Physics

James S. Walker

Chapter 31

Atomic Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

The electron in a hydrogen atom is typically found at a distance of about $5.3 \times 10^{-11} \mathrm{m}$ from the nucleus, which has a diameter of about $1.0 \times 10^{-15} \mathrm{m} .$ If you assume the hydrogen atom to be a sphere of radius $5.3 \times 10^{-11} \mathrm{m},$ what fraction of its volume is occupied by the nucleus?

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Problem 2

Referring to Problem $1,$ suppose the nucleus of the hydrogen atom were enlarged to the size of a baseball (diameter $=7.3 \mathrm{cm} ) .$ At what typical distance from the center of the baseball would you expect to find the electron?

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Problem 3

In the Thomson model of the atom, the mass of an atom is distributed uniformly throughout its volume in a material analogous to pudding. (a) Calculate the density of the "pudding" of a gold atom that has a mass of $3.27 \times 10^{-25} \mathrm{kg}$ , modeling it as a sphere of radius 0.144 $\mathrm{nm}$ . (b) Compare the density in part (a) with the density of an alpha particle of mass $6.64 \times 10^{-27} \mathrm{kg}$ and radius $1.68 \times 10^{-15} \mathrm{m} .$

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Problem 4

Copper atoms have 29 protons in their nuclei. If the copper nucleus is a sphere with a diameter of $4.8 \times 10^{-15} \mathrm{m},$ find the work required to bring an alpha particle (charge $=+2 e )$ from rest at infinity to the "surface" of the nucleus.

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Problem 5

In Rutherford's scattering experiments, alpha particles (charge $=+2 e )$ were fired at a gold foil. Consider an alpha particle with an initial kinetic energy $K$ heading directly for the nucleus of a gold atom (charge $=+79 e ) .$ The alpha particle will come to rest when all its initial kinetic energy has been converted to electric potential energy. Find the distance of closest approach between the alpha particle and the gold nucleus for the case $K=5.3 \mathrm{MeV}.$

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Problem 6

Find the wavelength of the Balmer series spectral line corresponding to $n=12$ .

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Problem 7

What is the smallest value of $n$ for which the wavelength of a Balmer series line is less than 400 $\mathrm{nm}$ ?

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Problem 8

Find the wavelength of the three longest-wavelength lines of the Lyman series.

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Problem 9

Find the wavelength of the longest-wavelength lines of the Paschen series.

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Problem 10

Find (a) the longest wavelength in the Lyman series and (b) the shortest wavelength in the Paschen series.

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Problem 11

Predict/Explain (a) If the mass of the electron were magically doubled, would the ionization energy of hydrogen increase, decrease, or stay the same? (b) Choose the best explanation from
among the following:
$$\begin{array}{l}{\text { I. The ionization energy would increase because the increased }} \\ {\text { mass would mean the electron would orbit closer to the nucleus }} \\ {\text { and would require more energy to move to infinity. }}\end{array}$$
$$\begin{array}{l}{\text { II. The ionization energy would decrease because a more massive }} \\ {\text { electron is harder to hold in orbit, and therefore it is easier to }} \\ {\text { remove the electron and leave the hydrogen ionized. }}\end{array}$$
$$\begin{array}{l}{\text { III. The ionization energy would be unchanged because, just like }} \\ {\text { in gravitational orbits, the orbit of the electron is indepen- }} \\ {\text { dent of its mass. As a result, there is no change in the energy }} \\ {\text { required to move it to infinity. }}\end{array}$$

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Problem 12

Consider the Bohr model as applied to the following three atoms: ( A) neutral hydrogen in the state $n=2 ;(\mathrm{B})$ singly ionized helium in the state $n=1 ;(\mathrm{C})$ doubly ionized lithium in the state $n=3 .$ Rank these three atoms in order of increasing Bohr radius. Indicate ties where appropriate.

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Problem 13

Consider the Bohr model as applied to the following three atoms: (A) neutral hydrogen in the state $n=3 ;(\mathrm{B})$ singly ionized helium in the state $n=2 ;(\mathrm{C})$ doubly ionized lithium in the state $n=1 .$ Rank these three atoms in order of increasing energy. Indicate ties where appropriate.

