College Physics

Hugh D. Young

Chapter 29

Atoms, Molecules, and Solids - all with Video Answers

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Chapter Questions

Problem 1

$\cdot$ The orbital angular momentum of an electron has a magnitude of $4.716 \times 10^{-34} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}$ . What is the angular-momentum quantum number $/$ for this electron?

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Problem 2

Consider states with $l = 3 .$ (a) In units of $\hbar ,$ what is the largest possible value of $L _ { z } ?$ (b) In units of $\hbar ,$ what is the value of $L ?$ Which is larger, $L$ or the maximum possible $L _ { z } ?$ (c) Assume a model in which $\vec { \boldsymbol { L } }$ is described as a classical vector. For each allowed value of $L _ { z } ,$ what angle does the vector $\vec { L }$ make with the $+ z$ axis?

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Problem 3

An electron is in the hydrogen atom with $n = 3 .$ (a) Find the possible values of $L$ and $L _ { z }$ for this electron, in units of $\hbar .$ (b) For each value of $L ,$ find all the possible angles between $L$ and the $z$ axis.

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Problem 4

An electron is in the hydrogen atom with $n = 5 .$ (a) Find the possible values of $L$ and $L _ { z }$ for this electron, in units of \hbar. (b) For each value of $L ,$ find all the possible angles between $L$ and the $z$ axis. (d) What are the maximum and minimum values of the magnitude of the angle between $L$ and the $z$ axis?

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Problem 5

Consider an electron in the $N$ shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of $\hbar$ and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hbar$ and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hbar$ and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the $z$ direction to its orbital angular momentum in the $z$ direction?

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Problem 6

(a) How many different 3$d$ states does hydrogen have? Make a list showing all of them. (b) How many different 3f states does it have?

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Problem 7

(a) How many different 5$g$ states does hydrogen have? (b) Which of the states in part (a) has the largest angle between $\vec { L }$ and the $z$ axis and what is that angle? (c) Which of the states in part (a) has the smallest angle between $L$ and the $z$ axis, and what is that angle?

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Problem 8

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overline { L }$ and the $z$ axis is $\theta = 26.6 ^ { \circ } .$ (See Figure $29.2 . )$ If this is the smallest angle for this particular value of the angular momentum quantum number $l ,$ what is $l$ ?

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Problem 9

Make a list of the four quantum numbers $n , l , m _ { l } ,$ and $s$ for each of the 12 electrons in the ground state of the magnesium atom.

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Problem 10

(a) List the different possible combinations of quantum numbers $/$ and $m _ { l }$ for the $n = 5$ shell. (b) How many electrons can be placed in the $n = 5$ shell?

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Problem 11

For bromine $( Z = 35 ) ,$ make a list of the number of electrons in each subshell $( 1 s , 2 s , 2 p ,$ etc. $) .$

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Problem 12

(a) Write out the electron configuration $\left( 1 s ^ { 2 } 2 s ^ { 2 } ,$ etc. \right$)$ for Li and $\mathrm { Na }$ . (b) How many electrons does each of these atoms have in its outer shell?

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Problem 13

(a) Write out the ground-state electron configuration $\left( 1 s ^ { 2 } , 2 s ^ { 2 } ,$ etc. \right$)$ for the carbon atom. (b) What element of next-larger $Z$ has chemical properties similar to those of carbon? (See Example $29.3 . )$ Give the ground-state electron con- figuration for this element.

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Problem 14

(a) Write out the ground-state electron configuration $\left( 1 s ^ { 2 } 2 s ^ { 2 } ,$ etc. \right$)$ for the beryllium atom. (b) What element of next-larger $Z$ has chemical properties similar to those of beryllium? (See Example $29.3 .$ ) Give the ground-state electron con- figuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger $Z$ than in (b) will have
chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

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Problem 15

Write out the electron configuration $\left( 1 s ^ { 2 } 2 s ^ { 2 } ,$ etc. \right$)$ for Ne, Ar, and Kr. (b) How many electrons does each of these atoms have in its outer shell? (c) Predict the chemical behavior of these three atoms. Explain your reasoning.

