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College Physics

Hugh D. Young Philip W. Adams

Chapter 29

Atoms, Molecules, and Solids - all with Video Answers

Educators


Chapter Questions

01:40

Problem 1

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 $l$ for this electron?

Nathan Nowack
Nathan Nowack
Numerade Educator
05:43

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 $L$ 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?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
04:24

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.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
11:06

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.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
06:16

Problem 5

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:04

Problem 6

What is the ratio of the number of different $3 d$ states of the hydrogen atom to the number of $5 d$ states of the atom?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:27

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 $\overrightarrow{\boldsymbol{L}}$ and the $z$ axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between $\vec{L}$ and the $z$ axis, and what is that angle?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:57

Problem 8

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\boldsymbol{L}$ 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 $l ?$

Kyle Godbey
Kyle Godbey
Numerade Educator
03:48

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.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:07

Problem 10

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:34

Problem 11

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:42

Problem 12

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:32

Problem 13

(a) Write out the ground-state electron configuration $\left(1 s^{2} 2 s^{2},\right.$ and so on) 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 configuration for this element.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:58

Problem 14

(a) Write out the ground-state electron configuration $\left(1 s^{2} 2 s^{2},\right.$ and so on 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 configuration of this element. (c) Use the procedure of part (b) to predict what element of nextlarger $Z$ 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.

Zachary Warner
Zachary Warner
Numerade Educator
02:02

Problem 15

Write out the electron configuration $\left(1 s^{2} 2 s^{2},\right.$ and so on $)$ for $\mathrm{Ne}$ 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.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:56

Problem 16

In terms of $n$ and $\hbar$, derive an expression for the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom having a principal quantum number $n$. Compare this expression with the value of $n \hbar$ postulated in the Bohr model. Which is larger?

Kyle Godbey
Kyle Godbey
Numerade Educator
01:45

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?

Muhammad Ahsan
Muhammad Ahsan
Numerade Educator
01:34

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 $K_{\beta}$ X-rays of tungsten.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:54

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?

Kyle Godbey
Kyle Godbey
Numerade Educator
01:19

Problem 20

(a) A molecule decreases its vibrational energy by $0.250 \mathrm{eV}$ 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 $8.50 \mathrm{eV}$ 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 $3.20 \times 10^{-3} \mathrm{eV}$ 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?

Kai Chen
Kai Chen
Princeton University
04:31

Problem 21

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) What is the ratio of this energy to the ionization energy of the hydrogen atom?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:30

Problem 22

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:59

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 $\mathrm{KBr}$ crystal. (b) Compare the spacing for $\mathrm{KBr}$ with the spacing for $\mathrm{NaCl}$ (See the preceding problem.) Is the relationship between these two values qualitatively what you would expect? Explain your reasoning.

Kyle Godbey
Kyle Godbey
Numerade Educator
05:29

Problem 24

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. (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.

Kyle Godbey
Kyle Godbey
Numerade Educator
02:19

Problem 25

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
03:27

Problem 26

Sketch a qualitative (no numbers) graph of the resistance as a function of temperature between $100 \mathrm{~K}$ and $0 \mathrm{~K}$ for (a) an ordinary conductor, such as $\mathrm{Cu} ;$ (b) a superconductor with a transition temperature of $30 \mathrm{~K}$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:44

Problem 27

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 $80 \mathrm{eV}$. Why do these numbers keep increasing?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:56

Problem 28

The $z$ component of the angular momentum of an atom, $L_{z}$, is measured as a function of the angle that the angular momentum vector $\vec{L}$ makes with the positive $z$ axis, $\theta .$ The resulting data are given in the table, where the angular momentum is measured in units of $\hbar$.
$$
\begin{array}{cc}
\hline \boldsymbol{\theta}\left({ }^{\circ}\right) & \boldsymbol{L}_{z}(\hbar) \\
\hline 63.7 & 1.97 \\
79.2 & 1.05 \\
90.4 & -0.03 \\
102 & -0.95 \\
118 & -2.02 \\
\hline
\end{array}
$$
Make a plot of $L_{z}$ as a function of $\cos \theta$. Using a linear "best fit" to the data, determine (a) the magnitude of the angular momentum vector in units of $\hbar$ and (b) the corresponding angular momentum quantum number.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:28

Problem 29

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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:49

Problem 30

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is $4.48 \mathrm{eV}$. 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?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:14

Problem 31

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 electronvolts) 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?)

Kyle Godbey
Kyle Godbey
Numerade Educator
02:01

Problem 32

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:50

Problem 33

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.

Kyle Godbey
Kyle Godbey
Numerade Educator
00:58

Problem 34

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

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:57

Problem 35

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?

Kyle Godbey
Kyle Godbey
Numerade Educator
03:21

Problem 36

The energy of the van der Waals bond, which is responsible for a number of the characteristics of water, is about $0.50 \mathrm{eV}$. (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?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:31

Problem 37

(a) What is the lowest possible energy (in electronvolts) 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 $\hbar$ ) 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)?

