Ionization Energies of Atoms and Atomic Ions

Research: Science and Education

Peter F. Lang and Barry C. Smith* School of Biological and Chemical Sciences, Birkbeck College (University of London), Malet Street, London WC1E 7HX, England; *[email protected]

The ionization energy of an atom depends on its atomic and atomic ions. Ionization energies derived from optical and number and electronic configuration. Ionization energies tend mass spectroscopy and calculations ranging from crude ap- to decrease on descending groups in the s and p blocks (with proximations to complex equations based on quantum me- exceptions) and group 3 in the d block of the periodic table. chanical theory are accompanied by assessments of reliability Successive ionization energies increase with increasing charge and a bibliography (2). Martin, Zalubus, and Hagan reviewed on the cation. This paper describes some less familiar aspects energy levels and ionization limits for rare earth elements (3). of ionization energies of atoms and atomic ions. Apparently Handbook of Chemistry and Physics (4) contains authoritative irregular first and second ionization energies of transition data from these and later sources. For example, an experi- metals and rare earth metals are explained in terms of the mental value for the second ionization energy of cesium, electronic configurations of the ground states. A semiquan- 23.157 eV (5), supersedes 25.1 eV (2). titative treatment of pairing, exchange, and orbital energies accounts for discontinuities at half-filled p, d, and f electron Periodicity shells and the resulting zigzag patterns. Third ionization energies of atoms from lithium (Z = We begin with a reminder of the difference between ion- 3) to hafnium (Z = 72) are plotted against atomic number, ization potential and ionization energy. Ionization potential Z, in Figure 1. Values are from reference (4) except those for is the electric potential (measured in volts) required to sepa- Cs (Z = 55) and Ba (Z = 56) from the Journal of the Optical rate an electron from the orbital system in free space with Society of America (5, 6). The first peak occurs at Be2+ (Z = the kinetic energy remaining unchanged. Ionization energy 4), which has the electronic configuration 1s2. Peaks corre- is the work done in removing the electron at zero tempera- sponding to filled s, p, d, f, and half-filled d and f orbitals ture and is measured conveniently in electronvolts, where 1 ᎑ illustrate the shell model of the atom (7). Figure 1 provides eV = 1.6022 × 10 19 J. The molar ionization energy, or change ᎑ a more compelling demonstration of periodicity than plots in molar internal energy, is N eV = 96.485 kJ mol 1 where A of first ionization energy (8) where transition metals and rare N is the Avogadro constant. A earth metals do not show zigzag patterns. Ionization wavenumbers (reciprocal wavelengths) are For M2+ ions, 3d orbitals have lower energies than 4s derived from series limits of atomic spectral lines. The en- ᎑ ᎑ ᎑ orbitals (9), 4d orbitals have lower energies than 5s orbitals, ergy per cm 1 is 1.2398 × 10 4 eV or 1.9864 × 10 23 J. The ᎑ ᎑ and 5p orbitals have lower energies than 4f orbitals. molar energy per cm 1 is 11.962 J mol 1. s Electrons Sources of Data Figure 2 shows how first ionization energies decrease Three volumes containing wavenumbers and atomic en- from hydrogen to cesium and from helium to barium. ergy levels (1) preceded the critical survey by Moore of ion- Straight lines joining ionization energies from five pairs of ization limits from ground state to ground state for atoms group 1 and 2 atoms have intercepts of approximately 2.6

160

140

120

100

Figure 1. Third ionization 80 energies from lithium to hafnium. 60 Ionization Energy / eV 40

