GUEST AUTHOR R. N. Keller
University of Colorado Boulder
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Textbook Errors, 38
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Energy Level Diagrams and Extranuclear Building of the Elements
Simplified diagrams showing the approximate order of electronic energy levels in atoms and mnemonic devices to aid in predicting electronic configurations for atoms are often misleading with respect to the actual energy of binding of the electrons in atoms and ions of the transition element,^. Even though this subject is treated clearly in a number of sources (1-8) and at least one attempt (4) has been made to correct common misconceptions, errors and conflicting statements continue to appear in standard inorganic chemistry textbooks.' Perhaps hy unwittingly saying too little, most freshman chemistry textbooks leave the student with an erroneous picture which is oft,en not corrected until graduate school.
level still lacks its full compleme~ltof electrons. It also suggests that 4p electrons will not be involved until the 3d level is fully occupied. Equivalent information can be obtained from a mnemonic scheme such as that illustrated in Figure 2.
Energy Level Diagrams
Figure 1 is an example of s common energy level diagram which is uscd to explain why, in building the
Figure 2.
1,-
Figure 1. The approximote atomic orbitah
sequence of energierand stabilities for
electron clouds of atoms, the filling of available orbitals is not completely in accord with the order allowed by the Pauli principle (6,6). The order of filling of energy levels is presumed to take place in order of increasing height of the levels above the base line. This diagram makes it clear why the 4s orbital is occupied in potassium and calcium even though t,he M (n = 3) quantum Suggestions of material suitable for this column and guest columns suitable for publication directly are eagerly solicited. They should be sent with as many details as possible, and particularly with references to modern textbooks, to Karol J. Mysels, Department of Chemistry, LTniversity of Southern California, Loe Angeles 7, California. Since the purpose of this column is to prevent the spread and continuation of errors discussed, and not the evaluation of individual texts, the source of errors discussed will not be cited. The error muat occur in s t l e a ~ ttwo independent etandard textbooks to be presented.
Order of occupancy of otomic orbitair
Since many treatments of the electronic building of atoms stop a t this point, it is only natural for the student to assume that the order of addition of electrons as predicted by these figures is the reverse of their order of stabilities or tightness of binding. That is, in the case of scandium, for example, since the 3d electron "went in last" this electron will "come out first" if the atom is sufficiently excited. This, hox'ever, is contrary to the facts obtained experimentally. In a many-electron atom or ion the attractive forces acting on a given electron are the consequences of a number of factors, among these being the actual nuclear charge and the number and kinds of other electrons present. Figure 3 brings out the fact that the act,ual positions of the energy levels change as u-ell as their relative positions with respect to one another when the nuclear charge (Z) changes. This figure shows that, although the energy levels with values of 3 and 4, for example, for the quantum number n may not be grouped together when Z is low, all these levels are lined np in the expected order in the heavy atoms. That is to say, in a heavy atom the 3d level is below the 4s, the 4d as well as the 4f below a 5s, etc. The points of crossing of the energy levels are in the neighborhood of the Z valnes for the first members of the d- and f-transition elements. According to Figure 3 then, the "anomaly" of a 4s electron adding beyond the argon configuration rat,her Volume 39, Numbei 6 , June 1962
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than a 3d electron (as at potassium and calcium) is no longer observed in a heavy atom. If electrons could he added successively about the nucleus of a heavy atom, the 19th electron would be a 3d electron and not a 4s. In fact, spectroscopic evidence indicates (7) that the "normal" order of addition is already achieved at scandium, where the 19th electron added about a scandium nucleus is a 3d electron while the 20th and 21st are 4s electrons. In the case of titanium, the 19th and 20th electrons added are in the 3d level and the 2lst and 22nd electrons are in 4 s states.
can be properly appreciat,ed only when due recognition is given to the subtle int.erplay of factors affecting the energies of electrons. I t should be emphasized that any representation of electrons in specific orbitals and having individual allotments of energy is itself an approximation. The energies involved are those of the complete atom (aggregate of nucleus plus electrons). Just as the concept of simple Bohr "orbits" has to be stretched t,o represent "population densities," so does the idea of a discrete energy assignment have to admit participation in the wave function for the whole atom. Electrons Assume Slates of Lowest Energy
1-4
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I. .I Z-$0
Ifomlo number. 2
Figure 3. Approximala energies of atomic energy levels or o function of atomic number [adapted fram Reference 1211.
