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?

Yaqub Khan

Yaqub Khan

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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?

Yaqub Khan

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Problem 3

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.

Dading Chen

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Problem 4

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 electrical potential energy. Find the distance of closest approach between the alpha particle and the gold nucleus for the case $K=3.0 \mathrm{MeV}$.

Narayan Hari

Numerade Educator

Problem 5

Find the wavelength of the Balmer series spectral line corresponding to $n=15.$

Narayan Hari

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Problem 6

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

Yaqub Khan

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Problem 7

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

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Problem 8

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

Narayan Hari

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Problem 9

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

Yaqub Khan

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Problem 10

In Table $31-1$ we see that the Paschen series corresponds to $n^{\prime}=3$ in Equation $31-2,$ and that the Brackett series corresponds to $n^{\prime}=4$ (a) Show that the ranges of wavelengths of these two series overlap. (b) Is there a similar overlap between the Balmer series and the Paschen series? Verify your answer.

Linda Winkler

Numerade Educator

Problem 11

(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:
I. The ionization energy would increase because the increased mass would mean the electron would orbit closer to the nucleus and would require more energy to move to infinity.
II. The ionization energy would decrease because a more massive electron is harder to hold in orbit, and therefore it is easier to remove the electron and leave the hydrogen ionized.
III. The ionization energy would be unchanged because, just like in gravitational orbits, the orbit of the electron is independent of its mass. As a result, there is no change in the energy required to move it to infinity.

Yaqub Khan

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Problem 12

Consider the Bohr model as applied to the following three atoms: (A) neutral hydrogen in the state $n=2 ;$ (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.

Yaqub Khan

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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.

Yaqub Khan

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Problem 14

An electron in the $n=1 \mathrm{Bohr}$ 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?

Yaqub Khan

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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.

Yaqub Khan

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Problem 17

How much energy is required to ionize hydrogen when it is in the $n=4$ state?

Narayan Hari

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Problem 18

Find the energy of the photon required to excite a hydrogen atom from the $n=2$ state to the $n=5$ state.

Narayan Hari

Numerade Educator

Problem 19

In the Bohr model, the potential energy of a hydrogen atom in the $n$ th orbit has a value we will call $U_{n}$. What is the potential energy of a hydrogen atom when the electron is in the $(n+1)$ th Bohr orbit? Give your answer in terms of $U_{n}$ and $n$.

Narayan Hari

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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.

Yaqub Khan

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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.

Yaqub Khan

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Problem 22

Initially, an electron is in the $n=3$ state of hydrogen. If this electron acquires an additional $1.23 \mathrm{eV}$ of energy, what is the value of $n$ in the final state of the electron?

Narayan Hari

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Problem 23

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

Yaqub Khan

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Problem 24

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).

Guilherme Barros

Guilherme Barros

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Problem 25

Consider the following four transitions in a hydrogen atom:
(i) $n_{i}=2, n_{f}=6$
(ii) $n_{i}=2, n_{\mathrm{f}}=8$
(iii) $n_{i}=7, n_{f}=8$
(iv) $n_{i}=6, n_{f}=2$
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.

Guilherme Barros

Guilherme Barros

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Problem 26

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 $207 m_{e}$. (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.

Linda Winkler

Numerade Educator

Problem 27

(a) Find the radius of the $n=4$ Bohr orbit of a doubly ionized lithium atom $\left(\mathrm{Li}^{2+}, \mathrm{Z}=3\right)$. (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.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 28

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

Yaqub Khan

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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 $10^{-8} \mathrm{s}$ -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.

Yaqub Khan

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Problem 30

The kinetic energy of an electron in a particular Bohr orbit of hydrogen is $1.35 \times 10^{-19}$ 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).

Linda Winkler

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Problem 31

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).

Linda Winkler

Numerade Educator

Problem 32

Consider a head-on collision between two hydrogen atoms, both initially in their ground state and moving with the same speed. Find the minimum speed necessary to leave both atoms in their $n=2$ state after the collision.

Narayan Hari

Numerade Educator

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 $n_{f}=n-1 .$ (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 $n$.

Yaqub Khan

Numerade Educator

Problem 34

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

Linda Winkler

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Problem 35

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

Yaqub Khan

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Problem 36

Find an expression for the de Broglie wavelength of an electron in the $n$th state of the hydrogen atom.

Narayan Hari

Numerade Educator

Problem 37

What is the radius of the hydrogen-atom Bohr orbit shown in Figure $31-30 ?$

Yaqub Khan

Numerade Educator

Problem 38

(a) Find the kinetic energy (in eV) of an electron whose de Broglie wavelength is equal to $0.5 \mathrm{A},$ a typical atomic size. (b) Repeat part (a) for an electron with a wavelength equal to $10^{-15} \mathrm{m},$ a typical nuclear size.

