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 Faraday: physical chemistry courses. In recent years, however, this Electrolysis discussion has been given low priority and is frequently very limited. The last major article in THIS JOURNALwhich deals with Bohr's theory appeared in 1945 (1),and there have been Crookes tube: only a few short articles since (24).One reason for this may Cathode rays 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. Measured & noth her Teason fo; neglect of the Bohr theory may he an awareness that manv students find it ~articularlvfrustratina to study because iiseems rather arbitrary and pointless. i suggest that an historically accurate presentation which Rutherford: stresses the development of Bohr's ideas will he much more Measured e effective than what is commonly done. It is vital to give stu- Gold Foil Idea of the quantdm dents a sense of how Bohr arrived at ideas which to them (and Experiment: to Bohr's contemporaries) seem quite fantastic. discovery of It is frequently claimed that presenting ideas as they were nucleus actually developed tends to confuse students, and that con- cents should be arraneed for maximum claritv. It mav be eqhytrue that this ai~roach,which is partic&ly common Atomic Bohr: at the introductory level, misleads students into an overly Spectrum of Model of the Photoelectric simplistic view of the subject which is very hard to overcome Hydrogen effect; E = hv later on. It 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 Heisenberg: to either extreme. Uncertainty Wave-Particle In the case of the Bohr theory, the usual approach has the duality; characteristics of brevity and apparent simplicity, but pro- A = hlmv 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 Schriidinger: this involves expressing the Coulomb force in terms of the Quantum mechanics; centripetal acceleration and combining this with the expres- probability sion for the quantization of angular moment~m.~The diffi- culty arises from the "postulate" that angular momentum is quantized. Angular momentum is a very difficult concept for The role of the Bohr theory in the development of quantum mechanics. teachers to present and for students to grasp. It is therefore a singularlyunfortunate choice for one of the key concepts in The development of quantum mechanics is particularly rich the derivation. It is a choice, however, since Bohr did not in in good historical material. There are several very interesting fact postulate this at all. His original paper (10)very clearly individuals to focus on, not the least of whom is Niels Bohr. shows that the quantization of angular momentum was not There are hiographies3 available (12, 13) as well as hooks his startine-. ooint. . but was derived after the formula for the which discuss the history of quantum mechanics in general energies had been arrived at. The feeling expressed by many (14-16). Whatever source hook is chosen, two overwhelming students that this "postulate" seems particularly capricious impressions are received. One is that Bohr's interests and is thus intuitively correct. This objection has been raised he- genius were phenomenally broad. The other impression is of fore (11). hut it seems to have been virtually ignored. I would also argue for the integration of some purely his- Present address: Clorox Technical Center, P.O. Box 493, Pleas- torical and biographical material into the presentation. Stu- anion, CA 94566. This is ofien done incorrectly (9). dents frequently comment that chemistry as a subject, aside Unfortunately, Bohr wrote very little about his own life and work. from being difficult, is also cold, impersonal, analytical and A bibliography is included in reference ( 12). dry. Perhaps this results at least in part from the fact that we 'The discussion in this section is adapted directly from Bohr's original have succeeded so well in divorcing- our science from the men paper and from the book by Jammer ( 14) which traces the development and women who created it. of Bohr's ideas in detail.

372 Journal of Chemical Education Bohr's stature as a human heing. As his son has written: Bohr simplv and darinelv-. proposed. . that electrons do not ra- "However great he was as a scientist and however profound diate eneigi as they orbit the nucleus, but exist in states of his insight into life itself, for us it was as a human heing that constant enerm which he called stationary states. They can he was greatest" (17). If our students do not find Bohr's theory only change their energy by undergoing a transition from one precisely inspiring, perhaps they can he inspired by Bohr stationarv state to another. (Bohr said nothing ahout how this himself. transition was accomplished.) During the transition they ei- ther absorb or emit an amount of energy which is exactly equal The Development of Bohr's Ideas4 to the energy difference between the two states. This behavior There are two important aspects of the development of produces the characteristic spectral lines. Hohr'i theory. The ti& is his dei~tto the ideas of othrrs. Once the stareerinp. implications oi Rutherford's gnld foil experi- Derivation of the Energies of the stationary StatesB ment (18)hadhecome obvious to the commudty of physicists, What Bohr needed to do was to develop a formula for the the development of a reasonable atomic model became a entqirs of the statlonary stiltes whirh adulrl agree with the central problem in physics. It seemed to Bohr, newly arrived exuerirnentallv ohscn,ed surctral