0 10 20 30 4050 60 70 Atomic Number

938 Journal of Chemical Education • Vol. 80 No. 8 August 2003 • JChemEd.chem.wisc.edu Research: Science and Education eV. The intercept for lithium and beryllium is 1.46 eV but 30 there is no reason to believe that the Moore values are incor- rect. First ionization energies of francium and radium, not shown in Figure 2, are greater than those of cesium and 25 1s barium respectively as a result of poor shielding by 4f electrons. 20 First ionization energies of atoms from hydrogen to beryllium are plotted in Figure 3. The ionization energy of helium is greater than that of hydrogen but less than four times 15 as great (10) because the electrons provide some screening for each other as mutual repulsion pushes them away from the nucleus. The outer electron of lithium occupies a new 10 2s shell screened by two electrons and the ionization energy is Ionization Energy / eV 3s 4s lower than that of hydrogen or helium. Similarly, the first 5s ionization energy of beryllium is higher than that of lithium 5 6s but much lower than that of helium. Second ionization energies from helium to boron are higher and follow a similar 0 pattern. Other points correspond to third, fourth, and fifth 012 ionization energies of the respective atoms. Number of s Electrons Ionization energies of atomic hydrogen and one-electron Figure 2. First ionization energies of group 1 and group 2 atoms. atomic ions, at the left of Figure 3, are approximately pro- portional to the electron–nucleus attraction, Z 2. They are re- produced with reasonable accuracy by the following 400 expression, where RM is the appropriate Rydberg constant and α is the Sommerfeld fine structure constant (11): 350 V 2 2  α  RZM  11+ ZZ()−  300  4  250 Screening (electron–electron repulsion) reduces electron– IV nucleus attractions in helium and two-electron atomic ions 200 but ionization energies are not functions of simple squares, (Z − S)2, where S is a screening constant (12). The correct 150 expression takes account of relaxation by the remaining elec- III tron (13): Ionization Energy / eV 100 5Z 5 II Z 2 − + 50 4 16 I 0 The square roots of the first ionization energies plotted HHeLiBeB C N O against atomic number for six isoelectronic series are shown Atom in Figure 4. The one-electron plot falls close to a straight line Figure 3. Energies to remove 1s or 2s electrons from atoms or ions. through the origin. Differences between square roots of suc- Roman numerals denote the ionization number. cessive ionization energies for the other series confirm increasing curvature from left to right. Gradients for 2s series approach one half and gradients for 3s series approach one third of the gradient for the one-electron series. Their ion- 25 ization energies are based on quadratic expressions, where n 1s is the principal quantum number of the electron, and b and 20 c are constants characteristic of the series: 2 Z 15 2s − bZ+ c n 3s 10 p Electrons 5 First ionization energies of atoms from the first three Ionization Energy / eV periods of groups 13 to 18 (2p, 3p, and 4p electron series) form zigzag patterns in Figure 5. First and second ionization 0 HHeLiBeB C N O FNeNa Mg Al Si P S Cl energies of atoms from the next two periods (5p and 6p se- Atom ries) appear in Figure 6. Gallium has a slightly higher first ionization energy than aluminum because of relatively poor Figure 4. Square roots of energies to remove 1s, 2s, or 3s electrons.

JChemEd.chem.wisc.edu • Vol. 80 No. 8 August 2003 • Journal of Chemical Education 939 Research: Science and Education shielding by 3d electrons. Thallium and lead have higher first tion energy is correct (15). Condon and Shortley identified ionization energies than indium and tin, respectively, because differences between theoretical and experimental values for of poor shielding by 4f electrons. First ionization energies a number of atoms (16) but spin–orbit coupling effects of vary in the order B > Al < Ga > In < Tl and C > Si > Ge > Sn this magnitude were not observed. < Pb but decrease with increasing atomic number down First ionization energies at the bottom of Figure 7 in- groups 15 to 18. Lead and bismuth have greater second ion- crease from boron to nitrogen, decrease to oxygen, and in- ization energies than tin and antimony respectively. crease to neon. Second ionization energies increase from The first ionization energy of bismuth appears to be carbon to oxygen, decrease to fluorine, and increase to so- anomalous. The increase from thallium to lead is followed dium. Third, fourth, and fifth ionization energies of the re- by a decrease to bismuth rather than the expected increase spective atoms show similar patterns. to approximately 8 eV (14). It has been claimed that spin– Discontinuities at half-filled p orbitals are convention- orbit coupling by the Russell–Saunders scheme would lower ally attributed to repulsion between electrons of opposite spin the ground state of Bi+ by 0.8 eV and that the lower ioniza- occupying the same orbital in the second half of a subperiod.

25 25

2p II 3p 5p 20 20 4p 6p

15 15

I 10 10 Ionization Energy / eV Ionization Energy / eV

5 5

0 0 13 14 15 16 17 18 13 14 15 16 17 18 19 Group Group Figure 5. Energies to remove first p electron. Figure 6. Energies to remove first and second p electrons. Roman numerals denote the ionization number.

