Review. An electron moves in a three-dimensional box of

Is each one of the following statements true or false for a superconductor below its critical temperature? (a) It can carry infinite current. (b) It must carry some nonzero current. (c) Its interior electric field must be zero. (d) Its internal magnetic field must be zero. (e) No internal energy appears when it carries electric current.

An infrared absorption spectrum of a molecule is shown in Figure OQ43.2. Notice that the highest peak on either side of the gap is the third peak from the gap. After this spectrum is taken, the temperature of the sample of molecules is raised to a much higher value. Compared with Figure OQ43.2, in this new spectrum is the highest absorption peak (a) at the same frequency, (b) farther from the gap, or (c) closer to the gap?

What kind of bonding likely holds the atoms together in the following solids (i), (ii), and (iii)? Choose your answers from these possibilities: (a) ionic bonding, (b) covalent bonding, and (c) metallic bonding. (i) The solid is opaque, shiny, flexible, and a good electric conductor. (ii) The crystal is transparent, brittle, and soluble in water. It is a poor conductor of electricity. (iii) The crystal is opaque, brittle, very hard, and a good electric insulator

The Fermi energy for silver is 5.48 eV. In a piece of solid silver, free-electron energy levels are measured near 2 eV and near 6 eV. (i) Near which of these energies are the energy levels closer together? (a) 2 eV (b) 6 eV (c) The spacing is the same. (ii) Near which of these energies are more electrons occupying energy levels? (a) 2 eV (b) 6 eV (c) The number of electrons is the same.

As discussed in Chapter 27, the conductivity of metals decreases with increasing temperature due to electron collisions with vibrating atoms. In contrast, the conductivity of semiconductors increases with increasing temperature. What property of a semiconductor is responsible for this behavior? (a) Atomic vibrations decrease as temperature increases. (b) The number of conduction electrons and the number of holes increase steeply with increasing temperature. (c) The energy gap decreases with increasing temperature. (d) Electrons do not collide with atoms in a semiconductor.

(i) Should you expect an n-type doped semiconductor to have (a) higher, (b) lower, or (c) the same conductivity as an intrinsic (pure) semiconductor? (ii) Should you expect a p-type doped semiconductor to have (a) higher, (b) lower, or (c) the same conductivity as an intrinsic (pure) semiconductor?

Consider a typical material composed of covalently bonded diatomic molecules. Rank the following energies from the largest in magnitude to the smallest in magnitude. (a) the latent heat of fusion per molecule (b) the molecular binding energy (c) the energy of the first excited state of molecular rotation (d) the energy of the first excited state of molecular vibration

How do the vibrational and rotational levels of heavy hydrogen (D2) molecules compare with those of H2 molecules?

Discuss models for the different types of bonds that form stable molecules.

Discuss the differences between crystalline solids, amorphous solids, and gases.

Assume the distance between the protons in the H2 molecule is 0.750 3 10210 m. (a) Find the energy of the first excited rotational state, with J 5 1. (b) Find the wavelength of radiation emitted in the transition from J 5 1 to J 5 0

Why is the following situation impossible? The effective force constant of a vibrating HCl molecule is k 5 480 N/m. A beam of infrared radiation of wavelength 6.20 3 103 nm is directed through a gas of HCl molecules. As a result, the molecules are excited from the ground vibrational state to the first excited vibrational state.

The effective spring constant describing the potential energy of the HI molecule is 320 N/m and that for the HF molecule is 970 N/m. Calculate the minimum amplitude of vibration for (a) the HI molecule and (b) the HF molecule.

The rotational spectrum of the HCl molecule contains lines with wavelengths of 0.060 4, 0.069 0, 0.080 4, 0.096 4, and 0.120 4 mm. What is the moment of inertia of the molecule?

The atoms of an NaCl molecule are separated by a distance r 5 0.280 nm. Calculate (a) the reduced mass of an NaCl molecule, (b) the moment of inertia of an NaCl molecule, and (c) the wavelength of radiation emitted when an NaCl molecule undergoes a transition from the J 5 2 state to the J 5 1 state.

A diatomic molecule consists of two atoms having masses m1 and m2 separated by a distance r. Show that the moment of inertia about an axis through the center of mass of the molecule is given by Equation 43.3, I 5 mr 2

The nuclei of the O2 molecule are separated by a distance 1.20 3 10210 m. The mass of each oxygen atom in the molecule is 2.66 3 10226 kg. (a) Determine the rotational energies of an oxygen molecule in electron volts for the levels corresponding to J 5 0, 1, and 2. (b) The effective force constant k between the atoms in the oxygen molecule is 1 177 N/m. Determine the vibrational energies (in electron volts) corresponding to v 5 0, 1, and 2.

