Chapter #4 Solutions - Modern Physics for Scientists and Engineers - Andrew Rex, Stephen T. Thornton - 3rd Edition

 

1. In Thomson’s plum-pudding model, devise an atomic composition for carbon that consists of a pudding of charge -6e along with six electrons. Try to configure a system in which the charged particles move only about points in stable equilibrium. Get solution

1q. Thomson himself was perhaps the biggest critic of the model referred to as “plum pudding.” He tried for years to make it work. What experimental data could he not predict? Why couldn’t he make the planetary model of Rutherford-Bohr work? Get solution

2. How large an error (in percent) in the velocity do we make by treating the velocity of a 7.7-MeV alpha particle nonrelativistically? Get solution

2q. Does it seem fortuitous that most of the successful physicists who helped unravel the secrets of atomic structure (Thomson, Rutherford, Bohr, Geiger, and Moseley) worked either together or in close proximity in England? Why do you suppose we don’t hear names of physicists working on this idea in other European countries or in the United States? Get solution

3. In Example 4.1, show that the electron’s velocity must be ...in order to conserve energy and linear momentum. ... Get solution

3q. Could the Rutherford scattering of α particles past 90° be due to scattering from electrons collected together (say, 100 e-) in one place over a volume of diameter 10-15 m? Explain Get solution

4. Thomson worked out many of the calculations for multiple scattering. If we find an average scattering angle of 1° for alpha-particle scattering, what would be the probability that the alpha particle could scatter by as much as 80° because of multiple scattering? The probability for large-angle scattering is exp( - (θ/)2). Geiger and Marsden found that about 1 in 8000 α particles were deflected past 90°. Can multiple scattering explain the experimental results of Geiger and Marsden? Explain. Get solution

4q. In an intense electron bombardment of the hydrogen atom, significant electromagnetic radiation is produced in all directions upon decay. Which emission line would you expect to be most intense? Why? Get solution

5. Calculate the impact parameter for scattering a 7.7-MeV α particle from gold at an angle of (a) 1° and (b) 90°. Get solution

5q. Why are peaks due to higher-lying excited states in the Franck-Hertz experiment not more observable? Get solution

6. A beam of 8.0-MeV α particles scatters from a thin gold foil. What is the ratio of the number of α particles scattered to angles greater than 1° to the number scattered to angles greater than 2°? Get solution

6q. As the voltage increases above 5 V in the Franck-Hertz experiment, why doesn’t the current suddenly jump back up to the value it had below 5 V? Get solution

7. For aluminum (Z = 13) and gold (Z = 79) targets, what is the ratio of an alpha particle scattering at any angle for equal numbers of scattering nuclei per unit area? Get solution

7q. Using Hg gas in the Franck-Hertz experiment, approximately what range of voltages would you expect for the first peak? Explain. Get solution

8. What fraction of 5-MeV α particles will be scattered through angles greater than 8° from a gold foil (Z = 79, density = 19.3 g/cm3) of thickness 10-8 m? Get solution

8q. When are photons likely to be emitted in the Franck- Hertz experiment? Get solution

9. In an experiment done by scattering 5.5-MeV α particles from a thin gold foil, students find that 10,000 α particles are scattered at an angle greater than 50°. (a) How many of these α particles will be scattered greater than 90°? (b) How many will be scattered between 70° and 80°? Get solution

9q. Is an electron most strongly bound in an H, He+, or Li++ atom? Explain. Get solution

10. Students want to construct a scattering experiment using a powerful source of 5.5-MeV α particles to scatter from a gold foil. They want to be able to count 1 particle/s at 50°, but their detector is limited to a maximum count rate of 2000 particles/s. Their detector subtends a small angle. Will their experiment work without modifying the detector if the other angle they want to measure is 6°? Explain. Get solution

10q. Why do we refer to atoms as being in the “ground” state or “stationary”? What does an “excited” state mean? Get solution

11. The nuclear radii of aluminum and gold are approximately r = 3.6 fm and 7.0 fm, respectively. The radii of protons and alpha particles are 1.3 fm and 2.6 fm, respectively. (a) What energy α particles would be needed in head-on collisions for the nuclear surfaces to just touch? (This is about where the nuclear force becomes effective.) (b) What energy protons would be needed? In both (a) and (b), perform the calculation for aluminum and for gold. Get solution

11q. What lines would be missing for hydrogen in an absorption spectrum? What wavelengths are missing for hydrogen in an emission spectrum? Get solution

12. Consider the scattering of an alpha particle from the positively charged pan. of the Thomson plum-pudding model. Let the kinetic energy of the α particle be K (nonrelativistic) and let the atomic radius be R. (a) Assuming that the maximum transverse Coulomb force acts on the α particle for a time ...(where ... is the initial speed of the α particle), show that. the largest scattering angle we can expect from a single atom is ... (b) Evaluate ... for an 8.04MeV α particle scattering from a gold atom of radius ... Get solution