25 180 ionization energy V modified ionization energy 160 pairing energy 20 exchange energy 140

120 IV 15

100

10 80 III

Ionization Energy / eV 5

II Ionization Energy / eV 40

20 I 0

0 BCNOFNeNaMgAlSi -5 Atom BC NOFNe Figure 7. Energies to remove 2p electrons. Roman numerals de- Atom note the ionization number. Figure 8. First ionization energies from boron to neon.

940 Journal of Chemical Education • Vol. 80 No. 8 August 2003 • JChemEd.chem.wisc.edu Research: Science and Education

Less attention has been paid to quantum mechanical exchange Table 1. Pairing and Exchange Interactions interactions, where more energy is required to ionize an elec- on Ionization of Atomic Boron to Neon tron in a group with parallel spins as a result of increased Atom p e Ion p e ∆p ∆e electron–nuclear attractions (17). The semiquantitative ap- 0 0 1 1 proach developed here considers double occupancy and ex- B00 B+ 00 0 0 change interactions as discussed by Blake (18) and takes C01 C+ 00 0 ᎑1 account of the work of Johnson (19) and Cann (20). Table 1 summarizes numbers of pairing interactions N03 N+ 01 0 ᎑2 + (pairs of electrons occupying the same orbital), p0 and p1 O13 O03 ᎑1 0 (subcripts denote ionization numbers); numbers of exchange F24 F+ 13 ᎑1 ᎑1 interactions between pairs of electrons having parallel spins, N36e N e+ 24 ᎑1 ᎑2 e0 and e1; and changes on ionization, ∆p and ∆e, for boron to neon and their singly charged cations. We assume that in- dividual pairing energies, P, and exchange energies, Eex, are constant across the subperiod. Figure 8 shows how ionization energies from boron to neon are lowered by changes in pairing energy, P∆p, and raised by changes in exchange en- Table 2. Ground State Electronic Configurations ᎑ ergy, Eex∆e, where P = 2Eex = 1.778 eV. Modified ioniza- of Transition-Metal Atoms and Ions tion energies, adjusted for pairing and exchange interactions, MM+ M2+ M3+ lie on a curve that increases smoothly from boron to neon. Z Atom n Modified second ionization energies produce a curve that 3sd 4s 3sd 4 3sd 4 3d 4 increases from carbon through oxygen and fluorine to so- 2a0 C001 2000 ᎑ dium, where P = 2Eex = 2.354 eV. 2c1 S111 210100 Similar curves can be derived for other p electron series, 2i2 T221 220201 ᎑ where P = 2Eex = 1.038 eV for first ionization energies from 2V33 30 240302 ᎑ aluminum to argon and P = 2Eex = 0.784 eV for first ion- 2r4 C450 150403 ization energies from gallium to krypton. 2n5 M551 250504 2e6 F661 260605 Transition Metals 2o7 C770 280706 First ionization energies of calcium, strontium, barium, 2i8 N880 290807 and transition metals are plotted against group number in 2u9 C9100 1100 908 Figure 9. Apparent irregularities across the periods (21) are 3n0 Z012110 100 100 9 caused by different ground-state electronic configurations (1) 4sd 5s 4sd 5 4sd 5 4d 5 that are summarized in Table 2. Hafnium and other post- lanthanum atoms have higher first ionization energies than 3r8 S001 2000 3Y19 12 201000 4r0 Z221 220201 11 4b1 N340 140302 4o2 M450 150403

10 4c3 T551 250504 4u4 R670 170605 Ca to Zn 4h5 R780 180706 9 Sr to Cd Ba to Hg 4d6 P8100 090807 4g7 A9100 1100 908 8 4d8 C012110 100 100 9 5sd 6s 5sd 6 5sd 6 5d 6 7 5a6 B001 2000 5a7 L110 220100 Ionization Energy / eV 6 7f2 H222 210201 7a3 T330 24

5 7W44 40 25 7e5 R551 25 7s6 O661 26 4 23456789101112 7r7 I770 28 Group 7t8 P890 190807 7u9 A9100 1100 908 Figure 9. First ionization energies of alkaline earth and transition metals. 8g0 H012110 100 100 9

