Solved: Explain why the resistivity of metals increases

(I) Estimate the binding energy of a KCl molecule by calculating the electrostatic potential energy when the \(K^{+} \text {and } C l^{-}\) ions are at their stable separation of 0.28 nm. Assume each has a charge of magnitude 1.0e. Equation Transcription: Text Transcription: K^+ and C l^-

(II) The measured binding energy of KCl is 4.43 eV. From the result of Problem 1, estimate the contribution to the binding energy of the repelling electron clouds at the equilibrium distance \(r_{0}=0.28 \mathrm{~nm}\). Equation Transcription: Text Transcription: r_{0}=0.28 nm

(II) The equilibrium distance \(r_{0}\) between two atoms in a molecule is called the bond length. Using the bond lengths of homogeneous molecules (like \(H_{2}, O_{2}, \text { and } N_{2}\)), one can estimate the bond length of heterogeneous molecules (like CO, CN, and NO). This is done by summing half of each bond length of the homogenous molecules to estimate that of the heterogeneous molecule. Given the following bond lengths: \(H_{2}(=74 \mathrm{pm})\), \(N_{2}(=145 \mathrm{pm})\), \(O_{2}(=121 \mathrm{pm})\), \(C_{2}(=154 \mathrm{pm})\), estimate the bond lengths for: HN, CN, and NO. Equation Transcription: Text Transcription: r_{0} H_2 , O_2 , and N_2 H_2 (= 74 pm) N_2 (= 145 pm) O_2 (= 121 pm) C_2 (= 154 pm)

Binding energies are often measured experimentally in kcal per mole, and then the binding energy in eV per molecule is calculated from that result. What is the conversion factor in going from kcal per mole to eV per molecule? What is the binding energy of KCl (= 4.43 eV) in kcal per mole?

(III) Estimate the binding energy of the \(H_{2}\) molecule, assuming the two H nuclei are 0.074 nm apart and the two electrons spend 33% of their time midway between them. Equation Transcription: Text Transcription: H_{2}

(III) (a) Apply reasoning similar to that in the text for the S = 0 and S = 1 states in the formation of the \(H_{2}\) molecule to show why the molecule \(H e_{2}\) is not formed. (b) Explain why the \(H e_{2}^{+}\) molecular ion could form. (Experiment shows it has a binding energy of 3.1 eV at \(r_{0}=0.11 \mathrm{~nm}\).) Equation Transcription: Text Transcription: H_{2} He_2 He_2+ r_0=0.11 nm

(I) Show that the quantity \(\hbar^2/I\) has units of energy.

(II) (a) Calculate the "characteristic rotational energy," \(\hbar^2 / 2 I\), for the \(\mathrm{O}_2\) molecule whose bond length is \(0.121 \mathrm{~nm}\). (b) What are the energy and wavelength of photons emitted in an \(\ell = 3\) to \(\ell = 2\) transition?

(II) The "characteristic rotational energy," \(\hbar^2 / 2 I\), for \(\mathrm{N}_2\) is \(2.48 \times 10^{-4} \mathrm{eV}\). Calculate the \(\mathrm{N}_2\) bond length.

(II) The equilibrium separation of H atoms in the \(H_{2}\) molecule is 0.074 nm (Fig. 29–8). Calculate the energies and wave- lengths of photons for the rotational transitions (a) \(\ell=1\) to \(\ell=0\), (b) \(\ell=2\) to \(\ell=1\), and (c) \(\ell=3\) to \(\ell=2\). Equation Transcription: Text Transcription: H_{2} \ell=1 \ell=0 \ell=2 \ell=3

(II) Determine the wavelength of the photon emitted when the CO molecule makes the rotational transition \(\ell=5\) to \(\ell=4\) . [Hint: See Example 29–2.]

(II) Calculate the bond length for the NaCl molecule given that three successive wavelengths for rotational transitions are 23.1 mm, 11.6 mm, and 7.71 mm.

(a) Use the curve of Fig. 2917 to estimate the stiffness constant k for the molecule. (Recall that ) (b) Then estimate the fundamental wavelength for vibrational transitions using the classical formula (Chapter 11), but use only the mass of an H atom (because both H atoms move)

(II) Common salt, NaCl, has a density of \(2.165 \mathrm{~g} / \mathrm{cm}^{3}\). The molecular weight of NaCl is 58.44. Estimate the distance between nearest neighbor Na and Cl ions. [Hint: Each ion can be considered to be at the corner of a cube.] Equation Transcription: Text Transcription: 2.165 g/cm^3

(II) Repeat Problem 14 for KCl whose density is \(1.99 g/cm^3\).

The spacing between nearest neighbor Na and Cl ions in a NaCl crystal is 0.24 nm. What is the spacing between two nearest neighbor Na ions?

(I) A semiconductor is struck by light of slowly increasing frequency and begins to conduct when the wavelength of the light is 620 nm. Estimate the energy gap \(E_g\).

Calculate the longest-wavelength photon that can cause an electron in silicon to jump from the valence band to the conduction band

(II) The energy gap between valence and conduction bands in germanium is 0.72 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

(II) The band gap of silicon is 1.12 eV. (a) For what range of wavelengths will silicon be transparent? (See Example 29–5.) In what region of the electromagnetic spectrum does this transparent range begin? (b) If window glass is transparent for all visible wavelengths, what is the minimum possible band gap value for glass (assume \(\lambda=400 \mathrm{~nm}\) to 700 nm)? [Hint: If the photon has less energy than the band gap, the photon will pass through the solid without being absorbed.] Equation Transcription: Text Transcription: \lambda=400 nm

(II) The energy gap \(E_g\) in germanium is 0.72 eV. When used as a photon detector, roughly how many electrons can be made to jump from the valence to the conduction band by the passage of an 830-keV photon that loses all its energy in this fashion?