frequenrte.;. Hl, decided that at Cambridge from Denmark with his doctorate to study although classical electrddynamicshad failed to produce a under the ereat J. J. Thomson. that the solution to the di- successful atomic model, perhaps classical mechanics might l&&a layUina bold and daring approach. He thought the still apply. Bohr postulated elliptical orbits for the electrons, answer might lie with Planck's idea of the quantum (19) which similar to the orbits of the planets, hut the argument is the had been so brilliantly applied by Einstein (20) in his expla- same for the circular orbits and the mathematics is easier to nation of the experimental results observed by Lenard (21) follow. In the following derivation, I have used the same which are known as the photoelectric effect. symbols that Bohr used in his paper. Unfortunatelv. J. J. Thomson found Bohr's revolutionary For a body experiencing a force of attraction which is cen- ideas hard to accept. Sources disagree on the amount of fric- tral. a circular orbit is stable. If the radius of the orbit is des- tion hetween the two men, hut in March, 1912, Bohr left igniited hy a, the electrnstiltic forre ul'attrnctim is given by Cambridge to srrk an intellectual rlimate more suited to his ('oulmnb'i law ni &a2. \rhrrt.~and E are thr charreson the unusual ideas. His very fortunate choice was Rutherford's electron and nucleus, iespectively. This force produces a lahuratnn in Manchester. Althourh" Rutherford was initialls centripetal acceleration7 equal to u'la, where u is the velocity suspicious of Bohr because he was a theoretician, he appar- of the electron. Since force is the product of mass and accel- ently accepted him (partly because of his skill as a soccer eration, we can write 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- The potential energy of the system, V, may be found by in- standing" (22) Yet it is also clear that he was essentially on tegrating the expression for the Coulomb force. This energy his own as he worked on his theory, once complaining that is zero when a = a,so the potential energy at a is acquired by nobody else at the laboratory was really interested in funda- moving the electron in from infinity to the distance a. The mental theoretical problems (23). It was here in Manchester integral is then that Bohr began to develop the ideas which led to his land- eE mark paper "On the Constitution of Atoms and Molecules" (2) (10) which was published after his return to Copenhagen in a

1917&"AV. We can now write an expression for the total energy of the The second aspect of the development is Bohr's own mental svstem, U,which is the sum of the kinetic (T)and potential progression. Why should he have chosen Planck's idea of the energies: ouantum as the wav to eo? Bohr thoueht that the "solar" iodel 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 From eqn. (1)we have that (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 Since this quantity represents the total energy of the or- h2 (6.63 X 10-27 erg-se~)~ -= = 20.9 X 10@ crn hitinz electron. an amount of enerev eaual to this amount me2 (9.11 X g) (4.80 X lo-' wouli have to be supplied in orde;to remove the electron which is approximately the right order of magnitude for an comoletelv from the attractive null of the nucleus. Bohr called atomic dimension.5 This idea may seem rather sketchy, hut thisquantity the binding energy, W, so we have ~.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. These are today's values. Theones available to Bohr were some- Bohr was also guided in his thinking by a large body of ex- what less accurate. (1 esu = 1 ~m~'~g"~sec-'.) oerimental evidence which should relate directly to atomic The following derivation is suitable for presentation at the level of structure, hut which in Bohr's day was in such confusion that undergraduate physical chemistry courses. it is not intended that it be many scientists had despaired of ever making any sense of it. included in standard introductory courses; however, several crucial aspects of the development of Bohr's ideas are also described in this By Bohr's time, the idea that atomic spectra could provide a section. The derivation has been adapted directly from Bohr's paper key to the structure of the atom and that the electrons were with the aid of the book by Shamos (25. somehow responsible for producing the emitted light was A derivation of the centripetal acceleration may be found In ref- generally accepted, yet the precise connection could not be erence l26l.. or lZi7.,. found. Much attention was focused on the line spectrum of The use of the prime notation is to distingufshthe rotational fre- hydrogen, since it gave the simplest characteristic pattern. 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 and w, the frequency of rotation. The actual speed, v, with which the electron travels is related to this frequency as For hydrogen, where E = lei this becomes Now substituting eqn. (7) into eqn. (6) we have

The characteristic radii may be calculated from eqns. (6) and To eliminate a from this expression, we use a = eEI2W (from (12) 6) and substitute to get

