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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 42

Molecules and Condensed Matter - all with Video Answers

Educators

Chapter Questions

05:05

Problem 1

An Ionic Bond. (a) Calculate the electric potential energy for a $\mathrm{K}^{+}$ ion and a $\mathrm{Br}^{-}$ ion separated by a distance of $0.29 \mathrm{nm},$ the equilibrium separation in the KBr molecule. Treat the ions as point charges. (b) The ionization energy of the potassium atom is $4.3 \mathrm{eV}$. Atomic bromine has an electron affinity of $3.5 \mathrm{eV}$. Use these data and the results of part (a) to estimate the binding energy of the KBr molecule. Do you expect the actual binding energy to be higher or lower than your estimate? Explain your reasoning.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:59

Problem 2

During each of these processes, a photon of light is given up. In each process, what wavelength of light is given up, and in what part of the electromagnetic spectrum is that wavelength? (a) A molecule decreases its vibrational energy by $0.198 \mathrm{eV} ;$ (b) an atom decreases its energy by $7.80 \mathrm{eV} ;$ (c) a molecule decreases its rotational energy by $4.80 \times 10^{-3} \mathrm{eV}$.

Jaime Munoz
Jaime Munoz
Numerade Educator
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Problem 3

For the $\mathrm{H}_{2}$ molecule the equilibrium spacing of the two protons is $0.074 \mathrm{nm}$. The mass of a hydrogen atom is $1.67 \times 10^{-27} \mathrm{~kg}$. Calculate the wavelength of the photon emitted in the rotational transition $l=3$ to $l=1$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:54

Problem 4

The $\mathrm{H}_{2}$ molecule has a moment of inertia of $4.6 \times 10^{-48} \mathrm{~kg} \cdot \mathrm{m}^{2}$. What is the wavelength $\lambda$ of the photon absorbed when $\mathrm{H}_{2}$ makes a transition from the $l=3$ to the $l=4$ rotational level?

Bettina Hanlon
Bettina Hanlon
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Problem 5

A hypothetical NH molecule makes a rotational-level transition from $l=3$ to $l=1$ and gives off a photon of wavelength $1.710 \mathrm{nm}$ in doing so. What is the separation between the two atoms in this molecule if we model them as point masses? The mass of hydrogen is $1.67 \times 10^{-27} \mathrm{~kg},$ and the mass of nitrogen is $2.33 \times 10^{-26} \mathrm{~kg} .$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
00:50

Problem 6

Two atoms of cesium (Cs) can form a $\mathrm{Cs}_{2}$ molecule. The equilibrium distance between the nuclei in a $\mathrm{Cs}_{2}$ molecule is $0.447 \mathrm{nm}$. Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining them. The mass of a cesium atom is $2.21 \times 10^{-25} \mathrm{~kg}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
05:15

Problem 7

The rotational energy levels of $\mathrm{CO}$ are calculated in Example 42.2. If the energy of the rotating molecule is described by the classical expression $K=\frac{1}{2} I \omega^{2},$ for the $l=1$ level what are (a) the angular speed of the rotating molecule; (b) the linear speed of each atom; (c) the rotational period (the time for one rotation)?

Kai Chen
Kai Chen
Princeton University
06:43

Problem 8

The vibrational and rotational energies of the CO molecule are given by Eq. (42.9). Calculate the wavelength of the photon absorbed by CO in each of these vibration-rotation transitions: (a) $n=0$ $l=2 \rightarrow n=1, l=3 ;$ (b) $n=0, l=3 \rightarrow n=1, l=2 ;$ (c) $n=0$ $l=4 \rightarrow n=1, l=3$.

Keshav Singh
Keshav Singh
Numerade Educator
01:46

Problem 9

A lithium atom has mass $1.17 \times 10^{-26} \mathrm{~kg},$ and a hydrogen atom has mass $1.67 \times 10^{-27} \mathrm{~kg}$. The equilibrium separation between the two nuclei in the LiH molecule is $0.159 \mathrm{nm} .$ (a) What is the difference in energy between the $l=3$ and $l=4$ rotational levels? (b) What is the wavelength of the photon emitted in a transition from the $l=4$ to the $l=3$ level?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:03

Problem 10

If a sodium chloride $(\mathrm{NaCl})$ molecule could undergo an $n \rightarrow n-1$ vibrational transition with no change in rotational quantum number, a photon with wavelength $20.0 \mu \mathrm{m}$ would be emitted. The mass of a sodium atom is $3.82 \times 10^{-26} \mathrm{~kg},$ and the mass of a chlorine atom is $5.81 \times 10^{-26} \mathrm{~kg}$. Calculate the force constant $k^{\prime}$ for the interatomic force in $\mathrm{NaCl}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
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Problem 11

When a hypothetical diatomic molecule having atoms $0.8900 \mathrm{nm}$ apart undergoes a rotational transition from the $l=2$ state to the next lower state, it gives up a photon having energy $8.800 \times 10^{-4} \mathrm{eV}$. When the molecule undergoes a vibrational transition from one energy state to the next lower energy state, it gives up $0.2590 \mathrm{eV}$. Find the force constant of this molecule.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
00:30

Problem 12

Potassium bromide (KBr) has a density of $2.75 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and the same crystal structure as $\mathrm{NaCl}$. The mass of a potassium atom is $6.49 \times 10^{-26} \mathrm{~kg},$ and the mass of a bromine atom is $1.33 \times 10^{-25} \mathrm{~kg} .$ (a) Calculate the average spacing between adjacent atoms in a $\mathrm{KBr}$ crystal. (b) How does the value calculated in part (a) compare with the spacing in $\mathrm{NaCl}$ (see Exercise 42.13 )? Is the relationship between the two values qualitatively what you would expect? Explain.

