Presenting the Bohr Atom Blanca L. Haendleri Lafayette College, Easton, PA 18042

A discussion of the Bohr model of the hydrogen atom is included in many freshman chemistry courses and in some physical chemistry courses. In recent years, however, this discussion has been given low priority and is frequently very limited. The last major article in THIS JOURNAL which deals with Bohr's theory appeared in 1945 (1 ),and there have been only a few short articles since ( 2 4 ) . One reason for this may he a belief that the theory is simply not important enough to merit more attention. In my opinion, Bohr's ideas are too crucial to the development of quantum mechanics and too potentially valuable in introducing students to this area of studv to be eiven onlv a token treatment.

noth her Teason fo; neglect of the Bohr theory may he an awareness that manv students find it ~articularlv frustratina to study because iiseems rather arbitrary a n d pointless. i suggest that an historically accurate presentation which stresses the development of Bohr's ideas will he much more effective than what is commonly done. I t is vital to give stu- dents a sense of how Bohr arrived at ideas which to them (and to Bohr's contemporaries) seem quite fantastic.

I t is frequently claimed that presenting ideas as they were actually developed tends to confuse students, and that con- cents should be arraneed for maximum claritv. I t mav be e q h y true that this a i~roach, which is partic&ly common a t the introductory level, misleads students into an overly simplistic view of the subject which is very hard to overcome later on. I t may better to try to strike a balance which retains as much historical accuracy as possible without sacrificing essential claritv. This balanced amroach mav be more diffi- cult and time-consuming to achi&e but can also be superior to either extreme.

In the case of the Bohr theory, the usual approach has the characteristics of brevity and apparent simplicity, but pro- duces little understanding of Bohr's ideas or their significance. This is particularly apparent in the "derivation" which pro- duces the formula for the enerw and radii of the states. ~~~~~~~ ~ ~ ~

Judging from standard textboogat both introductory and more advanced levels (54, the almost universal approach to this involves expressing the Coulomb force in terms of the centripetal acceleration and combining this with the expres- sion for the quantization of angular m o m e n t ~ m . ~ The diffi- culty arises from the "postulate" that angular momentum is quantized. Angular momentum is a very difficult concept for teachers to present and for students to grasp. I t is therefore a singularlyunfortunate choice for one of the key concepts in the derivation. I t is a choice, however, since Bohr did not in fact postulate this a t all. His original paper (10) very clearly shows that the quantization of angular momentum was not his startine ooint. but was derived after the formula for the -. . energies had been arrived at. The feeling expressed by many students that this "postulate" seems particularly capricious is thus intuitively correct. This objection has been raised he- fore (11). hut it seems to have been virtually ignored.

I would also argue for the integration of some purely his- torical and biographical material into the presentation. Stu- dents frequently comment that chemistry as a subject, aside from being difficult, is also cold, impersonal, analytical and dry. Perhaps this results a t least in part from the fact that we have succeeded so well in divorcing our science from the men - and women who created it.

372 Journal of Chemical Education

Faraday: Electrolysis

Crookes tube: Cathode rays

Measured &

Measured e Rutherford: Gold Foil Experiment: discovery of nucleus

Atomic Bohr: Spectrum of Model of the Hydrogen

Heisenberg: Uncertainty

Idea of the quantdm

Photoelectric effect; E = hv

Wave-Particle duality; A = hlmv

Schriidinger: Quantum mechanics; probability

The role of the Bohr theory in the development of quantum mechanics.

The development of quantum mechanics is particularly rich in good historical material. There are several very interesting individuals to focus on, not the least of whom is Niels Bohr. There are hiographies3 available (12, 13) as well as hooks which discuss the history of quantum mechanics in general (14-16). Whatever source hook is chosen, two overwhelming impressions are received. One is that Bohr's interests and genius were phenomenally broad. The other impression is of

Present address: Clorox Technical Center, P.O. Box 493, Pleas- anion, CA 94566.

This is ofien done incorrectly (9). Unfortunately, Bohr wrote very little about his own life and work.

A bibliography is included in reference ( 12). ' The discussion in this section is adapted directly from Bohr's original paper and from the book by Jammer ( 14) which traces the development of Bohr's ideas in detail.

Bohr's stature as a human heing. As his son has written: "However great he was as a scientist and however profound his insight into life itself, for us it was as a human heing that he was greatest" (1 7). If our students do not find Bohr's theory precisely inspiring, perhaps they can he inspired by Bohr himself.