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Problem 14

An electron in the $n=1$ Bohr orbit has the kinetic energy $K_{1} .$ In terms of $K_{1},$ what is the kinetic energy of an electron in the $n=2$ Bohr orbit?

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Problem 15

Find the ratio $v / c$ for an electron in the first excited state $(n=2)$ of hydrogen.

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Problem 16

Find the magnitude of the force exerted on an electron in the ground-state orbit of the Bohr model.

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Problem 17

How much energy is required to ionize hydrogen when it is in the $n=3$ state?

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Problem 18

Find the energy of the photon required to excite a hydrogen atom from the $n=1$ state to the $n=4$ state.

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Problem 19

Atomic Clock Transition The global standard for time is based on a transition in cesium atoms that occurs when a microwave photon of frequency $9,192,631,770 \mathrm{Hz}$ absorbed. What is the energy difference in eV between the two levels of cesium that correspond with this transition?

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Problem 20

A hydrogen atom is in its second excited state, $n=3 .$ Using the Bohr model of hydrogen, find (a) the linear momentum and (b) the angular momentum of the electron in this atom.

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Problem 21

Referring to Problem 20 , find (a) the kinetic energy of the electron, (b) the potential energy of the atom, and (c) the total energy of the atom. Give your results in eV.

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Problem 22

Initially, an electron is in the $n=2$ state of hydrogen. If this electron acquires an additional 2.86 $\mathrm{eV}$ of energy, what is the value of $n$ in the final state of the electron?

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Problem 23

Identify the initial and final states if an electron in hydrogen emits a photon with a wavelength of 656 $\mathrm{nm.}$

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Problem 24

Predict/Calculate An electron in hydrogen absorbs a photon and jumps to a higher orbit. (a) Find the energy the photon must have if the initial state is $n=3$ and the final state is $n=5$ . (b) If the initial state was $n=5$ and the final state $n=7,$ would the energy of the photon be greater than, less than, or the same as that found in part (a)? Explain. (c) Calculate the photon energy for part (b).

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Problem 25

Predict/Calculate Consider the following four transitions in a hydrogen atom:
$$\begin{array}{ll}{\text { (i) } n_{\mathrm{i}}=2, n_{\mathrm{f}}=6} & {\text { (ii) } n_{\mathrm{i}}=2, n_{\mathrm{f}}=8} \\ {\text { (iii) } n_{\mathrm{i}}=7, n_{\mathrm{f}}=8} & {\text { (iv) } n_{\mathrm{i}}=6, n_{\mathrm{f}}=2}\end{array}$$
Find (a) the longest-and (b) the shortest-wavelength photon that can be emitted or absorbed by these transitions. Give the value of the wavelength in each case. (c) For which of these transitions does the atom lose energy? Explain.

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Problem 26

Predict/Calculate Muonium Muonium is a hydrogen-like atom in which the electron is replaced with a muon, a fundamental particle with a charge of $-e$ and a mass equal to 207$m_{\mathrm{e}} .$ (The muon is sometimes referred to loosely as a "heavy electron.") (a) What is the Bohr radius of muonium? (b) Will the wavelengths in the Balmer series of muonium be greater than, less than, or the same as the wavelengths in the Balmer series of hydrogen? Explain. (c) Calculate the
longest wavelength of the Balmer series in muonium.

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Problem 27

Predict/Calculate (a) Find the radius of the $n=4$ Bohr orbit of a doubly ionized lithium atom $\left(\mathrm{Li}^{2+}, Z=3\right) .$ (b) Is the energy required to raise an electron from the $n=4$ state to the $n=5$ state in $\mathrm{Li}^{2+}$ greater than, less than, or equal to the energy required to raise an electron in hydrogen from the $n=4$ state to the $n=5$ state? Explain. (c) Verify your answer to part (b) by calculating the relevant energies.

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Problem 28

Applying the Bohr model to a triply ionized beryllium atom $\left(\mathrm{Be}^{3+}, Z=4\right),$ find (a) the shortest wavelength of the Lyman series for $\mathrm{Be}^{3+}$ and (b) the ionization energy required to remove the final electron in $\mathrm{Be}^{3+}$.