Mayukh Banik
Mayukh Banik
Numerade Educator
04:44

Problem 13

The spacing of adjacent atoms in a crystal of sodium chloride is $0.282 \mathrm{nm}$. The mass of a sodium atom is $3.82 \times 10^{-26} \mathrm{~kg},$ and the mass of a chlorine atom is $5.89 \times 10^{-26} \mathrm{~kg} .$ Calculate the density of sodium chloride.

CM
Corbyn Mellinger
Numerade Educator
05:29

Problem 14

The gap between valence and conduction bands in diamond is $5.47 \mathrm{eV}$. (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond is transparent and colorless. (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

Kyle Godbey
Kyle Godbey
Numerade Educator
03:22

Problem 15

The maximum wavelength of light that a certain silicon photocell can detect is $1.11 \mu \mathrm{m}$. (a) What is the energy gap (in electron volts) between the valence and conduction bands for this photocell? (b) Explain why pure silicon is opaque.

Kai Chen
Kai Chen
Princeton University
03:27

Problem 16

The gap between valence and conduction bands in silicon is $1.12 \mathrm{eV}$. A nickel nucleus in an excited state emits a gamma-ray photon with wavelength $9.31 \times 10^{-4} \mathrm{nm}$. How many electrons can be excited from the top of the valence band to the bottom of the conduction band by the absorption of this gamma ray?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
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Problem 17

Calculate the density of states $g(E)$ for the free-electron model of a metal if $E=7.8 \mathrm{eV}$ and $V=1.0 \mathrm{~cm}^{3} .$ Express your answer in units of states per electron volt.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:03

Problem 18

The Fermi energy of sodium is $3.23 \mathrm{eV}$. (a) Find the average energy $E_{\mathrm{av}}$ of the electrons at absolute zero. (b) What is the speed of an electron that has energy $E_{\mathrm{av}}$ ? (c) At what Kelvin temperature $T$ is $k T$ equal to $E_{\mathrm{F}} ?$ (This is called the Fermi temperature for the metal. It is approximately the temperature at which molecules in a classical ideal gas would have the same kinetic energy as the fastest-moving electron in the metal.)

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:05

Problem 19

Consider the density of states in the free-electron model for an energy of $1.60 \mathrm{eV}$. At what energy will the density of states be (a) doubled and (b) halved?

CM
Corbyn Mellinger
Numerade Educator
01:47

Problem 20

For a metal with $k T$ equal to $25 \%$ of the Fermi energy, what is the probability that a state will be occupied by an electron if its energy is (a) $50 \%$ of the Fermi energy and (b) $50 \%$ greater than the Fermi energy? Are your results consistent with Fig. 42.23 ?

Keshav Singh
Keshav Singh
Numerade Educator
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Problem 21

Silver has a Fermi energy of $5.48 \mathrm{eV}$. Calculate the electron contribution to the molar heat capacity at constant volume of silver, $C_{V}$, at $282 \mathrm{~K}$. Express your result (a) as a multiple of $R$ and $(\mathrm{b})$ as a fraction of the actual value for silver, $C_{V}=25.3 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} .$ (c) Is the value of $C_{V}$ due principally to the electrons? If not, to what is it due?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:50

Problem 22

At the Fermi temperature $T_{\mathrm{F}}, E_{\mathrm{F}}=k T_{\mathrm{F}}$ (see Exercise 42.18). When $T=T_{\mathrm{F}}$, what is the probability that a state with energy $E=2 E_{\mathrm{F}}$ is occupied?

CM
Corbyn Mellinger
Numerade Educator
02:26

Problem 23

For a solid metal having a Fermi energy of $8.480 \mathrm{eV},$ what is the probability, at room temperature, that a state having an energy of $8.540 \mathrm{eV}$ is occupied by an electron?

Zhaojie Xu
Zhaojie Xu
Numerade Educator
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Problem 24

Pure germanium has a band gap of $0.67 \mathrm{eV}$. The Fermi energy is in the middle of the gap. (a) For temperatures of $255 \mathrm{~K}, 290 \mathrm{~K},$ and $355 \mathrm{~K},$ calculate the probability $f(E)$ that a state at the bottom of the conduction band is occupied. (b) For each temperature in part (a), calculate the probability that a state at the top of the valence band is empty.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:02

Problem 25

Germanium has a band gap of $0.67 \mathrm{eV}$. Doping with arsenic adds donor levels in the gap $0.01 \mathrm{eV}$ below the bottom of the conduction band. At a temperature of $300 \mathrm{~K},$ the probability is $4.4 \times 10^{-4}$ that an electron state is occupied at the bottom of the conduction band. Where is the Fermi level relative to the conduction band in this case?

