1. Consider again the rotational energy states of the N2 molecule as
described in Example 10.1. Find the energy involved in a transition (a)
from the / = 2 to / = 1 state, and (b) from the / = 10 to / = 9 state. Get solution
1q. Explain in your own words why the sky is blue. Get solution
2. (a) Use the data in Table 10.1 to find the approximate spacing between vibrational energy levels in CO. (b) What temperature would be needed to excite this vibration thermally? ... Get solution
2q. Explain why n ? m is required in Equation (10.1) in order to produce the potential energy curve shown in Figure 10.1. ... ... Get solution
3. Estimate the amplitude of the smallest vibration of the HCl molecule (see Example 10.2). ... ... Get solution
3q. Compare the force constants for diatomic molecules (Table 10.1) with those of common laboratory springs. (Remember your introductory college or high school lab experience.) ... Get solution
4. The distance between the centers of the H and Cl atoms in the HCl molecule is approximately 0.128 nm. (a) Find the angular velocity of the molecule about its center of mass when / = 1 and / = 5. (b) What is the speed of the H atom in each of the cases in (a)? (c) What value of / is required for the H atom to have a speed of 0.1c? (d) Estimate the classical temperature associated with the / you found in (c). Get solution
4q. Explain how to use the rotational spectra to determine the equilibrium separation between the two nuclei in a diatomic molecule. Get solution
5. Derive an expression for the allowed rotational levels in a homopolar diatomic molecule using the Bohr quantization rule for angular momentum. Discuss your result in comparison to the correct quantum mechanical result, Equation (10.2). ... Get solution
5q. Why do the gases He, Ne, and Xe tend to be monatomic rather than diatomic? Get solution
6. The wavelength of a microwave absorption line in CO corresponding to a transition from / = 0 to / = 1 is 2.60 mm. (a) Calculate the rotational inertia of the CO molecule. (b) Show that it is impossible for this amount of energy (corresponding to a photon of wavelength 2.60 mm) to be absorbed by CO in a vibrational transition. Get solution
6q. Explain the low melting points and boiling points for inert gases. (For example, Ne has a melting point 24.5 K and boiling point 27.1 K. Get solution
7. If the energy of a vibrational transition from the n = 0 state to the n = 1 state in CO could be absorbed in a rotational transition that begins in the ground state (/ = 0), what would be the value of / for the final state? Explain why such a rotational transition is impossible. Get solution
7q. Do you expect the fundamental vibrational frequency to be higher for HCl or NaCl? Explain. Get solution
8. Show that I = μR2 for a diatomic molecule, where R is the distance between the two atomic centers and μ= m1m2/(m1 + m2) is the reduced mass. (Note: You should assume that the atoms are point particles.) Get solution
8q. Is it necessary that the substance used in a laser have at least three energy levels? Why or why not? Get solution
9. The energy of a transition from the / = 2 to the / = 3 state in CO is 1.43 × 10-3 eV. (a) Compute the rotational inertia of the CO molecule. (b) What is the average separation between the centers of the C and O atoms? Get solution
9q. Critique the following statement: The Pauli exclusion principle is responsible for keeping solids from collapsing to zero volume. Get solution
10. Consider the model of the H2O molecule shown in the diagram. (a) Find the rotational inertia of H2O about the dashed line. (b) Estimate the energies of the first two rotational energy levels ... 2). (c) What is the wavelength of a photon required to excite a transition from ... ... Get solution
10q. The average nearest-neighbor distance between nuclei in solid NaCl is 0.282 nm, but the distance is 0.236 nm in a free NaCl molecule. How do you account for the difference? Get solution
11. Consider the rotational energy of a single helium atom. Assume that the electrons are uniformly distributed throughout a sphere of radius equal to the Bohr radius and that the nucleus is a uniform solid sphere of radius 1.9 × 10-15 m. (a) Estimate the energy of the first (nonzero) rotational energy level in helium. (b) Is this state likely to be observed? Why or why not? Get solution
11q. What patterns do you notice within the groups of salts in Table 10.2? (A group is defined as having the same metal, e.g., sodium.) Explain those patterns. ... Get solution
12. Consider the NaCl molecule, for which the rotational inertia is 1.30 × 10-45 kg . m2. If infrared radiation with wavelength 30 μm is Raman-scattered from a free NaCl molecule, what are the allowed wavelengths of the scattered radiation? Get solution
12q. What makes elements good candidates for paramagnetism? For diamagnetism? Get solution
13. This problem deals with the splitting of rotational energy levels of diatomic molecules. If one atom of the molecule has more than one stable isotope, then both isotopes are normally present in a sample. Show that the fractional change Δf/f in the observed frequency of a photon emitted in a transition between adjacent rotational states is equal to the fractional difference in the reduced mass Δ μ/ μ for molecules containing the two different isotopes. Get solution
13q. Explain why the paramagnetic susceptibilities of rare earth elements tend to be higher than those of the transition elements. Get solution
14. (a) At T = 293 K what are the relative Maxwell- Boltzmann factors for N2 molecules in the / = 0, / = 1, and / = 2 states? Use I = 1.4 × 10-46 kg . m2. (b) Use your answer to (a) along with the fact that there is also a degeneracy factor of 2/ + 1 on the /th angular momentum level to find the relative populations of the / = 0, / = 1, and / = 2 states of N2 at room temperature. (c) Explain why the 2/ = 1 degeneracy factor is more important for lower rotational states, but the Maxwell-Boltzmann factor dominates for higher states. Get solution
14q. Consider the electronic configurations of the five ferromagnetic elements and justify why they are ferromagnetic. Why are some elements in the same columns as these five, with similar electronic configurations, not ferromagnetic? Get solution
