In his recent commentary Lowe (1) does not challenge our quantum mechanical explanation of the H/He ionization energy ratio (2), but argues that our claim that consideration of kinetic energy is essential at the atomic level is too restrictive. To prove this he offers a potential-energy-only (PEO) model for the successive ionization energies of sulfur that he claims is "extremely simple", works well, and is suitable for introductory students. We suggest that this PEO model is not a suitable pedagogical model for two reasons. First, it is built (incorrectly) on the virial theorem, and therefore an added piece of theory needs to be explained to introductory students. Second, it is built on the incorrect assumption that atomic size does not change with ionization from a given subshell, or at least that such changes can be ignored. Lowe acknowledges both of these features, but we feel the second item deserves more attention as we show below.

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", and r_i is the distance of the ionized electron from the nucleus. Lowe says of the model he proposes that "It is theoretically supported by the virial theorem." However the virial theorem is valid only when the initial and final states are stable, equilibrium states. The product ion, under the model used in this calculation, is not in an equilibrium state, since the radius is assumed fixed, and therefore the application of the virial theorem will be compromised.

The next step is to write the ratio of ith and jth ionization energies based on the previous equation.

If we again assume that within a subshell r_i = r_j and then choose one of the ionization energies as a experimental data available for atomic and ionic radii. For example, sulfur has an atomic radius of 104 pm, whereas S⁴⁺, S⁶⁺, and S^2- have ionic radii of 37, 29, and 170 pm, respectively (3). These data indicate that adding or removing electrons has a significant effect on the radius of an element, and therefore presumably on the radii of the shells.

As noted above, this model as used by Lowe requires a reference point (and re-referencing for each new subshell), and it is very sensitive to the reference point chosen. For example, Lowe uses the second ionization energy of sulfur as the reference point, because using the first ionization energy does not give good results. Looking at the first four ionization energies and choosing I₂ as the reference yields a cumulative error of 14% (1). Using I₁ as the reference yields a cumulative error of 34% (I₁ [1000, ref]; I₂ [2000, 11.2%]; I₃ [3000, 10.6%]; I₄ [4000, 12.2%]), where the ionization energies are given in kJ/mol.

To explain this difference, Lowe invokes the special stability of the half-filled shell (we assume he means half-filled subshell). This argument is not convincing, since both ionization processes involve a half-filled subshell--I₁ on the product side (3p⁴ 3p³) and I₂ on the reactant side (3p³ 3p²). In any case, it seems his simple model requires guidance from a more advanced model, the orbital approximation of modern quantum theory, just to get started. In addition, the re-referencing required for each subshell in order to achieve modest quantitative success is tedious and challenges the pedagogical utility of this PEO model.

This PEO model gives noticeably worse predictions for oxygen than it does for sulfur. Again, using the cumulative error as a criterion of goodness of fit, oxygen yields a cumulative error of 42% compared to 14% for sulfur. We might expect an atom with fewer electrons to yield better results, but this is not the case.

We would also like to stress that we do not criticize the use of the shell model to teach the basic elements of atomic structure. The most important characteristic of electronic shells and subshells is their capacity for electrons. This can be seen clearly by a direct examination of graphs of the first ionization energies of the elements or tables of successive ionization energies, which can be found in any general chemistry text (4). Or one can follow Gillespie, Spencer, and Moog (5) and use the orbital ionization energies obtained from photoelectron spectroscopy. All these approaches clearly reveal the existence of electronic shells; one does not need to construct a classical PEO model with questionable foundations to demonstrate the validity and utility of the atomic shell model. In his article Lowe made reference to the fact that DeKock and Gray used an electrostatic-only model in their textbook (6). For the reasons described above, DeKock now wishes to distance himself from this model as a pedagogical tool.

Shells, subshells, quantized energy levels, and atomic stability 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. Rioux, F.; DeKock, R. L. J. Chem. Educ. 1998, 75, 537-539.
3. CRC Handbook of Chemistry and Physics, 80th ed.; Lide, D. R., Ed.; CRC: Boca Raton, FL, 1999; pp 12-15.
4. Moore, J. W.; Stanitski, C. L.; Jurs, P. C. Chemistry: The Molecular Science; Harcourt College Publishers: New York, 2002; pp 302-305.
5. Gillespie, R. J.; Spencer, J. N. Moog, R. S. J. Chem. Educ. 1996, 73, 617-622.
6. DeKock, R. L.; Gray, H. B. Chemical Structure and Bonding; University Science Books: Sausalito, CA, 1989; pp 74-75.

See Lowe's Reply.