Mayukh Banik
Mayukh Banik
Numerade Educator
04:36

Problem 26

(a) Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron-hole pairs. If each pair requires $0.67 \mathrm{eV}$ of energy, what is the maximum wavelength that can be detected? In what portion of the spectrum does it lie? (b) What are the answers to part (a) if the material is silicon, with an energy requirement of $1.12 \mathrm{eV}$ per pair, corresponding to the gap between valence and conduction bands in that element?

Kyle Godbey
Kyle Godbey
Numerade Educator
07:04

Problem 27

At a temperature of $290 \mathrm{~K}$, a certain $p-n$ junction has a saturation current $I_{\mathrm{S}}=0.500 \mathrm{~mA} .$ (a) Find the current at this temperature when the voltage is (i) $1.00 \mathrm{mV},(\mathrm{ii})-1.00 \mathrm{mV}$ (iii) $100 \mathrm{mV},$ and (iv) $-100 \mathrm{mV}$. (b) Is there a region of applied voltage where the diode obeys Ohm's law?

Kai Chen
Kai Chen
Princeton University
05:17

Problem 28

For a certain $p-n$ junction diode, the saturation current at room temperature $\left(20^{\circ} \mathrm{C}\right)$ is $0.950 \mathrm{~mA}$. What is the resistance of this diode when the voltage across it is (a) $85.0 \mathrm{mV}$ and (b) $-50.0 \mathrm{mV} ?$

Kyle Godbey
Kyle Godbey
Numerade Educator
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Problem 29

(a) A forward-bias voltage of $15.0 \mathrm{mV}$ produces a positive current of $9.25 \mathrm{~mA}$ through a $p-n$ junction at $300 \mathrm{~K}$. What does the positive current become if the forward-bias voltage is reduced to $10.0 \mathrm{mV}$ ? (b) For reverse-bias voltages of $-15.0 \mathrm{mV}$ and $-10.0 \mathrm{mV},$ what is the reverse-bias negative current?

Joshua Young
Joshua Young
Numerade Educator
08:28

Problem 30

A $p-n$ junction has a saturation current of $6.40 \mathrm{~mA}$. (a) At a temperature of $300 \mathrm{~K},$ what voltage is needed to produce a positive current of $40.0 \mathrm{~mA} ?$ (b) For a voltage equal to the negative of the value calculated in part (a), what is the negative current?

Kyle Godbey
Kyle Godbey
Numerade Educator
03:01

Problem 31

The saturation current for a junction diode is 10 microamps. Calculate the resistance of the diode at forward and reverse potential differences of $0.20 \mathrm{~V}$ when $T=17^{\circ} \mathrm{C}$.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:41

Problem 32

When a diatomic molecule undergoes a transition from the $l=2$ to the $l=1$ rotational state, a photon with wavelength $54.3 \mu \mathrm{m}$ is emitted. What is the moment of inertia of the molecule for an axis through its center of mass and perpendicular to the line connecting the nuclei?

Kyle Godbey
Kyle Godbey
Numerade Educator
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Problem 33

(a) The equilibrium separation of the two nuclei in an $\mathrm{NaCl}$ molecule is $0.24 \mathrm{nm}$. If the molecule is modeled as charges $+e$ and $-e$ separated by $0.24 \mathrm{nm},$ what is the electric dipole moment of the molecule (see Section 21.7$) ?(\mathrm{~b})$ The measured electric dipole mo- ment of an $\mathrm{NaCl}$ molecule is $3.0 \times 10^{-29} \mathrm{C} \cdot \mathrm{m} .$ If this dipole moment arises from point charges $+q$ and $-q$ separated by $0.24 \mathrm{nm},$ what is $q$ ? (c) A definition of the fractional ionic character of the bond is $q / e$. If the sodium atom has charge $+e$ and the chlorine atom has charge $-e$, the fractional ionic character would be equal to 1 . What is the actual fractional ionic character for the bond in $\mathrm{NaCl} ?$ (d) The equilibrium distance between nuclei in the hydrogen iodide (HI) molecule is $0.16 \mathrm{nm},$ and the measured electric dipole moment of the molecule is $1.5 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$. What is the fractional ionic character for the bond in HI? How does your answer compare to that for $\mathrm{NaCl}$ calculated in part (c)? Discuss reasons for the difference in these results.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
08:47

Problem 34

When a NaF molecule makes a transition from the $l=3$ to the $l=2$ rotational level with no change in vibrational quantum number or electronic state, a photon with wavelength $3.83 \mathrm{~mm}$ is emitted. A sodium atom has mass $3.82 \times 10^{-26} \mathrm{~kg},$ and a fluorine atom has mass $3.15 \times 10^{-26} \mathrm{~kg} .$ Calculate the equilibrium separation between the nuclei in a NaF molecule. How does your answer compare with the value for $\mathrm{NaCl}$ given in Section $42.1 ?$ Is this result reasonable? Explain.