(III) We saw that there are 2N possible electron states in the 3s band of Na, where N is the total number of atoms. How many possible electron states are there in the (a) 2s band, (b) 2p band, and (c) 3p band? (d) State a general formula for the total number of possible states in any given electron band.

(III) Suppose that a silicon semiconductor is doped with phosphorus so that one silicon atom in \(1.5 \times 10^6\) is replaced by a phosphorus atom. Assuming that the “extra” electron in every phosphorus atom is donated to the conduction band, by what factor is the density of conduction electrons increased? The density of silicon is \(2330 \ \mathrm {kg/m}^3\) and the density of conduction electrons in pure silicon is about \(10^{16} \ \mathrm m^{-3}\) at room temperature.

At what wavelength will an LED radiate if made from a material with an energy gap Eg = 1.3 eV?

(I) If an LED emits light of wavelength \(\lambda=730 \mathrm{~nm}\), what is the energy gap (in eV) between valence and conduction bands? Equation Transcription: Text Transcription: \lambda=730 \mathrm{~nm}

(I) A semiconductor diode laser emits \(1.3-\mu m\) light. Assuming that the light comes from electrons and holes recombining, what is the band gap in this laser material? Equation Transcription: Text Transcription: 1.3-\mu m

A silicon diode, whose currentvoltage characteristics are given in Fig. 2930, is connected in series with a battery and a resistor. What battery voltage is needed to produce a 14-mA current?

(II) An ac voltage of 120-V rms is to be rectified. Estimate very roughly the average current in the output resistor \(R\ (= 31\ k\Omega)\) for (a) a half-wave rectifier (Fig. 29-31), and (b) a full-wave rectifier (Fig.29-32) without capacitor.

(III) Suppose that the diode of Fig. 29–30 is connected in series to a \(180-\Omega\) resistor and a 2.0-V battery. What current flows in the circuit? [Hint: Draw a line on Fig. 29–30 representing the current in the resistor as a function of the voltage across the diode; the intersection of this line with the characteristic curve will give the answer.]

Sketch the resistance as a function of current, for for the diode shown in Fig. 2930

A 120-V rms 60-Hz voltage is to be rectified with a full-wave rectifier as in Fig. 2932, where and (a) Make a rough estimate of the average current. (b) What happens if [Hint: See Section 196.]

From Fig. 2941, write an equation for the relationship between the base current the collector current and the emitter current ( not labeled in Fig. 2941). Assume

Draw a circuit diagram showing how a pnp transistor can operate as an amplifier, similar to Fig. 2941 showing polarities, etc

(II) If the current gain of the transistor amplifier in Fig. 29–41 is \(\beta=i_\mathrm C/i_ \mathrm B=95\), what value must \(R_\mathrm C\) have if a \(1.0-\mu \mathrm A\) ac base current is to produce an ac output voltage of 0.42 V?

Suppose that the current gain of the transistor in Fig. 29-41 is \(\beta = i_C/i_B = 85\). If \(R_c = 3.8\) \(k \Omega\) calculate the ac output voltage for an ac input current of 2.0 \(\mu A\).

(II) An amplifier has a voltage gain of 75 and a \(25-k \Omega\) load (output) resistance. What is the peak output current through the load resistor if the input voltage is an ac signal with a peak of 0.080 V? Equation Transcription: Text Transcription: 25-k \Omega

(II) A transistor, whose current gain \(\beta=i_{\mathrm{C}} / i_{\mathrm{B}}=65\), is connected as in Fig. 29-41 with \(R_{\mathrm{B}}=3.8 \ \mathrm{k} \Omega\) and \(R_{\mathrm{C}}=7.8 \ \mathrm{k} \Omega\). Calculate (a) the voltage gain, and (b) the power amplification.

Use the uncertainty principle to estimate the binding energy of the molecule by calculating the difference in kinetic energy of the electrons between (i) when they are in separate atoms and (ii) when they are in the molecule. Take for the electrons in the separated atoms to be the radius of the first Bohr orbit, 0.053 nm, and for the molecule take to be the separation of the nuclei, 0.074 nm. [Hint: Let ]

The average translational kinetic energy of an atom or molecule is about KE = \(\frac{3}{2}kT\) (see Section 13–9), where \(k =\) \(1.38 \times 10^-23\) \(J/K\) is Boltzmann’s constant. At what temperature T will be on the order of the bond energy (and hence the bond easily broken by thermal motion) for (a) a covalent bond (say \(H_2\)) of binding energy 4.0 eV, and (b) a “weak” hydrogen bond of binding energy 0.12 eV?

A diatomic molecule is found to have an activation energy of 1.3 eV. When the molecule is disassociated, 1.6 eV of energy is released. Draw a potential energy curve for this molecule.

. In the ionic salt KF, the separation distance between ions is about 0.27 nm. (a) Estimate the electrostatic potential energy between the ions assuming them to be point charges (magnitude 1e). (b) When F grabs an electron, it releases 3.41 eV of energy, whereas 4.34 eV is required to ionize K. Find the binding energy of KF relative to free K and F atoms, neglecting the energy of repulsion.

The rotational absorption spectrum of a molecule displays peaks about \(8.9 \times 10^{11}\ Hz\) apart. Determine the moment of inertia of this molecule.

For \(\mathrm O_2\) with a bond length of 0.121 nm, what is the moment of inertia about the center of mass?

For with a bond length of 0.121 nm, what is the moment of inertia about the center of mass?

For a certain semiconductor, the longest wavelength radiation that can be absorbed is 2.06 mm. What is the energy gap in this semiconductor?