Narayan Hari

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Problem 39

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

Yaqub Khan

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Problem 40

How many different values of $m_{\ell}$ are possible when the principal quantum number is $n=4 ?$

Narayan Hari

Numerade Educator

Problem 41

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) $\sqrt{36}(h / 2 \pi)$.

Yaqub Khan

Numerade Educator

Problem 42

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 a tom'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. (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.

Linda Winkler

Numerade Educator

Problem 43

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 a tom can have, what is its energy?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 44

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.

Linda Winkler

Numerade Educator

Problem 45

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 $L_{\mathrm{II}}$ 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 $L_{\mathrm{II}}$ ? Explain. (b) Calculate the ratio $L_{\mathrm{I}} / L_{\mathrm{II}}$.

Narayan Hari

Numerade Educator

Problem 46

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

Yaqub Khan

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Problem 47

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

Yaqub Khan

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Problem 48

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?

Yaqub Khan

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Problem 49

Give the electronic configuration for the ground state of carbon.

Yaqub Khan

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Problem 50

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

Yaqub Khan

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Problem 51

Give the electronic configuration for the ground state of nitrogen.

Narayan Hari

Numerade Educator

Problem 52

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 $3 \mathrm{s}$ subshell.

Narayan Hari

Numerade Educator

Problem 53

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.

Yaqub Khan

Numerade Educator

Problem 54

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

Yaqub Khan

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Problem 55

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

Yaqub Khan

Numerade Educator

Problem 56

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

Yaqub Khan

Numerade Educator

Problem 57

Generalize the results of Problem 56 and show that the number of different sets of quantum numbers for the $n$ th shell is $2 n^{2}$.

Narayan Hari

Numerade Educator

Problem 58

Suppose that the $5 d$ subshell is filled in a certain atom. Write out the 10 sets of four quantum numbers $\left(n, \ell, m_{c}, m_{s}\right)$ for the electrons in this subshell.

Linda Winkler

Numerade Educator

Problem 59

(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 in creased? (b) Choose the best explanation from among the following:
I. Increasing the energy of the incoming electrons will in crease the wavelength of the emitted $X$
rays.
II. When the energy of the incoming electrons is increased, the energy of the $X$ -rays is also increased; this, in turn, decreases the wavelength.
III. The wavelength of characteristic X-rays depends only on the material used in the metal target, and does not change if the energy of incoming electrons is increased.

Linda Winkler

Numerade Educator

Problem 60

Is the wavelength of the radiation that excites a fluorescent material greater than, less than, or equal to the wavelength of the radiation the material emits? Explain.

Linda Winkler

Numerade Educator

Problem 61

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

Yaqub Khan

Numerade Educator

Problem 62

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

Yaqub Khan

Numerade Educator

Problem 63

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

Yaqub Khan

Numerade Educator

Problem 64

An electron drops from the L shell to the K shell and gives off an X-ray with a wavelength of 0.0205 nm. What is the atomic number of this atom?

Linda Winkler

Numerade Educator

Problem 65

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 $\mathrm{K}_{\alpha}$ X-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 $\mathrm{K}_{a} \mathrm{X}$ -rays.

Linda Winkler

Numerade Educator

Problem 66

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?

Narayan Hari

Numerade Educator

Problem 67

Consider the following three transitions in a hydrogen atom: (A) $n_{1}=5, n_{f}=2$ (B) $n_{1}=7, n_{f}=2 ;(C) n_{1}=7, n_{f}=6 .$ Rank the transitions in order of increasing (a) wavelength and (b) frequency of the emitted photon. Indicate ties where appropriate.

Narayan Hari

Numerade Educator

Problem 68

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.

Yaqub Khan

Numerade Educator

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.

Yaqub Khan

Numerade Educator

Problem 69

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.

Yaqub Khan

Numerade Educator

Problem 70

The electronic configuration of a particular potassium atom is $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{1}$. Is this atom in its ground state or in an excited state? Explain.

Linda Winkler

Numerade Educator

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.

Yaqub Khan

Numerade Educator

Problem 72

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

Yaqub Khan

Numerade Educator

Problem 73

It was pointed out in Section $31-3$ that intermolecular collisions at room temperature do not have enough energy to cause an excitation in hydrogen from the $n=1$ state to the $n=2$ state. Given that the average kinetic energy of a hydrogen atom in a high-temperature gas is $\frac{3}{2} \mathrm{KT}$ (where $k$ is Boltzmann's constant), find the minimum temperature required for atoms to have enough thermal energy to excite electrons from the ground state to the $n=2$ state.

Linda Winkler

Numerade Educator

Problem 74

The electron in a hydrogen atom makes a transition from the $n=4$ state to the $n=2$ state, as indicated in Figure $31-31$. (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.