Kyle Godbey
Kyle Godbey
Numerade Educator
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Problem 35

Consider a gas of diatomic molecules (moment of inertia $I$ ) at an absolute temperature $T$. If $E_{\mathrm{g}}$ is a ground-state energy and $E_{\mathrm{ex}}$ is the energy of an excited state, then the Maxwell-Boltzmann distribution (see Section 39.4 ) predicts that the ratio of the numbers of molecules in the two states is $n_{\mathrm{ex}} / n_{\mathrm{g}}=e^{-\left(E_{\mathrm{ex}}-E_{\mathrm{g}}\right) / k T}$. (a) Explain why the ratio of the number of molecules in the $l$ th rotational energy level to the number of molecules in the ground-state $(l=0)$ rotational level is
$$
\frac{n_{l}}{n_{0}}=(2 l+1) e^{-\left[l(l+1) \hbar^{2}\right] / 2 I k T}
$$
(b) Determine the ratio $n_{l} / n_{0}$ for a gas of CO molecules at $300 \mathrm{~K}$ for (i) $l=1 ;$ (ii) $l=2 ;$ (iii) $l=10 ;$ (iv) $l=20 ;$ (v) $l=50$. The moment of inertia of the CO molecule is given in Example 42.2 (Section 42.2). (c) Your results in part (b) show that as $l$ is increased, the ratio $n_{l} / n_{0}$ first increases and then decreases. Explain why.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:58

Problem 36

Part (a) of Problem 42.35 gives an equation for the number of diatomic molecules in the $l$ th rotational level to the number in the ground-state rotational level. (a) Derive an expression for the value of $l$ for which this ratio is the largest. (b) For the CO molecule at $T=300 \mathrm{~K},$ for what value of $l$ is this ratio a maximum? (The moment of inertia of the CO molecule is given in Example $42.2 .$ )

Keshav Singh
Keshav Singh
Numerade Educator
03:11

Problem 37

Semiconductor devices can radiate heat up to a maximum power rating. If the device dissipates more power than that limit, it will heat up and explode, melt, or simply break down. A typical power rating for a diode is $500 \mathrm{~mW}$. The power dissipated by any two-terminal device is the product of the current through it and the voltage across it. (a) Use Eq. (42.22) to estimate the power dissipated by a diode when it is forward biased at $V=0.6 \mathrm{~V}$ at a temperature of $300 \mathrm{~K}$. The saturation current is $1.2 \times 10^{-11} \mathrm{~A} .(\mathrm{b})$ Estimate to one significant figure the forward voltage at which the device will dissipate $500 \mathrm{~mW}$. (c) Then what is the current?

Keshav Singh
Keshav Singh
Numerade Educator
04:06

Problem 38

Our galaxy contains numerous molecular clouds, regions many light-years in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are $\mathrm{H}_{2}$, but a small fraction of the molecules are carbon monoxide (CO). Such a molecular cloud in the constellation Orion is shown in Fig. $\mathbf{P 4 2 . 3 8}$. The upper image was made with an ordinary visible-light telescope; the lower image shows the molecular cloud in Orion as imaged with a radio telescope tuned to a wavelength emitted by CO in a rotational transition. The different colors in the radio image indicate regions of the cloud that are moving either toward us (blue) or away from us (red) relative to the motion of the cloud as a whole, as determined by the Doppler shift of the radiation. (Since a molecular cloud has about 10,000 hydrogen molecules for each CO molecule, it might seem more reasonable to tune a radio telescope to emissions from $\mathrm{H}_{2}$ than to emissions from CO. Unfortunately, it turns out that the $\mathrm{H}_{2}$ molecules in molecular clouds do not radiate in either the radio or visible portions of the electromagnetic spectrum.) (a) Using the data in Example 42.2 (Section 42.2), calculate the energy and wavelength of the photon emitted by a CO molecule in an $l=1 \rightarrow l=0$ rotational transition. (b) As a rule, molecules in a gas at temperature $T$ will be found in a certain excited rotational energy level, provided the energy of that level is no higher than $k T$ (see Problem 42.35 ). Use this rule to explain why astronomers can detect radiation from CO in molecular clouds even though the typical temperature of a molecular cloud is a very low $20 \mathrm{~K}$.

Keshav Singh
Keshav Singh
Numerade Educator
03:21

Problem 39

The force constant for the internuclear force in a hydrogen mole- cule $\left(\mathrm{H}_{2}\right)$ is $k^{\prime}=576 \mathrm{~N} / \mathrm{m}$. A hydrogen atom has mass $1.67 \times 10^{-27} \mathrm{~kg} .$ Calculate the zero-point vibrational energy for $\mathrm{H}_{2}$ (that is, the vibrational energy the molecule has in the $n=0$ ground vibrational level). How does this energy compare in magnitude with the $\mathrm{H}_{2}$ bond energy of $-4.48 \mathrm{eV} ?$

Khaled Yasein
Khaled Yasein
Numerade Educator
04:24

Problem 40

When an OH molecule undergoes a transition from the $n=0$ to the $n=1$ vibrational level, its internal vibrational energy increases by $0.463 \mathrm{eV}$. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen atom is $2.66 \times 10^{-26} \mathrm{~kg},$ and the mass of a hydrogen atom is $\left.1.67 \times 10^{-27} \mathrm{~kg} .\right)$