The Development of Bohr's Ideas4 There are two important aspects of the development of

Hohr'i theory. The ti& is his dei~t to the ideas of othrrs. Once the stareerinp. implications oi Rutherford's gnld foil experi- ment (18) hadhecome obvious to the commudty of physicists, the development of a reasonable atomic model became a central problem in physics. I t seemed to Bohr, newly arrived a t Cambridge from Denmark with his doctorate to study under the ereat J. J. Thomson. that the solution to the di- l&&a layUin a bold and daring approach. He thought the answer might lie with Planck's idea of the quantum (19) which had been so brilliantly applied by Einstein (20) in his expla- nation of the experimental results observed by Lenard (21) which are known as the photoelectric effect.

Unfortunatelv. J. J. Thomson found Bohr's revolutionary ideas hard to accept. Sources disagree on the amount of fric- tion hetween the two men, hut in March, 1912, Bohr left Cambridge to srrk an intellectual rlimate more suited to his unusual ideas. His very fortunate choice was Rutherford's lahuratnn in Manchester. Althourh Rutherford was initialls "

suspicious of Bohr because he was a theoretician, he appar- ently accepted him (partly because of his skill as a soccer player) and gave him genuine encouragement. The congenial atmosphere was of crucial importance. A colleague has written, "Niels Bohr simply could not work if he did not find in his closest environment the most complete harmony and under- standing" (22) Yet it is also clear that he was essentially on his own as he worked on his theory, once complaining that nobody else a t the laboratory was really interested in funda- mental theoretical problems (23). I t was here in Manchester that Bohr began to develop the ideas which led to his land- mark paper "On the Constitution of Atoms and Molecules" (10) which was published after his return to Copenhagen in 1917 &"AV.

The second aspect of the development is Bohr's own mental progression. Why should he have chosen Planck's idea of the ouantum as the wav to eo? Bohr thoueht that the "solar" iode l of the atom, ihere'the electrons circled the nucleus like the planets around the sun, sounded promising. There had heen unsuccessful attempts to developsuch a model already (24). Bohr felt that any theory which tried to explain the stability of a solar atom would yield a constant which has the dimensions of length and would describe the distance of the electron from the center of the stable orbit. Bohr recognized that by combining the mass and charge of the electron with h, Planck's constant, he would obtain an expression which has the dimension of length

h2 -= (6.63 X 10-27 e r g - s e ~ ) ~ = 20.9 X 10@ crn

me2 (9.11 X g) (4.80 X lo-' which is approximately the right order of magnitude for an atomic dimension.5 This idea may seem rather sketchy, hut it can he nresented as an indication to Bohr that his ideas were ~. ---- -~

& ~ - ~ ~ ~ ~~~ ~

tending towards a fruitful approach, and as an interesting insight into a great theorist's methods of prohlem-solving.

Bohr was also guided in his thinking by a large body of ex- oerimental evidence which should relate directly to atomic structure, hut which in Bohr's day was in such confusion that many scientists had despaired of ever making any sense of it. By Bohr's time, the idea that atomic spectra could provide a key to the structure of the atom and that the electrons were somehow responsible for producing the emitted light was generally accepted, yet the precise connection could not be found. Much attention was focused on the line spectrum of hydrogen, since i t gave the simplest characteristic pattern.

Bohr simplv and darinelv proposed that electrons do not ra- -. . . diate eneigi as they orbit the nucleus, but exist in states of constant enerm which he called stationary states. They can only change their energy by undergoing a transition from one stationarv state to another. (Bohr said nothing ahout how this transition was accomplished.) During the transition they ei- ther absorb or emit an amount of energy which is exactly equal to the energy difference between the two states. This behavior produces the characteristic spectral lines.

Derivation of the Energies of the stationary StatesB What Bohr needed to do was to develop a formula for the

en tq i r s of the statlonary stiltes whirh adulrl agree with the exuerirnentallv ohscn,ed surctral frequenrte.;. Hl, decided that although classical electrddynamicshad failed to produce a successful atomic model, perhaps classical mechanics might still apply. Bohr postulated elliptical orbits for the electrons, similar to the orbits of the planets, hut the argument is the same for the circular orbits and the mathematics is easier to follow. In the following derivation, I have used the same symbols that Bohr used in his paper.