Although Figure 3 is an improvement over Figure 1, it leaves much to be desired. It does not show, for example, that all levels decrease in energy with increasing atomic number nor does it offer a satisfactory answer to the question of why the three electrons in scandium above the closed argon core configuration do not all go into the 3d level if this level is in fact below the 4s level in energy. The deficiencies of Figure 3 arc to some extent corrected by Figure 4 . This figure (8.9) shows calculated values of electron energies as a function of atomic number and indicates clearly the general lowering of all levels as the nuclear charge increases. However, as an inspection of this figure indicates, the crossings of the curves do not correlate \re11 with the points in the Periodic Table a t which the d- and f-transition elements begin. This figure is also just as incapable as Figure 3 of providing an answer to the question of numerical distribution of electrons between energy levels. The inadequacies of the above diagrams serve to emphasize that no simplified diagram is capable of representing the real situation for all atoms and ions. Inasmuch as the energies of all electrons are affected by a change in the atomic number in going from one element to another, or in going from a neutral atom to one of its ions, a separate and distinct energy level diagram for each atom and ionic species is required. The order of electron addition and the number of electrons entering each energy level in atomic building 290
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When electrons are added successively to the field of a nucleus, the electrons assume the quantum states or occupy the energy levels which confer the greatest stability on the system as a whole. Thisis the arrangement which also results in the electrons being bound most tightly by the field of the nucleus. I n the case of scandium, cited above, the 19th electron assumes a 3d state rather than a 4 s st,ate beca.use the Sc+l ion has lower total energy with a 3d' t,ban with a 4s' configuration. However, the Sc+ ion is more stable if the 20th electron occupies a 4s orbital rather than a 3 d ; or, a 3d14s' state is more stable for this ion than a 3d2 or a 4s2 st,ate. Similarly the lowest lying state for neutral scandium, 3d14s2,is a more favorable state energetically than any other state such as 3d3, 4s24p': etc. Or saying 'this in a slightly different way, the effective nuclear charge in tripositive scandium is such as to cause the 19th electron added (i.e., S C + ~ e- -+ Sc+=)to occupy the 3d level because this level lies lower energetically than the 4s. However, once the 19th electron has been added, the whole electronic energy level system for Scf2 is now slightly different from that for S C + ~ . The interaction of the nucleus of scandinm with the 19 electrons in S C +produces ~ a field which favors a 4s state for the 20th electron added (i.e., Sc+? e- -+Sc+). The resulting field in turn favors a 4s state for the 21st elect.ronadded
+
+
Figure 4. Colculmtcd energies of atomic energy levels or o fundion of otomicnumber [adapted fram References I81 ond 1911.
The effect of nuclear charge on the relative order of energy levels is well illustrated (10,11) by the spectra of the following isoelectronic particles: K, Ca+, Sc+%. I n the spectrum of neutral potassium the 3d level is higher than even the 4p level, whereas in singly ionized calcium the 3d level has dropped below the 4 p and is only slightly higher than the 4s level. As might he anticipated from t h ~ trend, s m S C +the ~ energy of the 3d level is now lower than the 4s level. As a consequence, the ground state of the 19th electron in Scf2 is no longer a 4s state, as it is in I< and Ca+, hut is a 3d state. I t is interesting in this connection that as early as 1921 Bohr had come to sim~larconclusions from the trends shown by the spectra of K and Ca+. Energy Level Chart of the Periodic Table
By the nse of ionization energies and spectroscopic data DeVault (18) has devised an energy level chart of the Periodic Table showing the order of binding of all the electrons in neutral atoms (Fig. 5). This somewhat elaborate chart. which is essentially a comp0sit.e of individual energy level diagrams for the ground state of each atom, is worthy of careful study. The lowering of each level ~vith increasing atomic number is clearly shown as well as the crossing of energy levels and the number of electrons in each orbital. In contrast to the approximate curves of Figure 4, the curves of Figure .5 show a~curat~ely the changes in relative energies of the levels and the points of crossing of the levels with changing atomic number. For most elements the order of removal of electrons during successive ionizations of the atoms is easily predicted from the chart; this order is the same as the order of the electrons in the vertical column corresponding to a specific value of the atomic number, commencing with the highest electron and proceeding downward. Changes in Conflgurotion Accompanying Ionization
Usually when the most loosely hound electron is removed from an atom or an ion there is no change of configuration of the remaining ion. Thus, for the series Ti, Ti+,Ti+', Tif3.TI+', the ground state configurations are respectively: Rd24s2.3d24s1,3d2,3d1,3d0. With some of the transition elements, however, changes in ground state configurations accompany ionization (13). For example, d i l e the ground state configuration for neutral vanadium is 3d34sZ,singly ionized vanadium has a ronfiguration 3d4 instead of the expected 3da4s1. Other examples include the following: Co, 3d74s2 CO+, 3d8 e-; Xi, 3d84s2 Nif, 3ds e-: La, .idlGs? + La+, 5dZ e-. These examples emphasize further that the actual electronic configuration of an atom or ion is the resultant of complex forces and cannot necessarily he predicted from an over-simplified energy level diagram.
+
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+
+
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Conclusion
I n the opinion of the author two common pedagogical shortcomings lead to the confusion regarding the order of entry of electrons into atoms, the order in which electrons leave when atoms are ionized, and the relative energies of electrons of various quantum states: one is the mis-use of over-simplified energy level diagrams; the other is the manner in which the Aufbau or Buildingup Principle (&9)is applied.