Figure P43.18 is a model of a benzene molecule. All atoms lie in a plane, and the carbon atoms (mC 5 1.99 3 10226 kg) form a regular hexagon, as do the hydrogen atoms (mH 5 1.67 3 10227 kg). The carbon atoms are 0.110 nm apart center to center, and the adjacent carbon and hydrogen atoms are 0.100 nm apart center to center. (a) Calculate the moment of inertia of the molecule about an axis perpendicular to the plane of the paper through the center point O. (b) Determine the allowed rotational energies about this axis.

(a) In an HCl molecule, take the Cl atom to be the isotope 35Cl. The equilibrium separation of the H and Cl atoms is 0.127 46 nm. The atomic mass of the H atom is 1.007 825 u and that of the 35Cl atom is 34.968 853 u. Calculate the longest wavelength in the rotational spectrum of this molecule. (b) What If? Repeat the calculation in part (a), but take the Cl atom to be the isotope 37Cl, which has atomic mass 36.965 903 u. The equilibrium separation distance is the same as in part (a). (c) Naturally occurring chlorine contains approximately three parts of 35Cl to one part of 37Cl. Because of the two different Cl masses, each line in the microwave rotational spectrum of HCl is split into a doublet as shown in Figure P43.19. Calculate the separation in wavelength between the doublet lines for the longest wavelength

Estimate the moment of inertia of an HCl molecule from its infrared absorption spectrum shown in Figure P43.19.

An H2 molecule is in its vibrational and rotational ground states. It absorbs a photon of wavelength 2.211 2 mm and makes a transition to the v 5 1, J 5 1 energy level. It then drops to the v 5 0, J 5 2 energy level while emitting a photon of wavelength 2.405 4 mm. Calculate (a) the moment of inertia of the H2 molecule about an axis through its center of mass and perpendicular to the HH bond, (b) the vibrational frequency of the H2 molecule, and (c) the equilibrium separation distance for this molecule.

Photons of what frequencies can be spontaneously emitted by CO molecules in the state with v 5 1 and J 5 0?

Most of the mass of an atom is in its nucleus. Model the mass distribution in a diatomic molecule as two spheres of uniform density, each of radius 2.00 3 10215 m and mass 1.00 3 10226 kg, located at points along the y axis as in Figure 43.5a, and separated by 2.00 3 10210 m. Rotation about the axis joining the nuclei in the diatomic molecule is ordinarily ignored because the first excited state would have an energy that is too high to access. To see why, calculate the ratio of the energy of the first excited state for rotation about the y axis to the energy of the first excited state for rotation about the x axis.

Use a magnifying glass to look at the grains of table salt that come out of a salt shaker. Compare what you see with Figure 43.11a. The distance between a sodium ion and a nearest-neighbor chlorine ion is 0.261 nm. (a) Make an order-of-magnitude estimate of the number N of atoms in a typical grain of salt. (b) What If? Suppose you had a number of grains of salt equal to this number N. What would be the volume of this quantity of salt?

Use Equation 43.18 to calculate the ionic cohesive energy for NaCl. Take a 5 1.747 6, r0 5 0.281 nm, and m 5 8.

Consider a one-dimensional chain of alternating singly-ionized positive and negative ions. Show that the potential energy associated with one of the ions and its interactions with the rest of this hypothetical crystal is U 1r2 5 2ake e 2 r where the Madelung constant is a 5 2 ln 2 and r is the distance between ions. Suggestion: Use the series expansion for ln (1 1 x)

Calculate the energy of a conduction electron in silver at 800 K, assuming the probability of finding an electron in that state is 0.950. The Fermi energy of silver is 5.48 eV at this temperature.

(a) State what the Fermi energy depends on according to the free-electron theory of metals and how the Fermi energy depends on that quantity. (b) Show that Equation 43.25 can be expressed as EF 5 (3.65 3 10219)ne 2/3, where EF is in electron volts when ne is in electrons per cubic meter. (c) According to Table 43.2, by what factor does the free-electron concentration in copper exceed that in potassium? (d) Which of these metals has the larger Fermi energy? (e) By what factor is the Fermi energy larger? (f) Explain whether this behavior is predicted by Equation 43.25.