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Problem 16

Calculate, in units of $\hbar ,$ the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2,20 ,$ and 200 . Compare each with the value of $n \hbar$ postulated in the Bohr model. What trend do you see?

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Problem 17

(a) What is the orbital angular momentum of any s-subshell electron? (b) If we try to model the atom classically as a scaled-down version of a solar system, with the electrons orbiting the nucleus the way the planets orbit the sun, what does the result in part (a) tell us would be the speed of an s-subshell electron? Is this result physically possible? What would happen to an electron with that speed?

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Problem 18

The energies for an electron in the $K , L ,$ and $M$ shells of the tungsten atom are $- 69,500 \mathrm { eV } , - 12,000 \mathrm { eV } ,$ and $- 2200 \mathrm { eV }$ , respectively. Calculate the wavelengths of the $K _ { \alpha }$ and $K _ { \beta } \mathrm { x }$ rays of tungsten.

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Problem 19

If the energy of the $\mathrm { H } _ { 2 }$ covalent bond is $- 4.48 \mathrm { eV } ,$ what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?

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Problem 20

(a) A molecule decreases its vibrational energy by 0.250 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (b) An atom decreases its energy by 8.50 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (c) A molecule decreases its rotational energy by $3.20 \times$ $10 ^ { - 3 }$ eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electro- magnetic spectrum does that wavelength of light lie?

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Problem 21

An ionic bond. (a) Calculate the electric potential energy for a $\mathrm { K } ^ { + }$ ion and a 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 eV. Atomic bromine has an electron affinity of 3.5 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.

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Problem 22

The spacing of adjacent atoms in a NaCl crystal is 0.282$\mathrm { nm }$ and the masses of the atoms are $3.82 \times 10 ^ { - 26 } \mathrm { kg } ( \mathrm { Na } )$ and $5.89 \times 10 ^ { - 26 } \mathrm { kg } ( \mathrm { Cl } ) .$ Use this information to calculate the density of sodium chloride.

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Problem 23

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 potassium is $6.49 \times 10 ^ { - 26 } \mathrm { kg } ,$ and that of bromine is $1.33 \times$ $10 ^ { - 25 } \mathrm { kg } .$ (a) Calculate the average spacing between adjacent atoms in a KBr crystal. (b) Compare the spacing for KBr with the spacing for NaCl. (See previous problem.) Is the relation between these two values qualitatively what you would expect? Explain your reasoning.

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Problem 24

Look at the graph for a diode in Figure 29.23$( \mathrm { b } )$ in the text. Why does this graph go below the horizontal axis to the left of the origin? Would it do this if the diode were replaced by an ordinary resistor?

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Problem 25

The gap between valence and conduction bands in diamond is 5.47 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. (Hint: Will photons of visible light that strike a diamond be absorbed or transmitted?) (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

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Problem 26

A variable dc power supply having reversible polarity is connected to a diode having a $p - n$ junction as shown in Figure 29.28 Starting with the power supply's polarity as shown in gradually decreased to zero gradually decreased to zero and then gradually increased in the reverse direction. (a) Sketch a graph of the reading in the ammeter as a function of the potential difference $V$ across the power supply. Make sign differences clear. (b) Suppose now that the terminals of the diode are reversed and the same procedure is followed. Sketch a graph of the reading in the ammeter as a function of the potential difference $V$ across the power supply. Make sign differences clear. (c) Suppose now that the diode is replaced by an ordinary resistor and the same procedure is followed. Sketch a graph of the reading in the ammeter as a function of the potential difference $V$ across the power supply. Make sign differences clear. (d) Explain the reasons for the differences between the graphsin (i) parts (a) and (b), (ii) parts (a) and (c).

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Problem 27

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?

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Problem 28

Sketch a qualitative (no numbers) graph of the resistance as a function of temperature for (a) an ordinary conductor, such as Cu, including temperatures approaching $0 \mathrm { K } ;$ (b) a superconductor; include temperatures above and below the critical temperature, and let the temperature approach 0$\mathrm { K }$ .

