1. (a) Use Equation (9.5) to show that the one-dimensional rms speed
is ... (b) Show that Equation (9.5) can be rewritten as ... ... Get solution
1q. How relevant is the Heisenberg uncertainty principle in frustrating Laplace’s goal of determining the behavior of an essentially classical system of particles (say, an ideal gas) through knowledge of the motions of individual particles? Get solution
2. The result of Problem 1 can be used to estimate the relative probabilities of various velocities. Pick a small interval Δvx = 0.002vx rms. For 1 mole of an ideal gas, compute the number of molecules within the range Δvx centered at (a) vx = 0.01vx rms, (b) vx = 0.20vx rms, (c) vx = vx rms, (d) vx = 5vx rms, and (e) vx = 100vx rms. Get solution
2q. Why might the measured molar heat capacity of Cl2 not match the prediction of the equipartition theorem as well as that of O2? Get solution
3. 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
3q. In a diatomic gas why should it be more difficult to excite the vibrational mode than the rotational mode? Get solution
4. Consider the model of the diatomic gas oxygen (O2) shown in Figure 9.3. (a) Assuming the atoms are point particles separated by a distance of 121 pm, find the rotational inertia Ix for rotation about the x axis. (b) Now compute the rotational inertia of the molecule about the z axis, assuming almost all of the mass of each atom is in the nucleus, a nearly uniform solid sphere of radius 3.0 × 10-15 m. (c) Compute the rotational energy associated with the first (/ = 1) quantum level for a rotation about the x axis. (d) Using the energy you computed in (c), find the quantum number/ needed to reach that energy level with a rotation about the z axis. Comment on the result in light of what the equipartition theorem predicts for diatomic molecules. ... Get solution
4q. What is the physical significance of the root-mean-square speed in an ideal gas? Get solution
5. Using the Maxwell speed distribution, (a) write an integral expression for the number of molecules in an ideal gas that would have speed v > c at T = 293 K. (b) Explain why the numerical result of the expression you found in (a) is negligible. Get solution
5q. An insulated container fi lled with an ideal gas moves through field-free space with a constant velocity. Describe the effect this has on the Maxwell velocity distributions and Maxwell speed distribution. Get solution
6. Use a computer to explore the numerical value of the definite integral you constructed in the previous problem. Get solution
6q. Maxwell conceived of a device (later called Maxwell’s demon) that could use measurements of individual molecular speeds in a gas to separate faster molecules from slower ones. This device would operate a trap door between two (initially identical) compartments, allowing only faster molecules to pass one way and slower molecules the other way, thus creating a temperature imbalance. Then the temperature difference could be used to run a heat engine, and the result would be the production of mechanical work with no energy input. Of course this would violate the laws of thermodynamics. Discuss reasons why you think Maxwell’s demon cannot work. Get solution
7. It is important for nuclear engineers to know the thermal properties of neutrons in a nuclear reactor. Assuming that a gas of neutrons is in thermal equilibrium, find ... and v* for neutrons at (a) 300 K and (b) 630 K (a typical temperature inside a modern light-water nuclear reactor). Get solution
7q. If the distribution function f(q) for some physical property q is even, it follows that q = 0. Does it also follow that the most probable value q*= 0? Get solution
8. Show that the Maxwell speed distribution function F(v) approaches zero by taking the limit as v ...0 and as v ...∞. Get solution
8q. If a collection of particles are identical, how can they be distinguishable? If a collection of particles are identical, how can they be distinguishable? Get solution
9. Find v* for N2 gas in air (a) on a cold day at T =-15°C and (b) on a hot day at T = 35°C. Get solution
9q. Which of the following act as fermions and which as bosons: hydrogen atoms, deuterium atoms, neutrinos, muons, table-tennis balls? Get solution
10. For an ideal gas O2 at T = 293 K find the two speeds v that satisfy the equation 2F(v) = F(v*). Which of the two speeds you found is closer to v*? Does this make sense? Get solution
10q. Theorists tend to believe that free quarks (with charge ± ...e and ±... e) do not exist, but the question is by no means decided. If free quarks are found to exist, what can you say about the distribution function they would obey? Get solution
11. For the ideal gas Ar at T = 293 K, use a computer to show that ... and thereby verify that ... Get solution
11q. Explain why you expect E to be greater than...EF [Equation (9.45)] Get solution
12. Consider the ideal gas H2 at T = 293 K. Use a numerical integration program on a computer to find the fraction of molecules with speeds in the following ranges: (a) 0 to 10 m/s, (b) 0 to 100 m/s, (c) 0 to 1000 m/s, (d) 1000 m/s to 2000 m/s, (e) 2000 m/s to 5000 m/s, and (f) 0 to 5000 m/s. Get solution