15. Use the HCl absorption spectrum shown in Figure 10.8 to (a) compute the rotational inertia I of the molecule and (b) compute the force constant κ and compare with the value given in Table 10.1. ... ... Get solution
15q. Notice that ferromagnetic elements tend to come from the middle of the rows of rare earth elements and transition elements in the periodic table. Explain. Get solution
16. The equilibrium separation between the two ions in the KCl molecule is 0.267 nm. (a) Assuming that the K+ and Cl- ions are point particles, compute the electric dipole moment of the molecule. (b) Compute the ratio of your result in (a) to the measured electric dipole moment of 5.41 × 10-29 C . m. This ratio is known as the fractional ionic character of the molecular bond. Get solution
16q. Why should elements and compounds with positive paramagnetic susceptibilities not be good candidates for superconductivity? Get solution
17. Find the energy of the photon required to excite the transition from the ground state to the first excited vibrational state in HI. In what part of the electromagnetic spectrum is this? Get solution
17q. Explain similarities and differences between the Meissner effect and Lenz’s law. Get solution
18. (a) How many photons are emitted each second from a 5.0-mW helium-neon laser (λ = 632.8 nm)? (b) If the laser contains 0.02 mole of neon gas, what fraction of the neon atoms in the tube participate in the lasing process during each second of operation? (c) Comment on the relatively low numerical result in (b). Get solution
18q. Consider the superconducting transition temperatures of the elements as shown in Table 10.5. In cases in which there is more than one superconductor in a column of the periodic table, are the transition temperatures consistent with the spirit of the isotope effect (that is, does the heavier element have a lower Tc)? ... Get solution
19. A laser emits 5.50 × 1018 photons per second, using a transition from an excited state with energy 1.15 eV to a ground state with energy 0 eV. (a) What is the laser’s power output? (b) What is the wavelength? Get solution
19q. A superconducting ring can carry a current for an indefinite length of time. Isn’t this a perpetual motion machine, which violates the first or second law of thermodynamics? Explain. Get solution
20. The NOVA laser at Lawrence Livermore National Lab produces a 40-kJ burst of 3.5 ns duration, with a wavelength of 351 nm. (a) How many atoms made a transition from the excited state to the ground state in order to create this pulse? (b) What is the laser’s average power output during the burst? Get solution
20q. Consider a sample at a temperature initially above its superconducting Tc and in a magnetic field. When the sample is cooled to below Tc, currents are generated to expel the magnetic flux from the interior. What is the source of the energy for these currents? Get solution
21. (a) For the helium-neon laser, estimate the Doppler broadening (see Chapter 9, Problem 3) of the output wavelength 632.8 nm at T = 293 K. (b) Estimate the broadening of the same wavelength due to the Heisenberg uncertainty principle, assuming that the metastable state has a lifetime of about 1 ms. Consider an ideal gas enclosed in a spectral tube. When a high voltage is placed across the tube, many atoms are excited, and all excited atoms emit electromagnetic radiation at characteristic frequencies. According to the Doppler effect, the frequencies observed in the laboratory depend on the velocity of the emitting atom. The nonrelativistic Doppler shift of radiation emitted in the x direction is .... The resulting wavelengths observed in the spectroscope are spread to higher and lower values because of the (respectively) lower and higher frequencies, corresponding to negative and positive values of vx. We say that the spectral line has been Doppler broadened. This is what allows us to see the lines easily in the spectroscope, because the Heisenberg uncertainty principle does not cause significant line broadening in atomic transitions. (a) What is the mean frequency of the radiation observed in the spectroscope? (b) To get an idea of how much the spectral line is broadened at particular temperatures, derive an expression for the standard deviation of frequencies, defined to be Standard deviation ... Your result should be a function of f 0, T, and constants. (c) Use your results from (b) to estimate the fractional line width, defined by the ratio of the standard deviation to f0, for hydrogen (H2) gas at T = 293 K. Repeat for a gas of atomic hydrogen at the surface of a star, with T = 5500 K. Get solution
22. Consider the problem of using laser light to measure the distance from the Earth to the moon. (a) What is the maximum uncertainty in timing the round trip for a light pulse in order to determine the distance with an uncertainty of 1 meter? (b) Estimate the effect of the Earth’s atmosphere on this experiment, using the fact that the speed of light in air (at sea level) is slower than the speed of light in vacuum by a factor of 1.0003. Assume an 8-km-high atmosphere of uniform sea-level density. Get solution
23. What is the minimum fraction of the lasing molecules in a three-level laser that must be in the excited state in order for the laser to operate? Answer the same question for a four-level laser. Get solution
22q. A superconducting wire and a low-resistance copper wire are connected together in parallel. When a potential difference is applied, explain why the copper wire carries no current. Get solution
23q. Explain on physical grounds why the ratio ρ /r0 in Equation (10.21) should be less than 1. ... Get solution
24. The 3s state of neon (see Figure 10.15) is 16.6 eV above the ground state. (a) Estimate the relative populations of the ground state and the 3s state at T = 293 K. (b) Repeat for T = 150 K. (c) Repeat for T = 600 K. (d) What implications (if any) do your answers for parts (a)–(c) have for the operation of a He-Ne laser at various temperatures? Get solution
25. The density of solid KCl is about 1980 kg/m3. Compute the nearest-neighbor distance in KCl, that is, the distance between neighboring K+ and Cl- ions. Note that KCl has the same lattice structure as NaCl. Get solution
26. Show that the Madelung constant for a one-dimensional lattice of alternating positive and negative ions is α= 2 ln 2. Get solution
