Modern Physics for Scientists and Engineers 4th Edition solutions

Modern Physics for Scientists and Engineers 4
How relevant is the Heisenberg uncertainty principle in frustrating Laplaces goal of determining the behavior of an essentially classical system of particles (say, an ideal gas) through knowledge of the motions of individual particles?

Modern Physics for Scientists and Engineers 4
Why might the measured molar heat capacity of Cl2 not match the prediction of the equipartition theorem as well as that of O2?

Modern Physics for Scientists and Engineers 4
In a diatomic gas why should it be more diffi cult to excite the vibrational mode than the rotational mode?

Modern Physics for Scientists and Engineers 4
What is the physical signifi cance of the root-meansquare speed in an ideal gas?

Modern Physics for Scientists and Engineers 4
An insulated container fi lled with an ideal gas moves through fi eld-free space with a constant velocity. Describe the effect this has on the Maxwell velocity distributions and Maxwell speed distribution.

Modern Physics for Scientists and Engineers 4
Maxwell conceived of a device (later called Maxwells 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 o

Modern Physics for Scientists and Engineers 4
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?

Modern Physics for Scientists and Engineers 4
If a collection of particles are identical, how can they be distinguishable?

Modern Physics for Scientists and Engineers 4
Which of the following act as fermions and which as bosons: hydrogen atoms, deuterium atoms, neutrinos, muons, table-tennis balls?

Modern Physics for Scientists and Engineers 4
Theorists tend to believe that free quarks (with charge 2 3e and 1 3e) 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?

Modern Physics for Scientists and Engineers 4
Explain why you expect E to be greater than 1 2EF [Equation (9.45)].

Modern Physics for Scientists and Engineers 4
Why doesnt the total energy of a collection of fermions approach zero as the temperature approaches zero?

Modern Physics for Scientists and Engineers 4
How would the behavior of metals be different if electrons were bosons rather than fermions?

Modern Physics for Scientists and Engineers 4
What would happen to the Planck distribution and the behavior of liquid helium if we let h S 0 in the Bose-Einstein density of states [Equation (9.53)]?

Modern Physics for Scientists and Engineers 4
If only the superfl uid component of liquid helium fl ows through a very fi ne capillary, is it possible to use capillary fl ow to separate completely the superfl uid component of a sample of liquid helium from the normal component?

Modern Physics for Scientists and Engineers 4
Why is BBE 1 the minimum allowed value in the integral in Equation (9.63)?

Modern Physics for Scientists and Engineers 4
Discuss similarities and differences between an atom laser and an optical laser.

Modern Physics for Scientists and Engineers 4
One way to decide whether Maxwell-Boltzmann statistics are valid is to compare the de Broglie wavelength of a typical particle with the average interparticle spacing d. If l V d then Maxwell-Boltzmann statistics are generally acceptable. (a) Using de Broglies relation l h/p, show that l h 13mkT2 1

Modern Physics for Scientists and Engineers 4
Use Equation (9.24) to turn the Maxwell speed distribution, Equation (9.14), into an energy distribution [Equations (9.25) and (9.26)].

Modern Physics for Scientists and Engineers 4
Consider an atom with a magnetic moment  and a total spin 1/2. The atom is placed in a uniform magnetic fi eld of magnitude B at temperature T. (a) Assuming Maxwell-Boltzmann statistics are valid at this temperature, fi nd the ratio of atoms with spins aligned with the fi eld to those aligned opposi

Modern Physics for Scientists and Engineers 4
The Fermi energy can be defi ned as the energy at which the Fermi factor FFD 0.5. Using this defi nition, show that the constant BFD in Equation (9.30) is equal to exp(EF) and that FFD 1 expb1E  EF 2  1 In other words, verify Equations (9.33) and (9.34).

Modern Physics for Scientists and Engineers 4
At T 0, what fraction of electrons have energy E  E?

Modern Physics for Scientists and Engineers 4
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 fi nd 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 distrib

Modern Physics for Scientists and Engineers 4
What fraction of the electrons in a good conductor have energies between 0.90 EF and EF at T 0?

Modern Physics for Scientists and Engineers 4
Use the data in Problem 23 to compute (a) EF and (b) uF for silver.

Modern Physics for Scientists and Engineers 4
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 temperature

Modern Physics for Scientists and Engineers 4
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 m3 to compute the number of conduction electrons per atom.

Modern Physics for Scientists and Engineers 4
Aluminum has a density of 2.70  103 kg/m3 at a temperature of 293 K, and its molar mass is 26.98 g. (a) Compute the number of aluminum atoms per unit volume at that temperature. (b) Use the fact that EF 11.63 eV for aluminum at 293 K to fi nd the number density of free electrons. (c) Combine your r

Modern Physics for Scientists and Engineers 4
Compute the Fermi speed for (a) Ca (EF 4.69 eV) and (b) Be (EF 14.3 eV).