When EM radiation is incident on diamond, it is found that light with wavelengths shorter than 226 nm will cause the diamond to conduct. What is the energy gap between the valence band and the conduction band for diamond?

The energy gap between valence and conduction bands in zinc sulfide is 3.6 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band

Most of the Suns radiation has wavelengths shorter than 1100 nm. For a solar cell to absorb all this, what energy gap ought the material have?

A TV remote control emits IR light. If the detector on the TV set is not to react to visible light, could it make use of silicon as a window with its energy gap What is the shortest-wavelength light that can strike silicon without causing electrons to jump from the valence band to the conduction band?

Green and blue LEDs became available many years after red LEDs were first developed. Approximately what energy gaps would you expect to find in green (525 nm) and in blue (465 nm) LEDs?

Consider a monatomic solid with a weakly bound cubic lattice, with each atom connected to six neighbors, each bond having a binding energy of \(3.4 \times 10^{-3} \ \mathrm {eV}\). When this solid melts, its latent heat of fusion goes directly into breaking the bonds between the atoms. Estimate the latent heat of fusion for this solid, in [Hint: Show that in a simple cubic lattice (Fig. 29–44), there are three times as many bonds as there are atoms, when the number of atoms is large.]

Problem 1COQ What holds a solid together? (a) Gravitational forces. (b) Magnetic forces. (c) Electric forces. (d) Glue. (e) Nuclear forces.

Problem 1MCQ What holds molecules together? (a) Gravitational forces. (b)Magnetic forces. (c) Electric forces. (d) Glue. (e) Nuclear forces.

Problem 1P (I) Estimate the binding energy of a KCl molecule by calculating the electrostatic potential energy when the K+ and Cl- ions are at their stable separation of 0.28 nm. Assume each has a charge of magnitude 1.0e.

What type of bond would you expect for (a) the \(N_{2}\) molecule, (b) the HCl molecule, (c) Fe atoms in a solid? Equation Transcription: Text Transcription: N_{2}

Problem 1SL Explain why metals are shiny. (See Section 29–5.)

(II) The measured binding energy of KCl is 4.43 eV. From the result of Problem 1, estimate the contribution to the binding energy of the repelling electron clouds at the equilibrium distance \(r_0 = 0.28\ nm\).

Which of the following is true for covalently bound diatomic molecules such as \(H_{2}\)? (a) All electrons in atoms have identical quantum numbers. (b) The molecule has fewer electrons than the two separate atoms do. (c) The molecule has less energy than two separate atoms. (d) The energy of the molecule is greatest when the atoms are separated by one bond length. Equation Transcription: Text Transcription: H_{2}

Describe how the molecule \(\mathrm{CaCl}_{2}\) could be formed. Equation Transcription: Text Transcription: CaCl_2

Problem 2SL Compare the potential energy diagram for an H2 molecule with the potential energy diagram for ATP formation from ADP and P. Explain the significance of the difference in shapes of the two diagrams. (See Section 29–2.)

A hydrogen atom \((Z=1)\) is bonded to a lithium atom \((Z=3)\) in lithium hydride, \(\mathrm{LiH}\). Which of the following are possible spin states of the two shared electrons? (a) \(+\frac{1}{2},+\frac{1}{2}\) (b) \(-\frac{1}{2},-\frac{1}{2}\) (c) \(+\frac{1}{2},-\frac{1}{2}\) () Both \(\text { (a) and }(b) \text {. }\) () Any of the above. Equation Transcription: Text Transcription: (Z=1) (Z=3) LiH (a)+1 over 2,+1 over 2 -1 over 2,+1 over 2 +1 over 2,+1 over 2 (a) and (b)

Does the \(H_{2}\) molecule have a permanent dipole moment? Does \(O_{2}\) ? Does \(H_{2} O\)? Explain. Equation Transcription: Text Transcription: H_{2} O_{2} H_{2} O

Problem 3P (II) The equilibrium distance r0 between two atoms in a molecule is called the bond length. Using the bond lengths of homogeneous molecules (like H2, O2 and N2 ), one can estimate the bond length of heterogeneous molecules (like CO, CN, and NO). This is done by summing half of each bond length of the homogenous molecules to estimate that of the heterogeneous molecule. Given the following bond lengths: H2 (= 74 pm), N2 (=145 pm), O2 (=121 pm) C2(=154 pm ) estimate the bond lengths for: HN, CN, and NO.

Problem 3SL (a) Why are weak bonds important in cells? (b) Explain why heating proteins too much may cause them to denature— that is, lose the specific shape they need to function. (See Section 29–3.) (c) What is the strongest weak bond, and why? (d) If this bond, and the other weak bonds, were stronger (that is, too strong), what would be the consequence for protein synthesis?

Problem 4MCQ Ionic bonding is related to (a) magnetic dipole interactions. (b) the transfer of one or more electrons from one atom to another. (c) the sharing of electrons between atoms. (d) the transfer of electrons to the solid. (e) oscillation dipoles.

(II) Binding energies are often measured experimentally in kcal per mole, and then the binding energy in eV per molecule is calculated from that result. What is the conversion factor in going from kcal per mole to eV per molecule?What is the binding energy of KCl (= 4.43 eV) in kcal per mole?

Although the molecule \(H_{3}\) is not stable, the ion \(H_{3}^{+}\) is. Explain, using the Pauli exclusion principle. Equation Transcription: Text Transcription: H_{3} H_{3}^{+}

Problem 4SL Assume conduction electrons in a semiconductor behave as an ideal gas. (This is not true for conduction electrons in a metal.) (a) Taking mass m = 9 X 10-31 kg and temperature T =300 k, determine the de Broglie wavelength of a semiconductor’s conduction electrons. (b) Given that the spacing between atoms in a semiconductor’s atomic lattice is on the order of 0.3 nm, would you expect room-temperature conduction electrons to travel in straight lines or diffract when traveling through this lattice? Explain.