For a body experiencing a force of attraction which is cen- tral. a circular orbit is stable. If the radius of the orbit is des- igniited hy a , the electrnstiltic forre ul'attrnctim is given by ('oulmnb'i law n i &a2. \ r h r r t . ~ and E are thr charreson the electron and nucleus, iespectively. This force produces a centripetal acceleration7 equal to u'la, where u is the velocity of the electron. Since force is the product of mass and accel- eration, we can write

The potential energy of the system, V, may be found by in- tegrating the expression for the Coulomb force. This energy is zero when a = a, so the potential energy at a is acquired by moving the electron in from infinity to the distance a. The integral is then

eE

a (2)

We can now write an expression for the total energy of the svstem, U , which is the sum of the kinetic ( T ) and potential energies:

From eqn. (1) we have that

Since this quantity represents the total energy of the or- hitinz electron. an amount of enerev eaual to this amount wouli have to be supplied in orde;to remove the electron comoletelv from the attractive null of the nucleus. Bohr called thisquantity the binding energy, W, so we have

These are today's values. Theones available to Bohr were some- what less accurate. (1 esu = 1 ~m~'~g"~sec- ' . )

The following derivation is suitable for presentation at the level of undergraduate physical chemistry courses. it is not intended that it be included in standard introductory courses; however, several crucial aspects of the development of Bohr's ideas are also described in this section. The derivation has been adapted directly from Bohr's paper with the aid of the book by Shamos ( 2 5 .

A derivation of the centripetal acceleration may be found In ref- erence l26l or lZi7. . . , .

The use of the prime notation is to distingufsh the rotational fre- quency from the frequency of the light. Bonr used u for both.

Volume 59 Number 5 May 1982 373

The next step is to develop a relationship between W and w, the frequency of rotation. The actual speed, v, with which the electron travels is related to this frequency as

Now substituting eqn. (7) into eqn. (6) we have

To eliminate a from this expression, we use a = eEI2W (from 6) and substitute to get

or

fi W 312 w=- eE&

(10)

So far, all of this was nothing more than classical (Newto- nian) mechanics, but now Bohr faced the problem of incor- porating Planck's quantum concept into his model. He knew that only certain values of W (and hence also of o and a) could be possible because of the existence of discrete spectral lines. Bohr postulated, as Planck had for black body radiation, that the energy emitted (or absorbed) by the electrons was equal to rhu'. where r is an inteeer. h is Planck's constant. and u'is ~ ~

a frequ& characteristirol;he electron's rotational motion and is not necessarilv the frenurncvof the emitted l i ~ h t . ~ The problem then became one of'relaGng this to W and"w.

Bohr started by considering a "free" electron which was completely removed from the nucleus and allowing it to un- dergo a transition to a stable orbit with binding energy W. Since the electron must radiate energy equal to the difference in energy between its initial and final states, and since the energy of the free electron is taken to he zero, the energy ra- diated must equal W, or

W = rhu' (11)

In Planck's model u'is the freauencvof oscillation, but Bohr argued that in his case of a transition between two states, the appropriate freauenw should be the aueraee of the rotational .. . frequencies in the two states. However, s&e he was consid- ering his initial state to he the free state with w = 0, the aver- age would be the final rotational frequency w, divided by 2. Thus we have

rho w = - 2

(12)

This argument seems rather contrived, and in fact it is. Bohr was trvina to make classical ideas such as circular orbits and rntati~~iaifrequencies fit where they redly could not fit nt all. It is at this point that the crucial importance of atomic spec- troscopy must be recognized. ~ c c o r d i n ~ to Jammer (28), ~ o h r was unable to arrive at a successful theory until he saw Bal- mer's formula for the hydrogen spectral lines and began working backward until he got his "fudge-factor" of 2 in eqn. (12). The amazing fact is that Bohr was not familiar with Balmer's work until well after he had returned from Man- chester to Copenhagen. In January of 1913 spectroscopist H. M. Hansen visited Bohr there, and asked him how his new atomic model could explain the spectroscopic results. Bhor's initial response was that he had not seriously considered the question, believing such an explanation to be impossibly complicated. At Hansen's insistence Bohr acquainted himself with the details of Balmer's work and soon had the answer he was looking for. As he repeatedly stated, "As soon as I saw Balmer's formula, the whole thing was immediately clear to me" (28).

Once eqn. (12) is arrived at, the rest follows easily. If we now substitute eqn. (10) into eqn. (121, we have

and

For hydrogen, where E = lei this becomes

The characteristic radii may be calculated from eqns. (6) and (12)

It should be pointed out that this expression contains (aside from dimensionless constants) precisely the function h21e2m which Bohr had originally recognized as having the dimen- sions of length.

