Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 39 Quantum Mechanics of Atoms - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

00:22

Problem 1

(I) For $n=7,$ what values can $\ell$ have?

João Gabriel Alencar Caribé

Numerade Educator

01:19

Problem 2

(I) For $n=6, \quad \ell=3,$ what are the possible values of $m_{\ell}$ and $m_{s} ?$

Rahul Nikhar

Numerade Educator

05:27

Problem 3

(I) How many different states are possible for an electron whose principal quantum number is $n=5 ?$ Write down the quantum numbers for each state.

João Gabriel Alencar Caribé

Numerade Educator

01:35

Problem 4

(I) If a hydrogen atom has $m_{\ell}=-4,$ what are the possible values of $n, \ell$, and $m_{s}$ ?

Anand Jangid

Numerade Educator

00:55

Problem 5

(I) A hydrogen atom has $\ell=5 .$ What are the possible values for $n, m_{\ell},$ and $m_{s} ?$

João Gabriel Alencar Caribé

Numerade Educator

01:26

Problem 6

(I) Calculate the magnitude of the angular momentum of an electron in the $n=5, \ell=3$ state of hydrogen.

Rahul Nikhar

Numerade Educator

02:55

(II) A hydrogen atom is in the $7 g$ state. Determine $(a)$ the principal quantum number, $(b)$ the energy of the state, (c) the orbital angular momentum and its quantum number $\ell$, and $(d)$ the possible values for the magnetic quantum number.

João Gabriel Alencar Caribé

Numerade Educator

02:28

Problem 8

(II) (a) Show that the number of different states possible for a given value of $\ell$ is equal to $2(2 \ell+1) .$ (b) What is this number for $\ell=0,1,2,3,4,5,$ and $6 ?$

Rahul Nikhar

Numerade Educator

04:29

Problem 9

(II) Show that the number of different electron states possible for a given value of $n$ is $2 n^{2}$. (See Problem 8.)

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 10

(II) An excited $\mathrm{H}$ atom is in a $5 d$ state. $(a)$ Name all the states to which the atom is "allowed" to jump with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

Lainey Roebuck

Numerade Educator

View

Problem 11

(II) The magnitude of the orbital angular momentum in an excited state of hydrogen is $6.84 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}$ and the $z$ component is $2.11 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} .$ What are all the possible values of $n, \ell,$ and $m_{\ell}$ for this state?

Eduard Sanchez

Numerade Educator

05:00

Problem 12

(I) Show that the ground-state wave function, Eq. $39-5,$ is normalized.

Zhaojie Xu

Numerade Educator

02:54

Problem 13

(II) For the ground state of hydrogen, what is the value of $(a) \psi,(b)|\psi|^{2},$ and $(c) P_{\mathrm{r}},$ at $r=1.5 r_{0} ?$

João Gabriel Alencar Caribé

Numerade Educator

04:44

Problem 14

(II) For the $n=2, \quad \ell=0$ state of hydrogen, what is the value of $(a) \psi,(b)|\psi|^{2},$ and $(c) P_{\mathrm{r}},$ at $r=4 r_{0} ?$

Rahul Nikhar

Numerade Educator

03:32

Problem 15

(II) By what factor is it more likely to find the electron in the ground state of hydrogen at the Bohr radius $\left(r_{0}\right)$ than at twice the Bohr radius $\left(2 r_{0}\right) ?$

João Gabriel Alencar Caribé

Numerade Educator

09:31

Problem 16

(II) $(a)$ Show that the probability of finding the electron in the ground state of hydrogen at less than one Bohr radius from the nucleus is $32 \%$ ( $b$ ) What is the probability of finding a $1 s$ electron between $r=r_{0}$ and $r=2 r_{0} ?$

Rahul Nikhar

Numerade Educator

16:26

Problem 17

(II) Determine the radius $r$ of a sphere centered on the nucleus within which the probability of finding the electron for the ground state of hydrogen is $(a) 50 \%,(b) 90 \%$ (c) $99 \%$.

