Orbit, Bohr circular

In his 1913 papers Bohr developed a theory of the stationary states of the hydrogen atom. According to his theory the electron was to be considered as moving in a circular orbit about the proton. The amount of angular momentum for a stationary tate was assumed by Bohr to be equal to nh/2w, with n = 1, 2, 3, . J n Appendix II there is given the derivation of the energy values fcr the Bohr circular orbits. For... 

Such considerations led to the early solar-system picture of the atom, in which an electron presumably revolved about the nucleus in one or another definite orbit but was unable to take a position between the orbits when the electron shifted from an outer orbit to an inner orbit, nearer the positively charged nucleus, it would give off the energy detected in the spectra. (It was an open question why an electron would be allowed in one orbit or another and yet be prohibited from taking an intermediate position between the orbits.) Bohr used such a picture and, with classical physical laws, drew up equations that described the supposed motion of the electron in its circular orbit. Using simple assumptions, he successfully explained the structure of the hydrogen spectrum. 

Each element has its own line spectrum. This is why the line spectrum for an element is also considered to be a fingerprint for that particular element. Because the amounts of light given off by the excited atoms were in fixed amounts, Bohr termed them quantized amounts of light. These fixed amounts of energy proved that the electrons could only make certain jumps between the orbits that were at fixed distances in the atom. Because of these fixed, circular orbits, Bohr s model of the atom is often referred to as the solar system model of the atom (see Figure 3.4). 

Without reference to electron spin, this result may also be deduced for atoms from Bohr circular orbit theory + Heisenberg uncertainty relationship . For principal quantum... 

I should also mention Sommerfeld, who extended Bohr s theory to try and account for the extra quantum numbers observed experimentally. Sommerfeld allowed the electrons to have an elliptic orbit rather than a circular one. 

According to the Bohr model, the radius of a circular orbit is given by the equation... 

In an early model of the hydrogen atom proposed by Niels Bohr, the electron traveled in a circular orbit of radius uncertainty principle rules out this model. 

Bohr theory, the radius of the circular orbit of the electron in the ground state of the hydrogen atom (Z = 1) with a stationary nucleus. Except in Section 6.5, where this substitution is not appropriate, we replace fx by and by ao in the remainder of this book. 

The units we use in daily life, such as kilogram (or pound) and meter (or inch) are tailored to the human scale. In the world of quantum mechanics, however, these units would lead to inconvenient numbers. For example, the mass of the electron is 9.1095 X J0 31 kg and the radius of the first circular orbit of the hydrogen atom in Bohr s theory, the Bohr radius, is 5.2918 X 10 11 m. Atomic units, usually abbreviated as au, are introduced to eliminate the need to work with these awkward numbers, which result from the arbitrary units of our macroscopic world. The atomic unit of length is equal to the length of the Bohr radius, that is, 5.2918 X 10 n m, and is called the bohr. Thus 1 bohr = 5.2918 X 10"11 m. The atomic unit of mass is the rest mass of the electron, and the atomic unit of charge is the charge of an electron. Atomic units for these and some other quantities and their values in SI units are summarized in the accompanying table. 

The first plausible theory of the electronic structure of the atom was proposed in 1914 by Niels Bohr (1885-1962), a Danish physicist. In order to explain the hydrogen spectrum (Fig. 17-1), he suggested that in each hydrogen atom, the electron revolves about the nucleus in one of several possible circular orbits, each having a definite radius corresponding to a definite energy for the electron. An electron in the orbit closest to the nucleus should have the lowest energy. With the... 

Following Sommerfeld s proposal of elliptical electron orbits in 1915, Bohr amended his original theory, which had included only circular orbits. 14 A 1922 paper in Zeitschriftfur Physik outlined the "Aufbauprinzip" by which electrons are fed into atomic subshells. There was a neat correlation between periodic groups containing 2, 8, 8, 18,... 

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

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. 

In the above calculation the system has been treated as though the nucleus were stationary and the electron moved in a circular orbit about the nucleus. The correct application of Newton s laws of motion to the problem of two particles with inverse-square force of attraction leads to the result that both particles move about their center of mass. The center of mass is the point on the line between the centers of the two particles such that the two radii are inversely proportional to the masses of the two particles. The equations for the Bohr orbits with consideration of motion of the nucleus are the same as those given above, except that the mass of the electron, m, is to be replaced by the reduced mass of the two particles, /, defined by the expression 1/m = 1/m + 1/M, where M is the mass of the nucleus. 

In the Bohr model of the hydrogen atom, an electron travels in a circular orbit about the nucleus at approximately 5 x 10 mlles Per hour How many rev°-lutions per second does the electron make if the radius of the orbit is 2 x 10"9 inches ... 

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... 

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... 

In the historical Bohr (Niels Bohr, 1885-1962) model the atom is pictured as having electrons in circular orbits surrounding a small, positively charged nucleus, to which the electrons are attracted by the electrostatic Coulomb force. These orbits have... 

An obvious possible improvement of the Bohr model was to bring it better into line with Kepler s model of the solar sxstem, which placed the planets in elliptical, rather than circular, orbits. Sommerfeld managed to solve this problem by the introduction of two extra quantum numbers in addition to the principal quantum number (n) of the Bohr model, and the formulation of general quantization rules for periodic systems, which contained the Bohr conjecture as a special case. 

In 1913 Bohr amalgamated classical and quantum mechanics in explaining the observation of not only the Balmer series but also the Lyman, Paschen, Brackett, Pfund, etc., series in the hydrogen atom emission spectrum, illustrated in Figure 1.1. Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that the angular momentum pe for an angle of rotation 9 is given by... 

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