In the Classroom
Trends in Ionization Energy of Transition-Metal Elements Paul S. Matsumoto Galileo Academy of Science and Technology, San Francisco, CA 94109;
[email protected] IE = (13.6 eV)
Z eff nq
2
=
1.31 MJ mol
Z eff nq
1500
IE / (kJ/mol)
Ionization energy (IE) is defined as the minimum energy required to remove an outer-shell electron from a gaseous atom or ion. The first IE is the minimum energy to remove the first electron from an atom (1–5). In this article, IE refers to the first IE. While introductory chemistry textbooks (1–3) and articles in this Journal (5–7) describe and rationalize some of the periodic trends in the IE, none of these sources rationalize the periodic trends in the IE of the transition-metal elements. In general, as the number of protons, Z, increase, the IE increases for the main-group elements; however, the IE is relatively steady for the transition-metal elements in the same period (or row) of the periodic table as shown in Figure 1 (1, 2). The purpose of this article is to provide a rationale for the lower rate of increase in the IE of the transition-metal elements compared to that of the main-group elements. IE may be approximated by the following equation
1000
500
0 15
25
35
45
55
Z Figure 1. The IE of period 4 and period 5 elements as a function of atomic number. The IE of the transition-metal elements (䉭) increases at a slower rate than the IE of the main-group elements (䊐).
2
(1) 1500
Zeff
4 500 2
0
0 15
25
35
45
55
Z Figure 2. The IE and effective nuclear charge (Zeff) of period 4 and period 5 elements as a function of its atomic number. The IE (䉭) and Zeff (䉱) of the transition-metal elements increases at a slower rate than the IE (䊐) and Zeff (䊏) of the main-group elements. The values of the Zeff were calculated using Slater’s rule, while the experimental values of the IE were obtained from a table (7).
•
1600
IE / (kJ/mol)
Notice that Zeff is a decreasing function of S. In concert with Zeff, the periodic trends in the IE are a consequence of the Aufbau principle. The Aufbau principle states that electrons are added from the lowest to highest energy-level orbital (2, 4). In the transition-metal elements, as Z increases in a period, the electrons enter an inner-shell electron orbital that shields the outer-shell electrons. In contrast, in the main-group elements, as Z increases in a period, the electrons enter an outer-shell electron orbital, which is less effective in shielding the outer-shell electrons. This difference results in an increase in the Zeff for the main-group elements (1, 2, 7), while the Zeff is relatively steady for the transition-metal elements. This rationale is supported by Slater’s rules (see appendix) and the predicted behavior of the Zeff was confirmed as shown in Figure 2. Journal of Chemical Education
6
1000
(2)
Z eff = Z − S
1660
IE / (kJ/mol)
8
where Zeff is the effective nuclear charge, n is the principal quantum number, and q is a correction factor depending on the element (6). This equation without the correction factor, q, is the same as the equation describing the electron energy levels in the Bohr model of the atom (8). In the same period, the value of q (in the p-block) and n does not change (6). In addition, there are only two values of q in the d-block (6), thus most of the variation in the calculated value of the IE depends on the value of Zeff. Consistent with this proposal, Figure 2 shows that the behavior of IE and Zeff as a function of Z are similar. The effective nuclear charge, Zeff, is the charge of the nucleus felt by the outer-shell electron. It depends on Z (the number of protons in the nucleus) and S (the shielding effect of the inner-shell electrons) and can be expressed as (1, 2, 4–6):
1200
800
400
0 20
25
30
35
40
Z Figure 3. A comparison of the IE as a function of Z based on eq 1 (䉬) and its actual value (䊐; 4). The values of Zeff were calculated using the Hartree–Fock approximation.