Kyle Godbey
Kyle Godbey
Numerade Educator
05:13

Problem 38

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 $\sqrt{L_{x}^{2}+L_{y}^{2}}=\hbar \sqrt{3}$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:44

Problem 39

Atoms of unusual size. In photosynthesis in plants, light is absorbed in light-harvesting complexes consisting of protein and pigment molecules. The energy absorbed is then transported to a specialized complex called the reaction center. Scientists believe that quantum mechanical effects may play an important role in this energy transfer. In a recent experiment, researchers used rubidium atoms cooled to a very low temperature to study a similar energy-transfer process in the laboratory. Laser light was used to excite an electron in each atom to a very highly excited state (large $n$ ). The excited electron behaves very much like the single electron in a hydrogen atom, with an effective $Z=1,$ but because $n$ is so large, the excited electron is quite far from the atomic nucleus, with an orbital radius of approximately $1 \mu \mathrm{m},$ and is weakly bound. Using these so-called Rydberg atoms, the researchers were able to study how energy is transported from one atom to the next in a process that may be a helpful model in understanding energy transport in photosynthesis.
In the Bohr model, what is the value of the quantum number $n$ at which the excited electron is at a radius of $1 \mu \mathrm{m} ?$
A. 140
B. 400
C. 20
D. 81

Kyle Godbey
Kyle Godbey
Numerade Educator
03:22

Problem 40

Atoms of unusual size. In photosynthesis in plants, light is absorbed in light-harvesting complexes consisting of protein and pigment molecules. The energy absorbed is then transported to a specialized complex called the reaction center. Scientists believe that quantum mechanical effects may play an important role in this energy transfer. In a recent experiment, researchers used rubidium atoms cooled to a very low temperature to study a similar energy-transfer process in the laboratory. Laser light was used to excite an electron in each atom to a very highly excited state (large $n$ ). The excited electron behaves very much like the single electron in a hydrogen atom, with an effective $Z=1,$ but because $n$ is so large, the excited electron is quite far from the atomic nucleus, with an orbital radius of approximately $1 \mu \mathrm{m},$ and is weakly bound. Using these so-called Rydberg atoms, the researchers were able to study how energy is transported from one atom to the next in a process that may be a helpful model in understanding energy transport in photosynthesis.
The size of a highly excited atom can be taken to be the diameter of the orbit of the excited electron. If the researchers want to do the experiment with the rubidium atoms in a gas, with atoms separated by 10 times their size, the density of the atoms should be about
A. $10^{5}$ atoms $/ \mathrm{cm}^{3}$.
B. $10^{8}$ atoms $/ \mathrm{cm}^{3}$.
C. $10^{11}$ atoms $/ \mathrm{cm}^{3}$.
D. $10^{21}$ atoms $/ \mathrm{cm}^{3}$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:47

Problem 41

Atoms of unusual size. In photosynthesis in plants, light is absorbed in light-harvesting complexes consisting of protein and pigment molecules. The energy absorbed is then transported to a specialized complex called the reaction center. Scientists believe that quantum mechanical effects may play an important role in this energy transfer. In a recent experiment, researchers used rubidium atoms cooled to a very low temperature to study a similar energy-transfer process in the laboratory. Laser light was used to excite an electron in each atom to a very highly excited state (large $n$ ). The excited electron behaves very much like the single electron in a hydrogen atom, with an effective $Z=1,$ but because $n$ is so large, the excited electron is quite far from the atomic nucleus, with an orbital radius of approximately $1 \mu \mathrm{m},$ and is weakly bound. Using these so-called Rydberg atoms, the researchers were able to study how energy is transported from one atom to the next in a process that may be a helpful model in understanding energy transport in photosynthesis.
Assume that the researchers place an atom in an $n=100, l=2$ state. What magnitude of the orbital angular momentum is associated with this state?
A. $\sqrt{2} \hbar$
B. $\sqrt{6} \hbar$
C. $\sqrt{100} \hbar$
D. $\sqrt{200} \hbar$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:00

Problem 42

Atoms of unusual size. In photosynthesis in plants, light is absorbed in light-harvesting complexes consisting of protein and pigment molecules. The energy absorbed is then transported to a specialized complex called the reaction center. Scientists believe that quantum mechanical effects may play an important role in this energy transfer. In a recent experiment, researchers used rubidium atoms cooled to a very low temperature to study a similar energy-transfer process in the laboratory. Laser light was used to excite an electron in each atom to a very highly excited state (large $n$ ). The excited electron behaves very much like the single electron in a hydrogen atom, with an effective $Z=1,$ but because $n$ is so large, the excited electron is quite far from the atomic nucleus, with an orbital radius of approximately $1 \mu \mathrm{m},$ and is weakly bound. Using these so-called Rydberg atoms, the researchers were able to study how energy is transported from one atom to the next in a process that may be a helpful model in understanding energy transport in photosynthesis.
How many different possible electron states are there in the $n=100, l=2$ subshell?
A. 2
B. 100
C. 10,000
D. 10

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:21

Problem 42

How many different possible electron states are there in the $n=100, l=2$ subshell?
A. 2
B. 100
C. 10,000
D. 10

Nathan Silvano
Nathan Silvano
Numerade Educator