Lowe uses the virial theorem to establish a mathematical relationship between the ionization energy and electron potential energy as shown below in atomic units (h/2p = m_e = e = 4pe₀ = 1),

where i is the index for the ionization potential and the charge on the "ion left behind". There are two significant approximations of questionable validity in the way Lowe uses the virial theorem to establish this relationship. The first is that the virial theorem applies to total potential energy, whereas Lowe's model considers only electron-nuclear potential energy and ignores electron-electron potential energy. Second, the virial theorem is valid as written only when the initial and final states are stable equilibrium states. The term i/2r_i is the energy that must be supplied to overcome the electron-nuclear attraction and form an ion of charge i. The product ion is not, under the model used in this calculation, in a stable equilibrium state, and therefore the application of the virial theorem is not strictly valid.

The next step in using Lowe's model is to write the ratio of ith and jth ionization energies based on the previous equation.

At this point Lowe assumes that within a sub-shell r_i = r_j, and after choosing one of the ionization energies as a reference proceeds to calculate the other ionization energies in the subshell. Unfortunately, this assumption, which is crucial to his model, is not consistent with experimental data available for atomic and ionic radii. There is a significant change in atomic and ionic radii, and therefore shell radii, as successive electrons are removed from an element. For example, sulfur has an atomic radius of 104 pm, but S⁴⁺ has an ionic radius of 37 pm, indicating a rather significant contraction of the 3p subshell as its four electrons are removed (4). S⁶⁺ has a radius of 29 pm and S^2- has a radius of 170 pm, indicating that adding or removing electrons has a significant impact on the radius of the element under consideration.

Furthermore, Lowe's model requires a reference point (and frequent re-referencing), and it is very sensitive to the reference point chosen. For example, Lowe uses the second ionization energy of sulfur as the reference point, whereas we (and, probably most everyone else) would instinctively use the first ionization energy as the reference ionization energy. The results for the two reference points (in kJ/mol) are shown in the table below.

Using the sum of the percent errors as a criterion, it is clear that choosing I₁ (34%) as a reference is not as good as using I₂ (14%). Lowe acknowledges that using the first ionization energy gives unsatisfactory results and he invokes the special stability of the half-filled shell as an explanation. This argument is not convincing, since both ionization processes involve a half-filled shell--I₁ on the product side (3p⁴ 3p³) and I₂ on the reactant side (3p³ 3p²). We can't imagine an introductory student understanding why I₂, rather than I₁, is chosen as the initial reference point, or being able to give that student a plausible explanation for that choice, other than it gives better results. In addition, the re-referencing required for each subshell in order to achieve modest quantitative success is tedious and challenges the pedagogical utility of Lowe's model. As we indicate below there are more direct ways to employ ionization energies, already in use, to demonstrate the shell model to introductory chemistry students.

We also found it curious that Lowe's model gives noticeably worse predictions for oxygen than it does for sulfur. Again, using the sum of the percent errors as a criterion of goodness-of-fit, oxygen yields a sum of 42% to 14% for sulfur. The selenium calculations are slightly better than sulfur, at 13%. Most of us would assume that the simpler the system, the better a simple model would be in explaining it. Not so for Lowe's potential-energy-only model. We believe that this is further evidence that his model is not a reasonable representation of the shell model or of atomic structure in general.

If the goal is in Lowe's words "to show what is rather than why it is", then a direct appeal to the experimental ionization energies is surely the best approach. Evidence for electronic shells and subshells is easily seen in a graph of the first ionization energies of the elements found in any general chemistry text (5). In addition, Gillespie, Spencer, and Moog have demonstrated that orbital ionization energies obtained from photoelectron spectroscopy also clearly show the electronic shell structure (6). Both of these approaches reveal that the existence of electronic shells within the atom is "data driven". One doesn't need to construct a classical potential-energy model built on unfirm foundations to demonstrate the validity of the atomic shell model.

Of course to find out why it is rather than what it is requires quantum mechanics. Shells, subshells, and quantized energy levels have their origin in the wave nature of the electron, which we all recognize is manifested in the electron's kinetic energy. That is why we think consideration of kinetic energy is essential.

Literature Cited

1. Lowe, J. P. J. Chem. Educ. 2000, 77, 155-156.
2. Gillespie, R. J.; Moog, R. S.; Spencer, J. N. J. Chem. Educ. 1998, 75, 539-540.
3. Rioux, F.; DeKock, R. L. J. Chem. Educ. 1998, 75, 537-539.
4. CRC Handbook of Chemistry and Physics, 80th ed.; Lide, D. R., Ed.; CRC: Boca Raton, FL, 1999; pp 12-15.
5. Moore, J. W.; Stanitski, C. L.; Wood, J. L. The Chemical World: Concepts and Applications, 2nd ed.; Saunders College Publishing: New York, 1998; p 338.
6. Gillespie, R. J.; Spencer, J. N.; Moog, R. S. J. Chem. Educ. 1996, 73, 617-622.

See the author's reply.