When solid silver starts to melt, what is the approximate fraction of the conduction electrons that are thermally excited above the Fermi level?

. (a) Find the typical speed of a conduction electron in copper, taking its kinetic energy as equal to the Fermi energy, 7.05 eV. (b) Suppose the copper is a currentcarrying wire. How does the speed found in part (a) compare with a typical drift speed (see Section 27.1) of electrons in the wire of 0.1 mm/s?

The Fermi energy of copper at 300 K is 7.05 eV. (a) What is the average energy of a conduction electron in copper at 300 K? (b) At what temperature would the average translational energy of a molecule in an ideal gas be equal to the energy calculated in part (a)?

Consider a cube of gold 1.00 mm on an edge. Calculate the approximate number of conduction electrons in this cube whose energies lie in the range 4.000 to 4.025 eV.

Sodium is a monovalent metal having a density of 0.971 g/cm3 and a molar mass of 23.0 g/mol. Use this information to calculate (a) the density of charge carriers and (b) the Fermi energy of sodium.

Why is the following situation impossible? A hypothetical metal has the following properties: its Fermi energy is 5.48 eV, its density is 4.90 3 103 kg/m3, its molar mass is 100 g/mol, and it has one free electron per atom.

For copper at 300 K, calculate the probability that a state with an energy equal to 99.0% of the Fermi energy is occupied

For a metal at temperature T, calculate the probability that a state with an energy equal to bEF is occupied where b is a fraction between 0 and 1

Review. An electron moves in a three-dimensional box of edge length L and volume L3. The wave function of the particle is c 5 A sin (kx x) sin (kyy) sin (kzz). Show that its energy is given by Equation 43.20, E 5 U2 p2 2meL2 1nx 2 1 ny 2 1 nz 2 2 where the quantum numbers (nx , ny, nz) are integers $ 1. Suggestion: The Schrdinger equation in three dimensions may be written U2 2m a '2 c 'x 2 1 '2 c 'y 2 1 '2 c 'z2 b 5 1U 2 E2c

(a) Consider a system of electrons confined to a threedimensional box. Calculate the ratio of the number of allowed energy levels at 8.50 eV to the number at 7.05 eV. (b) What If? Copper has a Fermi energy of 7.05 eV at 300 K. Calculate the ratio of the number of occupied levels in copper at an energy of 8.50 eV to the number at the Fermi energy. (c) How does your answer to part (b) compare with that obtained in part (a)?

Show that the average kinetic energy of a conduction electron in a metal at 0 K is Eavg 5 3 5EF. Suggestion: In general, the average kinetic energy is Eavg 5 1 ne 3 ` 0 EN 1E 2 dE where ne is the density of particles, N(E )dE is given by Equation 43.22, and the integral is over all possible values of the energy.

Most solar radiation has a wavelength of 1 mm or less. (a) What energy gap should the material in a solar cell have if it is to absorb this radiation? (b) Is silicon an appropriate solar cell material (see Table 43.3)? Explain your answer.

The energy gap for silicon at 300 K is 1.14 eV. (a) Find the lowest-frequency photon that can promote an electron from the valence band to the conduction band. (b) What is the wavelength of this photon?

Light from a hydrogen discharge tube is incident on a CdS crystal. (a) Which spectral lines from the Balmer series are absorbed and (b) which are transmitted?

A light-emitting diode (LED) made of the semiconductor GaAsP emits red light (l 5 650 nm). Determine the energy-band gap Eg for this semiconductor.

The longest wavelength of radiation absorbed by a certain semiconductor is 0.512 mm. Calculate the energy gap for this semiconductor

You are asked to build a scientific instrument that is thermally isolated from its surroundings. The isolation container may be a calorimeter, but these design criteria could apply to other containers as well. You wish to use a laser external to the container to raise the temperature of a target inside the instrument. You decide to use a diamond window in the container. Diamond has an energy gap of 5.47 eV. What is the shortest laser wavelength you can use to warm the sample inside the instrument?