JChemEd.chem.wisc.edu • Vol. 80 No. 8 August 2003 • Journal of Chemical Education 941 Research: Science and Education corresponding atoms in the first and second periods, presum- Table 3. Pairing and Exchange Interactions ably because of poor shielding by 4f electrons. Observed ion- and Term Symbols for Ions of the First Transition Period ization energies from ground state to ground state (4) will Ion p e Tnerm Io p e Term ∆p ∆e appear as filled circles in Figures 10–12. Other symbols rep- 2 2 3 3 resent ionization energies derived from wavenumbers of ex- Sc2+ 00 2DcS 3+ 001S00 cited states (1). Ti2+ 01 3FiT 3+ 002D0᎑1 2+ 4 3+ 3 ᎑ First Transition Period V 03 FV01 F02 Cr2+ 06 5DrC 3+ 034F0᎑3 Most atoms in the first transition period have outer elec- 2 6 3 5 Mn + 001 SnM + 06 D0᎑4 tronic configurations M(3dn4s2), where n represents the po- 2+ 5 3+ 6 sition of the atom in the d block. First and second ionization Fe 101 DeF 001 S ᎑1 0 energies are examined in Figure 10. Calcium, scandium, ti- Co2+ 211 4FoC 3+ 101 5D ᎑1 ᎑1 tanium, manganese, iron, and zinc lose both s electrons in Ni2+ 331 3FiN 3+ 211 4F ᎑1 ᎑2 processes M(3dn4s2) → M+(3dn4s1) → M2+(3dn4s0). Their 2+ 2 3+ 3 ᎑ ᎑ ionization energies, with energies derived from wavenumbers Cu 461 DuC 331 F 1 3 of excited states for other atoms (1), fall on curves IA and Zn2+ 502 1SnZ 3+ 461 2D ᎑1 ᎑4 IIA. Chromium (n = 4) and copper (n = 9) lose one s electron in the process M(3dn+14s1) → M+(3dn+14s0) and their first ionization energies form part of curve IB, which is slightly lower than curve IA. Vanadium, cobalt, and nickel from scandium to manganese, decreases to iron, and increases lose two s electrons and gain one d electron in processes to zinc. As with p electrons, discontinuities are convention- M(3dn4s2) → M+(3dn+14s0). Their first ionization energies ally attributed to repulsion between paired electrons of op- form part of zigzag pattern IC, which has a minimum at chro- posite spin in the second half of the subperiod. The energy mium and a maximum at manganese. This is the reverse of of removal of d electrons was discussed by Catalan et al. in zigzag pattern IIB formed by second ionization energies of 1954 (22). Our semiquantitative approach involves pairing atoms that lose one d electron in the process M+(3dn+14s0) and exchange energies as before and incorporates orbital en- → M2+(3dn4s0) and where chromium is higher than manga- ergies (19, 20, 23). nese. Zinc (n = 10) can not contain more than ten 3d elec- Table 3 summarizes numbers of pairing interactions, p2 trons and does not form part of curve IB or zigzag patterns and p3; exchange interactions, e2 and e3; changes on ioniza- IC or IIB. tion, ∆p and ∆e; and term symbols for doubly and triply Third ionization energies from Figure 1 are plotted on charged ions. Figure 11 shows the third ionization energies a larger scale in Figure 11. The systematic loss of one d elec- modified by pairing and exchange energies, P∆p and Eex∆e, tron in the process M2+(3dn4s0) → M3+(3dn−14s0) gives a zig- so that scandium, vanadium, manganese, iron, nickel, and zag pattern that excludes calcium (Z = 20, n = 0), increases zinc form a smooth, almost-linear curve with titanium and

42 25 third ionization energy 40 adjusted for exchange observed and pairing energy A adjusted for exchange, B 38 20 II pairing, and orbital C energies 36

15 34

10 30

I Ionization Energy / eV

Ionization Energy / eV 28 5

0 24 Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Sc Ti V Cr Mn Fe Co Ni Cu Zn Atom Atom

Figure 10. First and second ionization energies of calcium to zinc. Figure 11. Third ionization energies of scandium to zinc. Roman numerals denote the ionization number. See text for explanation of the curves A–C.