Problem 5MCQ Consider Fig. 29–10. As the last phosphate group approaches and then bonds to the ADP molecule, which of the following is true? Choose all that apply. (a) The phosphate group is first repelled and then attracted to the ADP molecule. (b) The phosphate group is always attracted to the ADP molecule. (c) The phosphate group is always repelled by the ADP molecule. (d) The system first loses and then stores potential energy. (e) Both binding energy and activation energy are negative. (f) Both binding energy and activation energy are positive.

Problem 5P (III) Estimate the binding energy of the H2 molecule, assuming the two H nuclei are 0.074 nm apart and the two electrons spend 33% of their time midway between them.

Would you expect the molecule \(H_{2}^{+}\) to be stable? If so, where would the single electron spend most of its time?

A strip of silicon \(1.6 \mathrm{~cm}\) wide and \(1.0 \mathrm{~mm}\) thick is immersed in a magnetic field of strength \(1.5 \mathrm{~T}\) perpendicular to the strip (Fig. 29–45). When a current of \(0.28 \mathrm{~mA}\) is run through the strip, there is a resulting Hall effect voltage of \(18 \mathrm{mV}\) across the strip (Section 20–4). How many electrons per silicon atom are in the conduction band? The density of silicon is \(2330 \mathrm{~kg} / \mathrm{m}^{3}\) Equation Transcription: Text Transcription: 1.6 cm 1.0 mm 1.5 T 0.28 mA 18 mV 2330 kg/m3

Problem 6P (III) (a) Apply reasoning similar to that in the text for the S =0 and S =1 states in the formation of the H2 molecule to show why the molecule is not formed. (b) Explain why the He2+ molecular ion could form. (Experiment shows it has a binding energy of 3.1 eV at r0 = 0.11 nm.)

Problem 6Q Explain why the carbon atom (Z = 6) usually forms four bonds with hydrogen-like atoms.

Problem 6SL For an arsenic donor atom in a doped silicon semiconductor, assume that the “extra” electron moves in a Bohr orbit about the arsenic ion. For this electron in the ground state, take into account the dielectric constant K = 12 of the Si lattice (which represents the weakening of the Coulomb force due to all the other atoms or ions in the lattice), and estimate (a) the binding energy, and (b) the orbit radius for this extra electron. [Hint: Substitute in E = KE 0 Coulomb’s law; see Section 17–8 and also 27–12.]

Problem 7MCQ In a p-type semiconductor, a hole is (a) a region in the molecular structure where an atom is missing. (b) an extra electron from one of the donor atoms. (c) an extra positively charged particle in the molecular structure. (d) a region missing an electron relative to the rest of the molecular structure.

(I) Show that the quantity \(h^2 /I\) has units of energy.

Problem 7Q The energy of a molecule can be divided into four categories. What are they?

Problem 8MCQ The electrical resistance of a semiconductor may decrease with increasing temperature because, at elevated temperature, more electrons (a) collide with the crystal lattice. (b) move faster. (c) are able to jump across the energy gap. (d) form weak van der Waals bonds.

Problem 8P (II) (a) Calculate the “characteristic rotational energy,” h2 /I For, the O2 molecule whose bond length is 0.121 nm. (b) What are the energy and wavelength of photons emitted in an l =3 to l= 2 transition?

If conduction electrons are free to roam about in a metal, why don’t they leave the metal entirely?

Which of the following would not be used as an impurity in doping silicon? (a) Germanium. (b) Gallium. (c) Boron. (d) Phosphorus. (e) Arsenic.

(II) The “characteristic rotational energy,” \(\hbar^{2} / 2 I\), for \(N_2\) is 2.48 \times \(10^{-4}\ eV\). Calculate the \(N_2\) bond length.

Explain why the resistivity of metals increases with increasing temperature whereas the resistivity of semiconductors may decrease with increasing temperature.

Why are metals good conductors? (a) Gaining a tiny bit of energy allows their electrons to move. (b) They have more electrons than protons, so some of the electrons are extra and free to move. (c) They have more protons than electrons, so some of the protons are extra and free to move. (d) Gaining a tiny bit of energy allows their protons to move. (e) Electrons are tightly bound to their atoms.

\((I I) \) The equilibrium separation of \(H\) atoms in the \(H_{2}\) molecule is \(0.074 \mathrm{~nm}\) (Fig. ). Calculate the energies and wavelengths of photons for the rotational transitions (a) \(l=1 \text { to } l=0\) (b) \(l=2 \text { to } l=1\) and (c) \(l=3 \text { to } l=2 Equation Transcription: Text Transcription: (II) H H2 0.074 nm l=1 to l=0 l=2 to l=1 l=3 to l=2

Problem 10Q Compare the resistance of a pn junction diode connected in forward bias to its resistance when connected in reverse bias.

(II) Determine the wavelength of the photon emitted when the CO molecule makes the rotational transition \(\ell=5\) to \(\ell=4\). [Hint: See Example 29–2.] Equation Transcription: Text Transcription: \ell=5 \ell=4

Explain how a transistor can be used as a switch.

Problem 12P (II) Calculate the bond length for the NaCl molecule given that three successive wavelengths for rotational transitions are 23.1 mm, 11.6 mm, and 7.71 mm.