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Problem 29

(a) Calculate the time required for an electron in the $n=2$ state of hydrogen to complete one orbit about the nucleus. (b) The typical "lifetime" of an electron in the $n=2$ state is roughly $10^{-8} s-$ after this time the electron is likely to have dropped back to the $n=1$ state. Estimate the number of orbits an electron completes in the $n=2$ state before dropping to the ground state.

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Problem 30

Predict/Calculate The kinetic energy of an electron in a particular Bohr orbit of hydrogen is $1.35 \times 10^{-19} \mathrm{J.}$ (a) Which Bohr orbit does the electron occupy? (b) Suppose the electron moves away from the nucleus to the next higher Bohr orbit. Does the kinetic energy of the electron increase, decrease, or stay the same? Explain. (c) Calculate the kinetic energy of the electron in the orbit referred to in part (b).

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Problem 31

Predict/Calculate The potential energy of a hydrogen atom in a particular Bohr orbit is $-1.20 \times 10^{-19} \mathrm{J}$ (a) Which Bohr orbit does the electron occupy in this atom? (b) Suppose the electron moves away from the nucleus to the next higher Bohr orbit. Does the potential energy of the atom increase, decrease, or stay the same? Explain. (c) Calculate the potential energy of the atom for
the orbit referred to in part (b).

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Problem 32

Consider a head-on collision between two hydrogen atoms, both initially in their ground state and moving with the same aspeed. Find the minimum speed necessary to leave both atoms in their $n=4$ state after the collision.

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Problem 33

A hydrogen atom is in the initial state $n_{i}=n,$ where $n>1$ (a) Find the frequency of the photon that is emitted when the electron jumps to state $n_{\mathrm{f}}=n-1 .$ (b) Find the frequency of the electron's orbital motion in the state $n$ . (c) Compare your results for parts (a) and (b) in the limit of large $n .$

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Problem 34

Predict/Explain (a) Is the de Broglie wavelength of an electron in the $n=2$ Bohr orbit of hydrogen greater than, less than, or equal to the de Broglit wavelength in the $n=1$ Bohr orbit?
(b) Choose the best explanation from among the following:
$$\begin{array}{l}{\text { I The de Broglie wavelength in the } n \text { th state is } 2 \pi r / n, \text { where } r \text { is }} \\ {\text { proportional to } n^{2} . \text { Therefore, the wavelength increases with }} \\ {\text { increasing } n, \text { and is greater for } n=2 \text { than for } n=1 .}\end{array}$$
$$\begin{array}{l}{\text { II. The de Broglie wavelength of an electron in the } n \text { th state is }} \\ {\text { such that } n \text { wavelengths fit around the circumference of the }} \\ {\text { orbit. Therefore, } \lambda=2 \pi r / n \text { and the wavelength for } n=2 \text { is }} \\ {\text { less than for } n=1}\end{array}$$
$$\begin{array}{l}{\text { III. The de Broglie wavelength depends on the mass of the electron, }} \\ {\text { and that is the same regardless of which state of the hydrogen }} \\ {\text { atom the electron occupies. }}\end{array}$$

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Problem 35

Find the de Broglie wavelength of an electron in the ground state of the hydrogen atom.

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Problem 36

The de Broglie wavelength of an electron in a hydrogen atom is 1.66 $\mathrm{nm} .$ Identify the integer $n$ that corresponds to its orbit.

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Problem 37

The orbital radius of an electron in a hydrogen atom is 0.846 $\mathrm{nm}$ . What is its de Broglie wavelength?

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Problem 38

Find an expression, in terms of $n$ and the fundamental constants, for the de Broglie wavelength of an electron in the $n$th state of the hydrogen atom.

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Problem 39

What is the radius of the hydrogen-atom Bohr orbit shown in FIGURE $31-40 ?$

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Problem 40

Predict/Calculate (a) An electron in a hydrogen atom jumps from the $n=6$ orbit to the $n=2$ orbit and emits a photon. Will its de Broglie wavelength increase, decrease, or stay the same after the jump? (b) Calculate its de Broglie wavelength before and after the jump.

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Problem 41

(a) Find the kinetic energy (in eV) of an electron whose de Broglie wavelength is equal to $0.5 \dot{\mathrm{A}}=0.05 \mathrm{nm},$ a typical atomic size. (b) If the wavelength of the electron is reduced to $10^{-15} \mathrm{m}, \mathrm{a}$ typical nuclear size, is its kinetic energy greater than, less than, or equal to the value found in part (a)? Explain.

