1. Give a reason why the Thomson plum-pudding model does not agree with experimental observations.

Transcription

1 [Problems] Walker, Physcs, 3 rd Edton Chapter 31 Conceptual Questons (Answers to odd-numbered Conceptual Questons can be ound n the back o the book, begnnng on page ANS-xx.) 1. Gve a reason why the Thomson plum-puddng model does not agree wth expermental observatons. 2. Gve a reason why the Rutherord solar-system model does not agree wth expermental observatons. 3. Cte one example o how the Bohr model dsagrees wth the quantum mechancal model o the hydrogen atom. 4. What observaton led Rutherord to propose that atoms have a small nucleus contanng most o the atom's mass? 5. Do you expect the lght gven o by (a) a neon sgn or (b) an ncandescent lghtbulb to be contnuous n dstrbuton or n the orm o a lne spectrum? Explan. 6. In prncple, how many spectral lnes are there n any gven seres o hydrogen? Explan. 7. Is there an upper lmt to the radus o an allowed Bohr orbt? Explan. 8. (a) Is there an upper lmt to the wavelength o lnes n the spectrum o hydrogen? Explan. (b) Is there a lower lmt? Explan. 9. The prncpal quantum number, n, can ncrease wthout lmt n the hydrogen atom. Does ths mean that the energy o the hydrogen atom also can ncrease wthout lmt? Explan. 10. For each o the ollowng conguratons o outermost electrons, state whether the conguraton s allowed by the rules o quantum mechancs. I the conguraton s not allowed, gve the rule or rules that are volated. (a) d, (b) 1 p, (c) 3 p, (d) 4 g. 11. (a) In the quantum mechancal model o the hydrogen atom, there s one value o n or whch the angular momentum o the electron must be zero. What s ths value o n? (b) Can the angular momentum o the electron be zero n states wth other values o n? Explan. 12. Would you expect characterstc X-rays to be emtted by (a) helum atoms or (b) lthum atoms n ther ground state? Explan. 1

2 13. The elements luorne, chlorne, and bromne are ound to exhbt smlar chemcal propertes. Explan. 2

3 Conceptual Exercses (Answers to odd-numbered Conceptual Exercses can be ound n the back o the book, begnnng on page ANS-xx.) 1. Consder the ollowng three transtons n a hydrogen atom: (A) n = 5, n = 2; (B) n = 7, n = 2; (C) n = 7, n = 6. Rank the transtons n order o ncreasng (a) wavelength and (b) requency o the emtted photon. Indcate tes where approprate. 2. I the mass o the electron were to be doubled, would the onzaton energy o hydrogen ncrease, decrease, or stay the same? Explan. 3. Consder the Bohr model as appled to the ollowng three atoms: (A) neutral hydrogen n the state n = 2; (B) sngly onzed helum n the state n = 1; (C) doubly onzed lthum n the state n = 3. Rank these three atoms n order o ncreasng Bohr radus. Indcate tes where approprate. 4. Consder the Bohr model as appled to the ollowng three atoms: (A) neutral hydrogen n the state n = 3; (B) sngly onzed helum n the state n = 2; (C) doubly onzed lthum n the state n = 1. Rank these three atoms n order o ncreasng energy. Indcate tes where approprate. 5. Is the de Brogle wavelength o an electron n the n = 2 Bohr orbt o hydrogen greater than, less than, or equal to the de Brogle wavelength n the n = 1 Bohr orbt? Explan. 6. In the Bohr model, the potental energy o a hydrogen atom n the nth orbt has a value we wll call U What s the potental energy o a hydrogen atom when the electron s n the ( n + 1)th Bohr orbt? Gve your answer n terms o U and n. n 7. An electron n the nth Bohr orbt has a knetc energy we wll call K n. In terms o K n and n, what s the knetc energy o an electron n the ( n + 1)th Bohr orbt? 8. Suppose an electron s n the ground state o hydrogen. (a) What s the hghest-energy photon ths system can absorb wthout dssocatng the electron rom the proton? Explan. (b) What s the lowest-energy photon ths system can absorb? Explan. 9. How many electrons can occupy (a) the 2p subshell and (b) the 3p subshell? 10. (a) How many electrons can occupy the 3d subshell? (b) How many electrons can occupy the n = 2 shell? The electronc conguraton o a gven atom s 1s 2 s 2p 3s 3p. How many electrons are n ths atom? n. 3