It should be pointed out that this expression contains (aside or from dimensionless constants) precisely the function h21e2m fiW 312 which Bohr had originally recognized as having the dimen- w=- (10) eE& sions of length. The great success of Bohr's theory was its ability to predict So far, all of this was nothing more than classical (Newto- correctly the spectral frequencies which were experimentally nian) mechanics, but now Bohr faced the problem of incor- observed for hydrogen. For a transition between two sta- porating Planck's quantum concept into his model. He knew tionary states, light corresponding to their energy difference that only certain values of W (and hence also of o and a) could should be emitted. From eqn. (15), this difference is 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 and a frequ& characteristirol;he electron's rotational motion and is not necessarilv the frenurncvof the emitted li~ht.~The problem then became one of'relaGng this to W and"w. Bohr started by considering a "free" electron which was where u here is the frequency of the emitted light and is not completely removed from the nucleus and allowing it to un- generally the same as the frequency of electron motion re- dergo a transition to a stable orbit with binding energy W. ferred to earlier. The subscripts i and f refer to initial and tinal Since the electron must radiate energy equal to the difference states. A comparison of this equation with the more general in energy between its initial and final states, and since the Rydherg equation (usually expressed in wavenumbers rather energy of the free electron is taken to he zero, the energy ra- than frequency) shows that they are identical in form and diated must equal W, or that 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 Using today's values we have 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. = 109,760 cm-I Thus we have The experimentally determined value of the Rydherg constant rho is 109,677.581 ~m-'.~ w = - (12) 2 It is worth deriving one more relationship from Bohr's equations. The angular momentum, M, for the eletron in its This argument seems rather contrived, and in fact it is. Bohr circular orbit would usually be expressed as mva, but this can was trvina to make classical ideas such as circular orbits and he rewritten as follows rntati~~iaifrequenciesfit where they redly could not fit nt all. It is at this point that the crucial importance of atomic spec- troscopy must be recognized. ~ccordin~to Jammer (28), ~ohr was unable to arrive at a successful theory until he saw Bal- Since T = Wand W = ~hwl2,we have 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- which is the quantization of angular momentum which is so chester to Copenhagen. In January of 1913 spectroscopist H. often presented as one of Bohr's postulates. It is clear from his M. Hansen visited Bohr there, and asked him how his new paper that he derived it as above from W = rhw12, although atomic model could explain the spectroscopic results. Bhor's the two expressions are mathematically equivalent. 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 Bohr's agreement was not quite so good, again due lo the inaccu- I rate values available to him, but was stll impressive. The main dis- was looking for. As he repeatedly stated, "As soon as saw crepancy between me two values is due to the fact that Bohr's treatment Balmer's formula, the whole thing was immediately clear to neglects nuclear motion. Taking nuclear motion into account leads to me" (28). the replacement of the electron mass by the reduced mass p. Using Once eqn. (12) is arrived at, the rest follows easily. If we now this value in eqn. (19)gives Fin = 109,700, which is much closer to the substitute eqn. (10) into eqn. (121, we have exDerimental value. 374 Journal of Chemical Education The Bohr Atom in the Chemistry Curriculum with certain theoretical postulates and thus occupies a unique position as a transition from the comfortable world of classical In presenting the Bohr atom at the freshman level it is im- physics where physical analogies are possible to the uncom- portant to describe the development,^ leading up to Bohr's fortable world of quantum mechanics where they are not. work and to show as far as oossible how his own ideas grew. It is certainlv necessarv for any discussion of further ad- Thiican he dune by strrsiin; I'lanck'scontrihution to thiidea vances inquantum mechanics to make students at 11~1th1w.t.k uf the suantum nnd Einstr~n'swork on the photoelectric ef- aware of the failure oiBohr's theow to dw:ril)ecorrectly any fect. 1t is necessary to present a sufficient background in this systems with more than one electron and to point out the material in order to make the development of Bohr's ideas limitations of the electron orbit model. Students may also seen reasonable, and the importance of these contributions benefit from learning that the same aspects of the theory cannot be overemphasized. However, it is neither necessary which seem arbitrary to them were also criticized by many of nor desirable to exnlain the ohenomenon of hlackbodv ra- Bohr's colleaeues and that acceotance was hv no means im- diation. This topic seems very confusing to students and is not mediate. ow ever, this information should be balanced by really needed for an understanding of the true importance of stressing the positive aspects of Bohr's contrihution, since the Planck's work to the development of quantum mechanics, impression that the study of Bohr's work is pointless may arise which is the idea of the quantum. If Planck's contrihution is from an overemphasis on the failed aspects of his theory. The presented as introducing the revolutionary idea that energy agreement with Rydberg's equationwas considered an as- is not distributed continuouslv but is in fact limited to bundles tounding achievement even by skeptics. A discussion