Angular Bohr atom model

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... 

Following the wave-mechanical reformulation of the quantum atomic model it became evident that the observed angular momentum of an s-state was not the result of orbital rotation of charge. As a result, the Bohr model was finally rejected within twenty years of publication and replaced by a whole succession of more refined atomic models. Closer examination will show however, that even the most refined contemporary model is still beset by conceptual problems. It could therefore be argued that some other hidden assumption, rather than Bohr s quantization rule, is responsible for the failure of the entire family of quantum-mechanical atomic models. Not only should the Bohr model be re-examined for some fatal flaw, but also for the valid assumptions that led on to the successful features of the quantum approach. 

The sound part of Bohr s atomic model, and its successors, appears to be the assumed quantization of electronic angular momentum and energy, as well as atomic size. Had Bohr gone one step further the proposed quantiza-... 

The main objection against the Bohr and Sommerfeld atomic models was the ad hoc definition of stationary states. Simply declaring these as quantum states offers no explanation for the failure of an accelerated charge to radiate energy. The quantization of neither energy nor angular momentum implies such an effect. 

Bohr s atomic model> Energy for classical atomic model E = —9 a (m mass, eielectric charge, 1 angular momentum)... 

Equations 1.8 and 1.13-1.15 are in stark opposition to the classical picture of matter now the energy, speed, radius, and angular momentum of the orbit can take on only certain values. These values together define a quantum state of the system. Each quantum state differs from any other quantum state by the value of at least one parameter such as these. Rather than label each quantum state by the precise values of each parameter, we can instead identify the quantum state by its quantum numbers. For example, in the Bohr model we can find the values of r , and once we know the principal quantum number, n. The orbital radius r would be a continuous variable in a classical system, but in the Bohr atom 7 has become quantized to values r = tt ao/Z, equal for example to Uo = 0.5292 A for the = 1 state of the H atom and 4ao = 2.117A for the n = 2 state. The quantization of properties such as size and energy, which are continuous in classical systems, is what gives quantum mechanics its name. 

This justifies Bohr s assumption of quantized angular momentum. Therefore, each quantum state of the Bohr model corresponds to a different number of wavelengths of the electron at the orbital radius. Bohr s model is extraordinarily tempting in its simplicity and its power to predict the atomic spectra. The hidden strength of the model is that it accurately portrays the relationship between the electron energy and its characteristic v and r values. It is all the more important, therefore, that we be aware of its shortcomings. 

In Bohr s model of the hydrogen atom, only one number, n, was necessary to describe the location of the electron. In quantum mechanics, three quantum numbers are required to describe the distribution of electron density- in an atom. These numbers are derived from the mathematical solution of Schrbdinger s equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. Each atomic orbital in an atom is characterized by a unique set of these three quantum numbers. 

Following Rutherford s experiments in 1911, Niels Bohr proposed in 1913 a dynamic model of the hydrogen atom that was based on certain assumptions. The first of these assumptions was that there were certain "allowed" orbits in which the electron could move without radiating electromagnetic energy. Further, these were orbits in which the angular momentum of the electron (which for a rotating object is expressed as mvr) is a multiple of h/2ir (which is also written as fi),... 

The first application of quantum theory to a problem in chemistry was to account for the emission spectrum of hydrogen and at the same time explain the stability of the nuclear atom, which seemed to require accelerated electrons in orbital motion. This planetary model is rendered unstable by continuous radiation of energy. The Bohr postulate that electronic angular momentum should be quantized in order to stabilize unique orbits solved both problems in principle. The Bohr condition requires that... 

The chemical properties of an element are functions of the number and distribution of its electrons around the nucleus. In 1913, Niels Bohr devised a model for the atom that successfully explained why atomic spectra consist of discrete lines, not only in X-rays, as discussed above, but also in visible light. Building on the ideas of quantum mechanics, he postulated that the angular momentum of an orbiting electron can only have certain fixed values. If so, then the orbital energy associated with any electron cannot vary continuously, but can only have discrete quantum values. He described a series of spherical shells at fixed... 

Apart from the assumed quantization of orbital angular momentum the Bohr model predicted the quantization of electronic energy, radius, velocity and magnetic moment of atoms ... 

Re-examination of the first quantitative model of the atom, proposed by Bohr, reveals that this theory was abandoned before it had received the attention it deserved. It provided a natural explanation of the Balmer formula that firmly established number as a fundamental parameter in science, rationalized the interaction between radiation and matter, defined the unit of electronic magnetism and produced the fine-structure constant. These are not accidental achievements and in reworking the model it is shown, after all, to be compatible with the theory of angular momentum, on the basis of which it was first rejected with unbecoming haste. 

It was the analysis of the line spectrum of hydrogen observed by J. J. Balmer and others that led Neils Bohr to a treatment of the hydrogen atom that is now referred to as the Bohr model. In that model, there are supposedly allowed orbits in which the electron can move around the nucleus without radiating electromagnetic energy. The orbits are those for which the angular momentum, mvr, can have only certain values (they are referred to as quantized). This condition can be represented by the relationship... 

The Bohr model of one-electron atoms Bohr postulated quantization of the angular momentum, L = m vr = nh/lir, substituted the result in the classical equations of motion, and correctly accounted for the spectrum of all one-electron atoms. E = —Z jrd (rydbergs). The model could not, however, account for the spectra of many-electron atoms. 

Orbital hybridization, like the Bohr model of the hydrogen atom in its ground state, is an effort to dress up a defective classical model by the assumption ad hoc of quantum features. The effort fails in both cases because the quantum-mechanics of angular momentum is applied incorrectly. The Bohr model assumes a unit of quantized angular momentum for the electron which is presumed to orbit the nucleus in a classical sense. Quantum-mechanically however, it has no orbital angular momentum. The hybridization model, in turn, spurns the commutation rules of quantized... 

---