Kyle Godbey
Kyle Godbey
Numerade Educator
03:31

Problem 41

The hydrogen iodide (HI) molecule has equilibrium separation $0.160 \mathrm{nm}$ and vibrational frequency $6.93 \times 10^{13} \mathrm{~Hz}$. The mass of a hydrogen atom is $1.67 \times 10^{-27} \mathrm{~kg},$ and the mass of an iodine atom is $2.11 \times 10^{-25} \mathrm{~kg} .$ (a) Calculate the moment of inertia of HI about a perpendicular axis through its center of mass. (b) Calculate the wavelength of the photon emitted in each of the following vibration-rotation transitions: (i) $n=1, l=1 \rightarrow n=0, l=0 ;$ (ii) $n=1, l=2 \rightarrow n=0$, $l=1 ;$ (iii) $n=2, l=2 \rightarrow n=1, l=3 .$

Mayukh Banik
Mayukh Banik
Numerade Educator
06:18

Problem 42

Suppose the hydrogen atom in HF (see the Bridging Problem for this chapter) is replaced by an atom of deuterium, an isotope of hydrogen with a mass of $3.34 \times 10^{-27} \mathrm{~kg} .$ The force constant is determined by the electron configuration, so it is the same as for the normal HF molecule. (a) What is the vibrational frequency of this molecule? (b) What wavelength of light corresponds to the energy difference between the $n=1$ and $n=0$ levels? In what region of the spectrum does this wavelength lie?

Kyle Godbey
Kyle Godbey
Numerade Educator
03:15

Problem 43

Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is $851 \mathrm{~kg} / \mathrm{m}^{3}$, and the mass of a single potassium atom is $6.49 \times 10^{-26} \mathrm{~kg}$.

Keshav Singh
Keshav Singh
Numerade Educator
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Problem 44

The one-dimensional calculation of Example 42.4 (Section 42.3) can be extended to three dimensions. For the threedimensional fec $\mathrm{NaCl}$ lattice, the result for the potential energy of a pair of $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$ ions due to the electrostatic interaction with all of the ions in the crystal is $U=-\alpha e^{2} / 4 \pi \epsilon_{0} r,$ where $\alpha=1.75$ is the Madelung constant. Another contribution to the potential energy is a repulsive interaction at small ionic separation $r$ due to overlap of the electron clouds. This contribution can be represented by $A / r^{8},$ where $A$ is a positive constant, so the expression for the total potential energy is
$$
U_{\mathrm{tot}}=-\frac{\alpha e^{2}}{4 \pi \epsilon_{0} r}+\frac{A}{r^{8}}
$$
(a) Let $r_{0}$ be the value of the ionic separation $r$ for which $U_{\text {tot }}$ is a minimum. Use this definition to find an equation that relates $r_{0}$ and $A,$ and use this to write $U_{\text {tot }}$ in terms of $r_{0}$. For $\mathrm{NaCl}, r_{0}=0.281 \mathrm{nm} .$ Obtain a numerical value (in electron volts) of $U_{\text {tot }}$ for $\mathrm{NaCl}$. (b) The quantity $-U_{\text {tot }}$ is the energy required to remove an $\mathrm{Na}^{+}$ ion and a $\mathrm{Cl}^{-}$ ion from the crystal. Forming a pair of neutral atoms from this pair of ions involves the release of $5.14 \mathrm{eV}$ (the ionization energy of $\mathrm{Na}$ ) and the expenditure of $3.61 \mathrm{eV}$ (the electron affinity of $\mathrm{Cl}$ ). Use the result of part (a) to calculate the energy required to remove a pair of neutral Na and $\mathrm{Cl}$ atoms from the crystal. The experimental value for this quantity is $6.39 \mathrm{eV} ;$ how well does your calculation agree?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:53

Problem 45

Metallic lithium has a bcc crystal structure. Each unit cell is a cube of side length $a=0.35 \mathrm{nm}$. (a) For a bcc lattice, what is the number of atoms per unit volume? Give your answer in terms of $a$. (Hint: How many atoms are there per unit cell?) (b) Use the result of part (a) to calculate the zero-temperature Fermi energy $E_{\mathrm{F} 0}$ for metallic lithium. Assume there is one free electron per atom.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
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Problem 46

To determine the equilibrium separation of the atoms in the $\mathrm{HCl}$ molecule, you measure the rotational spectrum of $\mathrm{HCl}$. You find that the spectrum contains these wavelengths (among others): (a) Use your mea$60.4 \mu \mathrm{m}, 69.0 \mu \mathrm{m}, 80.4 \mu \mathrm{m}, 96.4 \mu \mathrm{m},$ and $120.4 \mu \mathrm{m}$ sured wavelengths to find the moment of inertia of the $\mathrm{HCl}$ molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei. (b) The value of $l$ changes by ±1 in rotational transitions. What value of $l$ for the upper level of the transition gives rise to each of these wavelengths? (c) Use your result of part (a) to calculate the equilibrium separation of the atoms in the HCl molecule. The mass of a chlorine atom is $5.81 \times 10^{-26} \mathrm{~kg},$ and the mass of a hydrogen atom is $1.67 \times 10^{-27} \mathrm{~kg} .$ (d) What is the longest-wavelength line in the rotational spectrum of $\mathrm{HCl}$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:00

Problem 47

The table gives the occupation probabilities $f(E)$ as a function of the energy $E$ for a solid conductor at a fixed temperature $T$.
$$
\begin{array}{l|cccccc}
f(\boldsymbol{E}) & 0.064 & 0.173 & 0.390 & 0.661 & 0.856 & 0.950 \\
\hline \boldsymbol{E}(\mathrm{eV}) & 3.0 & 2.5 & 2.0 & 1.5 & 1.0 & 0.5
\end{array}
$$
To determine the Fermi energy of the solid material, you are asked to analyze this information in terms of the Fermi-Dirac distribution. (a) Graph the values in the table as $E$ versus $\ln \{[1 / f(E)]-1\}$. Find the slope and $y$ -intercept of the best-fit straight line for the data points when they are plotted this way. (b) Use your results of part (a) to calculate the temperature $T$ and the Fermi energy of the material.