4 12. The electronc conguraton o a partcular carbon atom s 1s 2 s 2p 3s. Is ths atom n ts ground state or n an excted state? Explan The electronc conguraton o a partcular potassum atom s 1s 2 s 2p 3s 3p 3d. Is ths atom n ts ground state or n an excted state? Explan. 14. In an X-ray tube, do you expect the wavelength o the characterstc X-rays to ncrease, decrease, or stay the same the energy o the electrons strkng the target s ncreased? Explan. 15. Do you expect the onzaton energy o sodum (Na) to be greater than, less than, or equal to the onzaton energy o lthum (L)? Explan. 16. Is the wavelength o the radaton that exctes a luorescent materal greater than, less than, or equal to the wavelength o the radaton the materal emts? Explan. 4

5 Problems: 4, 10, 19, 25, 28, 32 Note: IP denotes an ntegrated conceptual/quanttatve problem. BIO dentes problems o bologcal or medcal nterest. Red bullets (,, ) are used to ndcate the level o dculty o each problem. Secton 31-1 Early Models o the Atom 1. The electron n a hydrogen atom s typcally ound at a dstance o about m rom the nucleus, whch has a dameter o about 10. m. I you assume the hydrogen atom to be a sphere o radus m, what racton o ts volume s occuped by the nucleus? 2. Reerrng to Problem 1, suppose the nucleus o the hydrogen atom were enlarged to the sze o a baseball ( dameter = 73. cm). At what typcal dstance rom the center o the baseball would you expect to nd the electron? 3. Copper atoms have 29 protons n ther nucle. I the copper nucleus s a sphere wth a dameter o m, nd the work requred to brng an alpha partcle ( charge = +2e ) rom rest at nnty to the surace o the nucleus. 4. In Rutherord's scatterng experments, alpha partcles ( charge = + 2e ) were red at a gold ol. Consder an alpha partcle wth an ntal knetc energy K headng drectly or the nucleus o a gold atom ( charge = +79e ). The alpha partcle wll come to rest when all ts ntal knetc energy has been converted to electrcal potental energy. Fnd the dstance o closest approach between the alpha partcle and the gold nucleus or the case K = 30. MeV. Secton 31-2 The Spectrum o Atomc Hydrogen 5. Fnd the wavelength o the Balmer seres spectral lne correspondng to n = What s the smallest value o n or whch the wavelength o a Balmer seres lne s less than 400 nm? 7. Fnd the wavelength o the three longest-wavelength lnes o the Lyman seres. 8. Fnd the wavelength o the three longest-wavelength lnes o the Paschen seres. 9. Fnd (a) the longest wavelength n the Lyman seres and (b) the shortest wavelength n the Paschen seres. 10. In Table 31-1 we see that the Paschen seres corresponds to n = 3 n Equaton 31-2, and that the Brackett seres corresponds to n = 4. (a) Show that the ranges o wavelengths o these two seres overlap. (b) Is there a smlar overlap between the Balmer seres and the Paschen seres? Very your answer. 5

