Know the limit and teach within it! Part 3: Evolution and revolution

Know the limit and teach within it!

Part 3: Evolution and revolution

In part 1 of this series,1 I described how different representations of concepts and theories can be confusing for dualist (“right or wrong”) and concrete operational students. I also talked about the nature and limits of scientific models, and warned that we should never say, “So we throw this model out and get a new one!”  In part 2,2 I began illustrating this through the evolution of atomic theory, from Dalton’s “billiard ball” through to Rutherford’s nuclear model.

We saw how each model is a refinement of the previous one, both incorporating and extending it.  Further, while our visualisation of the atom changed dramatically, none of the primary concepts and related calculations were actually “thrown out” — they are all still there. This is summarized in Fig. 1:                                              

 Positively-charged highly dense nucleus at centre of atomic volume.

Up to this point, scientists knew that the atoms of each element had a unique characteristic mass. It was also suspected that the atomic number of each element (the numeric rank of each element in order of increasing mass) might be connected to its mass, but this was not confirmed until the work of Henry Moseley in 1915.3

Rutherford continued to probe the subatomic structure of the atom, particularly the nucleus. By firing hydrogen nuclei into other elements, he was able to show by 1920 that the nuclei of all elements contained hydrogen nuclei; this was named the proton and shown to have both mass and charge.4 The neutron would have to wait until Henry Chadwick’s 1932 discovery, however.5 

Right, wrong, or good enough?

This still left the question of electron behaviour within the atom. An obvious solution was to conceptualise electrons as orbiting the nucleus, much like planets orbit the sun — an idea that gained much attention in the popular science fiction of the day. The concept of fixed orbits implies that electrons can’t be forced closer to the nucleus (since this would require drastically changing their kinetic energy), explaining the apparent hard sphere nature of atoms.

But this creates an additional problem: negatively charged electrons orbiting a positively charged nucleus should, over time, spiral into the nucleus. Since non-radioactive elements had demonstrably stable atoms, this clearly could not be the case. Rutherford’s model provides no explanation for this, even though it (and the theory it is derived from) does properly explain other observed properties and behaviours. 

Suggested Activity 

Discuss with your students the following questions:

  • If a model fails in one point, does that make the model wrong? Why or why not?
  • If a model fails in one point, does that make the underlying theory wrong?
  • What alternative conclusions might you draw about the model and the theory?

Here it is important to emphasize two things — especially for students still operating with a dualist mentality.6 Firstly, if other predictions made by the model hold, then it is good enough for those purposes and can be used within the applicable limits. Secondly, one impossible prediction does not mean that the underlying theory is completely wrong: it could be that some key additional concept has been overlooked and awaits discovery; alternatively, the core idea may be correct but may have been visualised in an inappropriate or unhelpful manner.

From a teaching perspective, this means that we must make a clear distinction between theoretical concepts and the way we conceptualize them: that is, the models and analogies we use to represent the ideas we are trying to communicate. It also means that we need to be careful that our presentation — and especially our assessments — encourage students to move beyond a dualistic approach.  As David Finster noted6: “Dualism in science is but a convenience offered by situations upon which scientists agree; only then do we see right and wrong. Yet it is from this simplified vantage point that we see much of chemistry presented … Another insight [comes from] the ACS General Chemistry Exam… As a reflection of what chemical educators value, it presents chemistry as a collection of facts and equations along Dualistic perspectives.”  (Emphasis added)

In fact, we must recognise here the provisional nature of our knowledge — whether in the historical development or the current understanding of our subject. In other words, we need to challenge our students to shift their perspective towards a more multiplist — and even a relativist — view of science. And one way to do this is to deliberately engage with, rather than skip over, the historical development of the concepts and theories we teach.

This is where Niels Bohr and his 1913 model of the hydrogen atom come in. While Bohr’s model was certainly thinking “outside the box”, the way for his work was prepared first by Max Planck, and then by Albert Einstein. What’s interesting about this particular chain of developments is not just how the various theoretical insights came about, but the reaction of the scientists involved.*

Enlarging the foundation: Niels Bohr and the quantum atom

Most students are familiar with Planck’s constant (h); few remember where it came from. And fewer still realise just how uncomfortable Planck was with the idea of energy quanta. It was the inevitable outcome of, in his words, “an act of desperation… A theoretical interpretation [of the black body radiation formula] had to be found at any cost…”.7

But where Planck saw an uncomfortable idea (energy was discrete, not continuous), Einstein saw an opportunity: if the energy of light was discontinuous, maybe this could explain phenomena involving a discontinuity in energy.  Einstein published his light-quantum hypothesis in 1905, applying it (amongst other observations) to the photoelectric effect.8 Einstein’s predictions were subsequently proven by experiment, resulting in his being awarded the 1921 Nobel Prize in Physics.9 Einstein would subsequently have his own reservations about quantum theory and wave mechanics, but he was hardly alone.