12q. Why doesn’t the total energy of a collection of fermions approach zero as the temperature approaches zero?. Get solution
13. (a) Find vrms for H2 gas and N2 gas, both at T = 293 K. (b) Considering your answers to part (a), discuss why our atmosphere contains nitrogen but not hydrogen. Get solution
13q. How would the behavior of metals be different if electrons were bosons rather than fermions? Get solution
14. (a) Find the total translational kinetic energy of 1 mole of argon atoms at T = 273 K. (b) Would your answer be the same or different for 1 mole of oxygen (O2) molecules? Explain. Get solution
14q. What would happen to the Planck distribution and the behavior of liquid helium if we let h ... 0 in the Bose-Einstein density of states [Equation (9.53)]? ... Get solution
15. Use the Maxwell-Boltzmann energy distribution Equation (9.26) to (a) find the mean translational kinetic energy of an ideal gas and (b) compare your results with ... and.... ... Get solution
15q. If only the superfluid component of liquid helium flows through a very fi ne capillary, is it possible to use capillary fl ow to separate completely the superfluid component of a sample of liquid helium from the normal component? Get solution
16. From the Maxwell-Boltzmann energy distribution, find the most probable energy E*. Plot F(E) versus E and indicate the position of E* on your plot. Get solution
16q. Why is BBE = 1 the minimum allowed value in the integral in Equation (9.63)? ... Get solution
17. Near the surface of a certain kind of star, approximately one hydrogen atom per 10 million is in the first excited level (n = 2). Assume that the other atoms are in the n = 1 level. Use this information to estimate the temperature there, assuming that Maxwell-Boltzmann statistics are valid. (Hint: In this case, the density of states depends on the number of possible quantum states available on each level, which is 8 for n = 2 and 2 for n = 1.) Get solution
17q. Discuss similarities and differences between an atom laser and an optical laser. Get solution
18. One way to decide whether Maxwell-Boltzmann statistics are valid is to compare the de Broglie wavelength λ of a typical particle with the average interparticle spacing d. If λd then Maxwell-Boltzmann statistics are generally acceptable. (a) Using de Broglie’s relation λ = h/p, show that ... (b) Use the fact that N/V = 1/d3 to show that the inequality λ d can be expressed as ... (c) Use the result of (b) to determine whether Maxwell- Boltzmann statistics are valid for Ar gas at room temperature (293 K) and for the conduction electrons in pure silver at T = 293 K. Get solution
19. Use Equation (9.24) to turn the Maxwell speed distribution, Equation (9.14), into an energy distribution [Equations (9.25) and (9.26)]. ... ... ... Get solution
20. Consider an atom with a magnetic moment μ and a total spin 1/2. The atom is placed in a uniform magnetic field of magnitude B at temperature T. (a) Assuming Maxwell-Boltzmann statistics are valid at this temperature, find the ratio of atoms with spins aligned with the field to those aligned opposite the field. (b) Evaluate numerically with B = 5.0 T, for T = 77 K, T = 273 K, and T = 900 K. Get solution
21. The Fermi energy can be defined as the energy at which the Fermi factor FFD = 0.5. Using this definition, show that the constant BFD in Equation (9.30) is equal to exp(-βEF) and that ... In other words, verify Equations (9.33) and (9.34). ... ... ... Get solution
22. At T = 0, what fraction of electrons have energy E E? Get solution
23. Silver has exactly one conduction electron per atom. (a) Use the density of silver (1.05 × 104 kg/m3) and the mass of 107.87 g/mol to find the density of conduction electrons in silver. (b) At what temperature is A = 1 for silver (where A is the normalization constant in the Maxwell-Boltzmann distribution)? (c) At what temperature is A = 10-3? Get solution
24. What fraction of the electrons in a good conductor have energies between 0.90 EF and EF at T = 0? Get solution
25. Use the data in Problem 23 to compute (a) EF and (b) uF for silver. Silver has exactly one conduction electron per atom. (a) Use the density of silver (1.05 × 104 kg/m3) and the mass of 107.87 g/mol to find the density of conduction electrons in silver. (b) At what temperature is A = 1 for silver (where A is the normalization constant in the Maxwell-Boltzmann distribution)? (c) At what temperature is A = 10-3? Get solution
26. The Fermi energy for gold is 5.51 eV at T = 293 K. (a) Find the average energy of a conduction electron at that temperature. (b) Compute the temperature at which the average kinetic energy of an ideal gas molecule would equal the average energy you found in (a). (c) Comment on the relative temperatures in (a) and (b). Get solution
27. The density of pure copper is 8.92 × 103 kg/m3, and its molar mass is 63.546 grams. Use the experimental value of the conduction electron density, 8.47 × 1028 m-3 to compute the number of conduction electrons per atom. Get solution
28. F Get solution
29. Compute the Fermi speed for (a) Ca (EF = 4.69 eV) and (b) Be (EF = 14.3 eV). Get solution
30. Verify that Equation (9.35) is valid in the limit T ...0. ... Get solution
31. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
32. Use the result of Problem 31 with copper (EF = 7.0 eV) to sketch n(E) at (a) T = 0, (b) T = 293 K, and (c) T = 1500 K. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
33. Use numerical integration of the function given in Problem 31 to verify that ... Choose the parameters T = 300 K and use EF = 7.00 eV for copper. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
34. Use numerical integration of the function given in Problem 31 to find the fraction of conduction electrons with energies between 6.00 eV and 7.00 eV in copper at T = 293 K. Comment on your results. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
35. In a neutron star the entire star’s mass has collapsed essentially to nuclear density. For a neutron star with radius 10 km and mass 4.50 × 1030 kg, find the Fermi energy of the neutrons. Get solution
36. Consider a collection of fermions at T = 293 K. Find the probability that a single-particle state will be occupied if that state’s energy is (a) 0.1 eV less than EF; (b) equal to EF; (c) 0.1 eV greater than EF. Get solution
37. Suppose you have an ideal gas of fermions at room temperature (293 K). How large must E = EF be for Fermi-Dirac and Maxwell-Boltzmann statistics to agree to within 1%? Do you think the agreement is within 1% for ideal gases under normal conditions? Get solution
38. Gold is a dense metal, with a density of 1.93 × 104 kg/m3 at T × 300 K. (a) If gold has exactly one conduction electron per atom, what is its Fermi energy? Compare your result with the measured value in Table 9.4. (b) If the electrons in gold were treated as a classical ideal gas, what would be their mean thermal energy (also at 300 K)? Explain the large discrepancy between your answers in (a) and (b). ... Get solution
39. Next to helium, the lightest noble gas is neon. (a) Estimate the temperature at which neon should become a superfluid. (The density of the liquid is about 1200 kg/m3.) (b) Why doesn’t neon become a superfluid? Hint: its melting point at 1 atm is 24.7 K. Get solution
40. Consider the problem of photons in a spherical cavity at temperature T, as described in Section 9.7. (a) For the entire collection of photons, what is the number density (number of photons per unit volume)? (b) Evaluate your result from (a) numerically at T = 500 K and T = 5800 K (the approximate temperature of the sun’s surface). Get solution
41. Use numerical integration on a computer to verify the output of the definite integral in Equation (9.63). ... Get solution
42. Show that the wave functions in Equations (9.67) satisfy the requirements for symmetric and antisymmetric wave functions: Show that the wave functions in Equations (9.67) satisfy the requirements for symmetric and antisymmetric wave functions: ψS(1,2) = ψS(2,1) and ψA(1,2) = -ψA(2,1). ... Get solution
43g. Assume that air is an ideal gas under a uniform gravitational field, so that the potential energy of a molecule of mass m at altitude z is mgz. Show that the distribution of molecules varies with altitude as given by the distribution function f(z) dz = Cz exp(-βmgz) dz and that the normalization constant Cz = mg/kT. This distribution is referred to as the law of atmospheres. Get solution
44g. Use the law of atmospheres (Problem 43) to compare the air densities at sea level, Chicago (altitude 176 m), Denver (altitude 1610 m), and the summit of Mt. Rainier (altitude 4390 m), assuming the same temperature 273 K in each case. Assume that air is an ideal gas under a uniform gravitational field, so that the potential energy of a molecule of mass m at altitude z is mgz. Show that the distribution of molecules varies with altitude as given by the distribution function f(z) dz = Czexp(-βmgz) dz and that the normalization constant Cz = mg/kT. This distribution is referred to as the law of atmospheres. Get solution
45g. Consider the law of atmospheres (Problem 43). First show that the pressure difference ΔP corresponding to an altitude change Δz is approximately ... Next, assume that the temperature is constant over small altitude changes and then show that ..., where P0 is the pressure at z = 0. Get solution
46g. Consider a thin-walled, fixed-volume container of volume V that holds an ideal gas at constant temperature T. It can be shown by dimensional analysis that the number of particles striking the walls of the container per unit area per unit time is given by ..., where as usual n is the particle number density. The container has a small hole of area A in its surface through which the gas can leak slowly. Assume that A is much less than the surface area of the container. (a) Assuming that the pressure inside the container is much greater than the outside pressure (so that no gas will leak from the outside back in), estimate the time it will take for the pressure inside to drop to half the initial value. Your answer should contain A, V, and the mean molecular speed .... (b) Obtain a numerical result for a spherical container with a diameter of 40 cm containing air at 293 K, if there is a circular hole of diameter 1.0 mm in the surface. Get solution