of the of magnitude proportional to h, and it is then shown how work of Franck and Hertz (29) at an appropriate level will Einstein applied this idea to explain the photoelectric effect familiarize the student with the first experimental verification (which mist students find mudh less confusing than black- of Bohr's ideas. hodv radiation), an effective presentation will be accom- It should he pointed out that no one was more aware of the plished. defects in his theory than Bohr himself, and that he presided It is also essential to present a discussion of the atomic over many of the further developments in quantum mechanics spectrum of hydrogen, and to point out that Bohr's funda- as the director of the Institute for Theoretical Phvsics in Co- mental postulate was prohahly suggested to him by the fact penhagen. The role he played in nurturing these ideas in the that the spectral emissions were discrete rather than contin- minds of those who followed him is perhaps as crucially im- uous.'0 Then it can be shown how these concepts are all in- portant as his own first step. A flow chart similar to the one cor~oratedinto the model of the orbiting- electron and the shown in the figure can he very helpful in demonstrating the postulate of discrete energy levels with transitions occurring central importance of Bohr's work, its synthesis of previous only between these states. The formula for the energies of the discoveries, and its relationship to those that followed. stationary states can be presented and the comparison with The truly revolutionary idea of discrete electronic energy Rydberg's equations made. It should he explained that Bohr levels still stands at the heart of quantum mechanics today. used an almost chance encounter with Balmer's formula to As Bohr's friend and colleague Richard Courant has written, break an impasse in his thinking and that this breakthrough ". . . it was not good luck hut intuition which made him open allowed him-to complete his theory. just the right path,'against tradition, to an enormous inex- It is obviously beyond the scope of almost any freshman haustible new world of scientific understanding" (30). course to present a derivation similar to the one in the previous section. It seems equally pointless to present the "derivation" Conclusion found in many freshman texts since my experience has been A more historically accurate treatment of the Bohr atom that most students find it arbitrary and meaningless. It also at both the freshman and more advanced levels should have confuses any attempt to explain the development of Bohr's several benefits. It should increase the students' under- ideas, since it does not, in fact, follow this development. It is standing of the process by which scientific theories develop, preferable to state the results and explain that the mathe- remove some of the frustration thev feel at the arbitrary na- matics is not appropriate to the course level. If it is necessary ture of current presentations, help them rememher one of the for further work that the students know how angular mo- greatest of all physicists as more than the author of a failed mentum is quantized, it should be explained that the rela- theory, and ease their entry into the forbidding world of tionship can he derived from the energy formula. If an in- quantum mechancis. structor decides that it is desirable to do the angular mo- mentum "derivation," one can emphasize that it follows Acknowledgment Bohr's ideas in reverse order for mathematical simplicity, and Acknowledgment is made to the Andrew W. Mellon was not in fact his actual approach. Foundation for a grant in partial support of the research for

~~~ In nhvsical.~ , ~~ chemistrv~ courses. in addition to the aualitative this paper. presentatam of the develupmtntif ~ohr'sideas, his hwiration chr~uldhe included. Since it involve.; man\, euunti~~~~sand i~ Literature Cited not readily available in texts, it should he handed out to the ill Wirwesaer. W. J.. J. CHEM. E~uc.,22.370(1945). students so that thev can follow the discussion without the (2) Garrett. A.B.. J.,CHEM.EDUC..~~.~~~119621. pressure of having to copy it down. (1) Dankel, T. Jr., and Levy. J. R., J, CHEM, EDIIC.. 51.898 11974). 14) Kwh,H., J. CmM. Eouc..s4.208 119771. There are several reasons for including this. Modern (5) Roikess, R. S., and Eddson. E.. "Chemical Principles." Harper and Row, New Yark, quantum mechanics, as well as most theoretical chemistry, 1978, p, 186. 16) Mshan. 9. H.,"University Chemistry."3rd ed.. Addison Wesley, Ileadine. MA. 1975. is confusing and frustrating to students for two main reasons. p. 426. First, there is almost nev& time to go into the development (7) Pilar. F. L., "Chemistry." Addison Wesley. Reading, MA. 1979, p. 98. of these theories so that students can understand where the 18) Rarrow. G. M., "Physical Chemistry." 4th ed.. McGraw-Hill, New York. 1979. p. 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 Jammer also states that the lesser-known work of Whiddington with a reasonnlh invc>;ttnent ot time. It mn tw pointed out was important in suggesting the idea of energy levels to Bohr (see 14, that thr thwry combines a physic id model of electrun mutim 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. Gee. 16.457 11914);Phys. Zeii., 17,409 120) Einstein, A., Ann. der Phy~..17.132 11905). (1916); Phw Zeil., 20,132 (19191. 121) Lenard, P., Ann. derPhys., 8,149 (1902). (30) Refolence (12),p. 305. (22) Reference (12). p. 26.

376 Journal of Chemical Education