Vol. 82 No. 11 November 2005
•
www.JCE.DivCHED.org
In the Classroom
Figure 3 shows in more detail that the IE values generated from eq 1 can be used to predict the experimental IE, implying that the variation in the Zeff is responsible for the behavior of the IE. Another rationale for the importance of Zeff in determining the IE is as follows. (i) Coulomb’s law (8) implies that the force of attraction (Fatt) between the nucleus and outer-shell electron is an increasing function of Zeff. (ii) The definition of work (8) implies that the work needed to remove an outer-shell electron would be an increasing function of Fatt; hence, work would be an increasing function of Zeff. (iii) An application of the work–energy theorem (5, 8) implies that the energy needed to remove an outer-shell electron would be an increasing function of Zeff. This energy is the IE, thus IE is an increasing function of Zeff. Slater’s rules (see the appendix) provide a simple way to calculate Zeff, while the Hartree–Fock approximation of an atom is a more complex calculation (4, 7 ). Although using the Zeff values from Slater’s rules in eq 1 generates a poor fit between the calculated and actual IE values (data not shown), the calculated IE values had a similar pattern as the actual IE values. In contrast, using the Zeff values from the Hartree– Fock calculation (4) generates a much better fit (Figure 3), which implies that the variation in Zeff is the basis of the variation in IE. The calculated IE values for the period 5 elements were not shown, since the Hartree–Fock derived Zeff values were unavailable. The purpose of using Slater’s rule and the Hartree–Fock approximation of an atom in the article was to evaluate, in a semiquantitative manner, the validity of the rationale in this article regarding the different periodic trends in the IE between the transition-metal and the main-group elements.
Department of Physics). The impetus for this article is due to a remark by Luis Baker, a chemistry student, that the effective nuclear charge is stable in the transition-metal elements. Literature Cited 1. Brown, T. L.; LeMay, H. E.; Bursten, B. E. Chemistry. The Central Science, 7th ed.; Prentice Hall: Upper Saddle River, NJ, 1997. 2. Zumdahl, S. S.; Zumdahl, S. A. Chemistry, 6th ed.; Houghton Mifflin Company: Boston, MA, 2003. 3. Dorin, H.; Demmin, P. E.; Gabel, D. L. Chemistry. The Study of Matter, 4th ed.; Prentice-Hall; Needham, MA, 1992. 4. Huheey, J. E. Inorganic Chemistry, 2nd ed.; Harper & Row; New York, 1978. 5. Lang, P. F.; Smith, B. C. J. Chem. Educ. 2003, 80, 938–946. 6. Waldron, K. A.; Fehringer, E. M.; Streeb, A. E.; Trosky, J. E.; Pearson, J. J. J. Chem. Educ. 2001, 78, 635–639. 7. Cann, P. J. Chem. Educ. 2000, 77, 1056–1061. 8. Cutnell, J. D.; Johnson, K. W. Physics, 4th ed.; John Wiley & Sons; New York, 1998.
Appendix
Determination of Zeff Using Slater’s Rules Slater’s rules (4, 6) are a set of empirical rules to calculate Zeff by determining the value of S for electrons in either the ns or np orbital, where n is the energy level. First, write the electron configuration of an element in the following format, (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d,4f ) (5s,5p)…
Summary The arguments in this article provide a rationale for the difference in the periodic trends in the IE of the transitionmetal versus the main-group elements, a topic that is absent in introductory chemistry textbooks. In essence, the difference is that in the transition-metal elements, the electrons enter an inner-shell electron orbital, while in the main-group elements, the electrons enter an outer-shell electron orbital. As inner-shell electrons have a greater shielding effect than outer-shell electrons, transition-metal elements have a smaller rate of increase in their IE than main-group elements. The level of complexity of this article should be accessible to either advanced high school students or first-year college students.
which does not follow the Aufbau Principle (2, 4). Next, determine the value of S, which is the sum of (i)
35% of the number of electrons in the outer-shell minus 0.35
(ii)
85% of the number of electrons in the next lower electron shell
(iii)
100% of the number of electrons in the remaining lower electron shells
For example, to determine the Zeff of arsenic, write its electron configuration in the above format (1s2) (2s2 2p6) (3s2 3p6) (3d10) (4s2 4p3) then calculate S
Acknowledgments
S = [35%(5) − 0.35] + 85%(18) + 100%(10) = 26.7
The concepts in this article were discussed with Sergio Aragon (San Francisco State University; Department of Chemistry) and John Morrison (University of Louisville;
www.JCE.DivCHED.org
•
and finally
Vol. 82 No. 11 November 2005
Zeff = Z − S = 33.0 − 26.7 = 6.3
•
Journal of Chemical Education
1661