\(\text { (II) }\) Calculate the bond length for the \(\mathrm{NaCl}\)molecule given that three successive wavelengths for rotational transitions are \(23.1 \mathrm{~mm}, 11.6 \mathrm{~mm} \text {, and } 7.71 \mathrm{~mm}\) Equation Transcription: Text Transcription: (II) NaCl 23.1 mm, 11.6 mm, and 7.71 mm

Problem 13Q What is the main difference between n-type and p-type semiconductors?

\(\text { (II) }\) (a) Use the curve of Fig. 29–17 to estimate the stiffness constant \(\mathrm{k}\) for the \(\mathrm{H}_{2}\) molecule. (Recall that \(P E=\frac{1}{2} k x^{2}\).) (b) Then estimate the fundamental wavelength for vibrational transitions using the classical formula (Chapter 11), but use only \(\frac{1}{2}\) the mass of an H atom (because both H atoms move). Equation Transcription: Text Transcription: (II) k H2 PE=1 over 2kx2 1 over 2

Problem 14P (II) Common salt, NaCl, has a density of 2.165 g/cm3. The molecular weight of NaCl is 58.44. Estimate the distance between nearest neighbor Na and Cl ions. [Hint: Each ion can be considered to be at the corner of a cube.]

Explain on the basis of energy bands why the sodium chloride crystal is a good insulator. [Hint: Consider the shells of \(Na^+\) and \(CI^-\) ions.]

(II) Repeat Problem 14 for KCl whose density is 1.99 \(g/cm^3\).

\(\text { (II) }\) Repeat Problem 14 for \(\mathrm{KCl}\) whose density is \(1.99 \mathrm{~g} / \mathrm{cm}^{3}\) Equation Transcription: Text Transcription: (II) KCl 1.99 g/cm3

Problem 16P (II) The spacing between “nearest neighbor” Na and Cl ions in a NaCl crystal is 0.24 nm. What is the spacing between two nearest neighbor Na ions?

Problem 16Q A transistor can amplify an electronic signal, meaning it can increase the power of an input signal. Where does it get the energy to increase the power?

(I) A semiconductor is struck by light of slowly increasing frequency and begins to conduct when the wavelength of the light is 620 nm. Estimate the energy gap \(E_g\).

Problem 17Q A silicon semiconductor is doped with phosphorus. Will these atoms be donors or acceptors? What type of semiconductor will this be?

(I) Calculate the longest-wavelength photon that can cause an electron in silicon \((E_g=1.12 \ \mathrm {eV}\)) to jump from the valence band to the conduction band.

Problem 18Q Do diodes and transistors obey Ohm’s law? Explain.

Problem 19P (II) The energy gap between valence and conduction bands in germanium is 0.72 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

Can a diode be used to amplify a signal? Explain.

Problem 20P (II) The band gap of silicon is 1.12 eV. (a) For what range of wavelengths will silicon be transparent? (See Example 29–5.) In what region of the electromagnetic spectrum does this transparent range begin? (b) If window glass is transparent for all visible wavelengths, what is the minimum possible band gap value for glass (assume ? =400 NM to 700 nm)? [Hint: If the photon has less energy than the band gap, the photon will pass through the solid without being absorbed.]

(II) The energy gap \(E_g\) in germanium is 0.72 eV. When used as a photon detector, roughly how many electrons can be made to jump from the valence to the conduction band by the passage of an 830-keV photon that loses all its energy in this fashion?

Problem 22P (III) We saw that there are 2N possible electron states in the 3s band of Na, where N is the total number of atoms. How many possible electron states are there in the (a) 2s band, (b) 2p band, and (c) 3p band? (d) State a general formula for the total number of possible states in any given electron band.

Problem 23P (III) Suppose that a silicon semiconductor is doped with phosphorus so that one silicon atom in 1.5 X 106 is replaced by a phosphorus atom. Assuming that the “extra” electron in every phosphorus atom is donated to the conduction band, by what factor is the density of conduction electrons increased? The density of silicon is 2330 kg/m3 . and the density of conduction electrons in pure silicon is about 1016 m–3 at room temperature.

Problem 24P (I) At what wavelength will an LED radiate if made from a material with an energy gap Eg =1.3 eV?

(I) If an LED emits light of wavelength \(\lambda=730\ nm\), what is the energy gap (in eV) between valence and conduction bands?

\(\text { (II) }\) A silicon diode, whose current–voltage characteristics are given in Fig. 29–30, is connected in series with a battery and a \(\text { 960- } \Omega\) resistor. What battery voltage is needed to produce a \(14-m A\) current? Equation Transcription: Text Transcription: (II) 960- \Omega 14-mA

Problem 26P (I) A semiconductor diode laser emits 1.3-µm light. Assuming that the light comes from electrons and holes recombining, what is the band gap in this laser material?

\(\text { (II) }\) An ac voltage of \(120-\mathrm{V} \text { rms }\) is to be rectified. Estimate very roughly the average current in the output resistor \(R(=31 k \Omega)\) for (a) a half-wave rectifier (Fig. 29–31), and (b) a full-wave rectifier (Fig. 29–32) without capacitor. Equation Transcription: Text Transcription: (II) 120-Vrms R(=31 k \Omega)

Determine the three lowest rotational energy states (in eV) for a nitrogen molecule which has a moment of inertia \(I = 1.39 \times 10^{-46}\ kg \cdot m^2\).

Problem 29EB What are the wavelengths of the next three rotational transitions for CO?

Problem 29EC What is the energy of the first vibrational state above the ground state in the hydrogen molecule?

Problem 29ED Return to the Chapter-Opening Question, page 829, and answer it again now. Try to explain why you may have answered differently the first time. CHAPTER-OPENING QUESTION. What holds a solid together? 1. Gravitational forces. 2. Magnetic forces. 3. Electric forces. 4. Glue. 5. Nuclear forces.

