Understanding Atomic Structure: Is There a More Direct and Compelling Connection between Atomic Line Spectra and the Quantization of an Atom’s Energy?

Understanding Atomic Structure: Is There a More Direct and Compelling Connection between Atomic Line Spectra and the Quantization of an Atom’s Energy?

Robert C. Rittenhouse *

Department of Chemistry, Central Washington University, Ellensburg, Washington 98926, United States

J. Chem. Educ., Article ASAP

DOI: 10.1021/ed5007234

Publication Date (Web): March 25, 2015

Copyright © 2015 The American Chemical Society and Division of Chemical Education, Inc.

5 PAGES

Supporting Information (8 pages): A critical thinking exercise created by the author to guide students in deriving and using the mathematical expression for the energy of the atom.

ABSTRACT: http://pubs.acs.org/doi/abs/10.1021/ed5007234

In this article, the author presents an alternative method to the traditional historical path to discovery approach  used in many textbooks to introduce atomic structure and quantization of energy.  This approach which involves a “rapid progression through a list of discoveries and theoretical developments” often is a struggle to students because of the concepts that need to be integrated and logically connected, “many of them abstract and inconsistent with everyday experience”. [Personally, it is also a struggle to teach because of time constraint and the uncertain degree of explicitness required to relate the entire development.]

To illustrate his method, the author provides a brief but detailed synopsis of the historical path to discovery and the concepts and mathematical derivations by Bohr.  According to the author, because the path to discovery was not straightforward and involved some “working backward” assumptions (angular momentum conservation) to arrive at quantization, most textbooks have resorted to the simplifying the tale and leaving out the logical progression.  “Unfortunately, this approach misses the opportunity to draw together logically a few key discoveries and concepts to build a rigorous and compelling case for one of the most important characteristics of atoms; that energy is quantized.”

In the section, A More Direct Approach, the author presents an alternative approach focusing first on helping students understand the existence of atomic spectra, hydrogen in particular being the simplest.

First, the following concepts need to be “clearly defined and properly connected” for the students (taken verbatim from article to preserve accuracy):

(1) based on Einstein’s explanation of the photoelectric effect, the packets or photons of light emitted or absorbed by the atoms in line spectra experiments have energies given by Ephoton = hν;

(2) as a consequence of the Law of Conservation of Energy, any emission or absorption of energy in the form of light must be accompanied by a change in the energy of the atom that exactly matches the photon energy of the light;

(3) due to electrostatic attraction between the negative electron and the positive nucleus, the potential energy of the atom has a negative value that increases (becomes less negative) as the distance between the electron and nucleus increases, approaching zero as the distance approaches infinity.

Having established these with the students, the author then outlines step-by-step how the expression for the energy of the atom and its restriction to certain values can be derived mathematically for students.  The author refers to this as a hindsight approach, involving steps that Bohr did not connect at that time that he was working on this because his focus was on looking at the energy and behavior of individual electrons.  This approach does not explain the “why” of energy quantization but as the author suggests, one can then segue way into answering this question by invoking the unsuccessful planetary theory, de Broglie’s insight, and eventually modern quantum theory. In the conclusion, the author summarizes this approach as follows:

This approach carefully ties together (1) the Balmer-Rydberg law for the atomic line spectrum of hydrogen, (2) the particle nature of light and the link between photon energy and frequency, (3) the law of conservation of energy, and (4) the sign of the potential energy of the hydrogen atom arising from electrostatic attraction between electron and nucleus to build a more understandable and compelling case for the restriction of atomic energy to specific allowed values and to derive an expression that generates those values.

The author has tested and revised worksheets and an excel exercise to guide students through this mathematical progression and to give them an opportunity to try out the equation and predict wavelengths in the hydrogen spectrum.  These are provided in the supporting information.