6 Secton 31-3 Bohr's Model o the Hydrogen Atom 11. Fnd the rato v/c or an electron n the rst excted state ( n = 2 ) o hydrogen. 12. Fnd the magntude o the orce exerted on an electron n the ground-state orbt o the Bohr model. 13. How much energy s requred to onze hydrogen when t s n the n = 4 state? 14. Fnd the energy o the photon requred to excte a hydrogen atom rom the n = 2 state to the n = 5 state. 15. A hydrogen atom s n ts second excted state, n = 3. Usng the Bohr model o hydrogen, nd (a) the lnear momentum and (b) the angular momentum o the electron n ths atom. 16. Reerrng to Problem 15, nd (a) the knetc energy o the electron, (b) the potental energy o the atom, and (c) the total energy o the atom. Gve your results n ev. 17. Intally, an electron s n the n = 3 state o hydrogen. I ths electron acqures an addtonal 1.23 ev o energy, what s the value o n n the nal state o the electron? 18. Identy the ntal and nal states an electron n hydrogen emts a photon wth a wavelength o 656 nm. 19. IP An electron n hydrogen absorbs a photon and jumps to a hgher orbt. (a) Fnd the energy the photon must have the ntal state s n = 3 and the nal state s n = 5. (b) I the ntal state was n = 5 and the nal state n = 7, would the energy o the photon be greater than, less than, or the same as that ound n part (a)? Explan. (c) Calculate the photon energy or part (b). 20. IP Consder the ollowng our transtons n a hydrogen atom: () n () n = 2, n = 6 () n = 2, n = 8 = 7, n = 8 (v) n = 6, n = 2 Fnd (a) the longest-and (b) the shortest-wavelength photon that can be emtted or absorbed by these transtons. Gve the value o the wavelength n each case. (c) For whch o these transtons does the atom lose energy? Explan. 21. IP Muonum Muonum s a hydrogen-lke atom n whch the electron s replaced wth a muon, a undamental partcle wth a charge o e and a mass equal to 207m e. (The muon s sometmes reerred to loosely as a heavy electron. ) (a) What s the Bohr radus o muonum? (b) Wll the wavelengths n the Balmer seres o muonum be greater than, less than, or the same as the wavelengths n the Balmer seres o hydrogen? Explan. (c) Calculate the longest wavelength o the Balmer seres n muonum. 6

7 22. IP (a) Fnd the radus o the n = 4 Bohr orbt o a doubly onzed lthum atom ( L 2, Z = 3). (b) Is the energy requred to rase an electron rom the n = 4 state to the n = 5 state n L 2+ greater than, less than, or equal to the energy requred to rase an electron n hydrogen rom the n = 4 state to the n = 5 state? Explan. (c) Very your answer to part (b) by calculatng the relevant energes. 23. Applyng the Bohr model to a trply onzed beryllum atom ( Be 3, Z = 4), nd (a) the shortest wavelength 3+ o the Lyman seres or Be and (b) the onzaton energy requred to remove the nal electron n Be (a) Calculate the tme requred or an electron n the n = 2 state o hydrogen to complete one orbt about the nucleus. (b) The typcal letme o an electron n the n = 2 state s roughly 10 s ater ths tme the electron s lkely to have dropped back to the n = 1 state. Estmate the number o orbts an electron completes n the n = 2 state beore droppng to the ground state. 25. IP The knetc energy o an electron n a partcular Bohr orbt o hydrogen s J. (a) Whch Bohr orbt does the electron occupy? (b) Suppose the electron moves away rom the nucleus to the next hgher Bohr orbt. Does the knetc energy o the electron ncrease, decrease, or stay the same? Explan. (c) Calculate the knetc energy o the electron n the orbt reerred to n part (b). 26. IP The potental energy o a hydrogen atom n a partcular Bohr orbt s J. (a) Whch Bohr orbt does the electron occupy n ths atom? (b) Suppose the electron moves away rom the nucleus to the next hgher Bohr orbt. Does the potental energy o the atom ncrease, decrease, or stay the same? Explan. (c) Calculate the potental energy o the atom or the orbt reerred to n part (b). 27. Consder a head-on collson between two hydrogen atoms, both ntally n ther ground state and movng wth the same speed. Fnd the mnmum speed necessary to leave both atoms n ther n = 2 state ater the collson. 28. A hydrogen atom s n the ntal state n = n, where n > 1. (a) Fnd the requency o the photon that s emtted when the electron jumps to state n state n. (c) Compare your results or parts (a) and (b) n the lmt o large n. = Secton 31-4 de Brogle Waves and the Bohr Model n + 1. (b) Fnd the requency o the electron's orbtal moton n the 29. Fnd the de Brogle wavelength o an electron n the ground state o the hydrogen atom. 30. Fnd an expresson or the de Brogle wavelength o an electron n the nth state o the hydrogen atom

