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I Textbook Errors

3 8

GUEST

UTHOR

R N Keller

University of Colorado

Energy Level Diagrams and

Boulder

Extranuclear Building

o

the Elements

Simplified diagrams showing the ap-

proximate order of electronic energy levels in atoms and

mnemonic devices to aid in predicting electronic con-

figurations 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 state-

ments continue to appear in standard inorganic chemis-

try 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.

Energy

Level Diagrams

Figure

1

is an example of s common energy level

diagram which is uscd to explain why, in building the

1

Figure

1

The approximote

sequen e

of energierand stabilities

for

atomic orbitah

electron clouds of atoms, the filling of available orbitals

is not completely in accord with the order allowed by

the Pauli principle

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 potas-

sium and calcium even though t,heM

(n

3 quantum

Suggestions of material suitable

for

this column and guest col-

umns suitable for publication directly

are

eagerly solicited. They

should be sent with

s

many details

as

possible and particularly

with references to modern textbooks to Karol J Mysels Depart-

ment of Chemistry LTniversity of Southern California

Loe

Angeles

7

California.

Since the purpose of this column is to preve nt the spread and

continuation of errors discussed and not the evaluation of indi-

vidual texts the source

of errors

discussed will not be cited. The

error muat occur in s t

l e ~ t

wo independent etandard textbooks

to be presented.

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 in-

formation can be obtained from a mnemonic scheme

such as that illustrated in Figure 2.

Figure

2.

Order of

occupancy of otomic

orbitair

Since many treatments of the electronic building of

atoms stop at this point, it is only natural for the

student to assume tha t the order of addition of elec-

trons as predicted by these figures is the reverse of their

order of stabilities or tightness of binding. Tha t is, in

the case of scandium, for example, since the

3d

elec-

tron 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

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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

3 d

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

4 s

electrons. In the case of titanium, the

19th and 20th electrons added are in the

3d

level and

the 2ls t and 22nd electrons are in

4 s

states.

I .

1-4 - 0

Ifomlo number

Figure

3

pproximala 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 in-

creasing 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 s level in energy. The deficiencies of Figure

3

arc to some extent corrected by Figure

4 .

This figure

8.

) shows calculated values of electron energies as

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 be-

tween 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

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

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 arrange-

ment which also results in the electrons being bound

most tightly by the field of the nucleus. In 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

configura-

tion. However, the Sc+ ion is more stable if the 20th

electron occupies a

4 s

orbital rather than a

3 d ;

or,

a

3d14s

state is more stable for this ion than a

3d2

or a

4 s 2

st,ate. Similarly the lowest lying state for neutral

scandium,

3d14s2 ,

s a more favorable state energetically

than any other state such as

3 d 3 ,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+=) o 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 Sc f2 is now slightly different from tha t for SC + ~ .

The interaction of the nucleus of scandinm with the

19 electrons in S C + ~roduces a field which favors a

4 s

state for the 20th electron added (i.e., Sc+? e- -+Sc+).

The resulting field in turn favors a

4 s

state for the 21st

elect.ron added

Figure 4.

Colculmtcd energies of atomic energy levels

or

o fundion of

otomicnumber [adapted

fram References

I81

ond

1911

29

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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 4 p 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 an-

ticipated from t h ~ srend, m S C + ~he energy of the

3d

level is now lower than the 4s level. As a consequence,

the ground sta te of the 19th electron in Sc f2 s no longer

a

4s

state, as i t is in I< and Ca+, hut is a

3d

state. It

is interesting in this connection t ha t as early as 1921

Bohr had come to s im~lar onclusions from the t rends

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 com-

p0sit.e of individual energy level diagrams for the

ground stat e of each atom, is worthy of careful study.

The lowering of each level ~vith ncreasing 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~elyhe

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 pre-

dicted from the char t; this order is the same as the

order of the electrons in the vertical column correspond-

ing to a specific value of the atomic number, commenc-

ing with the highest electron and proceeding downward.

Changes in Conflgurotion ccompanying 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 I+', the ground state con-

figurations are respectively: Rd24s2.3d24s1, d2,3d1,3d0.

With some of the transit ion elements, however, changes

in ground state configurations accompany ionization

(13). For example, d i l e the ground state configura-

tion for neutral vanadium is 3d34sZ,singly ionized

vanadium has a ronfiguration 3d4 instead of the ex-

pected 3da4s1. Other examples include the following:

Co, 3d74s2

-

CO+, 3d8 e-; Xi, 3d84s2

-

Nif,

3ds

e-: La,

.idlGs?

La+,

5dZ

e-.

These ex-

amples 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.

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 quan tum state s: one is

the mis-use of over-simplified energy level diagrams;

the other is the manner in which the Aufbau or Building-

up Principle ( 9) s applied.

Simplified electronic energy level diagrams should bc

used judiciously in teaching the elertronic building of

the atoms. These diagrams are usefnl in indicating

th at t he building process is not controlled solely by the

major quantum number n , hu t 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

d

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 cns-

tomary 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 it s 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 tha t if t he 4s

level was already filled at 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 en try of the

electrons yet admitt ing of another order of t,heir removal

(the experimentally observed order) by ionizat,ion.

Complete parallelism between the order of ent ry 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 in to energy levels about

a

bare nu cln ~s ntil the

neutral atom in its ground state is oht,ained, or the

process of adding a proton and an electron to the pre-

ceding 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,

th at 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 ~vould mmediat,ely rear-

range into a

3d14s1

configuration, which is the ground

sta te 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

3d14s2 onfiguration 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 ap-

proach is used in the discussion of the electronic building

of the atoms, the order of successive removal of elec-

trons in the formation of ions among the transition type

elements will surely appear paradoxical t,he perceptive

student.

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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, M IT ,

for

reading the

manuscript and making certain t,hat violenrn had not.

brm

committed against n rr rp trd t,hnories.

Litemture Cited

1 ) GLASSTONE,

.,

Textbook of Physical Chemistry, 2nd ed.,

D. Van Nostrand Co., Inc., New York, 1946.

2 ) HERZBERG, ., Atomic Spwtra and Atomic Structure,

2nd ed., Dover Pohlirat~ons,New York, 1944.

3) WH~TE,.

E.,

Introdur tion t o Atomic

Spectra,

McGraw-

Hill

Book

Ca., h e . ,

ca

>-ark, 1934.

(4) SWINEHART,). F.,

J .

CHEM.

EUTC.,7 6 2 2 4

(1950)

(5) PAI~LISG,., The Nature

of

t,he Chemioal Bond, 3rd ed.,

Carnell University P r m ,

1960

1)&ge9 7 and 580.

(6) HERZBERC,p a t . Chap.

:

7 )

I b d P. 151.

8)

PAITLING,

p czt .

p. 56.

9) LATTER, .,

Phys

Rev., 99 510 (1955).

(10) WH~TE ,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,

).,

J. CHEM.EDVC. ,1 575-81 (1944).

(13) MOORE,

C. E.,

At,ornic Energy Levels, Circular

of

the

National Bureau o i Standard s 467, Vol. I, 1949; Vol.

11.

1952: Val. 111 1958.

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