Linda Winkler

Numerade Educator

Problem 75

Referring to Problem $74,$ find (a) the energy of the emitted photon and (b) the kinetic energy of the hydrogen atom after the photon is emitted. (c) Do you expect the sum of the energies in parts (a) and (b) to be greater than, less than, or the same as the difference in energy between the $n=4$ and $n=2$ states of hydrogen? Explain.

Linda Winkler

Numerade Educator

Problem 76

Laser Eye Surgery In laser eye surgery, the laser emits a 1.45 -ns pulse focused on a spot that is $34.0 \mu \mathrm{m}$ in diameter. (a) If the energy contained in the pulse is $2.75 \mathrm{mJ},$ what is the power per square meter (the irradiance) associated with this beam? (b) Suppose a molecule with a diameter of $0.650 \mathrm{nm}$ 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.)

Cyra Jelle Calleja

Cyra Jelle Calleja

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Problem 77

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.

Yaqub Khan

Numerade Educator

Problem 78

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?

Yaqub Khan

Numerade Educator

Problem 79

The element helium is named for the Sun because that is where it was first observed. (a) What is the shortest wavelength that one would expect to observe from a singly ionized helium atom in the atmosphere of the Sun? (b) Suppose light with a wavelength of $388.9 \mathrm{nm}$ is observed from singly ionized helium. What are the initial and final values of the quantum number $n$ corresponding to this wavelength?

Linda Winkler

Numerade Educator

Problem 80

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.

Yaqub Khan

Numerade Educator

Problem 81

Find the approximate wavelength of $\mathrm{K}_{\beta}$ X-rays emitted by molybdenum $(Z=42)$, and compare your result with Figure $31-22 .$ (Hint: An electron in the M shell is shielded from the nucleus by the single electron in the $\mathrm{K}$ shell, plus all the electrons in the L shell.)

Yaqub Khan

Numerade Educator

Problem 82

Referring to the hint given in Problem 81 , estimate the wavelength of $L_{\alpha}$ X-rays in molybdenum.

Linda Winkler

Numerade Educator

Problem 83

The Pickering Series ln 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 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.

Linda Winkler

Numerade Educator

Problem 84

Rydberg Atoms There is no limit to the size a hydrogen atom can attain, provided it is free from disruptive outside 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 $n-1$. (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.

Linda Winkler

Numerade Educator

Problem 85

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 $31-32$. 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: $m v r=n h,$ where $n=1,2,3, \ldots,$ and $\hbar=h / 2 \pi.$
A. Show that the radii of its allowed orbits have the following values:
$$
r_{n}=\sqrt{\frac{n h}{q B}}
$$
b. Find the speed of the particle in each allowed orbit.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 $31-32$. 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: $m v r=n h,$ where $n=1,2,3, \ldots,$ and $\hbar=h / 2 \pi.$
A. Show that the radii of its allowed orbits have the following values:
$$r_{n}=\sqrt{\frac{n h}{q B}}$$
b. Find the speed of the particle in each allowed orbit.

Linda Winkler

Numerade Educator

Problem 86

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
$$E_{n}=n^{2}\left(\frac{h^{2}}{8 m L^{2}}\right) \quad n=1,2,3, \ldots$$

Yaqub Khan

Numerade Educator

Problem 87

Show that the time required for an electron in the $n$ th Bohr orbit of hydrogen to circle the nucleus once is given by
$$T=T_{1} n^{3} \quad n=1,2,3, \ldots$$
where $T_{1}=h^{3} / 4 \pi^{2} m k^{2} e^{4}.$

Narayan Hari

Numerade Educator

Problem 88

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

Yaqub Khan

Numerade Educator

Problem 89

A different type of laser also emits $1.49 \times 10^{19}$ photons per second. If all of its photons have a wavelength of $414.0 \mathrm{nm},$ is its power output greater than, less than, or equal to the power output of the argon laser in Problem $88 ?$

Narayan Hari

Numerade Educator

Problem 90

What is the power output of the laser in Problem $89 ?$
A. $1.23 \mathrm{W}$
B. $2.39 \mathrm{W}$
C. $4.80 \mathrm{W}$
D. $7.16 W$

Narayan Hari

Numerade Educator

Problem 91

- What is the energy difference (in eV) between the states of an argon atom that are responsible for a photon with a wavelength of $514.5 \mathrm{nm} ?$
A. $2.13 \mathrm{eV}$
B. $2.42 \mathrm{eV}$
C. $3.87 \mathrm{eV}$
D. $6.40 \mathrm{eV}$

Narayan Hari

Numerade Educator

Problem 92

What is the energy difference (in eV) between the states of an argon atom that are responsible for a photon with a wavelength of $514.5 \mathrm{nm} ?$
A. $2.13 \mathrm{eV}$
B. $2.42 \mathrm{eV}$
C. $3.87 \mathrm{eV}$
D. $6.40 \mathrm{eV}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 93

Referring to Example $31-3$ (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?

Narayan Hari

Numerade Educator