8 31. What s the radus o the hydrogen-atom Bohr orbt shown n Fgure 31-30? 32. (a) Fnd the knetc energy (n ev) o an electron whose de Brogle wavelength s equal to 0.5 A, a typcal atomc sze. (b) Repeat part (a) or an electron wth a wavelength equal to Secton 31-5 The Quantum Mechancal Hydrogen Atom 33. What are the allowed values o l when the prncpal quantum number s n = 5? m, a typcal nuclear sze. 34. How many derent values o m l are possble when the prncpal quantum number s n = 4? 35. Gve the value o the quantum number l, one exsts, or a hydrogen atom whose orbtal angular momentum has a magntude o (a) 6( h 2π ), (b) 15( h 2π ), (c) 30( h 2π ), or (d) 36( h 2π ). 36. IP Hydrogen atom number 1 s known to be n the 4 state. (a) What s the energy o ths atom? (b) What s the magntude o ths atom's orbtal angular momentum? (c) Hydrogen atom number 2 s n the 5d state. Is ths atom's energy greater than, less than, or the same as that o atom 1? Explan. (d) Is the magntude o the orbtal angular momentum o atom 1 greater than, less than, or the same as that o atom 2? Explan. 37. IP A hydrogen atom has an orbtal angular momentum wth a magntude o 10 57( h 2π ). (a) Determne the value o the quantum number l or ths atom. (b) What s the mnmum possble value o ths atom's prncpal quantum number, n? Explan. (c) I 10 57( h 2π ). s the maxmum orbtal angular momentum ths atom can have, what s ts energy? 38. IP The electron n a hydrogen atom wth an energy o ev s n a subshell wth 18 states. (a) What s the prncpal quantum number, n, or ths atom? (b) What s the maxmum possble orbtal angular momentum ths atom can have? (c) Is the number o states n the subshell wth the next lowest value o l equal to 16, 14, or 12? Explan. 39. IP Consder two derent states o a hydrogen atom. In state I the maxmum value o the magnetc quantum number s m l = 3; n state II the correspondng maxmum value s ml = 2. Let L I magntudes o the orbtal angular momentum o an electron n states I and II, respectvely. (a) Is less than, or equal to L II? Explan. (b) Calculate the rato L L Secton 31-6 Multelectron Atoms and the Perodc Table I II. 40. Gve the electronc conguraton or the ground state o carbon. and L represent the II L I greater than, 41. Lst the values o the our quantum numbers ( n,, m, m ) or each o the electrons n the ground state o l l s 8