12q. Why can’t the Bohr model be applied to the neutral He atom? What difficulties do you think Bohr had in modifying his model for He? Get solution

13. Using the results of the previous problem, (a) find the average scattering angle of a 10-MeV α particle from a gold atom (R ... 10-10 m) for the positively charged part of the Thomson model. (b) How does this compare with the scattering from the electrons? Get solution

13q. Describe how the hydrogen atom might absorb a photon of energy less than 13.6 eV. Describe a process by which a 9.8-eV photon might be absorbed. What about a 15.2-eV photon? Get solution

14. The radius of a hydrogen nucleus is believed to be about 1.2 × 10-15 m. (a) If an electron rotates around the nucleus at that radius, what would be its speed according to the planetary model? (b) What would be the total mechanical energy? (c) Are these reasonable? Get solution

15. Make the (incorrect) assumption that the nucleus is composed of electrons and that the protons are outside. (a) If the size of an atom were about 10-10 m, what would be the speed of a proton? (b) What would be the total mechanical energy? (c) What is wrong with this model? Get solution

16. Calculate the speed and radial acceleration for a ground-state electron in the hydrogen atom. Do the same for the He+ ion and the Li++ ion. Get solution

17. What is the total mechanical energy for a ground-state electron in ... atoms? For which atom is the electron most strongly bound? Why? Get solution

18. Calculate the time, according to classical laws, it would take the electron of the hydrogen atom to radiate its energy and crash into the nucleus. [Hint: The radiated power P is given by ... ...where Q is the charge, c the speed of light, and r the position vector of the electron from the center of the atom.] Get solution

19. The Ritz combination rules express relationships between observed frequencies of the optical emission spectra. Prove one of the more important ones: ... where K α and K β refer to the Lyman series and L α to the Balmer series of hydrogen (Figure 4.18). ... Get solution

20. Calculate the angular Momentum in kg • m2/s for the lowest electron orbit in the hydrogen atom. Compare the result with Planck's Constant h. Get solution

22. What is the speed (ratio of v/c) of the electron in the first three Bohr orbits of the H atom? Get solution

23. A hydrogen atom in an excited state absorbs a photon of wavelength 434 nm. What were the initial and final states of the hydrogen atom? Get solution

24. A hydrogen atom in an excited state emits a photon of wavelength 95 nm. What are the initial and final states of the hydrogen atom? Get solution

25. What is the binding energy of the electron in the ground state of (a) deuterium, (b) He+, and (c) Be++? Get solution

26. The isotope shift of spectral lines refers to the shift in wavelengths (or frequencies) due to the different isotopic masses of given elements. Find the isotope shifts for each of the four visible Balmer series wavelengths for deuterium and tritium compared with hydrogen. Get solution

27. Find the isotope shift (see Problem 26) of the ground-state energy for deuterium and tritium compared with the ground-state energy of hydrogen. Express the answer in eV. Problem 26 The isotope shift of spectral lines refers to the shift in wavelengths (or frequencies) due to the different isotopic masses of given elements. Find the isotope shifts for each of the four visible Balmer series wavelengths for deuterium and tritium compared with hydrogen. Get solution

28. Describe the visible absorption spectra for (a) a hydrogen atom and (b) an ionized helium atom, He+. Get solution

29. A hydrogen atom exists in an excited state for typically 10-8 s. How many revolutions would an electron make in an n = 3 state before decaying? Get solution

30. Electromagnetic radiation of wavelength 100 nm is incident upon the ground-state hydrogen atom at rest What is the highest state to which hydrogen can be excited? Get solution

31. A muonic atom consists of a muon (mass m = 106 MeV/c2 and charge q = -e) in place of an electron. For the muon in a hydrogen atom, what is (a) the smallest radius and (b) the binding energy of the muon in the ground state? (c) Calculate the series limit of the wavelength for the first three series. Get solution

32. Positronium is an atom composed of an electron and a positron (mass m = me, charge q = +e). Calculate the distance between the particles and the energy of the lowest energy state of positronium. (Hint: what is the reduced mass of the two particles? See Problem 53.) Problem 53 The proton (mass M) and electron (mass m) in a hydrogen atom actually rotate about their common center of mass as shown in Figure 4.17. The distance r = re + rM is still defined to be the electron-nucleus distance. Show that Equation (4.24) is only modified by substituting for m by ... ... ... Get solution

33. (a) Find the Bohr radius of the positronium atom described in the previous problem. (b) Find the wavelength for the transition from nu = 2 to n/ = 1 for positronium. Get solution

34. What is the difference in the various bohr radii ra ... Get solution

35. Compare the Balmer series of hydrogen with the series where n/ = 4 for the ionized helium atom He+. What is the difference between the wavelengths of the L α and L β line of hydrogen and the nu = 6 and 8 of He+? Is there a wavelength of the Balmer series that is very similar to any wavelength values where n/ = 4 in He+? Explain. Get solution