27. Write the first five terms of the Madelung constant for a two-dimensional lattice of alternating positive and negative ions. Get solution
28. Use Equation (10.19) to evaluate the net force (F = -dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
29. Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... where ... and ... Problem 28 Use Equation (10.19) to evaluate the net force (F =-dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
30. If we consider again the linear term (r) in the result of Problem 29, we have a harmonic oscillator. (a) Find an expression for the frequency of oscillation and evaluate using the correct values of α, r0, and ρ for NaCl. (b) Find the photon wavelength corresponding to the frequency you computed in (a) and compare with the observed absorption wavelength for NaCl of about 61 μm. Problem 29 Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... where ... and ... Problem 28 Use Equation (10.19) to evaluate the net force (F =-dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
31. Refer again to the result of Problem 29. (a) Use the fact that the average net force on an ion must be zero (if there is no overall translation of the crystal) to show that the average values of r and (δr)2 are related by .... (b) According to the equipartition theorem the mean potential energy of an oscillator is ...Use this to show that ...and thereby show that the coefficient of thermal expansion is approximately .... (c) Use the result of (b) to evaluate the coefficient of thermal expansion for NaCl and compare with the experimental value of 4 × 10-6 K-1. Problem 29 Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... Where ... and ... Get solution
32. Silver has an electrical conductivity of 6.30 × 107Ω-1. m-1 at 293 K. Use the Wiedemann-Franz law without quantum corrections to compute the thermal conductivity of silver at the same temperature and compare with the experimental result of 429 W . K-1 . m-1. Get solution
33. Use the data in Figure 10.23 to estimate the numerical value of the constant b/a2 in the thermal expansion formula, Equation (10.26). Make sure you express the correct units. ... ... Get solution
34. (a) Derive Equation (10.26). (b) Evaluate the constant 3bk/4a2 in Equation (10.26) for copper, given that the coefficient of linear expansion [defined as ...]for copper is found experimentally to be 1.67 × 10-5 K-1 at T = 293 K. Get solution
35. (a) Explain why the parameter a in Equation (10.22) is essentially half the effective force constant of a spring connecting adjacent atoms. (b) Then use this result along with the result of Problem 34b to estimate a value for the parameter b in Equation (10.22), the coefficient of the x3 term in the potential energy. ... (a) Explain why the parameter a in Equation (10.22) is essentially half the effective force constant of a spring connecting adjacent atoms. (b) Then use this result along with the result of Problem 34b to estimate a value for the parameter b in Equation (10.22), the coefficient of the x3 term in the potential energy. ... Problem 34b (a) Derive Equation (10.26). (b) Evaluate the constant 3bk/4a2 in Equation (10.26) for copper, given that the coefficient of linear expansion [defined as ...]for copper is found experimentally to be 1.67 × 10-5 K-1 at T = 293 K. Get solution
36. (a) Show that the ideal gas law can be written as ... where N is the number of particles in the sample and ...is the mean energy. (b) Use the result of (a) to estimate the pressure of the conduction electrons in copper, assuming an ideal Fermi electron gas. Comment on the numerical result, noting that 1 atm = 1.01 ×105 Pa. Get solution
37. Show that the bulk modulus, defined as ... (where P is pressure and V is volume) can be written as ... for a Fermi electron gas with Fermi energy EF. (Hint: Use the relationship between P and V given in Problem 36a.) Problem 36a Show that the ideal gas law can be written as ... Get solution
38. (a) From the result of Problem 37, compute the bulk modulus of pure silver. (b) Compare your result with the experimental value of 1.01 × 1011 N/m2. Problem 37 Show that the bulk modulus, defined as ... (where P is pressure and V is volume) can be written as ... for a Fermi electron gas with Fermi energy EF. (Hint:Use the relationship between P and V given in Problem 36a.) Problem 36a Show that the ideal gas law can be written as ... Get solution
39. Retrace the derivation of the induced diamagnetic moment in Equation (10.38), assuming that (a) the electron orbits clockwise and the magnetic field points out of the page and (b) the electron orbits counterclockwise and the magnetic field points into the page. ... Get solution
40. Use the result of the preceding problem to determine the induced magnetic moment of a diamagnetic atom with an outer shell having three electrons in a p shell with m/ = 0, m/ = 1, and m/ = -1. Get solution
41. Start with Equation (10.41) and derive an expression for ... valid in the low-temperature limit kT B ... Get solution
42. (a) Plot ... versus B/kT over the range μ B/kT = 0 to μ B/kT = 4. (b) Compute ... at μ B/kT = 5 and compare your results with the approximation used for m in Problem 41. (c) Compute ... at μ B/kT = 0.10 and compare your results with the approximate value given in Section 10.4, .... Problem 41 Start with Equation (10.41) and derive an expression for ... valid in the low-temperature limit kT B ... Get solution
43. Prove that magnetic susceptibility ... is a dimensionless quantity. Note that the definition in Equation (10.37) presumes SI units. ... Get solution
44. (a) Compute the maximum magnetization of a bulk sample of iron, assuming perfect alignment of the spins and one unpaired spin per atom. (b) Compare with the observed maximum magnetization of about 1.6 × 106 A/m. (c) On the basis of your results in (a) and (b), what can you say about the actual number of unpaired spins per atom of iron? Get solution
45. At what temperature (expressed as a fraction of Tc) is Bc = 0.25Bc(0), according to the BCS theory? [Note: Bc(0) is the critical field at temperature T = 0.] Repeat for Bc = 0.50 Bc(0) and Bc = 0.75Bc(0). Get solution
46. It is found that for a given pure metal superconductor, photons of wavelength 0.568 mm are sufficient to break the Cooper pairs at T = 2.0 K. Identify the superconductor. Get solution
47. Compute Tc for the mercury isotopes 201Hg and 204Hg. Get solution