Simplified electronic energy level diagrams should bc used judiciously in teaching the elertronic building of the atoms. These diagrams are usefnl in indicating that the building process is not controlled solely by the major quantum number n , hut they can he misleading if applied too literally to the relatix energies of electrons in transition type atoms and ions. As shown in Figure 1 and similar diagrams, the 3d level is plared above the 4s level. However, once Sd electrons are present in an atom or ion these electrons are lower in energy than 4s electrons. Similarly, 4d electrons are lower than 5s electrons, and 5d lower than 6s. In an at,om such as gadolinium which contains valenre elect,rons of three different types, the 4 j electrons are lower in energy than the 5d, and the 5d in turn lo\r-er t,hm the (is, Examination of the DeVault chart will make this clear In the application of the hufhau prinriple it is cnstomary to imagine that any given dement ran be formed from the preceding element by the sim~rltanrous addition of a proton to the nucleus of this element and an elect,ron to its elect,ron cloud. This approach too often leaves the inlpression that the new element is identical in all details to the preceding element except for t,he added proton (and one or more neutrons) and the added electron, which is sometimrs t,ernled the differentiating electron. I t is easy to see why a student who is given the fact that the rlrctronir configuration for scandium is 3d'4s2 would conclude that if the 4s level was already filled a t calrinm. then t,he electron which was added t,o make scandium from ralrinm must certainly he a 3d electron; and, sinre this dectron as added after the t,wo 4s electrons, it. is the electron in scandium which is most loosely bound. This proredure, moreover, places t,he instructor in the awkward position of building a case for a hypothet,ical order of entry of the electrons yet admitting of another order of t,heir removal (the experimentally observed order) by ionizat,ion. Complete parallelism between the order of entry of electrons and their order of leaving (except. for the converse relationship) can he achieved easily hy one of two procedures. One can imagine either that. electrons are fed into energy levels about a bare nucln~snntil the neutral atom in its ground state is oht,ained, or the process of adding a proton and an electron to the preceding element is separated into two distinct steps (14)first, the addition of the proton and, then, the addition of the electron I t should he made clear, for example, that if a proton could he added to a calcium nucleus while keeping the electrons constant a t 20, t,he two 4s valence electrons of calcium ~vouldimmediat,ely rearrange into a 3d14s1configuration, which is the ground state for Scf. The next or last elect,ron added, then, to "make" scandium from calcium is actanally a 4s type. Also if a proton could he added in turn to a scandium nucleus with a cloud of 21 electrons, the 3d14s2configuration of the valence elect,ronsof scandium would a t once rearrange to a 3d24s1configuration, the ground stat.e of Ti+. Again, the next or last electron added to "convert" scandium t,o tit.anium is a 4s electron and not a 3d. Unless this or an equivalent approach is used in the discussion of the electronic building of the atoms, the order of successive removal of electrons in the formation of ions among the transition type elements will surely appear paradoxical t,heperceptive student. Volume 39, Number 6 , June 1962 ./ 291
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Journal of Chemical Education
Acknowledgment
The aut,hor is plrasrd to express his appreriation t,o Dr. William A. Rt-nsr, Ilepartment of Physics, University of Colorado, and t,o Dr. Charles D. Coryell, Depart.ment of Chemist,ry, MIT, for reading the manuscript and making certain t,hat violenrn had not. brm committed against nrrrptrd t,hnories.
Litemture Cited ( 1 ) GLASSTONE, S., "Textbook of Physical Chemistry," 2nd ed., D. Van Nostrand Co., Inc., New York, 1946.
(2) HERZBERG,G., "Atomic Spwtra and Atomic Structure," 2nd ed., Dover Pohlirat~ons,New York, 1944.
( 3 ) WH~TE, H. E., "Introdurtion to Atomic Spectra," McGrawHill Book Ca., h e . , Nca >-ark, 1934. (4) SWINEHART, I). F., J. CHEM.EUTC.,27, 6 2 2 4 (1950) (5) PAI~LISG, L., "The Nature of t,he Chemioal Bond," 3rd ed., Carnell University P r m , 1960, 1)&ge947 and 580. op. a t . , Chap. :1 ( 6 ) HERZBERC, ( 7 ) I b d , P. 151. (8) PAITLING,op. czt., p. 56. (9) LATTER,R., Phys. Rev., 99, 510 (1955). (10) W H ~ T Eo,p . eit., pp. !I$ 264. (11) REMI, H., "Treatise on Inorganic Chemistry," Elsevier Publishing Co., Nex- York, 1956, Vol. I, p. 252; Vol. 11, Introduction, p. nxii. (12) DEVAULT,I)., J. CHEM.EDVC.,21, 575-81 (1944). (13) MOORE,C. E., "At,ornic Energy Levels," Circular of the National Bureau o i Standards 467, Vol. I, 1949; Vol. 11. 1952: Val. 111. 1958.
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