Chemical Education Today
Commentary
Comments on Kinetic, Potential, and Ionization Energies by John P. Lowe
Rioux and DeKock (1) criticize a potential-energy-only model for atomic ionization energies put forward by Gillespie, Spencer, and Moog (2). They state that “interpreting the ionization of any atom or molecule requires quantum chemical tools and a consideration of both kinetic and potential energy.” They indicate that “what is missing in [the Gillespie et al.] potential energy argument is the fact that kinetic energy is an important factor in the quantum world of atoms and molecules and cannot be ignored.” Their article concludes with the observation that detailed quantum-mechanical arguments “are obviously too advanced for introductory students”, and that we are left “with the important task of deciding what we can say to introductory students about the details of the periodicity of physical properties, such as ionization energies, that is both correct and understandable.” I suggest that this view is too restrictive. It is true that kinetic and potential energies together determine the wave function for an atom or molecule: to understand why a wave function is as it is requires knowledge of both. However, if we wish only to probe the nature of the wave function—to show what it is rather than why it is—then a simpler approach may suffice. Demonstrating that atomic ionization energies are consistent with a shell structure is a case in point: kinetic energy changes can be ignored in an appropriately parameterized potential-energy-only model. This results from the virial theorem, which states that kinetic (T ), potential (V ), and total (E ) energies are related as V = ᎑2T = 2E in any stable state of an isolated atom or isolated ion having a single nucleus (3). This means that any ionization energy I = ∆E is equal to ∆V/2. Calculating ionization energies can therefore be carried out by calculating the corresponding potential energy changes and dividing by two. Therefore, if one seeks to demonstrate that ionization energies are consistent with a shell structure, one needs only to show that the potential energy changes of ionization are consistent with shell structure. Furthermore, if one parameterizes such potential-energy-only calculations by using a few ionization energies as reference points, one need not calculate ∆V explicitly: using a known ∆E effectively identifies a known ∆V/2, and one can proceed from that point using arguments about relative ∆V values.
considerably closer, that the 2s electrons are a little closer than that, and that the 1s electrons are again much closer. Our goal is to demonstrate to students that this assumed shell structure is consistent with observed ionization energies. Suppose we choose the second successive ionization energy, I2, as a reference point. (We show later why choosing I1 works less well for this element.) We picture an electron being removed from the 3p effective radial distance to infinity, leaving behind an ion of charge +2 au. The measured energy (5) is 2252 kJ/mol. According to the virial theorem, this means that ∆V /2 equals 2252 kJ/mol. The simple potentialenergy-only-model (4 ) predicts (using Coulomb’s law) that I1 is about half as great (since the electron is leaving from about the same distance but is leaving behind a charge of +1 au), that I3 is about 3/2 times as great, and I4 is about 4/2 times as great. It also predicts that I5 is significantly larger than 5/2 times as great because the effective distance of a 3s electron from the nucleus is assumed to be smaller than that for a 3p electron. These predictions work fairly well. In values in kJ/mol (predicted/observed//error) are: I1 (1126/1000//12.5%) [I2 (2252/2252//ref )] I3 (3378/3357//0.6%) I4 (4504/4556//1.1%) I5 (>5630/7004)
At this point we can re-reference our calculations to fit the 3s electrons, which are starting their journeys from a little closer in. Setting I5 = 7004 kJ/mol and using I6 = 6I5/5 gives I6 (8405/8500//1.1%). Again, we expect I7 to be much greater than what we would predict using a 3s distance parameter: I7 >> 7I5/5, I7 (>>9806/27107). Once again we recalibrate for 2p electrons by taking I7 = 27107 kJ/mol. Then I8 = 8I7 /7, etc. This yields: I8 (30979/31720//2.3%) I9 (34852/36621//4.8%) I10 (38724/43177//10.3%) I11 (42597/48706//12.5%)
A Simple Example It is remarkable how well an extremely simple potentialenergy-only model works. We focus on successive ionization energies to display the relation between such energies and shell structure. The approach we use here is based on one suggested in a textbook by DeKock and Gray (4 ). We take as our example an isolated sulfur atom. The ground state neutral atom has configuration 1s22s22p63s23p4. We assume a shell structure by imagining that the 3p electrons are all at about the same average distance from the nucleus, that the 3s electrons are slightly closer, that the 2p electrons are
I12 (46469/54460//14.7%)
This model is very simple and reveals shell structure nicely. It lends itself well to student exercises: for example, given (kJ/mol) I1 = 737.7 and I3 = 7733 for Mg, estimate I2, I4, and I5. Student answers can then be compared with the observed values: I2 = 1450, I4 = 10550, I5 = 13620. (This problem is similar to one given by DeKock and Gray [6 ], though their emphasis is on calculating and comparing effective radii from ionization energies rather than predicting ionization energies from an assumed shell structure.)