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Problem 42

What are the allowed values of $\ell$ when the principal quantum number is $n=5 ?$

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Problem 43

How many different values of $m_{\ell}$ are possible when the principal quantum number is $n=5 ?$

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Problem 44

Give the value of the quantum number $\ell,$ if one exists, for a hydrogen atom whose orbital angular momentum has a magnitude of (a) $\sqrt{6}(h / 2 \pi),($ b) $\sqrt{15}(h / 2 \pi),($ c) $\sqrt{30}(h / 2 \pi),$ or (d) $\sqrt{36}(h / 2 \pi) .$

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Problem 45

Predict/Calculate Hydrogen atom number 1 is known to be in the 4$f$ state. (a) What is the energy of this atom? (b) What is the magnitude of this atom's orbital angular momentum? (c) Hydrogen atom number 2 is in the 5$d$ state. Is this atom's energy greater than, less than, or the same as that of atom 1$?$ Explain. $(\mathrm{d})$ Is the magnitude of the orbital angular momentum of atom 1 greater than, less than, or the same as that of atom 2$?$ Explain.

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Problem 46

Predict/Calculate An electron in a hydrogen atom makes a transition from a 5$p$ state to a 4$d$ state. (a) Does the energy of the electron increase, decrease, or stay the same? Explain. (b) Does the magnitude of the electron's angular momentum increase, decrease, or stay the same? Explain. (c) Calculate the change in the electron's energy in eV. (d) Calculate the change in the magnitude of the electron's angular momentum.

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Problem 47

Predict/Calculate A hydrogen atom has an orbital angular momentum with a magnitude of 10$\sqrt{57}(h / 2 \pi) .$ (a) Determine the value of the quantum number $\ell$ for this atom. (b) What is the minimum possible value of this atom's principal quantum number, $n ?$ Explain. (c) If 10 $\sqrt{57}(h / 2 \pi)$ is the maximum orbital angular momentum this atom can have, what is its energy?

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Problem 48

Predict/Calculate The electron in a hydrogen atom with an energy of $-0.544 \mathrm{eV}$ is in a subshell with 18 states. (a) What is the principal quantum number, $n,$ for this atom? (b) What is the maximum possible orbital angular momentum this atom can have? (c) Is the number of states in the subshell with the next lowest value of $\ell$ equal to $16,14,$ or 12$?$ Explain.

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Problem 49

Predict/Calculate Consider two different states of a hydrogen atom. In state I the maximum value of the magnetic quantum number is $m_{\ell}=3$; in state II the corresponding maximum value is $m_{\ell}=2 .$ Let $L_{1}$ and $L_{11}$ represent the magnitudes of the orbital
angular momentum of an electron in states I and II, respectively. (a) Is $L_{1}$ greater than, less than, or equal to $L_{\mathrm{ll}} ?$ Explain. (b) Calculate the ratio $L_{1} / L_{11}$ .

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Problem 50

How many electrons can occupy (a) the 2$p$ subshell and (b) the 3$p$ subshell?

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Problem 51

(a) How many electrons can occupy the 3$d$ subshell? (b) How many electrons can occupy the $n=2$ shell?

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Problem 52

The electronic configuration of a given atom is 1$s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{1}$.
How many electrons are in this atom?

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Problem 53

Give the electronic configuration for the ground state of carbon.

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Problem 54

Give a list of all possible sets of the four quantum numbers $\left(n, \ell, m_{\ell}, m_{\mathrm{s}}\right)$ for electrons in the 3$p$ subshell.

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Problem 55

List the values of the four quantum numbers $\left(n, \ell, m_{\ell}, m_{s}\right)$ for each of the electrons in the ground state of neon.

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Problem 56

List the values of the four quantum numbers $\left(n, \ell, m_{\ell}, m_{s}\right)$ for each of the electrons in the ground state of magnesium.

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Problem 57

The configuration of the outer electrons in Ni is 3$d^{8} 4 s^{2} .$ Write out the complete electronic configuration for Ni.

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Problem 58

Determine the number of different sets of quantum numbers possible for each of the following shells: (a) $n=2,$ (b) $n=3,$ (c) $n=4$.