João Gabriel Alencar Caribé

Numerade Educator

06:56

Problem 18

(II) (a) Estimate the probability of finding an electron, in the ground state of hydrogen, within the nucleus assuming it to be a sphere of radius $r=1.1 \mathrm{fm} .$ ( $b$ ) Repeat the estimate assuming the electron is replaced with a muon, which is very similar to an electron (Chapter 43) except that its mass is 207 times greater.

Narayan Hari

Numerade Educator

06:10

Problem 19

(II) Show that the mean value of $r$ for an electron in the ground state of hydrogen is $\bar{r}=\frac{3}{2} r_{0},$ by calculating
$$
\overline{\boldsymbol{r}}=\int_{\text {all space }} \boldsymbol{r}\left|\psi_{100}\right|^{2} d \boldsymbol{V}=\int_{0}^{\infty} \boldsymbol{r}\left|\psi_{100}\right|^{2} 4 \boldsymbol{\pi} r^{2} d r
$$

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 20

(II) Show that $\psi_{200}$ as given by Eq. $39-8$ is normalized.

Lainey Roebuck

Numerade Educator

04:55

Problem 21

(II) Determine the average radial probability distribution $P_{\mathrm{r}}$ for the $n=2, \ell=1$ state in hydrogen by calculating
$$
P_{\mathrm{r}}=4 \pi r^{2}\left[\frac{1}{3}\left|\psi_{210}\right|^{2}+\frac{1}{3}\left|\psi_{211}\right|^{2}+\frac{1}{3}\left|\psi_{21-1}\right|^{2}\right] .
$$

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 22

(II) Use the result of Problem 21 to show that the most probable distance $r$ from the nucleus for an electron in the $2 p$ state of hydrogen is $r=4 r_{0},$ which is just the second Bohr radius (Eq. $37-11$, Fig. $37-25$ ).

Lainey Roebuck

Numerade Educator

03:47

Problem 23

(II) For the ground state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius $0.99 r_{0}$ and outer radius $1.01 r_{0} ?$

João Gabriel Alencar Caribé

Numerade Educator

03:47

Problem 24

(III) For the $n=2, \ell=0$ state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius $4.00 r_{0}$ and outer radius $5.00 r_{0} ?$

João Gabriel Alencar Caribé

Numerade Educator

07:59

Problem 25

(III) Show that $\psi_{100}$ (Eq. 39-5a) satisfies the Schrödinger equation (Eq. $39-1$ ) with the Coulomb potential, for energy $E=-m e^{4} / 8 \epsilon_{0}^{2} h^{2}$

Sam Stansfield

Numerade Educator

07:11

Problem 26

(III) Show that the probability of finding the electron within 1 Bohr radius of the nucleus in the hydrogen atom is (a) $3.4 \%$ for the $n=2, \ell=0$ state, and (b) $0.37 \%$ for the $n=2, \ell=1$ state. (See Problem 21.)

Rahul Nikhar

Numerade Educator

13:47

Problem 27

(III) The wave function for the $n=3, \quad \ell=0$ state in hydrogen is
$$
\psi_{300}=\frac{1}{\sqrt{27 \pi r_{0}^{3}}}\left(1-\frac{2 r}{3 r_{0}}+\frac{2 r^{2}}{27 r_{0}^{2}}\right) e^{-\frac{r}{3 r_{0}}}
$$
(a) Determine the radial probability distribution $P_{\mathrm{r}}$ for this state, and (b) draw the curve for it on a graph. (c) Determine the most probable distance from the nucleus for an electron in this state.

João Gabriel Alencar Caribé

Numerade Educator

03:40

Problem 28

(I) List the quantum numbers for each electron in the ground state of oxygen $(Z=8)$.

Rahul Nikhar

Numerade Educator

03:25

Problem 29

(I) List the quantum numbers for each electron in the ground state of $(a)$ carbon $(Z=6)$ (b) aluminum $(Z=13)$.