48. Estimate the BCS prediction for the change in Tc if all of the 16O atoms in YBa2Cu3O7 (assume Tc = 93.0 K) could be replaced by 18O. This change is not observed experimentally! Get solution
49. From the data given in the text, estimate Tc for Tl2Ba2Can-1O2n-4 with n = 4. Get solution
50. A solenoid made of superconducting wire has exactly 25 turns/cm of length and a diameter of 3.2 cm. If the wire carries a current of 4.5 A, what is the magnetic field strength near the center of the solenoid? What is the magnetic flux through a cross section of the solenoid taken near the center? To how many flux quanta does this correspond? Comment on the number of flux quanta. Get solution
51. Recall that the magnetic field at the surface of a uniform cylindrical wire of radius R carrying a current I is ... Find the minimum possible diameter for a wire of pure niobium so that the T = 0 critical field would not be exceeded if the wire carried a current of 2.5 A. Get solution
52. In a normal conductor heat is generated at a rate I 2R. Therefore a current-carrying conductor must dissipate heat effectively or it can melt or overheat the device in which it is used. Consider a long cylindrical copper wire (resistivity 1.72 × 10-8 Ω. m) of diameter 0.75 mm. If the wire can dissipate 80 W/m2 along its surface, what is the maximum current this wire can carry? Get solution
53. (a) Compute the maximum current that a 16-gauge (1.29-mm diameter) niobium wire can carry at T = 4.2 K. (b) Compare your result in (a) with the copper wire of the same diameter described in Problem 52. In a normal conductor heat is generated at a rate I 2R. Therefore a current-carrying conductor must dissipate heat effectively or it can melt or overheat the device in which it is used. Consider a long cylindrical copper wire (resistivity 1.72 × 10-8 Ω . m) of diameter 0.75 mm. If the wire can dissipate 80 W/m2 along its surface, what is the maximum current this wire can carry? Get solution
54. What is the maximum uncertainty in the measurement of the oscillation frequency in a Josephson junction if the voltage standard of 1 mV is to be maintained within 1 part in 1010? Assume a reference frequency of 483.6 GHz. Get solution
55. Find the minimum acceleration needed for a maglev train to reach a speed of 430 km/h in 3.0 km, one tenth of the length of the 30-km track in Shanghai. Express your answer as a fraction of g, the free-fall acceleration near Earth. Would this acceleration be noticeable? Get solution
56. (a) Compute the escape speed of a particle from the Earth’s surface. Earth’s radius is 6378 km, and its mass is 5.98 × 1024 kg. (b) Find the mean speed for a helium atom at a temperature of 293 K. (c) Comment on the fact that your answer to (b) is less than the answer to (a). Why then does helium not remain in the atmosphere in significant quantities? Get solution
57. A superconducting Nb3Sn magnet can achieve a peak magnetic field of 13.5 T in a magnet designed for use in the Large Hadron Collider. Find the maximum energy that a singly charged particle (for example, a proton or electron) can have if that field is maintained around a circular ring of circumference 27 km. (Note: In reality, particle energies are about 35% less because the peak field is not maintained throughout the ring.) Get solution
58g. Consider a model of a diatomic molecule with pointmass atoms of mass m1 and m2, separated by a distance R. (a) Show that the rotational inertia of the molecule is I = μR2, where the reduced mass μ=m1 m2/(m1 + m2). (b) Compute the rotational inertia of NaCl, which has a bond length of 0.236 nm. Assume the most common isotopes of sodium and chlorine. Get solution
59g. Rotational spectra are affected slightly by the fact that different isotopes have different masses. Suppose a sample of the common isotope 1H35Cl is changed to 1H37Cl. (a) By what fraction is the molecule’s rotational inertia different? (The bond length is 0.127 nm in each case.) (b) What is the change in energy of the / = 1 to the / = 0 transition if the isotope is changed? Get solution
60g. The transition from the / = 2 to the / = 1 state in CO is accompanied by the emission of a 9.55 × 10-4 eV photon. (a) Use this information to find the rotational inertia of the CO molecule. (b) What is the bond length between the C and O atoms? Get solution
61g. The National Ignition Facility (NIF), which became operational in 2009, uses 192 laser beams to stimulate nuclear fusion in a deuterium-tritium fuel pellet. The net output of the lasers is 1.8 MJ of 351-nm light, delivered in a brief (4.0-ns) pulse. (a) What is the average power delivered during the pulse? Compare your answer with the average power consumption in the United States, about 3 × 1012 W. (b) How many photons are produced in each pulse? Get solution
62g. Estimate the temperature at which the critical magnetic field in superconducting mercury is equal to Earth’s surface magnetic field, about 5 × 10-5 T. Is Earth’s magnetic field likely to be a factor in applications that use superconducting mercury? Get solution
63g. Tin has a number of stable isotopes, ranging in mass from 112 u to 124 u. Estimate the difference in the transition temperature between those two isotopes. Get solution
64g. In the persistent current experiment described in Section 10.5, let us assume that the current persisted without detectable reduction for exactly 2.5 years. Given that the inductance of the ring was approximately 3.14 × 10-8 H and the sensitivity of the current measurement was 1 part in 109, (a) estimate an upper bound for the resistance of the ring and (b) estimate how long at least 90% of the current would be certain to remain. Get solution
65g. From thermodynamics the entropy difference per unit volume between the normal and superconducting states is ... where B 2/2_0 is the magnetic energy density needed to return a superconductor to the normal state. Use this fact to compute the entropy difference between the normal and superconducting states in 1 mole of niobium at a temperature of 6.0 K. Get solution
66g. A 2.0-m length of copper wire with a resistance of 1.50 ... is placed in series with a 2.0-m length of superconducting wire. When a 12.0-V battery is placed across the series combination, find (a) the current in the circuit and (b) the potential difference across the two ends of the copper wire. Get solution