Keshav Singh
Keshav Singh
Numerade Educator
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Problem 48

A $p-n$ junction is part of the control mechanism for a wind turbine that is used to generate electricity. The turbine has been malfunctioning, so you are running diagnostics. You can remotely change the bias voltage $V$ applied to the junction and measure the current through the junction. With a forward-bias voltage of $+5.00 \mathrm{mV}$, the current is $I_{\mathrm{f}}=0.407 \mathrm{~mA}$. With a reverse-bias voltage of $-5.00 \mathrm{mV}$ the current is $I_{\mathrm{r}}=-0.338 \mathrm{~mA}$. Assume that Eq. (42.22) accurately represents the current-voltage relationship for the junction, and use these two results to calculate the temperature $T$ and saturation current $I_{\mathrm{S}}$ for the junction.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
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Problem 49

The circuit shown in Fig. P42.49 is called a half-wave rectifier. The triangular symbol with the flat lower edge represents a diode, such that a forward bias drives the current downward. Use Eq. (42.22) to study the behavior of this circuit. Use a typical value for the saturation current: $I_{\mathrm{S}}=5 \times 10^{-13} \mathrm{~A} .$ Use a temperature of $300 \mathrm{~K}$. (a) Any predicted current less than $1 \mu \mathrm{A}$ can be considered a zero current. Determine the minimum voltage $V_{\text {out }}$ for which the current $I$ exceeds $1 \mu \mathrm{A} .$ (b) The current increases dramatically for diode voltages that exceed that value. At a given diode voltage $V_{\text {out }},$ the diode has an effective resistance given by $R_{\mathrm{D}}=V_{\text {out }} / I .$ Use Eq. (42.22) to estimate the diode voltage $V_{\text {out }}=V_{\max }$ for which $R_{\mathrm{D}}=100 \Omega$. (c) What is the current $I$ in that case? (d) If the resistance $R$ is greater than $10 \mathrm{k} \Omega$, then $R_{\mathrm{D}} \ll R$ for larger currents, and there is a negligible additional voltage drop across the diode. This effectively pins $V_{\text {out }}=V_{\max }$ for all input voltages $V_{\text {in }}>V_{\max }$. With this perspective, estimate how this circuit would respond to a sinusoidal input voltage with amplitude $5 \mathrm{~V}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 50

A $p-n$ junction includes $p$ -type silicon, with donor atom density $N_{\mathrm{D}}$, adjacent to $n$ -type silicon, with acceptor atom density $N_{\mathrm{A}}$. Near the junction, free electrons from the $n$ side have diffused into the $p$ side while free holes from the $p$ side have diffused into the $n$ side, leaving a "depletion region" of width $W$ where there are no free charge carriers (Fig. P42.50). The width of the depletion region on the $n$ side is $a_{n},$ and the width of the depletion region on the $p$ side is $a_{p} .$ Each depleted region has a constant charge density with magnitude equal to the corresponding donor atom density. Assume there is no net charge outside the depleted regions. Define an $x$ -axis pointing from the $p$ side toward the $n$ side with the origin at the center of the junction. An electric field $\overrightarrow{\boldsymbol{E}}=-E \hat{\imath}$ has developed in the depletion region, stabilizing the diffusion. (a) Use Gauss's law to determine the electric field on the $p$ side of the junction, for $-a_{p} \leq x \leq 0$. (Hint: Use a cylindrical Gaussian surface parallel to the $x$ -axis, with the left end outside the depletion region (where $\overrightarrow{\boldsymbol{E}}=0$ ) and the right side inside the region.) Note that the dielectric constant of silicon is $K=11.7 .$ (b) Determine the electric field on the $n$ side of the junction, for $0 \leq x \leq a_{n}$ (c) Use continuity at $x=0$ to determine a relationship between the depths $a_{p}$ and $a_{n}$ and in terms of $N_{\mathrm{A}}$ and $N_{\mathrm{D}}$. (d) The electric potential $V(x)$ may be obtained by integrating the electric field, as shown in Eq. (23.18). Determine $V(x)$ in the region $-a_{p} \leq x \leq 0$ using the convention that $V(x)=0$ for $x<-a_{p}$. (e) Similarly, determine $V(x)$ in the region $0 \leq x \leq a_{n} .$ (f) What is the "barrier potential" $V_{b}=V\left(a_{n}\right)-V\left(-a_{p}\right) ?(\mathrm{~g})$ The $p$ side is doped with boron atoms with density $N_{\mathrm{A}}=1.00 \times 10^{16} \mathrm{~cm}^{-3}$. The $n$ side is doped with arsenic atoms with density $N_{\mathrm{D}}=5.00 \times 10^{16} \mathrm{~cm}^{-3}$. The $n$ side depletion depth is $a_{n}=55.0 \mathrm{nm}$. What is the $p$ side depletion depth $a_{p} ?$ (h) What is the peak magnitude of the electric field, at $x=0 ?$ (i) What is the value of the barrier potential $V_{\mathrm{b}}$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 51