João Gabriel Alencar Caribé

Numerade Educator

01:31

Problem 30

(I) How many electrons can be in the $n=6, \ell=4$ subshell?

Rahul Nikhar

Numerade Educator

01:42

Problem 31

(II) An electron has $m_{\ell}=2$ and is in its lowest possible energy state. What are the values of $n$ and $\ell$ for this electron?

João Gabriel Alencar Caribé

Numerade Educator

00:20

Problem 32

(II) If the principal quantum number $n$ were limited to the range from 1 to $6,$ how many elements would we find in nature?

Rahul Nikhar

Numerade Educator

View

Problem 33

(II) What is the full electron configuration for ( $a$ ) nickel (Ni), (b) silver (Ag), (c) uranium (U)?

Lainey Roebuck

Numerade Educator

02:59

Problem 34

(II) Estimate the binding energy of the third electron in lithium using Bohr theory. [Hint: This electron has $n=2$ and "sees" a net charge of approximately $+1 e .]$ The measured value is $5.36 \mathrm{eV}$.

Rahul Nikhar

Numerade Educator

01:13

Problem 35

(II) Using the Bohr formula for the radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innermost $(n=1)$ orbit in uranium $(Z=92)$. Approximately how much energy would be required to remove this innermost electron?

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 36

(II) Let us apply the exclusion principle to an infinitely high square well (Section $38-8$ ). Let there be five electrons confined to this rigid box whose width is $\ell$. Find the lowest energy state of this system, by placing the electrons in the lowest available levels, consistent with the Pauli exclusion principle.

Lainey Roebuck

Numerade Educator

02:49

Problem 37

(II) Show that the total angular momentum is zero for a filled subshell.

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 38

(I) If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have $\lambda=0.027 \mathrm{nm}$, what is the voltage across the tube?

Lainey Roebuck

Numerade Educator

04:08

Problem 39

(I) What are the shortest-wavelength X-rays emitted by electrons striking the face of a $32.5-\mathrm{kV}$ TV picture tube? What are the longest wavelengths?

João Gabriel Alencar Caribé

Numerade Educator

02:04

Problem 40

(I) Show that the cutoff wavelength $\lambda_{0}$ in an $X$ -ray spectrum is given by
$$
\lambda_{0}=\frac{1240}{V} \mathrm{nm}
$$
where $V$ is the X-ray tube voltage in volts.

Rahul Nikhar

Numerade Educator

02:31

Problem 41

(II) Estimate the wavelength for an $n=2$ to $n=1$ transition in iron $(Z=26)$.

João Gabriel Alencar Caribé

Numerade Educator

01:53

Problem 42

(II) Use the result of Example $39-6$ to estimate the X-ray wavelength emitted when a cobalt atom $(Z=27)$ makes a transition from $n=2$ to $n=1$.

Prabhat Tyagi

Numerade Educator

04:12

Problem 43

(II) A mixture of iron and an unknown material are bombarded with electrons. The wavelength of the $\mathrm{K}_{\alpha}$ lines are $194 \mathrm{pm}$ for iron and $229 \mathrm{pm}$ for the unknown. What is the unknown material?

João Gabriel Alencar Caribé

Numerade Educator

06:22

Problem 44

(II) Use Bohr theory to estimate the wavelength for an $n=3$ to $n=1$ transition in molybdenum $(Z=42) .$ The measured value is $0.063 \mathrm{nm}$. Why do we not expect perfect agreement?

Rahul Nikhar

Numerade Educator

04:38

Problem 45

(II) Use conservation of energy and momentum to show that a moving electron cannot give off an X-ray photon unless there is a third object present, such as an atom or nucleus.