1q. Explain in your own words why the sky is blue. Get solution
2. (a) Use the data in Table 10.1 to find the approximate spacing between vibrational energy levels in CO. (b) What temperature would be needed to excite this vibration thermally? ... Get solution
2q. Explain why n ? m is required in Equation (10.1) in order to produce the potential energy curve shown in Figure 10.1. ... ... Get solution
3. Estimate the amplitude of the smallest vibration of the HCl molecule (see Example 10.2). ... ... Get solution
3q. Compare the force constants for diatomic molecules (Table 10.1) with those of common laboratory springs. (Remember your introductory college or high school lab experience.) ... Get solution
4. The distance between the centers of the H and Cl atoms in the HCl molecule is approximately 0.128 nm. (a) Find the angular velocity of the molecule about its center of mass when / = 1 and / = 5. (b) What is the speed of the H atom in each of the cases in (a)? (c) What value of / is required for the H atom to have a speed of 0.1c? (d) Estimate the classical temperature associated with the / you found in (c). Get solution
4q. Explain how to use the rotational spectra to determine the equilibrium separation between the two nuclei in a diatomic molecule. Get solution
5. Derive an expression for the allowed rotational levels in a homopolar diatomic molecule using the Bohr quantization rule for angular momentum. Discuss your result in comparison to the correct quantum mechanical result, Equation (10.2). ... Get solution
5q. Why do the gases He, Ne, and Xe tend to be monatomic rather than diatomic? Get solution
6. The wavelength of a microwave absorption line in CO corresponding to a transition from / = 0 to / = 1 is 2.60 mm. (a) Calculate the rotational inertia of the CO molecule. (b) Show that it is impossible for this amount of energy (corresponding to a photon of wavelength 2.60 mm) to be absorbed by CO in a vibrational transition. Get solution
6q. Explain the low melting points and boiling points for inert gases. (For example, Ne has a melting point 24.5 K and boiling point 27.1 K. Get solution
7. If the energy of a vibrational transition from the n = 0 state to the n = 1 state in CO could be absorbed in a rotational transition that begins in the ground state (/ = 0), what would be the value of / for the final state? Explain why such a rotational transition is impossible. Get solution
7q. Do you expect the fundamental vibrational frequency to be higher for HCl or NaCl? Explain. Get solution
8. Show that I = μR2 for a diatomic molecule, where R is the distance between the two atomic centers and μ= m1m2/(m1 + m2) is the reduced mass. (Note: You should assume that the atoms are point particles.) Get solution
8q. Is it necessary that the substance used in a laser have at least three energy levels? Why or why not? Get solution
9. The energy of a transition from the / = 2 to the / = 3 state in CO is 1.43 × 10-3 eV. (a) Compute the rotational inertia of the CO molecule. (b) What is the average separation between the centers of the C and O atoms? Get solution
9q. Critique the following statement: The Pauli exclusion principle is responsible for keeping solids from collapsing to zero volume. Get solution
10. Consider the model of the H2O molecule shown in the diagram. (a) Find the rotational inertia of H2O about the dashed line. (b) Estimate the energies of the first two rotational energy levels ... 2). (c) What is the wavelength of a photon required to excite a transition from ... ... Get solution
10q. The average nearest-neighbor distance between nuclei in solid NaCl is 0.282 nm, but the distance is 0.236 nm in a free NaCl molecule. How do you account for the difference? Get solution
11. Consider the rotational energy of a single helium atom. Assume that the electrons are uniformly distributed throughout a sphere of radius equal to the Bohr radius and that the nucleus is a uniform solid sphere of radius 1.9 × 10-15 m. (a) Estimate the energy of the first (nonzero) rotational energy level in helium. (b) Is this state likely to be observed? Why or why not? Get solution
11q. What patterns do you notice within the groups of salts in Table 10.2? (A group is defined as having the same metal, e.g., sodium.) Explain those patterns. ... Get solution
12. Consider the NaCl molecule, for which the rotational inertia is 1.30 × 10-45 kg . m2. If infrared radiation with wavelength 30 μm is Raman-scattered from a free NaCl molecule, what are the allowed wavelengths of the scattered radiation? Get solution
12q. What makes elements good candidates for paramagnetism? For diamagnetism? Get solution
13. This problem deals with the splitting of rotational energy levels of diatomic molecules. If one atom of the molecule has more than one stable isotope, then both isotopes are normally present in a sample. Show that the fractional change Δf/f in the observed frequency of a photon emitted in a transition between adjacent rotational states is equal to the fractional difference in the reduced mass Δ μ/ μ for molecules containing the two different isotopes. Get solution
13q. Explain why the paramagnetic susceptibilities of rare earth elements tend to be higher than those of the transition elements. Get solution
14. (a) At T = 293 K what are the relative Maxwell- Boltzmann factors for N2 molecules in the / = 0, / = 1, and / = 2 states? Use I = 1.4 × 10-46 kg . m2. (b) Use your answer to (a) along with the fact that there is also a degeneracy factor of 2/ + 1 on the /th angular momentum level to find the relative populations of the / = 0, / = 1, and / = 2 states of N2 at room temperature. (c) Explain why the 2/ = 1 degeneracy factor is more important for lower rotational states, but the Maxwell-Boltzmann factor dominates for higher states. Get solution
14q. Consider the electronic configurations of the five ferromagnetic elements and justify why they are ferromagnetic. Why are some elements in the same columns as these five, with similar electronic configurations, not ferromagnetic? Get solution
15. Use the HCl absorption spectrum shown in Figure 10.8 to (a) compute the rotational inertia I of the molecule and (b) compute the force constant κ and compare with the value given in Table 10.1. ... ... Get solution
15q. Notice that ferromagnetic elements tend to come from the middle of the rows of rare earth elements and transition elements in the periodic table. Explain. Get solution