Modern Physics for Scientists and Engineers 4
Verify that Equation (9.35) is valid in the limit T S 0.

Modern Physics for Scientists and Engineers 4
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 n1E2 3N 2 E 3/2 F E1/2 expb1E  EF 2  1

Modern Physics for Scientists and Engineers 4
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.

Modern Physics for Scientists and Engineers 4
Use numerical integration of the function given in Problem 31 to verify that q 0 n1E2dE N Choose the parameters T 300 K and use EF 7.00 eV for copper.

Modern Physics for Scientists and Engineers 4
Use numerical integration of the function given in Problem 31 to fi nd 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.

Modern Physics for Scientists and Engineers 4
In a neutron star the entire stars mass has collapsed essentially to nuclear density. For a neutron star with radius 10 km and mass 4.50  1030 kg, fi nd the Fermi energy of the neutrons.

Modern Physics for Scientists and Engineers 4
Consider a collection of fermions at T 293 K. Find the probability that a single-particle state will be occupied if that states energy is (a) 0.1 eV less than EF; (b) equal to EF; (c) 0.1 eV greater than EF.

Modern Physics for Scientists and Engineers 4
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?

Modern Physics for Scientists and Engineers 4
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

Modern Physics for Scientists and Engineers 4
Next to helium, the lightest noble gas is neon. (a) Estimate the temperature at which neon should become a superfl uid. (The density of the liquid is about 1200 kg/m3.) (b) Why doesnt neon become a superfl uid? Hint: its melting point at 1 atm is 24.7 K.

Modern Physics for Scientists and Engineers 4
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 tempe

Modern Physics for Scientists and Engineers 4
Use numerical integration on a computer to verify the output of the defi nite integral in Equation (9.63).

Modern Physics for Scientists and Engineers 4
Show that the wave functions in Equations (9.67) satisfy the requirements for symmetric and antisymmetric wave functions: S(1,2) S(2,1) and A(1,2) A(2,1).

Modern Physics for Scientists and Engineers 4
Assume that air is an ideal gas under a uniform gravitational fi eld, 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(mgz) dz and that the normalization const

Modern Physics for Scientists and Engineers 4
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.

Modern Physics for Scientists and Engineers 4
Consider the law of atmospheres (Problem 43). First show that the pressure difference P corresponding to an altitude change z is approximately P rgz Nmg V z Next, assume that the temperature is constant over small altitude changes and then show that P  P0 exp 1bmgz2, where P0 is the pressure

Modern Physics for Scientists and Engineers 4
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 nv/4, where as usual n is the particle number

Modern Physics for Scientists and Engineers 4
For the situation described in Problem 46, show that the speed distribution of the escaping molecules is proportional to v3 exp(mv2) and that the mean energy of the escaping molecules is 2kT.

Modern Physics for Scientists and Engineers 4
Escape speed from near Earths 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 Earths atmosphere?

Modern Physics for Scientists and Engineers 4
The escape speed of a particle near the suns 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 suns surface temperature is 5800 K. Compare your answer with the escape speed.

Modern Physics for Scientists and Engineers 4
(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) Repea

Modern Physics for Scientists and Engineers 4
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.

Modern Physics for Scientists and Engineers 4
Use the method described in Appendix 6 to evaluate the integral q 0 x n exp 1ax 2 2dx in terms of the constant a for n 3, 4, and 5.

Modern Physics for Scientists and Engineers 4
For a (classical) simple harmonic oscillator with fi xed total energy E, fi nd the mean value of kinetic energy K and the mean value of potential energy V . Show that K V E/2.

Modern Physics for Scientists and Engineers 4
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 th

Modern Physics for Scientists and Engineers 4
An atoms 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  1015

Modern Physics for Scientists and Engineers 4
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 o

Modern Physics for Scientists and Engineers 4
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.

Modern Physics for Scientists and Engineers 4
Bose-Einstein condensates of rubidium have reached temperatures of 20 nK. Treating rubidium as an ideal gas, fi nd 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

Modern Physics for Scientists and Engineers 4
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 m3 for argon. (b) Why isnt argon observed to become a Bose-Einstein conde

Modern Physics for Scientists and Engineers 4
In one experiment done by Cornell and Wieman, a Bose-Einstein condensate contained 2000 rubidium-87 atoms within a volume of about 1015 m3. Estimate the temperature at which Bose-Einstein condensation should have occurred.