The Bohr Theory

In 1913, Bohr proposed a model for the hydrogen atom that appeared to explain the line spectra discussed in Section 6.2. The motion of the electron around the nucleus was considered to be similar to the motion of a planet around the sun, the gravitational attraction that keeps the planet in a circular or an elliptical orbit being replaced by the coulom-bic attraction between the electron and the positively charged nucleus. To account for the line spectra, Bohr postulated that the angular momentum of the electron was restricted to multiple values of fl. This was an arbitrary postulate at the time it was made, but it comes naturally from the quantum mechanical description of a particle moving in a circle, as we have already seen in Section 5.1.3. 

The Bohr theory has been mentioned here because the radius of the first Bohr orbit, known simply as the Bohr radius, is still widely used in quantum mechanics. It is given the symbol and has a value given by the formula  

In this equation, is the permittivity of free space, is the mass of the electron and e is the electronic charge. 

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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). 

Hafinia, Latin name for Copenhagen) Many years before its discovery in 1932 (credited to D. Coster and G. von Hevesey), Hafnium was thought to be present in various minerals and concentrations. On the basis of the Bohr theory, the new element was expected to be associated with zirconium. 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

Unlike the total energy, the quantum mechanical value Pi of the orbital angular momentum is significantly different from that in the Bohr theory given in Equation (1.8). It is now given by... 

Moseley photographed characteristic spectra for some 38 elements that could serve as x-ray tube targets. In two papers,37 he not only uncovered structure in the K and L spectra—he alscr established the atomic number as more fundamental than the atomic weight, and he provided brilliant support for- the Bohr theory of atomic structure. 

Draw a picture of the electron jump corresponding to the first line in the visible emission spectrum of hydrogen according to the Bohr theory. 

The carbon atom has, outside its nucleus, six electrons which, on the Bohr theory of atomic structure, were believed to be arranged in orbits at increasing distance from the nucleus. These orbits corres-... 

The Bohr theory also permits a calculation of the total energy, E, of the atom from Hamilton s equation ... 

The Bohr theory of atomic structure allotted to each extra-nuclear electron within the atom a definite geometrical orbit and, more important, associated with each orbit a fixed total energy value. 

The dependence of the elastic pressure on the density can be expressed approximately by a power function p = Bpn, usually called polytropic. It could alternatively be considered that the force centers are repelled according to the relationship F = a/(3n-2) as assumed in the Bohr theory of crystal lattices. The thermal motion, at this degree of compression, consists of small oscillations. To each vibrational degree of freedom there corresponds an energy RT (per mole). The total oscillatory energy equals cvT, where cv is independent of the volume in this approximation... 

Figure 8. A Bohr-Sommerfeld model of the xenon atom. (From H. A. Kramers and H. Horst, The Atom and the Bohr Theory of its Structure, 1924).

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. 

The quantity a0 is interpreted in the Bohr theory as the radius of the smallest orbit in the hydrogen atom its value is 0.530 A. 

Using the Bohr theory, calculate the ratio of the energies of the K X-rays of I and Xe. 

The Bohr theory can be summarized as follows. An electron of charge —e and mass m in a circular orbit of radius r about an infinitely heavy positive charge of Ze obeys Newton s law for uniform circular motion5... 

A further refinement of the Bohr theory would require the exact and simultaneous measurement of electronic positions and velocities so that corrections to the theory could possibly be inferred from the deviation between theory and experiment. As we shall see, such exact measurements can not be made in principle in the universe as we know it. 

The energy depends only on n, and is given by the same equation as in the Bohr theory... 

The hydrogen-atom solutions are called atomic orbitals, by analogy with the orbits of the Bohr theory. Orbitals with / = 1 are called s orbitals, and those with l = 2, 3, and 4, arc called p, d, and / orbitals, respectively. (This notation comes from the names used to describe different series of spectroscopic lines.) The value of n is also specified, so that, for example, 2p is used to denote orbitals with n = 2 and l = 1 in this case, there are three possible m values, and 2p refers to the whole set of these. Table 4.1 lists the orbitals with n up to three, showing their names and the number of possible m values in each case. 

The sizes of atomic orbitals also increase with the energy indeed as in the Bohr theory (see eqn 4.13), the average radius of an orbital is given approximately by... 

The limitations of the Bohr theory arise because it does not reflect a fundamental facet of nature, namely the fact that particles possess wave properties. These limitations were transcended by the wave mechanics of Schrodinger,16 when he devised his famous equation in 1926 [12, 13]. Actually, the year before the Schrodinger... 

A note of caution The Bohr theory, even when improved and amplified, applies only to hydrogen and hydrogen-like species, such as He+ and Li+. The theory explains neither the spectra of atoms containing even as few as two electrons, nor the existence and stability of chemical compounds. The next advance in the understanding of atoms requires an understanding of the wave nature of matter. 

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. 

Take the Is electron as an example. In the Bohr theory, the electron moves in a fixed orbit of radius l o- On the other hand, in the wave mechanical treatment, the electron can in principle be found at any distance from the nucleus, and the most probable nucleus-electron separation is lao- Here we can see... 

The Bohr theory of the atom was further developed with great ingenuity to explain the complexities of atomic line spectra, but the significant advance came with the formulation of wave mechanics. 

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