16. The equilibrium separation between the two ions in the KCl molecule is 0.267 nm. (a) Assuming that the K+ and Cl- ions are point particles, compute the electric dipole moment of the molecule. (b) Compute the ratio of your result in (a) to the measured electric dipole moment of 5.41 × 10-29 C . m. This ratio is known as the fractional ionic character of the molecular bond. Get solution
16q. Why should elements and compounds with positive paramagnetic susceptibilities not be good candidates for superconductivity? Get solution
17. Find the energy of the photon required to excite the transition from the ground state to the first excited vibrational state in HI. In what part of the electromagnetic spectrum is this? Get solution
17q. Explain similarities and differences between the Meissner effect and Lenz’s law. Get solution
18. (a) How many photons are emitted each second from a 5.0-mW helium-neon laser (λ = 632.8 nm)? (b) If the laser contains 0.02 mole of neon gas, what fraction of the neon atoms in the tube participate in the lasing process during each second of operation? (c) Comment on the relatively low numerical result in (b). Get solution
18q. Consider the superconducting transition temperatures of the elements as shown in Table 10.5. In cases in which there is more than one superconductor in a column of the periodic table, are the transition temperatures consistent with the spirit of the isotope effect (that is, does the heavier element have a lower Tc)? ... Get solution
19. A laser emits 5.50 × 1018 photons per second, using a transition from an excited state with energy 1.15 eV to a ground state with energy 0 eV. (a) What is the laser’s power output? (b) What is the wavelength? Get solution
19q. A superconducting ring can carry a current for an indefinite length of time. Isn’t this a perpetual motion machine, which violates the first or second law of thermodynamics? Explain. Get solution
20. The NOVA laser at Lawrence Livermore National Lab produces a 40-kJ burst of 3.5 ns duration, with a wavelength of 351 nm. (a) How many atoms made a transition from the excited state to the ground state in order to create this pulse? (b) What is the laser’s average power output during the burst? Get solution
20q. Consider a sample at a temperature initially above its superconducting Tc and in a magnetic field. When the sample is cooled to below Tc, currents are generated to expel the magnetic flux from the interior. What is the source of the energy for these currents? Get solution
21. (a) For the helium-neon laser, estimate the Doppler broadening (see Chapter 9, Problem 3) of the output wavelength 632.8 nm at T = 293 K. (b) Estimate the broadening of the same wavelength due to the Heisenberg uncertainty principle, assuming that the metastable state has a lifetime of about 1 ms. Consider an ideal gas enclosed in a spectral tube. When a high voltage is placed across the tube, many atoms are excited, and all excited atoms emit electromagnetic radiation at characteristic frequencies. According to the Doppler effect, the frequencies observed in the laboratory depend on the velocity of the emitting atom. The nonrelativistic Doppler shift of radiation emitted in the x direction is .... The resulting wavelengths observed in the spectroscope are spread to higher and lower values because of the (respectively) lower and higher frequencies, corresponding to negative and positive values of vx. We say that the spectral line has been Doppler broadened. This is what allows us to see the lines easily in the spectroscope, because the Heisenberg uncertainty principle does not cause significant line broadening in atomic transitions. (a) What is the mean frequency of the radiation observed in the spectroscope? (b) To get an idea of how much the spectral line is broadened at particular temperatures, derive an expression for the standard deviation of frequencies, defined to be Standard deviation ... Your result should be a function of f 0, T, and constants. (c) Use your results from (b) to estimate the fractional line width, defined by the ratio of the standard deviation to f0, for hydrogen (H2) gas at T = 293 K. Repeat for a gas of atomic hydrogen at the surface of a star, with T = 5500 K. Get solution
22. Consider the problem of using laser light to measure the distance from the Earth to the moon. (a) What is the maximum uncertainty in timing the round trip for a light pulse in order to determine the distance with an uncertainty of 1 meter? (b) Estimate the effect of the Earth’s atmosphere on this experiment, using the fact that the speed of light in air (at sea level) is slower than the speed of light in vacuum by a factor of 1.0003. Assume an 8-km-high atmosphere of uniform sea-level density. Get solution
23. What is the minimum fraction of the lasing molecules in a three-level laser that must be in the excited state in order for the laser to operate? Answer the same question for a four-level laser. Get solution
22q. A superconducting wire and a low-resistance copper wire are connected together in parallel. When a potential difference is applied, explain why the copper wire carries no current. Get solution
23q. Explain on physical grounds why the ratio ρ /r0 in Equation (10.21) should be less than 1. ... Get solution
24. The 3s state of neon (see Figure 10.15) is 16.6 eV above the ground state. (a) Estimate the relative populations of the ground state and the 3s state at T = 293 K. (b) Repeat for T = 150 K. (c) Repeat for T = 600 K. (d) What implications (if any) do your answers for parts (a)–(c) have for the operation of a He-Ne laser at various temperatures? Get solution
25. The density of solid KCl is about 1980 kg/m3. Compute the nearest-neighbor distance in KCl, that is, the distance between neighboring K+ and Cl- ions. Note that KCl has the same lattice structure as NaCl. Get solution
26. Show that the Madelung constant for a one-dimensional lattice of alternating positive and negative ions is α= 2 ln 2. Get solution
27. Write the first five terms of the Madelung constant for a two-dimensional lattice of alternating positive and negative ions. Get solution
28. Use Equation (10.19) to evaluate the net force (F = -dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
29. Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... where ... and ... Problem 28 Use Equation (10.19) to evaluate the net force (F =-dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
30. If we consider again the linear term (r) in the result of Problem 29, we have a harmonic oscillator. (a) Find an expression for the frequency of oscillation and evaluate using the correct values of α, r0, and ρ for NaCl. (b) Find the photon wavelength corresponding to the frequency you computed in (a) and compare with the observed absorption wavelength for NaCl of about 61 μm. Problem 29 Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... where ... and ... Problem 28 Use Equation (10.19) to evaluate the net force (F =-dV/dr) on an atom in a sodium chloride lattice. Then show that the force can be expressed as ... ... Get solution
31. Refer again to the result of Problem 29. (a) Use the fact that the average net force on an ion must be zero (if there is no overall translation of the crystal) to show that the average values of r and (δr)2 are related by .... (b) According to the equipartition theorem the mean potential energy of an oscillator is ...Use this to show that ...and thereby show that the coefficient of thermal expansion is approximately .... (c) Use the result of (b) to evaluate the coefficient of thermal expansion for NaCl and compare with the experimental value of 4 × 10-6 K-1. Problem 29 Starting with the result of Problem 28, approximate r ... r0 + δ r (where δ r r0) and show that ... Perform a Taylor series expansion about r = r0, keeping terms up to and including order (δ r)2. Show that ... Where ... and ... Get solution
32. Silver has an electrical conductivity of 6.30 × 107Ω-1. m-1 at 293 K. Use the Wiedemann-Franz law without quantum corrections to compute the thermal conductivity of silver at the same temperature and compare with the experimental result of 429 W . K-1 . m-1. Get solution
33. Use the data in Figure 10.23 to estimate the numerical value of the constant b/a2 in the thermal expansion formula, Equation (10.26). Make sure you express the correct units. ... ... Get solution
34. (a) Derive Equation (10.26). (b) Evaluate the constant 3bk/4a2 in Equation (10.26) for copper, given that the coefficient of linear expansion [defined as ...]for copper is found experimentally to be 1.67 × 10-5 K-1 at T = 293 K. Get solution
35. (a) Explain why the parameter a in Equation (10.22) is essentially half the effective force constant of a spring connecting adjacent atoms. (b) Then use this result along with the result of Problem 34b to estimate a value for the parameter b in Equation (10.22), the coefficient of the x3 term in the potential energy. ... (a) Explain why the parameter a in Equation (10.22) is essentially half the effective force constant of a spring connecting adjacent atoms. (b) Then use this result along with the result of Problem 34b to estimate a value for the parameter b in Equation (10.22), the coefficient of the x3 term in the potential energy. ... Problem 34b (a) Derive Equation (10.26). (b) Evaluate the constant 3bk/4a2 in Equation (10.26) for copper, given that the coefficient of linear expansion [defined as ...]for copper is found experimentally to be 1.67 × 10-5 K-1 at T = 293 K. Get solution
36. (a) Show that the ideal gas law can be written as ... where N is the number of particles in the sample and ...is the mean energy. (b) Use the result of (a) to estimate the pressure of the conduction electrons in copper, assuming an ideal Fermi electron gas. Comment on the numerical result, noting that 1 atm = 1.01 ×105 Pa. Get solution
37. Show that the bulk modulus, defined as ... (where P is pressure and V is volume) can be written as ... for a Fermi electron gas with Fermi energy EF. (Hint: Use the relationship between P and V given in Problem 36a.) Problem 36a Show that the ideal gas law can be written as ... Get solution
38. (a) From the result of Problem 37, compute the bulk modulus of pure silver. (b) Compare your result with the experimental value of 1.01 × 1011 N/m2. Problem 37 Show that the bulk modulus, defined as ... (where P is pressure and V is volume) can be written as ... for a Fermi electron gas with Fermi energy EF. (Hint:Use the relationship between P and V given in Problem 36a.) Problem 36a Show that the ideal gas law can be written as ... Get solution
39. Retrace the derivation of the induced diamagnetic moment in Equation (10.38), assuming that (a) the electron orbits clockwise and the magnetic field points out of the page and (b) the electron orbits counterclockwise and the magnetic field points into the page. ... Get solution
40. Use the result of the preceding problem to determine the induced magnetic moment of a diamagnetic atom with an outer shell having three electrons in a p shell with m/ = 0, m/ = 1, and m/ = -1. Get solution
41. Start with Equation (10.41) and derive an expression for ... valid in the low-temperature limit kT B ... Get solution
42. (a) Plot ... versus B/kT over the range μ B/kT = 0 to μ B/kT = 4. (b) Compute ... at μ B/kT = 5 and compare your results with the approximation used for m in Problem 41. (c) Compute ... at μ B/kT = 0.10 and compare your results with the approximate value given in Section 10.4, .... Problem 41 Start with Equation (10.41) and derive an expression for ... valid in the low-temperature limit kT B ... Get solution
43. Prove that magnetic susceptibility ... is a dimensionless quantity. Note that the definition in Equation (10.37) presumes SI units. ... Get solution
44. (a) Compute the maximum magnetization of a bulk sample of iron, assuming perfect alignment of the spins and one unpaired spin per atom. (b) Compare with the observed maximum magnetization of about 1.6 × 106 A/m. (c) On the basis of your results in (a) and (b), what can you say about the actual number of unpaired spins per atom of iron? Get solution
45. At what temperature (expressed as a fraction of Tc) is Bc = 0.25Bc(0), according to the BCS theory? [Note: Bc(0) is the critical field at temperature T = 0.] Repeat for Bc = 0.50 Bc(0) and Bc = 0.75Bc(0). Get solution
46. It is found that for a given pure metal superconductor, photons of wavelength 0.568 mm are sufficient to break the Cooper pairs at T = 2.0 K. Identify the superconductor. Get solution
47. Compute Tc for the mercury isotopes 201Hg and 204Hg. Get solution