36. Calculate the Rydberg constant for the single-electron (hydrogen-like) ions of helium, potassium, and uranium. Compare each of them with R∞ and determine the percentage difference. Get solution

37. What wavelengths for the L α lines did Moseley predict for the missing Z = 43, 61, and 75 elements? (See Example 4.10.) ... Get solution

38. What wavelengths for the L α lines did Moseley predict for the missing Z = 43, 61, and 75 elements? (See Example 4.10.) ... Get solution

39. If the resolution of a spectrograph is Δ λ = 10-12 m, would it be able to separate the Kα lines for lead and bismuth? Explain. Get solution

40. Determine the correct equation to describe the Kα frequencies measured by Moseley. Compare that with Moseley’s equation for K β frequencies. Does the result agree with the data in Figure 4.19? Explain. ... Get solution

41. Calculate the Kα and Kβ wavelengths for He and Li. Get solution

42. (a) Calculate the ratio of Kα wavelengths for uranium and carbon. (b) Calculate the ratio of Lα wavelengths for platinum and calcium. Get solution

43. Calculate the three longest wavelengths and the series limit for the molybdenum atom. Get solution

44. An unknown element is used as a target in an x-ray tube. Measurements show that the characteristic spectral lines with the longest wavelengths are 0.155 nm and 0.131 nm. What is the element? (Hint: you will find the answer to Problem 40 to be useful.) Problem 40 Determine the correct equation to describe the Kα frequencies measured by Moseley. Compare that with Moseley’s equation for K β frequencies. Does the result agree with the data in Figure 4.19? Explain. ... Get solution

45. If an electron of 40 eV had a head-on collision with a Hg atom at rest, what would be the kinetic energy of the recoiling Hg atom? Assume an elastic collision. Get solution

46. In the Franck-Hertz experiment, explain why the small potential difference between the grid and collector plate is useful. Redraw the data of Figure 4.21 the way the data would appear without this small retarding potential. ... Get solution

47. Calculate the value of Planck’s constant determined by Franck and Hertz when they observed the 254-nm ultraviolet radiation using Hg vapor. Get solution

48. Consider an element having excited states at 3.6 eV and 4.6 eV used as a gas in the Franck-Hertz experiment. Assume that the work functions of the materials involved cancel out. List all the possible peaks that might be observed with electron scattering up to an accelerating voltage of 18 V. Get solution

49g. The redshift measurements of spectra from magnesium and iron are important in understanding distant galaxies. What are the Kα and Lα wavelengths for magnesium and iron? Get solution

50g. In the early 1960s the strange optical emission lines from starlike objects that also produced tremendous radio signals confused scientists. Finally, in 1963 Maarten Schmidt of the Mount Palomar observatory discovered that the optical spectra were just those of hydrogen but redshifted because of the tremendous velocity of the object with respect to Earth. The object was moving away from Earth at a speed of 50,000 km/s! Compare the wavelengths of the normal and redshifted spectral lines for the Kα and K β lines of the hydrogen atom. Get solution

51g. A beam of 8.0-MeV α particles scatters from a gold foil of thickness 0.32 μm. (a) What fraction of the α particles is scattered between 1.0° and 2.0°? (b) What is the ratio of α particles scattered through angles greater than 1° to the number scattered through angles greater than 10°? Greater than 90°? Get solution

52g. In Rutherford scattering we noted that angular momentum is conserved. The angular momentum of the incident α particle relative to the target nucleus is mv0b where m is the mass, v0 is the initial velocity of the α particle, and b is the impact parameter. Start with ... and show that angular momentum is conserved, and the magnitude is given by mv0b along the entire path of the α particle while it is scattered by the Coulomb force from a gold nucleus. Get solution

53g. The proton (mass M) and electron (mass m) in a hydrogen atom actually rotate about their common center of mass as shown in Figure 4.17. The distance r = re + rM is still defined to be the electron-nucleus distance. Show that Equation (4.24) is only modified by substituting for m by ... ... ... Get solution

54g. In Bohr’s Assumption D, he assumed the mean value K of the kinetic energy of the electron-nucleus system to be nhforb/2 where forb is the orbital frequency of the electron around the nucleus. Calculate forb in the ground state in the following ways: (a) Use fclassical in Equation (4.34). (b) Use Equation (4.33a), but first determine v and r. (c) Show that the mean value K is equal to the absolute value of the electron-nucleus system total energy and that this is 13.6 eV. Use this value of K to determine forb from the relation for K stated above. ... ... Get solution

55g. Show that the quantization of angular momentum L =nh follows from Bohr’s Assumption D that the mean value K of the kinetic energy of the electron-nucleus system is given by K _ nhforb/2. Assume a circular orbit. Get solution

56g. (a) Calculate the energies of the three lowest states of positronium. (b) Determine the wavelengths of the Kα , K β, Lα and L β transitions. Get solution


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