47g. For the situation described in Problem 46, show that the speed distribution of the escaping molecules is proportional to v3 exp(-βmv2) and that the mean energy of the escaping molecules is 2kT. Problem 46 Consider a thin-walled, fi xed-volume container of volume V that holds an ideal gas at constant temperature T. It can be shown by dimensional analysis that the number of particles striking the walls of the container per unit area per unit time is given by ..., where as usual n is the particle number density. The container has a small hole of area A in its surface through which the gas can leak slowly. Assume that A is much less than the surface area of the container. (a) Assuming that the pressure inside the container is much greater than the outside pressure (so that no gas will leak from the outside back in), estimate the time it will take for the pressure inside to drop to half the initial value. Your answer should contain A, V, and the mean molecular speed .... (b) Obtain a numerical result for a spherical container with a diameter of 40 cm containing air at 293 K, if there is a circular hole of diameter 1.0 mm in the surface. Get solution
48g. Escape speed from near Earth’s surface is 1.1 × 104 m/s. (a) Find the temperature required for helium gas to have an rms speed equal to that escape speed. (b) Your answer to (a) is much higher than temperatures on Earth. Why then does helium escape from Earth’s atmosphere? Get solution
49g. The escape speed of a particle near the sun’s surface is 6.2 × 105 m/s. Most of the gas there is atomic hydrogen. Find the rms speed of a hydrogen atom, assuming the sun’s surface temperature is 5800 K. Compare your answer with the escape speed. Get solution
50g. (a) What density of conduction electrons in copper is needed in order for the Maxwell-Boltzmann normalization constant to be A = 1 at T = 273 K? Use your result to argue why Maxwell-Boltzmann statistics are not valid in this case. (b) Repeat the calculation for He gas at the same temperature. (c) Repeat for a neutron star, which is composed entirely of neutrons and has a temperature of 106 K. (d) The average density of a neutron star is on the order of 1017 kg/m3. Use your answer to part (c) to discuss whether Maxwell- Boltzmann statistics are valid in this case. Get solution
51g. Starting with the Fermi energy given in Table 9.4, estimate the number of conduction electrons per atom for aluminum, which has density 2.70 × 103 kg/m3 at T = 300 K. ... Get solution
52g. Use the method described in Appendix 6 to evaluate the integral ... in terms of the constant a for n = 3, 4, and 5. Get solution
53g. For a (classical) simple harmonic oscillator with fixed total energy E, find the mean value of kinetic energy ... Get solution
54g. Stars similar to our sun eventually become white dwarfs, in which the hydrogen and helium have fused to form carbon and oxygen. The star has collapsed to a much smaller radius, which is why it is described as a “dwarf.” The electrons are not bound to the nuclei and form a degenerate Fermi gas within the white dwarf. Consider a white dwarf with a mass equal to the sun’s mass (1.99 × 1030 kg) but a radius of only 6.96 × 106 m, which is just 1% of the sun’s present radius. (a) If the white dwarf consists of equal parts carbon and oxygen, how many electrons are present? (b) What is the number density of electrons? (c) Find the Fermi energy of the electrons (in units of eV) and comment on the result. Get solution
55g. An atom’s nucleus is a collection of fermions— protons and neutrons. (a) In calculating the Fermi energy in a nucleus, the protons and neutrons must be considered separately. Why? (b) Find the Fermi energy of (i) the protons and (ii) the neutrons in a uranium nucleus, which has a radius of 7.4 × 10-15 m and contains 92 protons and 146 neutrons. Get solution
56g. During World War II, physicists developed methods to separate the uranium-235 and -238 isotopes. One method involved converting the uranium metal to a gas, UF6, and then allowing the gas to diffuse through a porous barrier, with the lighter gas diffusing faster. What is the difference in the speeds of the two UF6 gas species at room temperature? Get solution
57g. Find the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure. Get solution
58g. Bose-Einstein condensates of rubidium have reached temperatures of 20 nK. Treating rubidium as an ideal gas, find the rms speed of a rubidium atom at that temperature. (Assume the most common isotope 85Rb.) Repeat for a Bose-Einstein condensate of sodium, using its lowest measured temperature of 450 pK. Get solution
59g. The 40Ar isotope of argon (its most common form) is a boson, like 4He. (a) Follow the methods of Section 9.7 and estimate the temperature at which argon should become a Bose-Einstein condensate. Use a number density 2.5 ×1028 m-3 for argon. (b) Why isn’t argon observed to become a Bose-Einstein condensate? (Hint: The freezing point of argon is 84 K.) Get solution
60g. In one experiment done by Cornell and Wieman, a Bose-Einstein condensate contained 2000 rubidium-87 atoms within a volume of about 10-15 m3. Estimate the temperature at which Bose-Einstein condensation should have occurred. Get solution