Problem 29EE Which of the following impurity atoms in silicon would produce a p-type semiconductor? (a) Ge; (b) Ne; (c) Al; (d) As; (e) Ga; (f) none of the above.

\(\text { (III) }\) Suppose that the diode of Fig. 29-30 is connected in series to a \(180-\Omega\) resistor and a \(2.0-V\) battery. What current flows in the circuit? [Hint: Draw a line on Fig. 29-30 representing the current in the resistor as a function of the voltage across the diode; the intersection of this line with the characteristic curve will give the answer.] Equation Transcription: Text Transcription: (III) 180-\Omega 2.0-V

(III) Sketch the resistance as a function of current, for V > 0, for the diode shown in Fig. 29-30.

Problem 31P (III) A 120-V rms 60-Hz voltage is to be rectified with a full-wave rectifier as in Fig. 29–32, where R =33 K?, and C =28 µF.(a) Make a rough estimate of the average current. (b) What happens if [Hint: See Section 19–6.]

(I) Draw a circuit diagram showing how a pnp transistor can operate as an amplifier, similar to Fig. 29–41 showing polarities, etc.

\((I I)\) If the current gain of the transistor amplifier in Fig. is \(\beta=i_{c} / i_{B}=95\), what value must \(R c\) have if a \(1.0-\mu A\) ac base current is to produce an ac output voltage of \(0.42 V\)? Equation Transcription: Text Transcription: (II) \beta =ic/iB=95 Rc 1.0 - \mu A 0.42 V

(II) Suppose that the current gain of the transistor in Fig. 29-41 is \(\beta=i_\mathrm{C} / i_\mathrm{B}=85\). If \(R_\mathrm{C}=3.8 \ \mathrm{k} \Omega\), calculate the ac output voltage for an ac input current of \(2.0 \ \mu \mathrm A\).

Problem 36P (II) An amplifier has a voltage gain of 75 and a 25-K? load (output) resistance. What is the peak output current through the load resistor if the input voltage is an ac signal with a peak of 0.080 V?

\(\text { (II) }\) A transistor, whose current gain \(\beta=i_{c} / i_{B}=65\), is connected as in Fig. with \(R_{B}=3.8 \mathrm{k} \Omega \text { and } R_{c}=7.8 \mathrm{k} \Omega\) Calculate the voltage gain, and the power amplification. Equation Transcription: Text Transcription: (II) \beta =ic/iB=65 R_B=3.8 k \Omega and R_{c}=7.8 k \Omega

Problem 38GP Use the uncertainty principle to estimate the bindingenergy of the H2 molecule by calculating the difference inkinetic energy of the electrons between (i) when they are inseparate atoms and (ii) when they are in the molecule. Take ?xfor the electrons in the separated atoms to be theradius of the first Bohr orbit, 0.053 nm, and for the molecule take ?x to be the separation of the nuclei, 0.074 nm.[Hint: Let ?p ? ?px.]

The average translational kinetic energy of an atom or molecule is about \(\mathrm{KE} =\frac{3}{2} k T\) (see Section 13-9), where \(k=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}\) is Boltzmann's constant. At what temperature T will KE be on the order of the bond energy (and hence the bond easily broken by thermal motion) for (a) a covalent bond (say \(\mathrm{H}_2\)) of binding energy 4.0 eV, and (b) a "weak" hydrogen bond of binding energy 0.12 eV?

Problem 40GP A diatomic molecule is found to have an activation energyof 1.3 eV. When the molecule is disassociated, 1.6 eV ofenergy is released. Draw a potential energy curve for thismolecule.

In the ionic salt KF, the separation distance between ions is about 0.27 nm. (a) Estimate the electrostatic potential energy between the ions assuming them to be point charges (magnitude 1e). (b) When F “grabs” an electron, it releases 3.41 eV of energy, whereas 4.34 eV is required to ionize K. Find the binding energy of KF relative to free K and F atoms, neglecting the energy of repulsion.

The rotational absorption spectrum of a molecule displays peaks about \(8.9 \times 10^{11} \ \mathrm {Hz}\) apart. Determine the moment of inertia of this molecule.

For \(\mathrm O_2\) with a bond length of 0.121 nm, what is the moment of inertia about the center of mass?

Must we consider quantum effects for everyday rotating objects? Estimate the differences between rotational energy levels for a spinning baton compared to the energy of the baton. Assume the baton consists of a uniform 32-cm-long bar with a mass of 230 g and two small end masses, each of mass 380 g, and it rotates at 1.8 rev/s about the bar’s center.

For a certain semiconductor, the longest wavelength radiation that can be absorbed is 2.06 mm. What is the energy gap in this semiconductor?

Problem 46GP When EM radiation is incident on diamond, it is foundthat light with wavelengths shorter than 226 nm will causethe diamond to conduct. What is the energy gap betweenthe valence band and the conduction band for diamond?

Problem 47GP The energy gap between valence and conduction bands inzinc sulfide is 3.6 eV. What range of wavelengths can aphoton have to excite an electron from the top of thevalence band into the conduction band?

Most of the Sun’s radiation has wavelengths shorter than 1100 nm. For a solar cell to absorb all this, what energy gap ought the material have?

Problem 49GP A TV remote control emits IR light. If the detector on theTV set is not to react to visible light, could it make use ofsilicon as a “window” with its energy gap Eg = 1.12 eV?What is the shortest-wavelength light that can strike siliconwithout causing electrons to jump from the valence band tothe conduction band?

Green and blue LEDs became available many years after red LEDs were first developed. Approximately what energy gaps would you expect to find in green (525 nm) and in blue (465 nm) LEDs?

