Limitations of the Bohr Model

Electrons exist only in certain discrete energy levels, which are described by quantum numbers. 

Energy is involved in the transition of an electron from one level to another. 

We will now start to develop the successor to the Bohr model, which requires that we take a closer look at the behavior of matter. 

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Following the triumph of the Bohr theory, let us consider some limitations of the Bohr model of the atom. First, there are only flat circular orbitals and you have probably seen orbitals in organic chemistry textbooks that have different 3D shapes due to further research since 1913. However, because the AE (eV) = E2 — E values are correct, at least for (n —> n + 1) transitions, the Bohr model was a breakthrough in understanding the energy levels of atoms. Note also that these flat orbitals do not give insight as to how atoms combine into molecules. 

One example that is often used in chemistry classrooms may illustrate this. In the core of learning about the nature of science is learning about scientific models. Among other characteristics it is important to understand that models in science are developed by scientists, these models are never fully true or false, and can be changed or replaced in the light of new evidence. Different historical models of atomic structure are a good example to reflect about the nature of models in chemistry education. Models of Democritus, Dalton, Thomson, Rutherford and Bohr can be compared in the chemistry classroom, e.g. in a drama play (see Chapter 7). Students can start reflecting about the predictive potential and limitations of the different models. But students can also learn about the time in which the models were developed and about the scientists behind them. Other examples are different models of oxidation and reduction or acid-base chemistry. 

The quantum mechanical theory of the hydrogen atom is given below (see Chapter 7.5) The Bohr model of the hydrogen atom is a transition from purely classical presentations to quantum mechanical ones the motion of electrons along the orbits is accepted however not all orbits are permitted, the angular momentum is accepted, though its values and orientations are subject to strict limitation. One can consider the Bohr model as the transition from classical mechanics to quantum mechanics with the preservation of many its attributes. As a result, many of the ideas of the Bohr model will often be met in order to simplify the students understanding. 

The essential point in binary stopping theory is the avoidance of an expansion of T p) in powers of Zi this is achieved by mapping the Bohr model on a binary-collision problem involving a screened interaction potential, following a suggestion by Lindhard [21] but with an additional term that generates exact equivalence in the limit of distant collisions. The theory has been implemented in the PASS code [33] which allows incorporation of several features that were either unknown or of no interest at Bohr s time. 

Fig. 1. Relevant regions of interest to characterize the interaction of ions with solids, in terms of the ion velocity v and atomic number Zj. The line v/vq = 22 is indicative of the intermediate region between weakly and strongly ionized atoms the line v/vq = 2Zi shows the limits of applicability of the Bohr and Bethe models. The transition across this line is given by the Bloch model. The lower region is the domain of low and intermediate velocities where non-linear quantum effects are dominant.

The Bohr model was an immensely important contribution to the understanding of atomic structure. The idea that electrons exist in specific energy states and that transitions between states involve quanta of energy provided the linkage between atomic structure and atomic spectra. However, some limitations of this model quickly became apparent. Although it explained the hydrogen spectrum, it provided only a crude approximation of the spectra for more complex atoms. Subsequent development of more sophisticated experimental techniques demonstrated that there are problems with the Bohr theory even in the case of hydrogen. 

An atomic orbital is usually described in terms of three integral quantum numbers. We have already encountered the principal quantum number, n, in the Bohr model of the hydrogen atom. The principal quantum number is a positive integer with values lying between the limits 1 < n < oo allowed values arise when the radial part of the wavefunction is solved. 

The limits of Bohr s model Bohr s model explained hydrogen s observed spectral lines. However, the model failed to explain the spectrum of any other element. Moreover, Bohr s model did not fully account for the chemical behavior of atoms. In fact, although Bohr s idea of quantized energy levels laid the groundwork for atomic models to come, later experiments demonstrated that the Bohr model was fundamentally incorrect. The movements of electrons in atoms are not completely understood even now however, substantial evidence indicates that electrons do not move around the nucleus in circular orbits. 

The Paschen series of lines in the line spectrum of hydrogen occur in the near-IR. (a) Calculate the wavelength (in nm) of the series limit for the Paschen series of lines in the line spectrum of hydrogen, (b) The frequency of one line in the Paschen series of hydrogen is 2.34 x 10 Hz. Using the Bohr model of the atom with its circular orbits, sketch this specific electronic transition. 

In fact the concept of electronic configuration as a causally explanatory feature has become very much the domain of chemistry or to be more precise it is the dominant paradigm in modem chemistry. Conversely, physicists are only too aware of the limitations of the electronic configuration model and they only draw upon it as a zero order approximation. Hettema and Kuipers further state that Bohr s theory of the atom, despite its level of approximation, is to be regarded as a physical theory because the explanation of the periodic table was only a spin-off from its development. But given Heilbron and Kuhn s detailed version of the historical development, it was precisely the explanation of the periodic table which provided the initial impetus for Bohr s famous theory of the atom, whereas the explanation of the hydrogen spectmm only arose later. (Heilbron and Kuhn, 1969). 

I like to emphasize that Fig. 1 is not meant to indicate any fundamental limitation of quantum mechanics both Bohr s and Bethe s formulae invoke mathematical approximations to the underlying physical models, and Bethe s formula in particular relies on first-order perturbation theory for both distant and close collisions. 

In Bohr s model of the hydrogen atom, the circular orbits were determined by the quantum number more accurately, by the square of the quantum number n. No other orbits were allowed. By changing the orbits from circles to ellipses, Sommerfeld introduced a second radius, which gave him another variable to play with. So it was that Sommerfeld generalized Bohr s quantum condition for electron orbits in terms of the two quantum numbers n and k. His analysis led him to establish a relationship between the two quantum numbers namely, the quantum number n set the upper limit on the quantum number k, but k could have smaller values as follows ... 

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