1q. How relevant is the Heisenberg uncertainty principle in frustrating Laplace’s goal of determining the behavior of an essentially classical system of particles (say, an ideal gas) through knowledge of the motions of individual particles? Get solution
2. The result of Problem 1 can be used to estimate the relative probabilities of various velocities. Pick a small interval Δvx = 0.002vx rms. For 1 mole of an ideal gas, compute the number of molecules within the range Δvx centered at (a) vx = 0.01vx rms, (b) vx = 0.20vx rms, (c) vx = vx rms, (d) vx = 5vx rms, and (e) vx = 100vx rms. Get solution
2q. Why might the measured molar heat capacity of Cl2 not match the prediction of the equipartition theorem as well as that of O2? Get solution
3. 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
3q. In a diatomic gas why should it be more difficult to excite the vibrational mode than the rotational mode? Get solution
4. Consider the model of the diatomic gas oxygen (O2) shown in Figure 9.3. (a) Assuming the atoms are point particles separated by a distance of 121 pm, find the rotational inertia Ix for rotation about the x axis. (b) Now compute the rotational inertia of the molecule about the z axis, assuming almost all of the mass of each atom is in the nucleus, a nearly uniform solid sphere of radius 3.0 × 10-15 m. (c) Compute the rotational energy associated with the first (/ = 1) quantum level for a rotation about the x axis. (d) Using the energy you computed in (c), find the quantum number/ needed to reach that energy level with a rotation about the z axis. Comment on the result in light of what the equipartition theorem predicts for diatomic molecules. ... Get solution
4q. What is the physical significance of the root-mean-square speed in an ideal gas? Get solution
5. Using the Maxwell speed distribution, (a) write an integral expression for the number of molecules in an ideal gas that would have speed v > c at T = 293 K. (b) Explain why the numerical result of the expression you found in (a) is negligible. Get solution
5q. An insulated container fi lled with an ideal gas moves through field-free space with a constant velocity. Describe the effect this has on the Maxwell velocity distributions and Maxwell speed distribution. Get solution
6. Use a computer to explore the numerical value of the definite integral you constructed in the previous problem. Get solution
6q. Maxwell conceived of a device (later called Maxwell’s demon) that could use measurements of individual molecular speeds in a gas to separate faster molecules from slower ones. This device would operate a trap door between two (initially identical) compartments, allowing only faster molecules to pass one way and slower molecules the other way, thus creating a temperature imbalance. Then the temperature difference could be used to run a heat engine, and the result would be the production of mechanical work with no energy input. Of course this would violate the laws of thermodynamics. Discuss reasons why you think Maxwell’s demon cannot work. Get solution
7. It is important for nuclear engineers to know the thermal properties of neutrons in a nuclear reactor. Assuming that a gas of neutrons is in thermal equilibrium, find ... and v* for neutrons at (a) 300 K and (b) 630 K (a typical temperature inside a modern light-water nuclear reactor). Get solution
7q. If the distribution function f(q) for some physical property q is even, it follows that q = 0. Does it also follow that the most probable value q*= 0? Get solution
8. Show that the Maxwell speed distribution function F(v) approaches zero by taking the limit as v ...0 and as v ...∞. Get solution
8q. If a collection of particles are identical, how can they be distinguishable? If a collection of particles are identical, how can they be distinguishable? Get solution
9. Find v* for N2 gas in air (a) on a cold day at T =-15°C and (b) on a hot day at T = 35°C. Get solution
9q. Which of the following act as fermions and which as bosons: hydrogen atoms, deuterium atoms, neutrinos, muons, table-tennis balls? Get solution
10. For an ideal gas O2 at T = 293 K find the two speeds v that satisfy the equation 2F(v) = F(v*). Which of the two speeds you found is closer to v*? Does this make sense? Get solution
10q. Theorists tend to believe that free quarks (with charge ± ...e and ±... e) do not exist, but the question is by no means decided. If free quarks are found to exist, what can you say about the distribution function they would obey? Get solution
11. For the ideal gas Ar at T = 293 K, use a computer to show that ... and thereby verify that ... Get solution
11q. Explain why you expect E to be greater than...EF [Equation (9.45)] Get solution
12. Consider the ideal gas H2 at T = 293 K. Use a numerical integration program on a computer to find the fraction of molecules with speeds in the following ranges: (a) 0 to 10 m/s, (b) 0 to 100 m/s, (c) 0 to 1000 m/s, (d) 1000 m/s to 2000 m/s, (e) 2000 m/s to 5000 m/s, and (f) 0 to 5000 m/s. Get solution
12q. Why doesn’t the total energy of a collection of fermions approach zero as the temperature approaches zero?. Get solution