Consider a monatomic solid with a weakly bound cubic lattice, with each atom connected to six neighbors, each bond having a binding energy of \(3.4 \times 10^{-3} \mathrm{eV}\). When this solid melts, its latent heat of fusion goes directly into breaking the bonds between the atoms. Estimate the latent heat of fusion for this solid, in \(\mathrm{J} / \mathrm{mol}\). [Hint: Show that in a simple cubic lattice (Fig. ), there are three times as many bonds as there are atoms, when the number of atoms is large.] Equation Transcription: Text Transcription: 3.4 x 10-3eV J/mol

Estimate the binding energy of a KCl molecule by calculating the electrostatic potential energy when the and ions are at their stable separation of 0.28 nm. Assume each has a charge of magnitude 1.0e

The measured binding energy of KCl is 4.43 eV. From the result of Problem 1, estimate the contribution to the binding energy of the repelling electron clouds at the equilibrium distance r0 = 0.28 nm.

(II) The equilibrium distance \(r_0\) between two atoms in a molecule is called the bond length. Using the bond lengths of homogeneous molecules (like \(\mathrm H_2, \mathrm O_2\), and \(\mathrm N_2\)), one can estimate the bond length of heterogeneous molecules (like CO, CN, and NO). This is done by summing half of each bond length of the homogenous molecules to estimate that of the heterogeneous molecule. Given the following bond lengths: \(\mathrm H_2 (=74 \ \mathrm {pm}, \mathrm N_2 (=145 \ \mathrm {pm}, \mathrm O_2 (=121 \ \mathrm {pm}, \mathrm C_2 (=154 \ \mathrm {pm})\), estimate the bond lengths for: HN, CN, and NO.

Binding energies are often measured experimentally in kcal per mole, and then the binding energy in eV per molecule is calculated from that result. What is the conversion factor in going from kcal per mole to eV per molecule? What is the binding energy of in kcal per mole?

Estimate the binding energy of the molecule, assuming the two H nuclei are 0.074 nm apart and the two electrons spend 33% of their time midway between them.

(a) Apply reasoning similar to that in the text for the and states in the formation of the molecule to show why the molecule is not formed. (b) Explain why the molecular ion could form. (Experiment shows it has a binding energy of 3.1 eV at )

Show that the quantity has units of energy.

(II) (a) Calculate the “characteristic rotational energy,” \(\hbar^2/2I\), for the \(\mathrm O_2\) molecule whose bond length is 0.121 nm. (b) What are the energy and wavelength of photons emitted in an \(\ell=3\) to \(\ell=2\) transition?

The characteristic rotational energy, for is Calculate the bond length.

The equilibrium separation of H atoms in the molecule is 0.074 nm (Fig. 298). Calculate the energies and wavelengths of photons for the rotational transitions (a) to (b) to and

Determine the wavelength of the photon emitted when the CO molecule makes the rotational transition to . [Hint: See Example 292.]

Calculate the bond length for the NaCl molecule given that three successive wavelengths for rotational transitions are 23.1 mm, 11.6 mm, and 7.71 mm.

(a) Use the curve of Fig. 2917 to estimate the stiffness constant k for the molecule. (Recall that ) (b) Then estimate the fundamental wavelength for vibrational transitions using the classical formula (Chapter 11), but use only the mass of an H atom (because both H atoms move)

Common salt, NaCl, has a density of The molecular weight of NaCl is 58.44. Estimate the distance between nearest neighbor Na and Cl ions. [Hint: Each ion can be considered to be at the corner of a cube.

Repeat Problem 14 for KCl whose density is 1.99 gcm3

(II) The spacing between “nearest neighbor” Na and Cl ions in a NaCl crystal is 0.24 nm. What is the spacing between two nearest neighbor Na ions?

A semiconductor is struck by light of slowly increasing frequency and begins to conduct when the wavelength of the light is 620 nm. Estimate the energy gap

Calculate the longest-wavelength photon that can cause an electron in silicon to jump from the valence band to the conduction band

(II) The energy gap between valence and conduction bands in germanium is 0.72 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

(II) The band gap of silicon is 1.12 eV. (a) For what range of wavelengths will silicon be transparent? (See Example 29–5.) In what region of the electromagnetic spectrum does this transparent range begin? (b) If window glass is transparent for all visible wavelengths, what is the minimum possible band gap value for glass (assume \(\lambda=400 \ \mathrm {nm}\) to 700 nm)? [Hint: If the photon has less energy than the band gap, the photon will pass through the solid without being absorbed.]

The energy gap in germanium is 0.72 eV. When used as a photon detector, roughly how many electrons can be made to jump from the valence to the conduction band by the passage of an 830-keV photon that loses all its energy in this fashion?

We saw that there are 2N possible electron states in the 3s band of Na, where N is the total number of atoms. How many possible electron states are there in the (a) 2s band, (b) 2p band, and (c) 3p band? (d) State a general formula for the total number of possible states in any given electron band.

Suppose that a silicon semiconductor is doped with phosphorus so that one silicon atom in is replaced by a phosphorus atom. Assuming that the extra electron in every phosphorus atom is donated to the conduction band, by what factor is the density of conduction electrons increased? The density of silicon is and the density of conduction electrons in pure silicon is about at room temperature.

At what wavelength will an LED radiate if made from a material with an energy gap Eg = 1.3 eV?

(I) If an LED emits light of wavelength \(\lambda = 730\ nm\), what is the energy gap (in eV) between valence and conduction bands?

A semiconductor diode laser emits light. Assuming that the light comes from electrons and holes recombining, what is the band gap in this laser material?

A silicon diode, whose currentvoltage characteristics are given in Fig. 2930, is connected in series with a battery and a resistor. What battery voltage is needed to produce a 14-mA current?