48. Estimate the BCS prediction for the change in Tc if all of the 16O atoms in YBa2Cu3O7 (assume Tc = 93.0 K) could be replaced by 18O. This change is not observed experimentally! Get solution
49. From the data given in the text, estimate Tc for Tl2Ba2Can-1O2n-4 with n = 4. Get solution
50. A solenoid made of superconducting wire has exactly 25 turns/cm of length and a diameter of 3.2 cm. If the wire carries a current of 4.5 A, what is the magnetic field strength near the center of the solenoid? What is the magnetic flux through a cross section of the solenoid taken near the center? To how many flux quanta does this correspond? Comment on the number of flux quanta. Get solution
51. Recall that the magnetic field at the surface of a uniform cylindrical wire of radius R carrying a current I is ... Find the minimum possible diameter for a wire of pure niobium so that the T = 0 critical field would not be exceeded if the wire carried a current of 2.5 A. Get solution
52. In a normal conductor heat is generated at a rate I 2R. Therefore a current-carrying conductor must dissipate heat effectively or it can melt or overheat the device in which it is used. Consider a long cylindrical copper wire (resistivity 1.72 × 10-8 Ω. m) of diameter 0.75 mm. If the wire can dissipate 80 W/m2 along its surface, what is the maximum current this wire can carry? Get solution
53. (a) Compute the maximum current that a 16-gauge (1.29-mm diameter) niobium wire can carry at T = 4.2 K. (b) Compare your result in (a) with the copper wire of the same diameter described in Problem 52. In a normal conductor heat is generated at a rate I 2R. Therefore a current-carrying conductor must dissipate heat effectively or it can melt or overheat the device in which it is used. Consider a long cylindrical copper wire (resistivity 1.72 × 10-8 Ω . m) of diameter 0.75 mm. If the wire can dissipate 80 W/m2 along its surface, what is the maximum current this wire can carry? Get solution
54. What is the maximum uncertainty in the measurement of the oscillation frequency in a Josephson junction if the voltage standard of 1 mV is to be maintained within 1 part in 1010? Assume a reference frequency of 483.6 GHz. Get solution
55. Find the minimum acceleration needed for a maglev train to reach a speed of 430 km/h in 3.0 km, one tenth of the length of the 30-km track in Shanghai. Express your answer as a fraction of g, the free-fall acceleration near Earth. Would this acceleration be noticeable? Get solution
56. (a) Compute the escape speed of a particle from the Earth’s surface. Earth’s radius is 6378 km, and its mass is 5.98 × 1024 kg. (b) Find the mean speed for a helium atom at a temperature of 293 K. (c) Comment on the fact that your answer to (b) is less than the answer to (a). Why then does helium not remain in the atmosphere in significant quantities? Get solution
57. A superconducting Nb3Sn magnet can achieve a peak magnetic field of 13.5 T in a magnet designed for use in the Large Hadron Collider. Find the maximum energy that a singly charged particle (for example, a proton or electron) can have if that field is maintained around a circular ring of circumference 27 km. (Note: In reality, particle energies are about 35% less because the peak field is not maintained throughout the ring.) Get solution
58g. Consider a model of a diatomic molecule with pointmass atoms of mass m1 and m2, separated by a distance R. (a) Show that the rotational inertia of the molecule is I = μR2, where the reduced mass μ=m1 m2/(m1 + m2). (b) Compute the rotational inertia of NaCl, which has a bond length of 0.236 nm. Assume the most common isotopes of sodium and chlorine. Get solution
59g. Rotational spectra are affected slightly by the fact that different isotopes have different masses. Suppose a sample of the common isotope 1H35Cl is changed to 1H37Cl. (a) By what fraction is the molecule’s rotational inertia different? (The bond length is 0.127 nm in each case.) (b) What is the change in energy of the / = 1 to the / = 0 transition if the isotope is changed? Get solution
60g. The transition from the / = 2 to the / = 1 state in CO is accompanied by the emission of a 9.55 × 10-4 eV photon. (a) Use this information to find the rotational inertia of the CO molecule. (b) What is the bond length between the C and O atoms? Get solution
61g. The National Ignition Facility (NIF), which became operational in 2009, uses 192 laser beams to stimulate nuclear fusion in a deuterium-tritium fuel pellet. The net output of the lasers is 1.8 MJ of 351-nm light, delivered in a brief (4.0-ns) pulse. (a) What is the average power delivered during the pulse? Compare your answer with the average power consumption in the United States, about 3 × 1012 W. (b) How many photons are produced in each pulse? Get solution
62g. Estimate the temperature at which the critical magnetic field in superconducting mercury is equal to Earth’s surface magnetic field, about 5 × 10-5 T. Is Earth’s magnetic field likely to be a factor in applications that use superconducting mercury? Get solution
63g. Tin has a number of stable isotopes, ranging in mass from 112 u to 124 u. Estimate the difference in the transition temperature between those two isotopes. Get solution
64g. In the persistent current experiment described in Section 10.5, let us assume that the current persisted without detectable reduction for exactly 2.5 years. Given that the inductance of the ring was approximately 3.14 × 10-8 H and the sensitivity of the current measurement was 1 part in 109, (a) estimate an upper bound for the resistance of the ring and (b) estimate how long at least 90% of the current would be certain to remain. Get solution
65g. From thermodynamics the entropy difference per unit volume between the normal and superconducting states is ... where B 2/2_0 is the magnetic energy density needed to return a superconductor to the normal state. Use this fact to compute the entropy difference between the normal and superconducting states in 1 mole of niobium at a temperature of 6.0 K. Get solution
66g. A 2.0-m length of copper wire with a resistance of 1.50 ... is placed in series with a 2.0-m length of superconducting wire. When a 12.0-V battery is placed across the series combination, find (a) the current in the circuit and (b) the potential difference across the two ends of the copper wire. Get solution