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Problem 29

For magnesium, the first ionization potential is 7.6$\mathrm { eV }$ ; the second (the additional energy required to remove a second electron) is almost twice this, $15 \mathrm { eV } ,$ and the third ionization potential is much larger, about 80 eV. Why do these numbers keep increasing?

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Problem 30

Failure of the classical model. (a) What is the spin angular momentum of an electron along the $z$ direction, in SI units? (b) In the rest of this problem, we shall try to understand the behavior of an electron by modeling it as a classical ball of matter. We shall also use the nonrelativistic formulas, although a thorough treatment would require use of the more complicated relativistic ones. If we model this electron classically as a solid uniform sphere of diameter $1.0 \times 10 ^ { - 15 } \mathrm { m } ,$ what would be its angular velocity in $\mathrm { rad } / \mathrm { s } ?$ (c) What would be the speed of the surface of this classical sphere? Is there anything suspicious about this result? (d) In part (c), the result cannot be correct because the surface of the electron cannot move faster than the speed of light. Suppose we try to "correct" this model by assuming that the surface is moving at the speed of light the upper limit of its speed. What would have to be the diameter of an electron in that case? Given that atomic nuclei are about $10 \times 10 ^ { - 15 } \mathrm { m }$ in diameter, does this result seem plausible for the size of an electron? (e) Would the bizarre results in parts (c) and (d) be affected appreciably if we instead modeled the electron as a hollow spherical shell instead of a solid?
Why? Notice that we encounter results that contradict observation when we try to think of the electron simply as a spinning sphere. We simply cannot understand the behavior of an electron (and other subatomic particles) by using our ordinary classical models.

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Problem 31

An electron has spin angular momentum and orbital angular momentum. For the 3$d$ electron in scandium, what percent of its total orbital angular momentum is its spin angular momentum in the $z$ direction?

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Problem 32

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is 4.48 eV. In the gas phase (treated as an ideal gas), at what temperature is the average translational kinetic energy of a molecule equal to this energy?

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Problem 33

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. (Hint: Will visible light that strikes silicon be transmitted or absorbed?

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Problem 34

Use the electron configurations of He, Ne, and Ar to explain why these atoms normally do not combine chemically with other atoms.

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Problem 35

Use the electron configurations of $\mathrm { H }$ and $\mathrm { O }$ to explain why these atoms combine chemically in a two-to-one ratio to form water.

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Problem 36

Use the electron configurations of Si and O to explain why these atoms combine chemically in a one-to-two ratio to form sand.

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Problem 37

Consider an electron in hydrogen having total energy $- 0.5440 \mathrm { eV } .$ (a) What are the possible values of its orbital angular momentum (in terms of $\hbar$ ? (b) What wavelength of light would it take to excite this electron to the next higher shell? Is this photon visible to humans?

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Problem 38

The energy of the van der Waals bond, which is responsible for a number of the characteristics of water, is about 0.50 eV. (a) At what temperature would the average translational kinetic energy of water molecules be equal to this energy? (b) At that temperature, would water be liquid or gas? Under ordinary everyday conditions, do van der Waals forces play a role in the behavior of water?

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Problem 39

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is $\sqrt { 12 } \hbar ?$ (b) What are the largest and smallest values of the $z$ component of the orbital angular momentum (in terms of $\hat { h }$ ) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of $\hbar$ ) for the electron in part (a)?(d) What are the largest and smallest values of the orbital angular momentum (in terms of $\hbar$ ) for an electron in the $M$ shell of hydrogen?

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Problem 40

An electron in hydrogen is in the 5$f$ state. (a) Find the largest possible value of the $z$ component of its angular momentum. (b) Show that for the electron in part (a), the corresponding $x$ and $y$ components of its angular momentum satisfy the equation $\sqrt { L _ { x } ^ { 2 } + L _ { y } ^ { 2 } } = \hbar \sqrt { 3 }$

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