13. (a) Find vrms for H2 gas and N2 gas, both at T = 293 K. (b) Considering your answers to part (a), discuss why our atmosphere contains nitrogen but not hydrogen. Get solution
13q. How would the behavior of metals be different if electrons were bosons rather than fermions? Get solution
14. (a) Find the total translational kinetic energy of 1 mole of argon atoms at T = 273 K. (b) Would your answer be the same or different for 1 mole of oxygen (O2) molecules? Explain. Get solution
14q. What would happen to the Planck distribution and the behavior of liquid helium if we let h ... 0 in the Bose-Einstein density of states [Equation (9.53)]? ... Get solution
15. Use the Maxwell-Boltzmann energy distribution Equation (9.26) to (a) find the mean translational kinetic energy of an ideal gas and (b) compare your results with ... and.... ... Get solution
15q. If only the superfluid component of liquid helium flows through a very fi ne capillary, is it possible to use capillary fl ow to separate completely the superfluid component of a sample of liquid helium from the normal component? Get solution
16. From the Maxwell-Boltzmann energy distribution, find the most probable energy E*. Plot F(E) versus E and indicate the position of E* on your plot. Get solution
16q. Why is BBE = 1 the minimum allowed value in the integral in Equation (9.63)? ... Get solution
17. Near the surface of a certain kind of star, approximately one hydrogen atom per 10 million is in the first excited level (n = 2). Assume that the other atoms are in the n = 1 level. Use this information to estimate the temperature there, assuming that Maxwell-Boltzmann statistics are valid. (Hint: In this case, the density of states depends on the number of possible quantum states available on each level, which is 8 for n = 2 and 2 for n = 1.) Get solution
17q. Discuss similarities and differences between an atom laser and an optical laser. Get solution
18. One way to decide whether Maxwell-Boltzmann statistics are valid is to compare the de Broglie wavelength λ of a typical particle with the average interparticle spacing d. If λd then Maxwell-Boltzmann statistics are generally acceptable. (a) Using de Broglie’s relation λ = h/p, show that ... (b) Use the fact that N/V = 1/d3 to show that the inequality λ d can be expressed as ... (c) Use the result of (b) to determine whether Maxwell- Boltzmann statistics are valid for Ar gas at room temperature (293 K) and for the conduction electrons in pure silver at T = 293 K. Get solution
19. Use Equation (9.24) to turn the Maxwell speed distribution, Equation (9.14), into an energy distribution [Equations (9.25) and (9.26)]. ... ... ... Get solution
20. Consider an atom with a magnetic moment μ and a total spin 1/2. The atom is placed in a uniform magnetic field of magnitude B at temperature T. (a) Assuming Maxwell-Boltzmann statistics are valid at this temperature, find the ratio of atoms with spins aligned with the field to those aligned opposite the field. (b) Evaluate numerically with B = 5.0 T, for T = 77 K, T = 273 K, and T = 900 K. Get solution
21. The Fermi energy can be defined as the energy at which the Fermi factor FFD = 0.5. Using this definition, show that the constant BFD in Equation (9.30) is equal to exp(-βEF) and that ... In other words, verify Equations (9.33) and (9.34). ... ... ... Get solution
22. At T = 0, what fraction of electrons have energy E E? Get solution
23. Silver has exactly one conduction electron per atom. (a) Use the density of silver (1.05 × 104 kg/m3) and the mass of 107.87 g/mol to find the density of conduction electrons in silver. (b) At what temperature is A = 1 for silver (where A is the normalization constant in the Maxwell-Boltzmann distribution)? (c) At what temperature is A = 10-3? Get solution
24. What fraction of the electrons in a good conductor have energies between 0.90 EF and EF at T = 0? Get solution
25. Use the data in Problem 23 to compute (a) EF and (b) uF for silver. Silver has exactly one conduction electron per atom. (a) Use the density of silver (1.05 × 104 kg/m3) and the mass of 107.87 g/mol to find the density of conduction electrons in silver. (b) At what temperature is A = 1 for silver (where A is the normalization constant in the Maxwell-Boltzmann distribution)? (c) At what temperature is A = 10-3? Get solution
26. The Fermi energy for gold is 5.51 eV at T = 293 K. (a) Find the average energy of a conduction electron at that temperature. (b) Compute the temperature at which the average kinetic energy of an ideal gas molecule would equal the average energy you found in (a). (c) Comment on the relative temperatures in (a) and (b). Get solution
27. The density of pure copper is 8.92 × 103 kg/m3, and its molar mass is 63.546 grams. Use the experimental value of the conduction electron density, 8.47 × 1028 m-3 to compute the number of conduction electrons per atom. Get solution
28. F Get solution
29. Compute the Fermi speed for (a) Ca (EF = 4.69 eV) and (b) Be (EF = 14.3 eV). Get solution
30. Verify that Equation (9.35) is valid in the limit T ...0. ... Get solution
31. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