The great success of Bohr's theory was its ability to predict correctly the spectral frequencies which were experimentally observed for hydrogen. For a transition between two sta- tionary states, light corresponding to their energy difference should be emitted. From eqn. (15), this difference is

and

where u here is the frequency of the emitted light and is not generally the same as the frequency of electron motion re- ferred to earlier. The subscripts i and f refer to initial and tinal states. A comparison of this equation with the more general Rydherg equation (usually expressed in wavenumbers rather than frequency) shows that they are identical in form and that

Using today's values we have

= 109,760 cm-I The experimentally determined value of the Rydherg constant is 109,677.581 ~ m - ' . ~

I t is worth deriving one more relationship from Bohr's equations. The angular momentum, M, for the eletron in its circular orbit would usually be expressed as mva, but this can he rewritten as follows

Since T = Wand W = ~hwl2, we have

which is the quantization of angular momentum which is so often presented as one of Bohr's postulates. It is clear from his paper that he derived i t as above from W = rhw12, although the two expressions are mathematically equivalent.

Bohr's agreement was not quite so good, again due lo the inaccu- rate values available to him, but was stll impressive. The main dis- crepancy between me two values is due to the fact that Bohr's treatment neglects nuclear motion. Taking nuclear motion into account leads to the replacement of the electron mass by the reduced mass p. Using this value in eqn. (19) gives Fin = 109,700, which is much closer to the exDerimental value.

374 Journal of Chemical Education

The Bohr Atom in the Chemistry Curriculum

In presenting the Bohr atom a t the freshman level i t is im- portant to describe the development,^ leading up to Bohr's work and to show as far as oossible how his own ideas grew. Thiican he dune by strrsiin; I'lanck'scontrihution to thiidea uf the suantum nnd Einstr~n's work on the photoelectric ef- fect. 1t is necessary to present a sufficient background in this material in order to make the development of Bohr's ideas seen reasonable, and the importance of these contributions cannot be overemphasized. However, i t is neither necessary nor desirable to exnlain the ohenomenon of hlackbodv ra- diation. This topic seems very confusing to students and is not really needed for an understanding of the true importance of Planck's work to the development of quantum mechanics, which is the idea of the quantum. If Planck's contrihution is presented as introducing the revolutionary idea that energy is not distributed continuouslv but is in fact limited to bundles of magnitude proportional to h , and i t is then shown how Einstein applied this idea to explain the photoelectric effect (which mist students find mudh less confusing than black- hodv radiation), an effective presentation will be accom- plished.

I t is also essential to present a discussion of the atomic spectrum of hydrogen, and to point out that Bohr's funda- mental postulate was prohahly suggested to him by the fact that the spectral emissions were discrete rather than contin- uous.'0 Then it can be shown how these concepts are all in- cor~orated into the model of the orbiting electron and the - postulate of discrete energy levels with transitions occurring only between these states. The formula for the energies of the stationary states can be presented and the comparison with Rydberg's equations made. I t should he explained that Bohr used an almost chance encounter with Balmer's formula to break an impasse in his thinking and that this breakthrough allowed him-to complete his theory.

It is obviously beyond the scope of almost any freshman course to present a derivation similar to the one in the previous section. I t seems equally pointless to present the "derivation" found in many freshman texts since my experience has been that most students find it arbitrary and meaningless. It also confuses any attempt to explain the development of Bohr's ideas, since it does not, in fact, follow this development. I t is preferable to state the results and explain that the mathe- matics is not appropriate to the course level. If it is necessary for further work that the students know how angular mo- mentum is quantized, it should be explained that the rela- tionship can he derived from the energy formula. If an in- structor decides that it is desirable to do the angular mo- mentum "derivation," one can emphasize that it follows Bohr's ideas in reverse order for mathematical simplicity, and was not in fact his actual approach.

In nhvsical chemistrv courses. in addition to the aualitative ~~~ .~ , ~~ ~

presentatam of the develupmtntif ~ o h r ' s ideas, his hwiration chr~uld he included. Since i t involve.; man\, e u u n t i ~ ~ ~ ~ s and i~ not readily available in texts, it should he handed out to the students so that thev can follow the discussion without the pressure of having to copy it down.

There are several reasons for including this. Modern quantum mechanics, as well as most theoretical chemistry, is confusing and frustrating to students for two main reasons. First, there is almost nev& time to go into the development of these theories so that students can understand where the ideas come from and how thev are expressed in mathematical terms. Second, it is the apparent nourelatiou of quantum mechanics to the macroscooic phvsical world which students find particularly difficult td accept. The derivation presented here can be followed reasonably well by the average student with a year of physics and calculus. This affords the students the opportunitv to experience the working through of a theory . . with a reasonnlh invc>;ttnent ot time. I t mn tw pointed out that thr thwry combines a physic id model of electrun mutim

with certain theoretical postulates and thus occupies a unique position as a transition from the comfortable world of classical physics where physical analogies are possible to the uncom- fortable world of quantum mechanics where they are not.