942 Journal of Chemical Education • Vol. 80 No. 8 August 2003 • JChemEd.chem.wisc.edu Research: Science and Education cobalt above the curve and chromium and copper below. Third ionization energies involve loss of one d electron These irregularities are the result of changes in orbital en- in M2+(4dn5s0) → M3+(4dn−15s0) transitions. The zigzag pat- ergy, Eorb, since F → D transitions are higher and D → F tern in Figure 1, excluding strontium (Z = 38, n = 0), in- transitions are lower in energy. Application to titanium, chro- creases from yttrium to technetium, decreases to ruthenium, mium, cobalt, and copper gives a nearly linear curve that in- and increases to cadmium. Modification by pairing, exchange, ᎑ creases from scandium to zinc, where P = 4Eex = ±4Eorb = and orbital energies gives a smooth curve increasing from yt- 2.32 eV. Orbital energy changes were not relevant for p elec- trium through technetium and ruthenium to cadmium, trons and cannot be identified for D → S and S → D transi- where P = −4Eex = ±4Eorb = 1.46 eV. tions involving scandium, manganese, iron, and zinc, the first and last members of each arm of the zigzag pattern. Third Transition Period Fourth ionization energies increase from titanium to iron, decrease to cobalt, and increase to gallium. Similar Barium and lanthanum precede the lanthanides and have modification by pairing, exchange, and orbital energies gives lower first ionization energies than other atoms in the third an almost linear curve increasing from titanium to gallium, transition period. Hafnium loses one d electron and then loses ᎑ where P = 4Eex = ±4Eorb = 2.80 eV. two s electrons and gains one d electron in the processes M(5d26s2) → M+(5d16s2) → M2+(5d26s0). Tantalum and Second Transition Period tungsten lose two s electrons and gain one d electron in the process M(5dn6s2) → M+(5dn+16s0); rhenium, osmium, iri- First and second ionization energies of the second tran- dium, and mercury lose one s electron in the process sition period are examined in Figure 12. Strontium, zirco- M(5dn6s2) → M+(5dn6s1); and platinum and gold lose one s nium, technetium, and cadmium lose both s electrons in electron and then one d electron in the processes M(4dn+16s1) processes M(4dn5s2) → M+(4dn5s1) → M2+(4dn5s0). Their → M+(5dn+16s0) → M2+(5dn6s0). Mercury loses two s elec- ionization energies, with energies derived from wavenumbers trons in the processes M(5d106s2) → M+(5d106s1) → of excited states for other atoms (1), fall on curves IA and M2+(5d106s0). IIA. Niobium, molybdenum, ruthenium, rhodium, and sil- ver lose one s and then one d electron in the processes Rare Earth Metals M(4dn+15s1) → M+(4dn+15s0) → M2+(4dn5s0). Their ionization energies form part of curve IB and of zigzag pattern IIB. Table 4 summarizes outer electronic configurations of Yttrium follows the pathway M(4d15s2) → M+(4d05s2) → atoms and ions from the lanthanum and actinium series (2– M2+(4d05s1). The first ionization energy falls near curve IA 4). Eleven pairs of atoms have similar configurations. First and the second on IIB. Palladium loses two d electrons in ionization energies are plotted in Figure 13 against number the processes M(4d105s0) → M+(4d95s0) → M2+(4d85s0). The of post-lanthanum or post-actinium electrons, n, which rep- first ionization energy falls near curve IA and the second resents the position of the atom in the f block. Most actinides forms part of curve IB. Cadmium (n = 10) does not form have higher ionization energies than corresponding lan- part of curve IB or zigzag pattern IIB. thanides.

observed 7.0 20 A B II lanthanum series actinium series 6.5 15

10 6.0

I Ionization Energy / eV

5 5.5 Ionization Energy / eV

0 5.0 Sr Y Zr Nb Mo Tc RuRhPdAgCd 01234567891011 12 13 14 Atom “Ideal” Number of f Electrons in Series

Figure 12. First and second ionization energies of strontium to cad- Figure 13. First ionization energies of lanthanum and actinium series. mium. Roman numerals denote the ionization number. See text for explanation of the curves A and B.

JChemEd.chem.wisc.edu • Vol. 80 No. 8 August 2003 • Journal of Chemical Education 943 Research: Science and Education

Table 4. Ground State Electronic Configurations for the Lanthanum and Actinium Series MM+ M2+ M3+ nZAtom 4df 5s6f4d5s6f4d5s6f4d5s6