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Problem 59

Predict/Explain (a) In an X-ray tube, do you expect the wavelength of the characteristic X-rays to increase, decrease, or stay the same if the energy of the electrons striking the target is increased?
(b) Choose the best explanation from among the following:
$$\begin{array}{l}{\text { I. Increasing the energy of the incoming electrons will increase }} \\ {\text { the wavelength of the emitted X-rays. }}\end{array}$$
$$\begin{array}{l}{\text { II. When the energy of the incoming electrons is increased, the }} \\ {\text { energy of the X-rays is also increased; this, in turn, decreases }} \\ {\text { the wavelength. }}\end{array}$$
$$\begin{array}{l}{\text { III. The wavelength of characteristic X-rays depends only on the }} \\ {\text { material used in the metal target, and does not change if the }} \\ {\text { energy of incoming electrons is increased. }}\end{array}$$

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Problem 60

Using the Bohr model, estimate the wavelength of the $\mathrm{K}_{\alpha}$ X-ray in nickel $(Z=28) .$

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Problem 61

Using the Bohr model, estimate the energy of a $\mathrm{K}_{\alpha} \ \mathrm{X}$-ray emitted by lead $(Z=82) .$

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Problem 62

The K-shell ionization energy of iron is $8500 \mathrm{eV},$ and its L-shell ionization energy is 2125 $\mathrm{eV} .$ What is the wavelength of $\mathrm{K}_{\alpha} \ \mathrm{X}$ -rays emitted by iron?

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Problem 63

An electron drops from the L shell to the K shell and gives off an X-ray with a wavelength of 0.0279 $\mathrm{nm} .$ What is the atomic number of this atom?

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Problem 64

Consider an $X$ -ray tube that uses platinum $(Z=78)$ as its target. (a) Use the Bohr model to estimate the minimum kinetic energy electrons must have in order for $K_{\alpha}$ -rays to just appear in the X-ray spectrum of the tube. (b) Assuming the electrons are accelerated from rest through a voltage $V$ , estimate the minimum voltage necessary to produce the $\mathrm{K}_{\alpha} \mathrm{X}$ -rays.

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Problem 65

BIO Photorefractive Keratectomy A person's vision may be improved significantly by having the cornea reshaped with a laser beam, in a procedure known as photorefractive keratectomy. The
excimer laser used in these treatments produces ultraviolet light with a wavelength of 193 $\mathrm{nm}$ . (a) What is the difference in energy between the two levels that participate in stimulated emission in the excimer laser? (b) How many photons from this laser are
required to deliver an energy of $1.58 \times 10^{-13} \mathrm{J}$ to the cornea?

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Problem 66

Predict/Calculate An ultraviolet photon is absorbed by a fluorescent molecule, which then emits two photons. (a) Are the wavelengths of the emitted photons longer than, shorter than, or the same as the wavelength of the ultraviolet photon? Explain. (b) If the ultraviolet photon has a wavelength of 254 $\mathrm{nm}$ and one of the photons emitted by the fluorescent material has a wavelength of $415 \mathrm{nm},$ what is the wavelength of the other emitted photon? Assume that none of the energy of the other emitted let photon is converted to other forms.

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Problem 67

Fluorescent lightbulbs The most common fluorescent lightbulb converts ultraviolet light from mercury atoms to white light. If an ultraviolet photon of wavelength 185 $\mathrm{nm}$ is emitted by a mercury atom and absorbed by the phosphor, what is the maximum number of red 655 -nm photons that the phosphor can emit?

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Problem 68

Consider the following three transitions in a hydrogen atom: (A) $n_{\mathrm{i}}=5, n_{\mathrm{f}}=2 ;$ (B) $n_{\mathrm{i}}=7, n_{\mathrm{f}}=2 ;(C) n_{\mathrm{i}}=7, n_{\mathrm{f}}=6 .$ Rank the transitions in order of increasing (a) wavelength and (b) frequency of the emitted photon. Indicate ties where appropriate.

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Problem 69

Suppose an electron is in the ground state of hydrogen. (a) What is the highest-energy photon this system can absorb without dissociating the electron from the proton? Explain. (b) What is the lowest-energy photon this system can absorb? Explain.