An ac voltage of 120-V rms is to be rectified. Estimate very roughly the average current in the output resistor R for (a) a half-wave rectifier (Fig. 2931), and (b) a full-wave rectifier (Fig. 2932) without capacitor.

Suppose that the diode of Fig. 2930 is connected in series to a resistor and a 2.0-V battery. What current flows in the circuit? [Hint: Draw a line on Fig. 2930 representing the current in the resistor as a function of the voltage across the diode; the intersection of this line with the characteristic curve will give the answer.]

(III) Sketch the resistance as a function of current, for \(V>0\), for the diode shown in Fig. 29–30.

(III) A 120-V rms 60-Hz voltage is to be rectified with a full-wave rectifier as in Fig. 29–32, where \(R = 33\ k\Omega\), and \(C = 28\ \mu F\). (a) Make a rough estimate of the average current. (b) What happens if \(C = 0.10\ \mu F\)? [Hint: See Section 19-6.]

(I) From Fig. 29–41, write an equation for the relationship between the base current \(\left(I_{B}\right)\), the collector current \(\left(I_{C}\right)\), and the emitter current (\(I_{E}\) , not labeled in Fig. 29–41). Assume \(i_{B}=i_{C}=0\). Equation Transcription: Text Transcription: (I_B) (I_C) I_E i_B = i_C = 0

Draw a circuit diagram showing how a pnp transistor can operate as an amplifier, similar to Fig. 29-41 showing polarities, etc.

If the current gain of the transistor amplifier in Fig. 2941 is what value must have if a ac base current is to produce an ac output voltage of 0.42 V?

Suppose that the current gain of the transistor in Fig. 2941 is If calculate the ac output voltage for an ac input current of 2.0 mA

An amplifier has a voltage gain of 75 and a load (output) resistance. What is the peak output current through the load resistor if the input voltage is an ac signal with a peak of 0.080 V?

A transistor, whose current gain is connected as in Fig. 2941 with and Calculate (a) the voltage gain, and (b) the power amplification

Use the uncertainty principle to estimate the binding energy of the \(H_2\) molecule by calculating the difference in kinetic energy of the electrons between (i) when they are in separate atoms and (ii) when they are in the molecule. Take \(\Delta x\) for the electrons in the separated atoms to be the radius of the first Bohr orbit, 0.053 nm, and for the molecule take \(\Delta x\) to be the separation of the nuclei, 0.074 nm. [Hint: Let \(\Delta p \approx \Delta p_x\).]

The average translational kinetic energy of an atom or molecule is about \(K E=\frac{3}{2} k T\) (see Section 13–9), where \(k=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}\) is Boltzmann’s constant. At what temperature T will KE be on the order of the bond energy (and hence the bond easily broken by thermal motion) for (a) a covalent bond (say \(H_{2}\)) of binding energy 4.0 eV, and (b) a “weak” hydrogen bond of binding energy 0.12 eV? Equation Transcription: Text Transcription: K E=\frac{3}{2} k T

A diatomic molecule is found to have an activation energy of 1.3 eV. When the molecule is disassociated, 1.6 eV of energy is released. Draw a potential energy curve for this molecule.

In the ionic salt KF, the separation distance between ions is about 0.27 nm. (a) Estimate the electrostatic potential energy between the ions assuming them to be point charges (magnitude 1e). (b) When F “grabs” an electron, it releases 3.41 eV of energy, whereas 4.34 eV is required to ionize K. Find the binding energy of KF relative to free K and F atoms, neglecting the energy of repulsion.

The rotational absorption spectrum of a molecule displays peaks about \(8.9 \times 10^{11} \mathrm{~Hz}\) apart. Determine the moment of inertia of this molecule. Equation Transcription: Text Transcription: 8.9 \times 10^{11} Hz

For \(\mathrm{O}_{2}\) with a bond length of 0.121 nm, what is the moment of inertia about the center of mass? Equation Transcription: Text Transcription: \mathrm{O}_{2}

Must we consider quantum effects for everyday rotating objects? Estimate the differences between rotational energy levels for a spinning baton compared to the energy of the baton. Assume the baton consists of a uniform 32-cm-long bar with a mass of 230 g and two small end masses, each of mass 380 g, and it rotates at 1.8 rev/s about the bar’s center.

For a certain semiconductor, the longest wavelength radiation that can be absorbed is 2.06 mm. What is the energy gap in this semiconductor?

When EM radiation is incident on diamond, it is found that light with wavelengths shorter than 226 nm will cause the diamond to conduct. What is the energy gap between the valence band and the conduction band for diamond?

The energy gap between valence and conduction bands in zinc sulfide is 3.6 eV. What range of wavelengths can a photon have to excite an electron from the top of the valence band into the conduction band?

Most of the Suns radiation has wavelengths shorter than 1100 nm. For a solar cell to absorb all this, what energy gap ought the material have?

A TV remote control emits IR light. If the detector on the TV set is not to react to visible light, could it make use of silicon as a “window” with its energy gap Eg = 1.12 eV? What is the shortest-wavelength light that can strike silicon without causing electrons to jump from the valence band to the conduction band?

Green and blue LEDs became available many years after red LEDs were first developed. Approximately what energy gaps would you expect to find in green (525 nm) and in blue (465 nm) LEDs?

Consider a monatomic solid with a weakly bound cubic lattice, with each atom connected to six neighbors, each bond having a binding energy of When this solid melts, its latent heat of fusion goes directly into breaking the bonds between the atoms. Estimate the latent heat of fusion for this solid, in [Hint: Show that in a simple cubic lattice (Fig. 2944), there are three times as many bonds as there are atoms, when the number of atoms is large.]