Physcs 30 Lesson 31 The Bohr Model o the Atom I. Planetary models o the atom Ater Rutherord s gold ol scatterng experment, all models o the atom eatured a nuclear model wth electrons movng around a tny, massve nucleus. A smple way to vsualse the nuclear model was as planets orbtng a central Sun. As the Sun o our solar system attracted the planets, the postve nucleus o the atom would attract the negatve electrons. Whle the Sun and planets nvolve gravtaton, the nucleus and electrons nvolve electrostatc orces. Planet Sun Nucleus electron Gravtatonal Attracton Planetary Model o Atom The nuclear atom was a nce combnaton o gravtatonal deas and sub atomc partcles, but t had several major laws: 1. Dd all o the electrons travel n the same orbt? Why dd they not bump nto one another? What was the electron structure? 2. From the known bondng characterstcs o derent chemcal compounds, how are the electrons nvolved n the bondng process? 3. Why do the postve protons stay together n the nucleus? Ther strong mutual repulson should tear the nucleus apart. 4. The nal, and most mportant law, concerned the nature o acceleratng charges. James Maxwell had shown that acceleratng electrc charges radate EM radaton (Lesson 24). I the electrons were n crcular orbts, they would contnually experence a centrpetal acceleraton and should contnually radate energy n the orm o electromagnetc waves. Further, snce ther knetc energy s beng converted nto radant energy, the electrons should spral nto the nucleus. From observaton, we know that atoms have stable structures or long perods o tme and they do not radate energy on ther own. A good model o the atom must nclude Rutherord s ndngs.e. that the majorty o an atom s volume s empty space contanng the electrons o the atom and there s a small, massve, postvely charged nucleus. In addton, any model o the atom would have to account or the absorpton and emsson spectra o elements and molecules. Reer to Pearson pages 773 to 778. Dr. Ron Lcht 31 1 www.structuredndependentlearnng.com
II. Regulartes n the hydrogen spectrum Over many years, derent people worked on measurng and understandng the propertes o the lghtest and smplest element hydrogen. The vsble brght lne spectrum ormed by excted hydrogen gas conssts o our lnes a red lne ( = 656.21 nm), a green lne ( = 486.07 nm), a blue lne ( = 434.01 nm) and a volet lne ( = 410.12 nm). Red Green Blue Volet In 1885, a mathematcan named Johann Jakob Balmer ound a smple emprcal ormula that would gve the wavelengths o these our lnes (an emprcal ormula descrbes the phenomenon, but t does not explan the phenomenon). Hs equaton s: 1 1 1 2 RH n where n = any nteger greater than 2 and less than 7 (.e. 3, 4, 5, 6) R H = 1.0972 x 10 7 / m (later called the Rydberg constant) When, or example, we calculate the wavelength or n = 3: 1 1 1 2 RH n 1 7 1 1 6 1.097210 / m 1.52410 / m 2 3 1 = 656.2 nm (red spectral lne) 6 1.52410 /m When n = 4, 5 and 6 are substtuted nto the equaton, we obtan the wavelengths o the green, blue and volet lnes respectvely. Balmer s values were accurate to wthn 0.02%. However, beng a mathematcan, Balmer wanted to know why values above n = 6 could not be used. When values above n = 6 are substtuted, the calculated wavelengths produced are not n the vsble part o the spectrum they all nto the ultravolet secton o the spectrum. It was several years beore scentsts, usng ultravolet senstve lm, were able to detect that the lnes n the ultravolet secton actually exsted. Once the UV lnes were vered, the Balmer equaton was moded to handle other possbltes. The new equaton was called the Rydberg equaton: 1 1 1 RH n n where n = nal nteger (1, 2, 3 ) n I = ntal nteger (1, 2, 3 ) Note: Ths equaton works or hydrogen only. For other atoms a derent R constant s requred. Dr. Ron Lcht 31 2 www.structuredndependentlearnng.com
When, or example, we calculate the wavelength when n = 3 and n = 1 or the Rydberg equaton: 1 1 1 RH n n 1 7 1 1 6 1.097210 / m 9.75310 / m 1 3 1 = 102.5 nm (lght s ultra volet) 6 9.75310 /m The work by Balmer and Rydberg encouraged other scentsts to search or spectral lnes outsde the normal vsble lght range. In 1908, or example, F. Paschen ound a whole seres o spectral lnes generated n the nrared secton o the spectrum or excted hydrogen gas. The lnes corresponded to the values that you obtan when n = 3 and n s gve values o 4, 5, 6, 7 etc. In 1914, Lyman ound spectral lnes n the ultra volet part o the spectrum. Lyman s lnes corresponded to n = 1 and n s gve values o 2, 3, 4, 5, etc. Further research by other scentsts led to the detecton o other spectral lne seres. In each case the detecton o the seres came ater the mathematcal calculaton o the values usng the Rydberg equaton. (See Pearson page 776.) The act that a precse, and arly smple, mathematcal relatonshp descrbed the wavelengths o spectral lnes generated by hydrogen gas ndcated that the spectral lnes were governed by some physcal relatonshp or law. Nels Bohr would use the producton o spectral lnes or hydrogen to enhance and rene hs model o the atom. Note, you are not requred to know the Balmer or Rydberg equatons, they were used above or nstructve purposes only. You are requred to know that hydrogen has a spectrum that can be descrbed by a smple mathematcal relatonshp. Dr. Ron Lcht 31 3 www.structuredndependentlearnng.com