9 neon. 42. Gve the electronc conguraton or the ground state o ntrogen. 43. Gve a lst o all possble sets o the our quantum numbers ( n,, m, m ) or electrons n the 3s subshell. l l s 44. Gve a lst o all possble sets o the our quantum numbers ( n,, m, m ) or electrons n the 3p subshell. l l s 45. Lst the values o the our quantum numbers ( n,, m, m ) or each o the electrons n the ground state o magnesum. l l s 46. The conguraton o the outer electrons n N s 3d 4 s. Wrte out the complete electronc conguraton or N. 47. Determne the number o derent sets o quantum numbers possble or each o the ollowng shells; (a) n = 2, (b) n = 3, (c) n = Generalze the results o Problem 47 and show that the number o derent sets o quantum numbers or the 2 nth shell s 2 n. 49. Suppose that the 5d subshell s lled n a certan atom. Wrte out the 10 sets o our quantum numbers ( n, l, ml, ms ) Secton 31-7 Atomc Radaton or the electrons n ths subshell Usng the Bohr model, estmate the wavelength o the K α X-ray n nckel ( Z = 28 ). 51. Usng the Bohr model, estmate the energy o a K α X-ray emtted by lead ( Z = 82 ). 52. The K-shell onzaton energy o ron s 8500 ev, and ts L-shell onzaton energy s 2125 ev. What s the wavelength o K α X-rays emtted by ron? 53. An electron drops rom the L shell to the K shell and gves o an X-ray wth a wavelength o nm. What s the atomc number o ths atom? 54. Consder an X-ray tube that uses platnum ( Z = 78 ) as ts target. (a) Use the Bohr model to estmate the mnmum knetc energy electrons must have n order or K α X-rays to just appear n the X-ray spectrum o the tube. (b) Assumng the electrons are accelerated rom rest through a voltage V, estmate the mnmum voltage necessary to produce the K α X-rays. 55. BIO Photoreractve Keratectomy A person's vson may be mproved sgncantly by havng the cornea reshaped wth a laser beam, n a procedure known as photoreractve keratectomy. The excmer laser used n 9

10 these treatments produces ultravolet lght wth a wavelength o 193 nm. (a) What s the derence n energy between the two levels that partcpate n stmulated emsson n the excmer laser? (b) How many photons rom ths laser are requred to delver an energy o J to the cornea? 56. IP BIO Weldng a Detached Retna A detached retna can be welded back nto place usng a laser beam that emts photons wth a wavelength o 514 nm. (a) I such a laser emts photons per second, what s ts power output n watts? (b) Suppose a second laser emts the same number o photons per second. I the wavelength o photons rom the second laser s 414 nm, s ts power output greater than, less than, or the same as the power ound n part (a)? Explan. (c) Calculate the power output o the second laser. General Problems 57. Fnd the mnmum requency a photon must have t s to onze the ground state o the hydrogen atom. 58. It was ponted out n Secton 31-3 that ntermolecular collsons at room temperature do not have enough energy to cause an exctaton n hydrogen rom the n = 1 state to the n = 2 state. Gven that the average knetc energy o a hydrogen atom n a hgh-temperature gas s 3 2 kt (where k s Boltzmann's constant), nd the mnmum temperature requred or atoms to have enough thermal energy to excte electrons rom the ground state to the n = 2 state. 59. IP Reerrng to Problem 60, nd (a) the energy o the emtted photon and (b) the knetc energy o the hydrogen atom ater the photon s emtted. (c) Do you expect the sum o the energes n parts (a) and (b) to be greater than, less than, or the same as the derence n energy between the n = 4 and n = 2 states o hydrogen? Explan. 60. The electron n a hydrogen atom makes a transton rom the n = 4 state to the n = 2 state, as ndcated n Fgure (a) Determne the lnear momentum o the photon emtted as a result o ths transton. (b) Usng your result to part (a), nd the recol speed o the hydrogen atom, assumng t was at rest beore the photon was emtted. 10