João Gabriel Alencar Caribé

Numerade Educator

01:23

Problem 46

(I) Verify that the Bohr magneton has the value $\mu_{\mathrm{B}}=9.27 \times 10^{-24} \mathrm{~J} / \mathrm{T}$ (see Eq. 39-12)

Jacob Shpiece

Numerade Educator

03:10

Problem 47

(I) If the quantum state of an electron is specified by $\left(n, \ell, m_{\ell}, m_{s}\right),$ estimate the energy difference between the states $\left(1,0,0,-\frac{1}{2}\right)$ and $\left(1,0,0,+\frac{1}{2}\right)$ of an electron in the $1 s$ state of helium in an external magnetic field of $2.5 \mathrm{~T}$

João Gabriel Alencar Caribé

Numerade Educator

00:10

Problem 48

(II) Silver atoms $\left(\right.$ spin $\left.=\frac{1}{2}\right)$ are placed in a 1.0-T magnetic field which splits the ground state into two close levels. (a) What is the difference in energy between these two levels, and $(b)$ what wavelength photon could cause a transition from the lower level to the upper one? (c) How would your answer differ if the atoms were hydrogen?

Rahul Nikhar

Numerade Educator

07:09

Problem 49

(II) In a Stern-Gerlach experiment, Ag atoms exit the oven with an average speed of $780 \mathrm{~m} / \mathrm{s}$ and pass through a magnetic field gradient $d B / d z=1.8 \times 10^{3} \mathrm{~T} / \mathrm{m}$ for a distance of $5.0 \mathrm{~cm} .(a)$ What is the separation of the two beams as they emerge from the magnet? $(b)$ What would the separation be if the $g$ -factor were 1 for electron spin?

João Gabriel Alencar Caribé

Numerade Educator

04:05

Problem 50

(II) For an electron in a $5 g$ state, what are all the possible values of $j, m_{j}, J,$ and $J_{z} ?$

Zhaojie Xu

Numerade Educator

03:53

Problem 51

(II) What are the possible values of $j$ for an electron in (a) the $4 p,$ (b) the $4 f,$ and $(c)$ the $3 d$ state of hydrogen? (d) What is $J$ in each case?

Ali Nazim Aslan

Numerade Educator

12:26

Problem 52

(II) (a) Write down the quantum numbers for each electron in the gallium atom. (b) Which subshells are filled? (c) The last electron is in the $4 p$ state; what are the possible values of the total angular momentum quantum number, $j,$ for this electron? $(d)$ Explain why the angular momentum of this last electron also represents the total angular momentum for the entire atom (ignoring any angular momentum of the nucleus). $(e)$ How could you use $a$ Stern-Gerlach experiment to determine which value of $j$ the atom has?

Rahul Nikhar

Numerade Educator

View

Problem 53

(III) The difference between the $2 \mathrm{P}_{3 / 2}$ and $2 \mathrm{P}_{1 / 2}$ energy levels in hydrogen is about $5 \times 10^{-5} \mathrm{eV},$ due to the spinorbit interaction. (a) Taking the electron's (orbital) magnetic moment to be 1 Bohr magneton, estimate the internal magnetic field due to the electron's orbital motion. (b) Estimate the internal magnetic field using a simple model of the nucleus revolving in a circle about the electron.

Lainey Roebuck

Numerade Educator

05:09

Problem 54

(II) A laser used to weld detached retinas puts out 23 -mslong pulses of 640 -nm light which average 0.63 -W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?

Aparna Shakti

Numerade Educator

01:52

Problem 55

(II) Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.6-mm-diameter mirror. Assume that $\lambda=694 \mathrm{nm} .$ What would be the diameter of this beam if it struck $(a)$ a satellite $380 \mathrm{~km}$ above the Earth,
(b) the Moon?

João Gabriel Alencar Caribé

Numerade Educator

01:55

Problem 56

(II) A low-power laser used in a physics lab might have a $\begin{array}{lllll}\text { power of } 0.50 \mathrm{~mW} & \text { and a beam diameter of } 3.0 \mathrm{~mm} .\end{array}$ Calculate $(a)$ the average light intensity of the laser beam, and $(b)$ compare it to the intensity of a lightbulb emitting $15 \mathrm{~W}$ of light viewed from a distance of $2.0 \mathrm{~m}$

Jacob Shpiece

Numerade Educator

02:12

Problem 57

(II) Calculate the wavelength of a He-Ne laser.