Review. When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound. You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom is attracted to this single nuclear charge 1e. The energy levels of the extra electron are similar to those of the electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1/k from what it would be in free space (see Eq. 26.21), where k is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the electron to move as if it had an effective mass m*, which is quite different from the mass me of a free electron. You can use the Bohr model of hydrogen to obtain relatively accurate values for the allowed energy levels of the extra electron. We wish to find the typical energy of these donor states, which play an important role in semiconductor devices. Assume k 5 11.7 for silicon and m* 5 0.220me. (a) Find a symbolic expression for the smallest radius of the electron orbit in terms of a0, the Bohr radius. (b) Substitute numerical values to find the numerical value of the smallest radius. (c) Find a symbolic expression for the energy levels En9 of the electron in the Bohr orbits around the donor atom in terms of me, m*, k, and En, the energy of the hydrogen atom in the Bohr model. (d) Find the numerical value of the energy for the ground state of the electron.

Assuming T 5 300 K, (a) for what value of the bias voltage DV in Equation 43.27 does I 5 9.00I 0? (b) What If? What if I 5 20.900I 0?

A diode, a resistor, and a battery are connected in a series circuit. The diode is at a temperature for which kBT 5 25.0 meV, and the saturation value of the current is I0 5 1.00 mA. The resistance of the resistor is R 5 745 V, and the battery maintains a constant potential difference of e 5 2.42 V between its terminals. (a) Use Kirchhoffs loop rule to show that e 2 DV 5 I 0R 1ee DV/k BT 2 12 where DV is the voltage across the diode. (b) To solve this transcendental equation for the voltage DV, graph the left-hand side of the above equation and the righthand side as functions of DV and find the value of DV at which the curves cross. (c) Find the current I in the circuit. (d) Find the ohmic resistance of the diode, defined as the ratio DV/I, at the voltage in part (b). (e) Find the dynamic resistance of the diode, which is defined as the derivative d(DV)/dI, at the voltage in part (b).

You put a diode in a microelectronic circuit to protect the system in case an untrained person installs the battery backward. In the correct forward-bias situation, the current is 200 mA with a potential difference of 100 mV across the diode at room temperature (300 K). If the battery were reversed, so that the potential difference across the diode is still 100 mV but with the opposite sign, what would be the magnitude of the current in the diode?

A diode is at room temperature so that kBT 5 0.025 0 eV. Taking the applied voltages across the diode to be 1 1.00 V (under forward bias) and 21.00 V (under reverse bias), calculate the ratio of the forward current to the reverse current if the diode is described by Equation 43.27.

A thin rod of superconducting material 2.50 cm long is placed into a 0.540-T magnetic field with its cylindrical axis along the magnetic field lines. (a) Sketch the directions of the applied field and the induced surface current. (b) Find the magnitude of the surface current on the curved surface of the rod.

A direct and relatively simple demonstration of zero DC resistance can be carried out using the four-point probe method. The probe shown in Figure P43.52 consists of a disk of YBa2Cu3O7 (a high-Tc superconductor) to which four wires are attached. Current is maintained through the sample by applying a DC voltage between points a and b, and it is measured with a DC ammeter. The current can be varied with the variable resistance R. The potential difference DVcd between c and d is measured with a digital voltmeter. When the probe is immersed in liquid nitrogen, the sample quickly cools to 77 K, below the critical temperature of the material, 92 K. The current remains approximately constant, but DVcd drops abruptly to zero. (a) Explain this observation on the basis of what you know about superconductors. (b) The data in the accompanying table represent actual values of DVcd for different values of I taken on the sample at room temperature in the senior authors laboratory. A 6-V battery in series with a variable resistor R supplied the current. The values of R ranged from 10V to 100V. Make an IDV plot of the data and determine whether the sample behaves in a linear manner. (c) From the data, obtain a value for the DC resistance of the sample at room temperature. (d) At room temperature, it was found that DVcd 5 2.234 mV for I 5 100.3 mA, but after the sample was cooled to 77 K, DVcd 5 0 and I 5 98.1 mA. What do you think might have caused the slight decrease in current? Current Versus Potential Difference DVcd Measured in a Bulk Ceramic Sample of YBa2Cu3O72d at Room Temperature I (mA) DVcd (mV) 57.8 1.356 61.5 1.441 68.3 1.602 76.8 1.802 87.5 2.053 102.2 2.398 123.7 2.904 155 3.61

A superconducting ring of niobium metal 2.00 cm in diameter is immersed in a uniform 0.020 0-T magnetic field directed perpendicular to the ring and carries no current. Determine the current generated in the ring when the magnetic field is suddenly decreased to zero. The inductance of the ring is 3.10 3 1028 H.