I t is certainlv necessarv for any discussion of further ad- vances inquantum mechanics to make students at 11~1th 1w.t.k aware of the failure oiBohr's theow to dw:ril)e correctly any systems with more than one electron and to point out the limitations of the electron orbit model. Students may also benefit from learning that the same aspects of the theory which seem arbitrary to them were also criticized by many of Bohr's colleaeues and that acceotance was hv no means im- mediate. ow ever, this information should be balanced by stressing the positive aspects of Bohr's contrihution, since the impression that the study of Bohr's work is pointless may arise from an overemphasis on the failed aspects of his theory. The agreement with Rydberg's equationwas considered an as- tounding achievement even by skeptics. A discussion of the work of Franck and Hertz (29) a t an appropriate level will familiarize the student with the first experimental verification of Bohr's ideas.

I t should he pointed out that no one was more aware of the defects in his theory than Bohr himself, and that he presided over many of the further developments in quantum mechanics as the director of the Institute for Theoretical Phvsics in Co- penhagen. The role he played in nurturing these ideas in the minds of those who followed him is perhaps as crucially im- portant as his own first step. A flow chart similar to the one shown in the figure can he very helpful in demonstrating the central importance of Bohr's work, its synthesis of previous discoveries, and its relationship to those that followed.

The truly revolutionary idea of discrete electronic energy levels still stands at the heart of quantum mechanics today. As Bohr's friend and colleague Richard Courant has written, ". . . it was not good luck hut intuition which made him open just the right path,'against tradition, to an enormous inex- haustible new world of scientific understanding" (30).

Conclusion A more historically accurate treatment of the Bohr atom

a t both the freshman and more advanced levels should have several benefits. It should increase the students' under- standing of the process by which scientific theories develop, remove some of the frustration thev feel at the arbitrary na- ture of current presentations, help them rememher one of the greatest of all physicists as more than the author of a failed theory, and ease their entry into the forbidding world of quantum mechancis.

Acknowledgment Acknowledgment is made to the Andrew W. Mellon

Foundation for a grant in partial support of the research for this paper.

Literature Cited i l l Wirwesaer. W. J.. J. CHEM. E~uc.,22.370(1945). (2) Garrett. A.B.. J . , C H E M . E D U C . . ~ ~ . ~ ~ ~ 119621. (1) Dankel, T. Jr., and Levy. J. R., J, CHEM, EDIIC.. 51.898 11974). 14) K w h , H., J. CmM. Eouc..s4.208 119771. (5) Roikess, R. S., and Eddson. E.. "Chemical Principles." Harper and Row, New Yark,

1978, p, 186. 16) Mshan. 9. H.,"University Chemistry." 3rd ed.. Addison Wesley, Ileadine. MA. 1975.

p. 426. (7) Pilar. F. L., "Chemistry." Addison Wesley. Reading, MA. 1979, p. 98. 18) Rarrow. G. M., "Physical Chemistry." 4th ed.. McGraw-Hill, New York. 1979. p.

Jammer also states that the lesser-known work of Whiddington was important in suggesting the idea of energy levels to Bohr (see 14, p. 77).

Volume 59 Number 5 May 1982 375

(14) Jammer, M., "The ConceptuaiDevelapment of Quantum Meehaniq"McGraw-Hill, (23) Reference U2), p. 46. New York. 1964pp. 62-88. (24) Reference (14). pp. 71-73.

(15) Gamov. G., "Thirty Years That Shmk Physics." Doubleday, New York, 1966, pp. (25) Shamos. M. H.. "Great Experiments in Physi~s:Henry Holt and Co.New York. 1959, 1-61. PP. 329-347.

(16) Cropper. W. H.. "The Quantum Physicists and an Introduction to Their Physier." (261 Simon, K. R., "Moehanics." 2nd ed., Addison Waley, Reading, MA, 1960. p. 90. Oxford University Press, New York, 1970, pp. 3-70. (27) Resnick, R.. and Hallidsy, D., "Phy8ics; 3rd ed., John Wiiey, New York. 1977, pp.

(17) Reference (141, p. 339. 59-63. (18) Rutherford, E., Phil. Mag., 21, €69 (1911). 128) Reference 1141, p. 77. (19) Planck,M., Vsrh.Dlseh.Phys. Crs.,2,202 (19Wl. (29) Franck, J.,snd Henz,G., Verh. Dtsch. Phya. G e e . 16.457 11914);Phys. Zeii., 17,409 120) Einstein, A., Ann. der Phy~. . 17.132 11905). (1916); P h w Zeil., 20,132 (19191. 121) Lenard, P., Ann. derPhys., 8,149 (1902). (30) Refolence (12),p. 305. (22) Reference (12). p. 26.

376 Journal o f Chemical Education