III. Bohr s postulates Nels Bohr receved hs PhD n 1911 n the eld o Physcs rom the Unversty o Copenhagen. In 1912, he spent a year workng under Ernest Rutherord at the Unversty o Manchester n England. In the same year, Bohr would propose a model o the hydrogen atom that ncorporated: J. J. Thomson s cathode ray tube experments Rutherord s nuclear model Planck s quantum theory Ensten s photon theory the emsson and absorpton spectra or hydrogen Rydberg s equaton or predctng the absorpton wavelengths o hydrogen the Franck-Hertz expermental results In order to account or the exstence o stable electron orbts and separate emsson spectra, Bohr made three major assumptons called the postulates. (A postulate s an assumed dea upon whch a theory s based.) The postulates are: 1. Electrons move n crcular orbts, but they do not radate energy. These orbts are called statonary states. 2. Electrons can jump rom one statonary state to another, but they cannot exst n between them. When an electron absorbs energy t jumps up n energy level. When an electron releases energy t jumps down n energy level. The energy emtted or absorbed has a requency determned by the relaton h = E E 3. O all possble orbts around the nucleus, only a ew are allowed. Each orbt has a characterstc energy and radus gven by the ollowng equatons. (See Pearson pages 774 and 775.) E E n 1 n 2 r n = n 2 r 1 Where n = 1, 2, 3, 4, n s the prncpal quantum number. For the ground state n = 1. For hydrogen: E 1 = 13.6 ev or 2.18 x 10-18 J r 1 = 5.29 x 10-11 m Bohr s postulates were ntended to work or all atoms, not just hydrogen, but t dd not work out that way. For mult-electron atoms, the nteractons between electrons requre a ar more sophstcated and nvolved dea see Lesson 34. It may be o nterest to note that Bohr was not very good at wrtng down hs deas. Hs PhD was delayed by many years because he had developed such a ear o wrtng. Fnally, he dctated hs deas to hs we who then wrote the necessary thess. Through out hs proessonal le, Bohr ound wrtng to be very panul. When you read Albert Ensten s papers there s an eloquent low o words and deas. When you try to read Bohr s papers, you are ortunate you get beyond the rst paragraph. When asked why he had such dcultes, he sad, Grammar teachers. Dr. Ron Lcht 31 4 www.structuredndependentlearnng.com
Below s an energy level dagram or hydrogen. Note that the sgn o the energy depends on where we place our zero value. I our zero pont s the onzaton state, then the energy levels are negatve relatve to the onzaton level. 0 ev 0.54 ev 0.85 ev 1.51 ev 3.40 ev energy measured rom onzaton onzaton n = n = 5 n = 4 n = 3 n = 2 13.6 ev 13.1 ev 12.8 ev 12.1 ev 10.2 ev energy measured rom ground state I we choose our zero pont to be the ground state, then the energes are postve relatve to the ground state. 13.6 ev n = 1 ground state 0 ev Note, ths s a very mportant dagram to understand. In order to jump rom one energy level to another, the atom had to absorb or emt the derence n the energy levels. E E E h E E hc E Example 1 E An electron drops rom the ourth energy level o hydrogen to the second energy level. a) What s the energy released? E E E E 3.4eV ( 0.85eV) E 2.55 ev b) What s the requency and wavelength o the emtted photon? E = h E 2.55 ev 15 h 4.1410 ev s = 6.16 x 1014 Hz c 8 3.0010 m/ s 14 6.1710 Hz = 487 nm whch corresponds to the green lne n the emsson spectrum o the Balmer seres Dr. Ron Lcht 31 5 www.structuredndependentlearnng.com