JChemEd.chem.wisc.edu • Vol. 77 No. 2 February 2000 • Journal of Chemical Education
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Percent errors increase as extrapolations lengthen: for sulfur, I8 predicted from I 7 is 2.3% too small, whereas I12 is 14.5% too small. The errors tend to grow in accord with a picture of shells contracting as electrons are removed, consistent with loss of screening of nuclear charge as electrons are lost—an effect that this simple reference point model does not explicitly take into account. Half-filled shell stabilities can be seen as well, as in the unexpectedly small observed value for I1 in sulfur cited above. (This is the reason that I2 is a better choice than I1 for the initial parameterization of sulfur— I1 corresponds to creating a half-filled shell and therefore requires less-than-“normal” energy, and if used as a parameterization point, consistently underestimates other ionization energies from the same shell.) The reparameterization at each new shell or subshell allows the model to work fairly well despite its lack of consideration of the detailed changes in potential interactions. Earlier Applications of Potential Energy Models Potential-energy-only models have been used to relate shell structure to ionization energies for at least 30 years. Pimentel and Sprately (7) used a model that includes effective nuclear charge (hence electron-electron repulsion effects) explicitly. Agmon (8) and Marone (9) have also discussed this model as it applies to their analyses of ionization data. Theoretical studies of the behavior of electron repulsion in successive ions have been reported in efforts to understand the accuracy of this method (10, 11). The much simpler (though less accurate) relationship illustrated above—a relationship that is well within the grasp of introductory chemistry students—was pointed out in 1973 by Liebman (12), though his discussion was limited to successive ionizations from the same shell. DeKock and Gray (4 ) later applied this method to the ionization energies of lithium, a situation where ionizations occur from two different shells. None of these authors made mention or explicit use of kinetic energy changes. The potential energy change being calculated by any of these methods takes no explicit account of potential energy changes occurring as the newly created ion relaxes. However, the fact that these methods work fairly well in relating measured ionization energies to shell structure suggests that these relaxation energies scale to some extent with ionic charge, hence can be successfully accommodated in the parameterizations. Conclusion Obviously this simple potential-energy-only model is not perfect, but it has the advantage of correlating observed ion-
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ization energies with shell structure in a manner students can comprehend. It is theoretically supported by the virial theorem. It avoids running afoul of the kinetic energy argument of Rioux and DeKock because we are really showing what the electronic structure of atoms is, and not how it comes about as the result of basic physical principles. Furthermore, the model falls into the tradition of semiempirical methods wherein some of the major elements of a process are explicitly treated while the more complicated details are sidestepped through use of parameters that tie the predictions to a few reference points. It even allows one to detect irregularities or trends that raise questions and encourage thought about the model’s deficiencies. This is very similar to the manner in which errors in the ideal gas model’s predictions lead to recognition of effects due to molecular size and stickiness. Indeed, creation and subsequent refinement of such simple models is precisely the way human minds over time have constructed our present understanding of the physical universe. That a model is simple in comparison to state-of-the-art science is no argument against its having a useful place in the educational process. That such models make predictions that differ from observation should be seen as an opportunity for exploring the next step in understanding. Literature Cited 1. Rioux, R.; DeKock, R. L. J. Chem. Educ. 1998, 75, 537–539. 2. Gillespie, R. J.; Spencer, J. N.; Moog, R. S. J. Chem. Educ. 1996, 73, 617–621. 3. Lowe, J. P. Quantum Chemistry, 2nd ed.; Academic: New York, 1993; pp 638–641. 4. DeKock, R. L.; Gray, H. B. Chemical Structure and Bonding; Benjamin/Cummings: Menlo Park, CA, 1980; p 75. 5. CRC Handbook of Chemistry and Physics, 75th ed.; CRC: Boca Raton, FL, 1994; pp 10-205–10-207. 6. DeKock, R. L.; Gray, H. B. Op. cit., p 129. 7. Pimentel, G. C.; Spratley, R. D. Chemical Bonding Clarified through Quantum Mechanics; Holden-Day: San Francisco, 1969; pp 53–58. 8. Agmon, N. J. Chem. Educ. 1988, 65, 42–44. 9. Mirone, P. J. Chem. Educ. 1991, 68, 132–133. 10. Pyper, N. C.; Grant, I. P. Proc. R. Soc. London 1978, A359, 525–543. 11. Benson, S. W. J. Phys. Chem. 1989, 93, 4457–4462. 12. Liebman, J. F. J. Chem. Educ. 1973, 50, 831–834.
John P. Lowe is in the Department of Chemistry, The Pennsylvania State University, University Park, PA 16802; email:
[email protected].
Journal of Chemical Education • Vol. 77 No. 2 February 2000 • JChemEd.chem.wisc.edu