32. Use the result of Problem 31 with copper (EF = 7.0 eV) to sketch n(E) at (a) T = 0, (b) T = 293 K, and (c) T = 1500 K. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
33. Use numerical integration of the function given in Problem 31 to verify that ... Choose the parameters T = 300 K and use EF = 7.00 eV for copper. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
34. Use numerical integration of the function given in Problem 31 to find the fraction of conduction electrons with energies between 6.00 eV and 7.00 eV in copper at T = 293 K. Comment on your results. Show that in general (that is, T ≠ 0) the energy distribution of N electrons in a conductor with Fermi energy EF at temperature T is ... Get solution
35. In a neutron star the entire star’s mass has collapsed essentially to nuclear density. For a neutron star with radius 10 km and mass 4.50 × 1030 kg, find the Fermi energy of the neutrons. Get solution
36. Consider a collection of fermions at T = 293 K. Find the probability that a single-particle state will be occupied if that state’s energy is (a) 0.1 eV less than EF; (b) equal to EF; (c) 0.1 eV greater than EF. Get solution
37. Suppose you have an ideal gas of fermions at room temperature (293 K). How large must E = EF be for Fermi-Dirac and Maxwell-Boltzmann statistics to agree to within 1%? Do you think the agreement is within 1% for ideal gases under normal conditions? Get solution
38. Gold is a dense metal, with a density of 1.93 × 104 kg/m3 at T × 300 K. (a) If gold has exactly one conduction electron per atom, what is its Fermi energy? Compare your result with the measured value in Table 9.4. (b) If the electrons in gold were treated as a classical ideal gas, what would be their mean thermal energy (also at 300 K)? Explain the large discrepancy between your answers in (a) and (b). ... Get solution
39. Next to helium, the lightest noble gas is neon. (a) Estimate the temperature at which neon should become a superfluid. (The density of the liquid is about 1200 kg/m3.) (b) Why doesn’t neon become a superfluid? Hint: its melting point at 1 atm is 24.7 K. Get solution
40. Consider the problem of photons in a spherical cavity at temperature T, as described in Section 9.7. (a) For the entire collection of photons, what is the number density (number of photons per unit volume)? (b) Evaluate your result from (a) numerically at T = 500 K and T = 5800 K (the approximate temperature of the sun’s surface). Get solution
41. Use numerical integration on a computer to verify the output of the definite integral in Equation (9.63). ... Get solution
42. Show that the wave functions in Equations (9.67) satisfy the requirements for symmetric and antisymmetric wave functions: Show that the wave functions in Equations (9.67) satisfy the requirements for symmetric and antisymmetric wave functions: ψS(1,2) = ψS(2,1) and ψA(1,2) = -ψA(2,1). ... Get solution
43g. Assume that air is an ideal gas under a uniform gravitational field, so that the potential energy of a molecule of mass m at altitude z is mgz. Show that the distribution of molecules varies with altitude as given by the distribution function f(z) dz = Cz exp(-βmgz) dz and that the normalization constant Cz = mg/kT. This distribution is referred to as the law of atmospheres. Get solution
44g. Use the law of atmospheres (Problem 43) to compare the air densities at sea level, Chicago (altitude 176 m), Denver (altitude 1610 m), and the summit of Mt. Rainier (altitude 4390 m), assuming the same temperature 273 K in each case. Assume that air is an ideal gas under a uniform gravitational field, so that the potential energy of a molecule of mass m at altitude z is mgz. Show that the distribution of molecules varies with altitude as given by the distribution function f(z) dz = Czexp(-βmgz) dz and that the normalization constant Cz = mg/kT. This distribution is referred to as the law of atmospheres. Get solution
45g. Consider the law of atmospheres (Problem 43). First show that the pressure difference ΔP corresponding to an altitude change Δz is approximately ... Next, assume that the temperature is constant over small altitude changes and then show that ..., where P0 is the pressure at z = 0. Get solution
46g. Consider a thin-walled, fixed-volume container of volume V that holds an ideal gas at constant temperature T. It can be shown by dimensional analysis that the number of particles striking the walls of the container per unit area per unit time is given by ..., where as usual n is the particle number density. The container has a small hole of area A in its surface through which the gas can leak slowly. Assume that A is much less than the surface area of the container. (a) Assuming that the pressure inside the container is much greater than the outside pressure (so that no gas will leak from the outside back in), estimate the time it will take for the pressure inside to drop to half the initial value. Your answer should contain A, V, and the mean molecular speed .... (b) Obtain a numerical result for a spherical container with a diameter of 40 cm containing air at 293 K, if there is a circular hole of diameter 1.0 mm in the surface. Get solution