João Gabriel Alencar Caribé

Numerade Educator

02:54

Problem 58

(II) Suppose that the energy level system in Fig. $39-20$ is not being pumped and is in thermal equilibrium. Determine the fraction of atoms in levels $E_{2}$ and $E_{1}$ relative to $E_{0}$ at $T=300 \mathrm{~K}$

João Gabriel Alencar Caribé

Numerade Educator

02:53

Problem 59

(II) To what temperature would the system in Fig. $39-20$ have to be raised (see Problem 58 ) so that in thermal equilibrium the level $E_{2}$ would have half as many atoms as $E_{0} ?$ (Note that pumping mechanisms do not maintain thermal equilibrium.)

João Gabriel Alencar Caribé

Numerade Educator

03:59

Problem 60

(II) Show that a population inversion for two levels (as in a pumped laser) corresponds to a negative Kelvin temperature in the Boltzmann distribution. Explain why such a situation does not contradict the idea that negative Kelvin temperatures cannot be reached in the normal sense of temperature.

Rahul Nikhar

Numerade Educator

03:28

Problem 61

The ionization (binding) energy of the outermost electron in boron is $8.26 \mathrm{eV}$. $(a)$ Use the Bohr model to estimate the "effective charge," $Z_{\text {eff }}$, seen by this electron. (b) Estimate the average orbital radius.

João Gabriel Alencar Caribé

Numerade Educator

02:06

Problem 62

How many electrons can there be in an " $h$ " subshell?

Rahul Nikhar

Numerade Educator

01:59

Problem 63

What is the full electron configuration in the ground state for elements with $Z$ equal to $(a) 25,(b) 34,(c) 39 ?[$ Hint: See the Periodic Table inside the back cover.]

Jacob Shpiece

Numerade Educator

01:27

Problem 64

What are the largest and smallest possible values for the angular momentum $L$ of an electron in the $n=6$ shell?

Rahul Nikhar

Numerade Educator

06:03

Problem 65

Estimate $(a)$ the quantum number $\ell$ for the orbital angular momentum of the Earth about the Sun, and $(b)$ the number of possible orientations for the plane of Earth's orbit.

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 66

Use the Bohr theory (especially Eq. $37-15$ ) to show that the Moseley plot (Fig. $39-12$ ) can be written
$$
\sqrt{\frac{1}{\lambda}}=a(Z-b)
$$
where $b \approx 1,$ and evaluate $a$

Lainey Roebuck

Numerade Educator

08:08

Problem 67

Determine the most probable distance from the nucleus of an electron in the $n=2, \ell=0$ state of hydrogen.

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 68

Show that the diffractive spread of a laser beam, $\approx \lambda / D$ as described in Section $39-9,$ is precisely what you might expect from the uncertainty principle. [Hint: Since the beam's width is constrained by the dimension of the aperture $D$, the component of the light's momentum perpendicular to the laser axis is uncertain.]

Lainey Roebuck

Numerade Educator

View

Problem 69

In the so-called vector model of the atom, space quantization of angular momentum (Fig. $39-3$ ) is illustrated as shown in Fig. $39-28 .$ The angular momentum vector of magnitude $L=\sqrt{\ell(\ell+1)} \hbar$ is thought of as precessing around the $z$ axis (like a spinning top or gyroscope) in such a way that the $z$ component of angular momentum, $L_{z}=m_{\ell} \hbar,$ also stays constant. Calculate the possible values for the angle $\theta$ between $\overrightarrow{\mathbf{L}}$ and the $z$ axis $(a)$ for $\ell=1,$ (b) $\ell=2,$ and (c) $\ell=3$ (d) Determine the minimum value of $\theta$ for $\ell=100$ and $\ell=10^{6}$. Is this consistent with the correspondence principle?