The effective spring constant associated with bonding in the N2 molecule is 2 297 N/m. The nitrogen atoms each have a mass of 2.32 3 10226 kg, and their nuclei are 0.120 nm apart. Assume the molecule is rigid. The first excited vibrational state of the molecule is above the vibrational ground state by an energy difference DE. Calculate the J value of the rotational state that is above the rotational ground state by the same energy difference DE.

The hydrogen molecule comes apart (dissociates) when it is excited internally by 4.48 eV. Assuming this molecule behaves like a harmonic oscillator having classical angular frequency v 5 8.28 3 1014 rad/s, find the highest vibrational quantum number for a state below the 4.48-eV dissociation energy.

(a) Starting with Equation 43.17, show that the force exerted on an ion in an ionic solid can be written as F 5 2ake e 2 r 2 c1 2 a r0 r b m21 d here a is the Madelung constant and r0 is the equilibrium separation. (b) Imagine that an ion in the solid is displaced a small distance s from r0. Show that the ion experiences a restoring force F 5 2Ks, where K 5 ake e 2 r 0 3 1m 2 12 (c) Use the result of part (b) to find the frequency of vibration of a Na1 ion in NaCl. Take m 5 8 and use the value a 5 1.747 6.

Under pressure, liquid helium can solidify as each atom bonds with four others, and each bond has an average energy of 1.74 3 10223 J. Find the latent heat of fusion for helium in joules per gram. (The molar mass of He is 4.00 g/mol.)

The dissociation energy of ground-state molecular hydrogen is 4.48 eV, but it only takes 3.96 eV to dissociate it when it starts in the first excited vibrational state with J 5 0. Using this information, determine the depth of the H2 molecular potential-energy function.

. Starting with Equation 43.17, show that the ionic cohesive energy of an ionically bonded solid is given by Equation 43.18.

The FermiDirac distribution function can be written as f 1E 2 5 1 e 1E 2E F2/k BT 1 1 5 1 e 1E/E F212TF/T 1 1 where TF is the Fermi temperature, defined according to kBTF ; EF (a) Write a spreadsheet to calculate and plot f(E) versus E/E F at a fixed temperature T. (b) Describe the curves obtained for T 5 0.1TF, 0.2TF, and 0.5TF.

A particle moves in one-dimensional motion through a field for which the potential energy of the particle field system is U 1x 2 5 A x 3 2 B x where A 5 0.150 eV ? nm3 and B 5 3.68 eV ? nm. The shape of this function is shown in Figure P43.61. (a) Find the equilibrium position x0 of the particle. (b) Determine the depth U0 of this potential well. (c) In moving along the x axis, what maximum force toward the negative x direction does the particle experience?

A particle of mass m moves in one-dimensional motion through a field for which the potential energy of the particlefield system is U 1x 2 5 A x 3 2 B x where A and B are constants. The general shape of this function is shown in Figure P43.61. (a) Find the equilibrium position x 0 of the particle in terms of m, A, and B. (b) Determine the depth U0 of this potential well. (c) In moving along the x axis, what maximum force toward the negative x direction does the particle experience?

As you will learn in Chapter 44, carbon-14 (14C) is an unstable isotope of carbon. It has the same chemical properties and electronic structure as the much more abundant isotope carbon-12 (12C), but it has different nuclear properties. Its mass is 14 u, greater than that of carbon-12 because of the two extra neutrons in the carbon-14 nucleus. Assume the CO molecular potential energy is the same for both isotopes of carbon and the examples in Section 43.2 contain accurate data and results for carbon monoxide with carbon-12 atoms. (a) What is the vibrational frequency of 14CO? (b) What is the moment of inertia of 14CO? (c) What wavelengths of light can be absorbed by 14CO in the (v 5 0, J 5 10) state that cause it to end up in the v 5 1 state?

As an alternative to Equation 43.1, another useful model for the potential energy of a diatomic molecule is the Morse potential U 1r2 5 B 3e 2a 1r 2 r0 2 2 14 2 where B, a, and r0 are parameters used to adjust the shape of the potential and its depth. (a) What is the equilibrium separation of the nuclei? (b) What is the depth of the potential well, defined as the difference in energy between the potentials minimum value and its asymptote as r approaches infinity? (c) If m is the reduced mass of the system of two nuclei and assuming the potential is nearly parabolic about the well minimum, what is the vibrational frequency of the diatomic molecule in its ground state? (d) What amount of energy needs to be supplied to the groundstate molecule to separate the two nuclei to infinity?