11 61. BIO Laser Eye Surgery In laser eye surgery, the laser emts a 1.25-ns pulse ocused on a spot that s μm n dameter. (a) I the energy contaned n the pulse s 2.75 mj, what s the power per square meter (the rradance) assocated wth ths beam? (b) Suppose a molecule wth a dameter o nm s rradated by the laser beam. How much energy does the molecule receve n one pulse rom the laser? (Ths much energy s lkely to dssocate the molecule.) 62. Consder an electron n the ground-state orbt o the Bohr model o hydrogen. (a) Fnd the tme requred or the electron to complete one orbt about the nucleus. (b) Calculate the current (n amperes) correspondng to the electron's moton. 63. A partcular Bohr orbt n a hydrogen atom has a total energy o 085. ev. What s (a) the knetc energy o the electron n ths orbt and (b) the electrc potental energy o the system? 64. The element helum s named or the Sun because that s where t was rst observed. (a) What s the shortest wavelength that one would expect to observe rom a sngly onzed helum atom n the atmosphere o the Sun? (b) Suppose lght wth a wavelength o nm s observed rom sngly onzed helum. What are the ntal and nal values o the quantum number n correspondng to ths wavelength? 65. An onzed atom has only a sngle electron. The n = 6 Bohr orbt o ths electron has a radus o m. Fnd (a) the atomc number Z o ths atom and (b) the total energy E o ts n = 3 Bohr orbt. 66. Fnd the approxmate wavelength o K β X-rays emtted by molybdenum ( Z = 42 ), and compare your result wth Fgure (Hnt: An electron n the M shell s shelded rom the nucleus by the sngle electron n the K shell, plus all the electrons n the L shell.) 67. Reerrng to the hnt gven n Problem 66, estmate the wavelength o L α X-rays n molybdenum. 11

12 68. IP The Pckerng Seres In 1896, the Amercan astronomer Edward C. Pckerng ( ) dscovered an unusual seres o spectral lnes n lght rom the hot star Zeta Pupps. Ater some tme, t was determned that these lnes are produced by sngly onzed helum. In act, the Pckerng seres s produced when electrons drop rom hgher levels to the n = 4 level o He +. Spectral lnes n the Pckerng seres have wavelengths gven by C F H I K λ = 16 n In ths expresson, n = 567,,, K. (a) Do you expect the constant C to be greater than, less than, or equal to the Rydberg constant R? Explan. (b) Fnd the numercal value o C. (c) Pckerng lnes wth n correspond to Balmer lnes n hydrogen wth n = 6810,,,K = 345,,, K. Very ths asserton or the n = 6 Pckerng lne. 69. IP Rydberg Atoms There s no lmt to the sze a hydrogen atom can attan, provded t s ree rom dsruptve outsde nluences. In act, rado astronomers have detected radaton rom large, so-called Rydberg atoms n the duse hydrogen gas o nterstellar space. (a) Fnd the smallest value o n such that the Bohr radus o a sngle hydrogen atom s greater than 8.0 mcrons, the sze o a typcal sngle-celled organsm. (b) Fnd the wavelength o radaton ths atom emts when ts electron drops rom level n to level n 1. (c) I the electron drops one more level, rom n 1 to n 2, s the emtted wavelength greater than or less than the value ound n part (b)? Explan. 70. Consder a partcle o mass m, charge q, and constant speed v movng perpendcular to a unorm magnetc eld o magntude B, as shown n Fgure The partcle ollows a crcular path. Suppose the angular momentum o the partcle about the center o ts crcular moton s quantzed n the ollowng way: mvr where n = 123,,, K, and h = h 2π. a. Show that the rad o ts allowed orbts have the ollowng values: = nh, r n = nh qb b. Fnd the speed o the partcle n each allowed orbt. 12

13 71. Consder a partcle o mass m conned n a one-dmensonal box o length L. In addton, suppose the matter wave assocated wth ths partcle s analogous to a wave on a strng o length L that s xed at both ends. Usng the de Brogle relatonshp, show that (a) the quantzed values o the lnear momentum o the partcle are and (b) the allowed energes o the partcle are nh pn = n 2L = 123,,,K 2 Fh En = n H G I 2 n ml K J = 123,,,K Show that the tme requred or an electron n the nth Bohr orbt o hydrogen to crcle the nucleus once s gven by where T = h 4π mk e. Interactve Problems T = Tn n = 123,,,K 73. IP Reerrng to Example 31-3 Suppose the electron s n a state whose standng wave consstng o two wavelengths. (a) Is the wavelength o ths standng wave greater than or less than m? (b) Fnd the wavelength o ths standng wave. 74. Reerrng to Example 31-3 (a) Whch state has a de Brogle wavelength o m? (b) What s the Bohr radus o ths state? 13