47g. For the situation described in Problem 46, show that the speed distribution of the escaping molecules is proportional to v3 exp(-βmv2) and that the mean energy of the escaping molecules is 2kT. Problem 46 Consider a thin-walled, fi xed-volume container of volume V that holds an ideal gas at constant temperature T. It can be shown by dimensional analysis that the number of particles striking the walls of the container per unit area per unit time is given by ..., where as usual n is the particle number density. The container has a small hole of area A in its surface through which the gas can leak slowly. Assume that A is much less than the surface area of the container. (a) Assuming that the pressure inside the container is much greater than the outside pressure (so that no gas will leak from the outside back in), estimate the time it will take for the pressure inside to drop to half the initial value. Your answer should contain A, V, and the mean molecular speed .... (b) Obtain a numerical result for a spherical container with a diameter of 40 cm containing air at 293 K, if there is a circular hole of diameter 1.0 mm in the surface. Get solution
48g. Escape speed from near Earth’s surface is 1.1 × 104 m/s. (a) Find the temperature required for helium gas to have an rms speed equal to that escape speed. (b) Your answer to (a) is much higher than temperatures on Earth. Why then does helium escape from Earth’s atmosphere? Get solution
49g. The escape speed of a particle near the sun’s surface is 6.2 × 105 m/s. Most of the gas there is atomic hydrogen. Find the rms speed of a hydrogen atom, assuming the sun’s surface temperature is 5800 K. Compare your answer with the escape speed. Get solution
50g. (a) What density of conduction electrons in copper is needed in order for the Maxwell-Boltzmann normalization constant to be A = 1 at T = 273 K? Use your result to argue why Maxwell-Boltzmann statistics are not valid in this case. (b) Repeat the calculation for He gas at the same temperature. (c) Repeat for a neutron star, which is composed entirely of neutrons and has a temperature of 106 K. (d) The average density of a neutron star is on the order of 1017 kg/m3. Use your answer to part (c) to discuss whether Maxwell- Boltzmann statistics are valid in this case. Get solution
51g. Starting with the Fermi energy given in Table 9.4, estimate the number of conduction electrons per atom for aluminum, which has density 2.70 × 103 kg/m3 at T = 300 K. ... Get solution
52g. Use the method described in Appendix 6 to evaluate the integral ... in terms of the constant a for n = 3, 4, and 5. Get solution
53g. For a (classical) simple harmonic oscillator with fixed total energy E, find the mean value of kinetic energy ... Get solution
54g. Stars similar to our sun eventually become white dwarfs, in which the hydrogen and helium have fused to form carbon and oxygen. The star has collapsed to a much smaller radius, which is why it is described as a “dwarf.” The electrons are not bound to the nuclei and form a degenerate Fermi gas within the white dwarf. Consider a white dwarf with a mass equal to the sun’s mass (1.99 × 1030 kg) but a radius of only 6.96 × 106 m, which is just 1% of the sun’s present radius. (a) If the white dwarf consists of equal parts carbon and oxygen, how many electrons are present? (b) What is the number density of electrons? (c) Find the Fermi energy of the electrons (in units of eV) and comment on the result. Get solution
55g. An atom’s nucleus is a collection of fermions— protons and neutrons. (a) In calculating the Fermi energy in a nucleus, the protons and neutrons must be considered separately. Why? (b) Find the Fermi energy of (i) the protons and (ii) the neutrons in a uranium nucleus, which has a radius of 7.4 × 10-15 m and contains 92 protons and 146 neutrons. Get solution
56g. During World War II, physicists developed methods to separate the uranium-235 and -238 isotopes. One method involved converting the uranium metal to a gas, UF6, and then allowing the gas to diffuse through a porous barrier, with the lighter gas diffusing faster. What is the difference in the speeds of the two UF6 gas species at room temperature? Get solution
57g. Find the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure. Get solution
58g. Bose-Einstein condensates of rubidium have reached temperatures of 20 nK. Treating rubidium as an ideal gas, find the rms speed of a rubidium atom at that temperature. (Assume the most common isotope 85Rb.) Repeat for a Bose-Einstein condensate of sodium, using its lowest measured temperature of 450 pK. Get solution
59g. The 40Ar isotope of argon (its most common form) is a boson, like 4He. (a) Follow the methods of Section 9.7 and estimate the temperature at which argon should become a Bose-Einstein condensate. Use a number density 2.5 ×1028 m-3 for argon. (b) Why isn’t argon observed to become a Bose-Einstein condensate? (Hint: The freezing point of argon is 84 K.) Get solution
60g. In one experiment done by Cornell and Wieman, a Bose-Einstein condensate contained 2000 rubidium-87 atoms within a volume of about 10-15 m3. Estimate the temperature at which Bose-Einstein condensation should have occurred. Get solution