The Bohr model o the atom explans why absorpton and emsson lne spectra occur or hydrogen and other elements. The atom can only absorb requences/energes o lght that correspond to derences between the atom s energy levels, resultng n dark lne absorpton spectra. Smlarly, atoms can only emt photons that have energes that correspond to energy transtons rom hgher to lower energy states. We wll occasonally work wth other cttous elements whch we wll treat n a Bohr-lke ashon. IV. Strengths o the Bohr model o the atom 1. Bohr s model o the atom could explan the sze o the hydrogen atom. The radus calculated corresponded to known values o the hydrogen atom. 2. Bohr s model gave an onzaton value ( E 1 ) that corresponded to the known onzaton value or the hydrogen atom. 3. Bohr s model could account or the ormaton o the spectral lnes n the hydrogen spectra. Whle Balmer, Lyman, Paschen, and others had used a mathematcal relatonshp to calculate the known wavelengths, the relatonshp had no scentc bass. Bohr s atom gave the same results and a strong scentc bass or the observed emsson and absorpton spectral lnes. 4. Bohr s model could be expanded wth some modcatons to account or a) larger charges n the nucleus and b) possble sheldng by nner electrons on outer electrons n larger atoms n the perodc table. When Bohr appled hs theores to the perodc table as a whole, he was able to explan why elements were grouped vertcally accordng to chemcal and physcal propertes. He ound all the elements n a vertcal column had the same number o electrons n ther outer most energy level. Bohr also ound that the energy level quantum number corresponded to the horzontal row (perod) o the perodc table. Bohr s atom provded greater understandng or the workngs o the perodc table desgned by Mendeleev and Moseley. 5. Bohr s atom used the nucleus dea o Rutherord but now gave an explanaton or the structure o the electron cloud. Dr. Ron Lcht 31 6 www.structuredndependentlearnng.com
V. Problems wth the Bohr model o the atom 1. Bohr s model only works well or hydrogen. It must be tremendously moded to accommodate other elements wth larger nucle and more electrons. 2. Bohr s model could not explan why the number o spectral lnes ncreased or some elements when they were placed n electrc and magnetc elds. 3. Bohr s model was not able to explan the relatve ntensty o some o the spectral lnes (.e. why some lnes were brghter than others). Nevertheless Bohr s model o the atom was a spectacular success. Hs model s stll used as the startng pont or teachng the atom n the eld o chemstry. Hs model wll explan many o the known propertes o the group 1,2, 13-18 elements. The Bohr model breaks down or transton elements and members o the Lanthande and Actnde seres. VI. Hand-n assgnment 1. What s the major problem wth Rutherord's dea o the planetary atom? 2. Photon emsson: A. What happens to electrons when an electrc current s passed through a gas? B. What s meant by an electron beng n an excted state? C. What happens when an electron jumps down to a lower energy level? D. How does photon emsson explan the emsson spectrum o a gas? 3. Descrbe how the emsson and absorpton spectra o hydrogen are explaned by the Bohr model o the atom. 4. What are the strengths o the Bohr model? What are the weaknesses o the Bohr model? Use the energy level dagram or hydrogen to answer questons 5 to 11. 5. How much energy must a hydrogen electron n energy level n = 1 absorb n order to jump to energy level n = 2 and n = 3? (10.2 ev, 12.1 ev) 6. How does the total energy o photons emtted by an electron as t jumps back down to ts ground state compare wth the energy absorbed by the electron when t jumped to the hgher level? 7. Why s a blue photon emtted when a hydrogen electron jumps rom energy level n = 5 to n = 2, but a red photon s emtted when an electron jumps rom level n = 3 to n = 2? 0 ev 0.54 ev 0.85 ev 1.51 ev 3.40 ev 13.6 ev n = n = 5 n = 4 n = 3 n = 2 n = 1 Dr. Ron Lcht 31 7 www.structuredndependentlearnng.com
8. What s the onzaton energy or the electron n hydrogen rom ts ground state? 9. What s the wavelength o a photon emtted when an electron n a hydrogen atom alls: A. From n = 4 to n = 2. (487 nm) B. From n = 5 to n = 1. (95.1 nm) 10. Electrons are accelerated through hydrogen gas at room temperature n a Franck- Hertz experment by a potental derence o 12.3 V. What wavelengths o lght can be expected to be emtted by the hydrogen? (122 nm, 103 nm, 657 nm) 11. What energy s needed to onze hydrogen rom the n=2 state? How lkely s ths to occur? Explan. (3.40 ev) 12. Use the energy level dagram below to answer the ollowng questons. n = 0.000 ev n = 9-0.450 ev n = 8-0.570 ev n = 7-0.744 ev n = 6 n = 5-1.013 ev -1.458 ev n = 4 n = 3-2.278 ev -4.050 ev n = 2-9.113 ev n = 1-36.450 ev A. What s the wavelength o the photon emtted when an electron alls rom the sxth to second energy level? (153 nm) B. What s the requency o the photon emtted when an electron alls rom n = 7 to n = 3? (7.987 x 10 14 Hz) C. What s the energy absorbed by an electron to jump rom n = 1 to n = 8? (35.880 ev) D. What s the onzaton energy o ths element? Dr. Ron Lcht 31 8 www.structuredndependentlearnng.com