19 was applied to the Bohr atom a most amazing answer was obtained. Combination of Eqs. 2.7 and 2.17 gave, for the electron velocity... 

The hydrogen atom with the symbol H is the smallest atom. It contains one proton and one electron in the K-shell. The model of the atom in Fig. 3.1 which meets all 6 requirements for building an atom is called the Bohr atomic model after Niels Henrik David Bohr, the... 

In order to explain the colour of glazes we first have to return to the atomic model, which is much more sophisticated than the Bohr atomic model mentioned in chapter 3, Chemistry. In that chapter we saw that all electrons move around a nucleus in regular orbits, the K shell, L shell, M shell etc. This model gave the impression that the distance between the electron and the nucleus is always the same. However, in reality the atomic model is much more complicated than that which means for instance that ... 

Students will demonstrate an understanding of the five basic atomic theories—the Dalton atom, the Thomson atom, the Rutherford atom, the Bohr atom, and the Schrodinger electron cloud model—and illustrate this understanding in a two-dimensional work of art. 

Figure 2.2 The Bohr atom was proposed by Niels Bohr. He believed that electrons moved around the nucleus similar to the way planets orbit the Sun.

QM grew out of studies of blackbody radiation and of the photoelectric effect. Besides QM, radioactivity and relativity contributed to the transition from classical to modem physics. The classical Rutherford nuclear atom, the Bohr atom, and the Schrodinger wave-mechanical atom are discussed. Hybridization, wavefunctions, Slater determinants and other basic concepts are explained. For obtaining eigenvectors and eigenvalues from the secular equations the elegant and simple matrix diagonalization method is explained and used. All the necessary mathematics is explained. 

The calculation yields v = 2.19 x 10s ms The value of v is correct for hydrogenlike atoms (one electron), because for these the Bohr atom is a correct model, at least mathematically if not conceptually. It should be approximately right for atoms with more than one electron, because we are considering n = 1, an s electron, and the effect of outer-shell electrons on the first shell is not large. This velocity is 2.19 x 108/3.00 x 10s = 0.73 of the speed of fight. 

The expressions for the Bohr atom technically should use the reduced mass // = electron nucleus/ ( electron T- nucleus) instead of the electronmass, as noted in Equation 5.22. This alters the calculated value of the Bohr radius do, and therefore also alters the radius R and the total energy E = —Ze2/2R. 

A simple demonstration of the correspondence principle is provided by the Bohr atomic model that allows for the transfer of an electron between neigh-... 

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