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Problem 70

The electronic configuration of a particular carbon atom is 1$s^{2} 2 s^{2} 2 p^{1} 3 s^{1} .$ Is this atom in its ground state or in an excited state? Explain.

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Problem 71

Do you expect the ionization energy of sodium (Na) to be greater than, less than, or equal to the ionization energy of lithium (Li)? Explain.

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Problem 72

Find the minimum frequency a photon must have if it is to ionize the ground state of the hydrogen atom.

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Problem 73

The electron in a hydrogen atom makes a transition from the $n=4$ state to the $n=2$ state, as indicated in FlGURE $31-41$ . (a) Determine the linear momentum of the photon emitted as a result of this transition. (b) Using your result to part (a), find the recoil speed of the hydrogen atom, assuming it was at rest before the photon was emitted.

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Problem 74

BIO Laser Eye Surgery In laser eye surgery, the laser emits a $1.45-$ ns pulse focused on a spot that is 32.5$\mu \mathrm{m}$ in diameter. (a) If the energy contained in the pulse is $2.95 \mathrm{mJ},$ what is the power per square meter (the irradiance) associated with this beam? (b) Suppose a molecule with a diameter of 0.500 $\mathrm{nm}$ is irradiated by the laser beam. How much energy does the molecule receive in one pulse from the laser? (The energy obtained in part (b) is more than
enough to dissociate a molecule.)

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Problem 75

Consider an electron in the ground-state orbit of the Bohr model of hydrogen. (a) Find the time required for the electron to complete one orbit about the nucleus. (b) Calculate the current (in amperes) corresponding to the electron's motion.

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Problem 76

A particular Bohr orbit in a hydrogen atom has a total energy of $-0.85$ eV. What are (a) the kinetic energy of the electron in this orbit and (b) the electric potential energy of the system?

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Problem 77

An ionized atom has only a single electron. The $n=6$ Bohr orbit of this electron has a radius of $2.72 \times 10^{-10} \mathrm{m} .$ Find (a) the atomic number $Z$ of this atom and (b) the total energy $E$ of its $n=3$ Bohr orbit.

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Problem 78

Find the approximate wavelength of $\mathrm{K}_{\beta}$ X-rays emitted by molybdenum $(Z=42),$ and compare your result with Figure $31-27$ . (Hint: An electron in the M shell is shielded from the nucleus by the single electron in the K shell, plus all the electrons in the L shell.)

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Problem 79

Diamagnetism Suppose FlGURE $31-42$ shows an electron in an atom that is orbiting a nucleus counterclockwise (that is, the charge is moving to the right at the moment pictured in the diagram). The indicated magnetic field has just been turned on from an initial value of zero. (a) Treating the electron as a circular ring of electric current, use Faraday's law to determine whether the electron will speed up or slow down in response to turning on the magnetic field. (b) The circulation of the electron produces a magnetic field at the center of its orbit. Will the magnitude of its magnetic field increase or decrease in response to turning on the external magnetic field? (c) Does the magnetic field produced by the orbiting electron point in the same direction as the external magnetic field, or in the opposite direction? (This question offers a crude explanation for the effect of diamagnetism, presented in Chapter $22 . )$

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Problem 80

Predict/Calculate Rydberg Atoms There is no limit to the size a hydrogen atom can attain, provided it is free from disruptive out- side influences. In fact, radio astronomers have detected radiation from large, so-called "Rydberg atoms" in the diffuse hydrogen gas of interstellar space. (a) Find the smallest value of $n$ such that the Bohr radius of a single hydrogen atom is greater than 8.0 microns,
the size of a typical single-celled organism. $[$ b) Find the wavelength of radiation this atom emits when its electron drops from level $n$ to level $n-1 .(\mathrm{c})$ If the electron drops one more level, from $n-1$ to $n-2,$ is the emitted wavelength greater than or less than the value found in part (b)? Explain.