Consider the $\mathrm{CO}_{2}$ molecule shown in Fig. 42.10c. The oxygen molecules have mass $M_{\mathrm{O}}$ and the carbon atom has mass $M_{\mathrm{C}}$. Parameterize the positions of the left oxygen atom, the carbon atom, and the right oxygen atom using $x_{1}, x_{2},$ and $x_{3}$ as the respective rightward deviations from equilibrium. Treat the bonds as Hooke's-law springs with common spring constant $k^{\prime}$. (a) Use Newton's second law to obtain expressions for $M_{i} \ddot{x}_{i}=M_{i} d^{2} x / d t^{2}$ in each case $i=1,2,3,$ where $M_{1,2,3}=\left(M_{\mathrm{O}}, M_{\mathrm{C}}, M_{\mathrm{O}}\right) .($ Note: We represent time derivatives using dots.) Assume $X_{\mathrm{C}}=0$ at $t=0 .$ (b) To ascertain the motion of the asymmetric stretching mode, set $x_{1}=x_{3} \equiv X_{\mathrm{O}}$ and set $x_{2} \equiv X_{\mathrm{C}} .$ Write the two independent equations that remain from your previous result. (c) Eliminate the sum $X_{\mathrm{O}}+X_{\mathrm{C}}$ from your equations. Use what remains to ascertain $X_{\mathrm{C}}$ in terms of $X_{O}$. (d) Substitute your expression for $X_{C}$ into your equation for $X_{\mathrm{O}}$ to derive a harmonic oscillator equation $M_{\mathrm{eff}} X_{\mathrm{O}}=-k X_{\mathrm{O}}$. What is $M_{\text {eff }} ?$ (e) This equation has the solution $X_{\mathrm{O}}(t)=A \cos (\omega t) .$ What is the angular frequency $\omega ?$ (f) Using the experimentally determined spring constant $k^{\prime}=1860 \mathrm{~N} / \mathrm{m}$ and the atomic masses $M_{\mathrm{C}}=12 \mathrm{u}$ and $M_{\mathrm{O}}=16 \mathrm{u}$, where $\mathrm{u}=1.6605 \times 10^{-27} \mathrm{~kg},$ to determine the oscillation frequency $f=\omega / 2 \pi$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 52

A colorless, odorless gas in a tank needs to be identified. The gas is nitrogen $\left(\mathrm{N}_{2}\right),$ carbon monoxide (CO), or nitric oxide (NO). (See Fig. P42.52.) White light is passed through a sample to obtain an absorption spectrum. Among others, the following wavelengths present dark bands on the spectrum: $\lambda=4.2680 \mu \mathrm{m}, 4.2753 \mu \mathrm{m}, 4.2826 \mu \mathrm{m}, 4.2972 \mu \mathrm{m},$ and $4.3046 \mu \mathrm{m} .$ These represent transitions from the lowest rotational states. (a) What photon energies correspond to these wavelengths? Specify your answers to five significant figures. Note that the speed of light is $2.9979 \times 10^{8} \mathrm{~m} / \mathrm{s}$ and $h=4.1357 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s}$ (b) Which one of the absorbed energies corresponds to transitions from an $l=0$ state of the gas? (Hint: Use Eq. (42.9) along with the transition rules as a guide.) (c) Which one of the absorbed energies corresponds to a transition $t o$ an $l=0$ state? (d) Determine the value of $\hbar \sqrt{k^{\prime} / m_{\mathrm{r}}}$, where $k^{\prime}$ is the effective spring constant of the molecule and $m_{\mathrm{r}}$ is its reduced mass. (Hint: Use the previous two results and Eq. (42.9).) (e) Determine the molecule's moment of inertia $I$. (f) The atomic masses and the bond

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:18

Problem 53

Consider a system of $N$ free electrons within a volume $V$. Even at absolute zero, such a system exerts a pressure $p$ on its surroundings due to the motion of the electrons. To calculate this pressure, imagine that the volume increases by a small amount $d V$. The electrons will do an amount of work $p d V$ on their surroundings, which means that the total energy $E_{\mathrm{tot}}$ of the electrons will change by an amount $d E_{\mathrm{tot}}=-p d V .$ Hence $p=-d E_{\mathrm{tot}} / d V .$ (a) Show that the pressure of the electrons at absolute zero is
$$
p=\frac{3^{2 / 3} \pi^{4 / 3} \hbar^{2}}{5 m}\left(\frac{N}{V}\right)^{5 / 3}
$$
(b) Evaluate this pressure for copper, which has a free-electron concentration of $8.45 \times 10^{28} \mathrm{~m}^{-3}$. Express your result in pascals and in atmospheres. (c) The pressure you found in part (b) is extremely high. Why, then, don't the electrons in a piece of copper simply explode out of the metal?