Enter Niels Bohr who, following his PhD, had spent time in the labs of Thomson and then Rutherford. Bohr had been struggling to develop a model to describe the inner electronic structure of Rutherford’s 1911 atom, 
based on a classical model involving electrons in elliptical orbits around the nucleus (much like planetary orbits). His breakthrough came when he was pointed to the visible light line emission spectrum of the hydrogen atom, first observed in 1885 by Johann Jakob Balmer.10 

a hydrogen atomic emission lines imaged from the sun’s chromosphere during a eclipse – there is an eclipse sun with a tube with all the rainbow of colour starting from purple to red

Fig. 2.  Hydrogen atomic emission lines imaged from the sun's chromosphere during an eclipse using a telephoto lens and diffraction grating.11  Image: Yujing Qin

By proposing that the electron in a hydrogen atom could have only specific quantized values of energy, Bohr’s model accounted for both the stability and line emission spectra of hydrogen atoms: since the electron could only undergo very specific, discrete changes in energy, only photons of similarly discrete energies could be emitted or absorbed. By associating each possible ‘allowed’ energy with an integer (our modern principal quantum number), Bohr’s model correctly described not only Balmer’s visible line series, but Freiderich Paschen’s infrared series as well.12

– two diagrams – one of the Rutherford atom with three small red circles as the centre and three blue circles orbiting around this centre. The second diagram is the Bohr hydrogen atom with a small red circle inside and a four larger circular lines circling around the centre – the circle closest to the centre has a small blue circle.

Fig. 3. Comparison of original Rutherford and  Bohr atomic models (not to scale)

Bohr’s model was thus descriptive and predictive1 — his predictions of both ultraviolet and additional infrared line series were later shown to be correct — and he consequently received the 1922 Nobel Prize in Physics.13 But his original model was not without problems. Firstly, the electronic transition energies for elliptical orbits differed systematically from the corresponding light energies, a problem that was solved by adopting circular electron orbits (for which the energies are the same).

More importantly, while the quantum numbers were implicit in the Balmer equation and necessary to provide stable electron orbits, it was not clear where they actually came from. They were, like Planck’s quantum of energy, necessary to make the model work, but their physical significance was unclear. The model was thus unsatisfactory in terms of its explanatory power. Indeed, it took Erwin Schrödinger’s 1926 development of wave mechanics before quantum numbers were seen to be intrinsically related to electron energies as a direct result of the conditions required for solutions to the wave equation.

Lessons learned from atomic theory

When discussing the development of any theoretical concept, it is always tempting to focus on what gets left out at each stage. I believe this to be a poor teaching strategy, however, for various reasons. Firstly, it creates a false view of how science actually progresses: “acts of desperation” and the subsequent struggle to understand why that particular approach actually works are far more common than students realise! Secondly, it reinforces a false dualist “right or wrong” perspective in our students, which can hinder their scientific development. Thirdly, it leaves students questioning why they should bother with these “wrong” ideas in the first place, with unsurprising results.

photos of Dalton, Thomson, Rutherford, Bohr and Schodinger -- each with the models of the atom that they developedMore importantly, the highly formal nature of quantum theory makes it much harder to relate to concrete realities, meaning it is also harder to understand and retain. By emphasizing how each development still contains key ideas from the preceding model, it is easier to connect the abstract to the everyday, building and connecting concepts rather than merely displacing them. Doing so will also help us discern good from bad analogies, along with their limits, leading to improved understanding. Finally, it will better prepare those students who will continue deeper into the subject once they leave our classrooms.


  1. D.C. Stone, “Know the limit and teach within it Part 1: Analogies and models”, Chem 13 News, 2017, December 2017/January 2018, pages 22-25.
  2. D.C. Stone, “Know the limit and teach within it Part 2: The evolution of models”, Chem 13 News, March 2018, pages 16-18.
  3. Moseley and atomic number:
  4. Rutherford and proton
  5. Chadwick and the neutron:
  6.  D.C. Finster, “Developmental instruction Part I: Perry’s model of intellectual development.” Journal of Chemical Education, 1989, August, pages 659-661.
  7. Max Planck, letter to R.W. Wood, 1931, as cited in Jim Baggott, Beyond Measure, 2004, Oxford University Press, Oxford UK, page 13.
  8. A.B. Arons and M.B. Peppard, “Einstein’s proposal of the photon concept — a translation of the Annalen der Physik paper of 1905”, American Journal of Physics., 1965, 33, pages 367-374. On-line (2017):
  9. laureates/1921/
  10. J.J. Balmer, Annalen der Physik, 1885, 261, pages 80–87. (in German). See also
  11. NASA, “The Flash Spectrum of the Sun”,
  12. Friederich Paschen, Annalen der Physik, 1908, 332,pages 537-570. (in German). See also
  13. laureates/1922/
  14. Page 16 — Photos of Dalton, Rutherford and Bohr taken from Creative Commons. Photo of Schrödinger taken from portrait in 1933 when he received the Nobel. Credits: Nobel Foundation.


*For an excellent yet accessible history of atomic and quantum theory, see either: Jim Baggott, The Meaning of Quantum Theory, 1992, Oxford University Press, Oxford UK; or Jim Baggott, Beyond Measure, 2004, Oxford University Press, Oxford UK.