Lainey Roebuck

Numerade Educator

03:22

Problem 70

The vector model (Problem 69 ) gives some insight into the uncertainty principle for angular momentum, which is $\Delta L_{z} \Delta \phi \geq \hbar$
for the $z$ component. Here $\phi$ is the angular position measured in the plane perpendicular to the $z$ axis. Once $m_{\ell}$ for an atom is known, $L_{z}$ is known precisely, so $\Delta L_{z}=0$. (a) What does this tell us about $\phi ?(b)$ What can you say about $L_{x}$ and $L_{y},$ which are not quantized (only $L$ and $L_{z}$ are $) ?$ (c) Show that although $L_{x}$ and $L_{y}$ are not quantized, nonetheless $\left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar$ is.

Jacob Shpiece

Numerade Educator

View

Problem 71

(a) Show that the mean value for $1 / r$ of an electron in the ground state of hydrogen equals $1 / r_{0},$ and from this conclude that the mean value of the potential energy is
$$
\bar{U}=-\frac{1}{4 \pi \epsilon_{0}} \frac{e^{2}}{r_{0}}
$$
(b) Using $E=\bar{U}+\bar{K},$ find a relationship between the average kinetic energy and the average potential energy in the ground state. [Hint: For $(a)$, see Problem 19 or Example $38-9 .$ ]

Lainey Roebuck

Numerade Educator

02:50

Problem 72

The angular momentum in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results for $n=2$

Rahul Nikhar

Numerade Educator

02:35

Problem 73

For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: $(a) 4 p \rightarrow 3 p$; (b) $3 p \rightarrow 1 s ;(c) 4 d \rightarrow 3 d ;(d) 4 d \rightarrow 3 s$ (e) $4 s \rightarrow 2 p$.

João Gabriel Alencar Caribé

Numerade Educator

01:45

Problem 74

It is possible for atoms to be excited into states with very high values of the principal quantum number. Electrons in these socalled Rydberg states have very small ionization energies and huge orbital radii. This makes them particularly sensitive to external perturbation, as would be the case if the atom were in an electric field. Consider the $n=45$ state of the hydrogen atom. Determine the binding energy, the radius of the orbit, and the effective cross-sectional area of this Rydberg state.

Jacob Shpiece

Numerade Educator

21:35

Problem 75

Suppose that the spectrum of an unknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an ion with $Z$ protons and one electron, determine $Z$ and the element in question.

João Gabriel Alencar Caribé

Numerade Educator

09:55

Problem 76

Suppose that the splitting of energy levels shown in Fig. $39-4$ was produced by a 1.6 -T magnetic field. $(a)$ What is the separation in energy between adjacent $m_{\ell}$ levels for the same $\ell ?(b)$ How many different wavelengths will there be for $3 d$ to $2 p$ transitions, if $m_{\ell}$ can change only by ±1 or $0 ?$ (c) What is the wavelength for each of these transitions?

João Gabriel Alencar Caribé

Numerade Educator

View

Problem 77

Populations in the $H$ atom. Use the Boltzmann factor (Eq. $39-16$ ) to estimate the fraction of $\mathrm{H}$ atoms in the $n=2$ and $n=3$ levels (relative to the ground state) for thermal equilibrium at $(a) T=300 \mathrm{~K}$ and $(b) T=6000 \mathrm{~K}$ [Note: Since there are eight states with $n=2$ and only two with $n=1,$ multiply your result for $n=2$ by $\frac{8}{2}=4 ;$ do similarly for $n=3 .](c)$ Given $1.0 \mathrm{~g}$ of hydrogen, estimate the number of atoms in each state at $T=6000 \mathrm{~K} .(d)$ Estimate the number of $n=3$ to $n=1$ and $n=2$ to $n=1$ photons that will be emitted per second at $T=6000 \mathrm{~K}$. Assume that the lifetime of each excited state is $10^{-8} \mathrm{~s}$.

Lainey Roebuck

Numerade Educator