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Problem 81

Predict/Calculate The Pickering Series In $1896,$ the American astronomer Edward C. Pickering $(1846-1919)$ discovered an unusual series of spectral lines in light from the hot star Zeta Puppis. After some time, it was determined that these lines are produced by singly ionized helium. In fact, the "Pickering series" is produced when electrons drop from higher levels to the $n=4$ level of $\mathrm{He}^{+} .$ Spectral lines in the Pickering series have wavelengths given by
$$\frac{1}{\lambda}=C\left(\frac{1}{16}-\frac{1}{n^{2}}\right).$$
In this expression, $n=5,6,7, \ldots$ (a) Do you expect the constant $C$ to be greater than, less than, or equal to the Rydberg constant $R ?$ Explain. (b) Find the numerical value of $C .$ (c) Pickering lines with $n=6,8,10, \ldots$ correspond to Balmer lines in hydrogen with $n=3,4,5, \ldots$ Verify this assertion for the $n=6$ Pickering line.

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Problem 82

Consider a particle of mass $m,$ charge $q,$ and constant speed $v$ moving perpendicular to a uniform magnetic field of magnitude $B,$ as shown in Figure $31-42 .$ The particle follows a circular path. Suppose the angular momentum of the particle about the center of its circular motion is quantized in the following way: $m v r=n \hbar$ where $n=1,2,3, \ldots,$ and $\hbar=h / 2 \pi$
a. Show that the radii of its allowed orbits have the following values:
$$r_{n}=\sqrt{\frac{n \hbar}{q B}}$$
b. Find the speed of the particle in each allowed orbit.

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Problem 83

Consider a particle of mass $m$ confined in a one-dimensional box of length $L .$ In addition, suppose the matter wave associated with this particle is analogous to a wave on a string of length $L$ that is fixed at both ends. Using the de Broglie relationship, show that (a) the quantized values of the linear momentum of the particle are
$$p_{n}=\frac{n h}{2 L} \quad n=1,2,3, \ldots$$
and (b) the allowed energies of the particle are
$$E_{n}=n^{2}\left(\frac{h^{2}}{8 m L^{2}}\right) \quad n=1,2,3, \ldots$$

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Problem 84

Suppose an argon laser emits $1.49 \times 10^{19}$ photons per second, half with a wavelength of 488.0 nm and half with a wavelength of 514.5 $\mathrm{nm} .$ What is the power output of this laser in watts?
$$\begin{array}{ll}{\text { A. } 1.49 \mathrm{W}} & {\text { B. } 5.76 \mathrm{W}} \\ {\text { C. } 5.92 \mathrm{W}} & {\text { D. } 6.07 \mathrm{W}}\end{array}$$

Yaqub Khan

Numerade Educator

Problem 85

A red krypton laser also emits $1.49 \times 10^{19}$ photons per second. Is its power output greater than, less than, or equal to the power output of the argon laser in Problem 84 ?

Yaqub Khan

Numerade Educator

Problem 86

The energy difference (in eV) between two states of a particular atom is 2.541 eV. What are the type and wavelength of the laser that corresponds to this transition?
$$\begin{array}{ll}{\text { A. } \mathrm{Kr}, 647.1 \mathrm{nm}} & {\text { B. Ar, } 514.5 \mathrm{nm}} \\ {\text { C. } \mathrm{Kr}, 568.2 \mathrm{nm}} & {\text { D. Ar, } 488.0 \mathrm{nm}}\end{array}$$

Yaqub Khan

Numerade Educator

Problem 87

How would the number of photons emitted per second by a yellow Kr laser compare to the number emitted per second by a red Kr laser that has the same power output?
\begin{equation}
\begin{array}{l}{\text { A. The yellow Kr laser emits more photons per second. }} \\ {\text { B. The yellow Kr laser emits fewer photons per second. }} \\ {\text { C. The yellow Kr laser emits the same number of photons per }} \\ {\text { second. }}\end{array}
\end{equation}

Yaqub Khan

Numerade Educator

Problem 88

Predict/Calculate REFFERING TO EXAMPLE $31-8$ Suppose the electron is in a state whose standing wave consists of two wavelengths. (a) Is the wavelength of this standing wave greater than or less than $1.33 \times 10^{-9} \mathrm{m} ?$ (b) Find the wavelength of this standing wave.

Yaqub Khan

Numerade Educator

Problem 89

Referring to Example $31-8$ (a) Which state has a de Broglie wavelength of $3.99 \times 10^{-9} \mathrm{m} ?$ (b) What is the Bohr radius of this state?

Yaqub Khan

Numerade Educator