075aL012020010 000 185eC112120200 100 295rP302301300 200 306dN402401400 300 416mP502501500 400 526mS602601600 500 636uE702701700 600 746dG712711710 700 856bT902901900 800 966yD0102101 0 100 0 900 170 6oH1102101 1 100 1 100 0 181 6rE2102101 2 100 2 100 1 192 6mT3102101 3 100 3 100 2 103 7bY4102101 4 100 4 100 3 114 7uL4102102 4 101 4 100 4 5df 6s7f5d6s7 098cA012002 109hT022021 219aP212 329U 312 439pN412 549uP602 659mA702 769mC712 879kB902 989fC0102 190 9sE1102 101 1m0 F2102 112 1d0 M3102 123 1o0 N4102 134 1r0 L4112

Table 5. Pairing and Exchange Interactions and Term Symbols for Ions of the Lanthanum Series

Ion p2 e2 Tnerm Io p3 e3 Term ∆p ∆e

La2+ 002FaL 3+ 001S00 Ce2+ 013HeC 3+ 002F0᎑1 Pr2+ 034IrP 3+ 013H0᎑2 Nd2+ 065IdN 3+ 034I0᎑3 Pm2+ 001 6HmP 3+ 065I0᎑4 Sm2+ 051 7FmS 3+ 001 6H0᎑5 Eu2+ 012 8SuE 3+ 051 7F0᎑6 Gd2+ 112 7FdG 3+ 012 8S ᎑1 0 Tb2+ 222 6HbT 3+ 112 7F ᎑1 ᎑1 Dy2+ 342 5IyD 3+ 222 6H ᎑1 ᎑2 Ho2+ 472 4IoH 3+ 342 5I ᎑1 ᎑3 Er2+ 513 3HrE 3+ 472 4I ᎑1 ᎑4 Tm2+ 663 2FmT 3+ 513 3H ᎑1 ᎑5 Yb2+ 724 1SbY 3+ 663 2F ᎑1 ᎑6 Lu22+ SuL 31+ S

944 Journal of Chemical Education • Vol. 80 No. 8 August 2003 • JChemEd.chem.wisc.edu Research: Science and Education

28 third ionization energy 16 adjusted for exchange observed 27 and pairing energy A adjusted for exchange, 26 14 B pairing, and orbital C energies D 25 II 12 E 24

23 10 22

8 21 Ionization Energy / eV

Ionization Energy / eV 20 I 6 19

4 18 La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Atom Atom Figure 14. First and second ionization energies of lanthanum series. Figure 15. Third ionization energies of lanthanum series.

Lanthanum Series bium giving a distorted zigzag pattern. These energies derived for excited states of lanthanum and gadolinium (3) differ First and second ionization energies of the lanthanum slightly from ground-state ionization energies in Figure 1. series are examined in Figure 14. Filled circles represent NBS Table 5 summarizes numbers of pairing interactions (National Bureau of Standards) values (3, 4). Other symbols (pairs of electrons occupying the same orbital), p2 and p3; represent ionization energies derived from wavenumbers of numbers of exchange interactions between pairs of electrons excited states for neighboring atoms (3). Most atoms lose both having parallel spins, e2 and e3; changes on ionization, ∆p s electrons during processes M(4f n+15d06s2) → and ∆e; and term symbols for ions of the lanthanide series. M+(4f n+15d06s1) → M2+(4f n+15d06s0). Their ionization en- Corrections for exchange and pairing energies remove the ergies fall on curves IA and IIA with derived energies for lan- principal discontinuity but as with d electrons, orbital ener- thanum, cerium, and gadolinium but not lutetium (n = 14). gies must be considered (19, 23). Corrections involving H Gadolinium loses both s electrons during the processes → F transitions, Eorb1, and