Mayukh Banik
Mayukh Banik
Numerade Educator
02:40

Problem 54

When the pressure $p$ on a material increases by an amount $\Delta p,$ the volume of the material will change from $V$ to $V+\Delta V$, where $\Delta V$ is negative. The bulk modulus $B$ of the material is defined to be the ratio of the pressure change $\Delta p$ to the absolute value $|\Delta V / V|$ of the fractional volume change. The greater the bulk modulus, the greater the pressure increase required for a given fractional volume change, and the more incompressible the material(see Section 11.4). Since $\Delta V<0,$ the bulk modulus can be written as $B=-\Delta p /\left(\Delta V / V_{0}\right) .$ In the limit that the pressure and volume changes are very small, this becomes
$$
B=-V \frac{d p}{d V}
$$
(a) Use the result of Challenge Problem 42.53 to show that the bulk modulus for a system of $N$ free electrons in a volume $V$ at low temperatures is $B=\frac{5}{3} p$. (Hint: The quantity $p$ in the expression $B=$ $-V(d p / d V)$ is the external pressure on the system. Can you explain why this is equal to the internal pressure of the system itself, as found in Challenge Problem $42.53 ?)(\mathrm{b})$ Evaluate the bulk modulus for the electrons in copper, which has a free-electron concentration of $8.45 \times 10^{28} \mathrm{~m}^{-3}$. Express your result in pascals. (c) The actual bulk modulus of copper is $1.4 \times 10^{11} \mathrm{~Pa}$. Based on your result in part (b), what fraction of this is due to the free electrons in copper? (This result shows that the free electrons in a metal play a major role in making the metal resistant to compression.) What do you think is responsible for the remaining fraction of the bulk modulus?

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:13

Problem 55

In the discussion of free electrons in Section $42.5,$ we assumed that we could ignore the effects of relativity. This is not a safe assumption if the Fermi energy is greater than about $\frac{1}{100} m c^{2}$ (that is, more than about $1 \%$ of the rest energy of an electron). (a) Assume that the Fermi energy at absolute zero, as given by Eq. (42.19), is equal to $\frac{1}{100} m c^{2}$. Show that the electron concentration is
$$
\frac{N}{V}=\frac{2^{3 / 2} m^{3} c^{3}}{3000 \pi^{2} \hbar^{3}}
$$
and determine the numerical value of $N / V$. (b) Is it a good approximation to ignore relativistic effects for electrons in a metal such as copper, for which the electron concentration is $8.45 \times 10^{28} \mathrm{~m}^{-3} ?$ Explain. (c) A white dwarf star is what is left behind by a star like the sun after it has ceased to produce energy by nuclear reactions. (Our own sun will become a white dwarf star in another $8 \times 10^{9}$ years or so.) A typical white dwarf has mass $2 \times 10^{30} \mathrm{~kg}$ (comparable to the sun) and radius $6000 \mathrm{~km}$ (comparable to that of the earth). The gravitational attraction of different parts of the white dwarf for each other tends to compress the star; what prevents it from compressing is the pressure of free electrons within the star (see Challenge Problem 42.53). Use both of the following assumptions to estimate the electron concentration within a typical white dwarf star: (i) the white dwarf star is made of carbon, which has a mass per atom of $1.99 \times 10^{-26} \mathrm{~kg} ;$ and (ii) all six of the electrons from each carbon atom are able to move freely throughout the star. (d) Is it a good approximation to ignore relativistic effects in the structure of a white dwarf star? Explain.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
03:22

Problem 56

The sensitivity of a diode thermometer depends on how much the voltage changes for a given temperature change, with the current remaining constant. What is the sensitivity for this diode thermometer, operated at $100 \mathrm{~mA},$ for a temperature change from $25^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C} ?(\mathrm{a})+0.2 \mathrm{mV} /{ }^{\circ} \mathrm{C} ;(\mathrm{b})+2.0 \mathrm{mV} /{ }^{\circ} \mathrm{C} ;(\mathrm{c})-0.2 \mathrm{mV} /{ }^{\circ} \mathrm{C}$ (d) $-2.0 \mathrm{mV} /{ }^{\circ} \mathrm{C}$.

Kyle Godbey
Kyle Godbey
Numerade Educator
04:02

Problem 57

Which statement best explains the temperature dependence of the current-voltage characteristics that the graph shows? At higher temperatures: (a) The band gap is larger, so the electron-hole pairs have more energy, which causes the current at a given voltage to be larger. (b) More electrons can move to the conduction band, which causes the current at a given voltage to be larger. (c) All of the electrons in the valence band move to the conduction band, and the diode behaves like a metal and follows Ohm's law. (d) The acceptor and donor impurity atoms are free to move through the material, which causes the current at a given voltage to be larger.

Kai Chen
Kai Chen
Princeton University
02:28

Problem 58

If the voltage rather than the current is kept constant, what happens as the temperature increases from $25^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C} ?$ (a) At first the current increases, then it decreases. (b) The current increases. (c) The current decreases, eventually approaching zero. (d) The current does not change unless the voltage also changes.

Kyle Godbey
Kyle Godbey
Numerade Educator