I → H transitions, Eorb2, and cor- M(4f 75d16s2) → M+(4f 75d16s1) → M2+(4f 75d16s0) and ion- responding negative transitions produce a surprisingly smooth ᎑ ization energies fall on curves IB and IIB with energies de- curve, where P = 6Eex = ±4Eorb1 = ±3Eorb2 = 2.625 eV. Esti- rived from ref 3 for most atoms from terbium to lutetium. mated experimental errors for third ionization energies in- The A and B curves increase across the series and are approxi- clude ±0.4 eV for Pm and ±0.3 eV for Nd, Sm, and Dy (3). mately parallel with differences that might depend on shield- Fourth ionization energies of the lanthanides show a ing by 6s or 4f electrons. similar distorted zigzag pattern. Fourteen atoms from cerium Lutetium has filled 4f orbitals and loses one d electron through lutetium but not lanthanum (n = 0), lose one f elec- followed by one s electron in M(4f145d16s2) → tron in M3+(4f n5d06s0) → M4+(4f n-15d06s0) transitions. Cor- M+(4f145d06s2) → M2+(4f145d06s1) transitions. These ion- rections for pairing, exchange, and orbital energies produce ᎑ ization energies fall on curves IC and IIC (approximately par- a reasonably smooth curve when P = 6Eex = ±4Eorb1 = ±3Eorb2 allel to IIA and IIB) with energies derived from wavenumbers = 2.6 eV, despite estimated experimental errors of ±0.7 eV (3) for erbium, thulium, and ytterbium. Lanthanum and ce- (Sm and Gd), ±0.6 eV (Pm, Eu, and Ho), and ±0.4 eV (Nd, rium lose two s electrons and gain one d electron in Dy, Er, and Tm). M(4f n5d16s2) → M+(4f n5d26s0) transitions. Their ionization energies fall on curve ID with derived energies for praseody- Actinium Series mium and neodymium. Lanthanum loses secondly one d Electronic configurations of lanthanum and actinium are electron and falls on curve IID with derived energies for ce- similar for atoms but different for ions, and actinium has the rium, praseodymium, and neodymium. Cerium loses sec- lower ionization energy. An irregular pattern from thorium ondly two d electrons while gaining one f electron and forms to neptunium and americium and curium in Figure 13 sug- part of reverse zigzag pattern IIE with derived energies for gests that cations have different configurations. Ionization lanthanum, praseodymium, and neodymium. energies of plutonium and berkelium to nobelium fall on a Third ionization energies in Figure 15 refer to loss of smooth curve in the second half of the series that is approxi- one f electron in M2+(4f n+15d06s0) → M3+(4f n5d06s0) tran- mately parallel to the curve from europium to ytterbium and sitions. They increase from lanthanum to praseodymium, flat- suggests loss of one s electron in M(5f n+16d07s2) → ten to promethium, increase to europium, and decrease to M+(5f n+16d07s1) transitions. We are unable to find a reliable gadolinium with similar behavior from gadolinium to ytter- ionization energy for lawrencium.

JChemEd.chem.wisc.edu • Vol. 80 No. 8 August 2003 • Journal of Chemical Education 945 Research: Science and Education

Conclusion 24. Series, G. W. Spectrum of Atomic Hydrogen; Oxford University Press: Oxford, England, 1957; Chapter 3. This paper reviews the ionization energies of atoms and 25. Hertzberg, G. Atomic Spectra and Atomic Structure, 2nd ed.; atomic ions in the s, p, d, and f blocks of the periodic table. Dover: New York, 1944; Chapter 1. The apparently irregular behavior of the first and second ion- 26. Sanders, J. H. The Fundamental Atomic Constants; Oxford Uni- ization energies across the transition-metal and rare earth versity Press: Oxford, England, 1961; Chapter 2. periods is explained in terms of electronic configurations of 27. Born, M. (revised Blin-Stoyle, R. J., Radcliffe, J. M.) Atomic the atoms and ions. A semiquantitative treatment of pairing Physics, 8th ed.; Blackie: London, 1969; Chapter 4. energies, P, exchange energies, Eex, and orbital energies, Eorb, 28. Sommerfeld, A. Atomic Structure and Spectral Lines; Methuen: explains discontinuities at half-filled p, d, and f electron shells. London, 1934; Chapter 2. Some of the great advances of the last century in the 29. Heisenberg, W. Physical Principals of the Quantum Theory; Do- understanding of atomic spectra and ionization energies are ver: New York, 1930. summarized in the Appendix at the suggestion of a reviewer. 30. Atkins, P. W. Molecular Quantum Mechanics; Oxford Univer- sity Press: Oxford, England, 1970; Chapter 8. Acknowledgments 31. Bethe H. A.; Salpeter, E. E. Quantum Mechanics of One and Two Electron Atoms; Plenum: New York, 1977. We thank C. D. Flint and P. J. Heard for helpful dis- cussions. Appendix Literature Cited In the late 19th century, it was shown that atomic spectra consist of discrete lines rather than continuous emission 1. Moore, C. E. Atomic Energy Levels; NBS Circular 467; U.S. or absorption. Paschen, Bracket, Pfund, and others made no- Department of Commerce: Washington DC; (a) 1949; Vol. table contributions (24), and Balmer identified four lines as 1. (b) 1952; Vol. 2. (c) 1958; Vol. 3. members of a converging series (25). Rydberg discovered the 2. Moore, C. E. Ionization Potentials and Ionization Limits De- constant that bears his name (26). At the beginning of the rived from the Analyses of Optical Spectra; NSRDS-NBS 34; 20th century, Bohr proposed that electrons in an atom oc- U.S. Department of Commerce: Washington DC, 1970. cupy discrete energy states and that radiation is emitted or 3. Martin, W. C.; Zalubus, R.; Hagan, L. Atomic Energy Levels– absorbed on transition from one discrete (quantum) state to The Rare Earth Elements; NSRDS-NBS 60; U.S. Department another (27). of Commerce: Washington DC, 1978. The energy change, ∆E, for hydrogen or a hydrogen- 4. Handbook of Chemistry and Physics, 81st ed.; Lide, D. R., Ed.; like ion is given by the following expression, where R is the CRC: Boca Raton, FL, 2000. Rydberg constant, Z is the atomic number, and n1 and n0 5. Reader, J. J. Opt. Soc. Am. 1975, 65, 638. are integers: 6. Reader, J.; Epstein, G. L. J. Opt. Soc. Am. 1976, 66, 590. 111 7. Gillespie, R. J.; Moog, R. S.; Spencer, J. N. J. Chem. Educ. E RZ2 ∆ = 2 − 2 1998, 75, 539–540. n1 n0 8. Ahrens, L. H. Ionization Potentials; Pergamon: Oxford, En- gland, 1983. Agreement between theory and experiment was well within 9. Pilar, F. L. J. Chem. Educ. 1978, 55, 2–6. the limits of error in the measurement of atomic constants 10. Rioux, F.; DeKock, R. L. J. Chem. Educ. 1998, 75, 537–539. at that time. Sommerfeld and others developed general laws 11. Lang, P. F.; Smith, B. C. Inorg. Nucl. Chem. Letters 1981, 17, of atomic spectroscopy (28). 27–29. Further advances came from the development of wave 12. Agmon, N. J. J. Chem. Educ. 1988, 65, 42–44. mechanics and matrix mechanics (29). The Schrodinger equa- 13. Ali, M. E. S.; Lang, P. F.; Smith B. C. J. Chem. Soc., Faraday tion can be solved exactly only for one-electron ions. Approxi- 2 1984, 80, 1089–1091. mate methods were used to determine energy levels and 14. Born, M. (revised Blin-Stoyle, R. J.; Radcliffe, J. M.) Atomic ionisation energies in other systems. In 1930, Slater devised Physics, 8th ed.; Blackie: London, 1969; Chapter 6. an approximate method for estimating the extent to which 15. Smith, D. W. J. Chem. Educ. 1975, 52, 576–577. electrons are shielded from the nucleus (30). He assumed that 16. Condon, E. U.; Shortley, G. H. Theory of Atomic Spectra; Cam- each electron is situated in the field of an effective nuclear bridge University: Cambridge, 1955; Chapter 7. charge, Z *, and that an effective quantum number, n*, could 17. Atkins, P. J. Quanta: A Handbook of Concepts, 2nd ed.; Ox- be assigned to each electron. The energy required to ionize ford University: Oxford, England, 1980; p 113. an electron, I, could be estimated as: 18. Blake, A. B. J. Chem. Educ. 1981, 58, 393–398. 2 19. Johnson, D. A. Some Thermodynamic Aspects of Inorganic Z * IR= Chemistry, 2nd ed.; Cambridge University: Cambridge, N * England, 1982; Chapter 6. 20. Cann, P. J. Chem. Educ. 2000, 77, 1056–1061. Slater’s rules are based on simple assumptions about 21. Lang, P. F.; Smith B. C. Educ. Chem. 1986, 23, 50–53. shielding and effective nuclear charge. They are unable to ac- 22. Catalan, M. A.; Rohrlich, R.; Shenstone, A. G. Proc. Roy. Soc. count for the zigzag patterns of ionization energies illustrated 1954, 221A, 421–437. in Figure 1. More sophisticated equations allow ionisation en- 23. Bills, J. L. J. Chem. Educ. 1998, 75, 589–593. ergies in multielectron atoms to be calculated (31).

946 Journal of Chemical Education • Vol. 80 No. 8 August 2003 • JChemEd.chem.wisc.edu