College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 27

Atomic Physics

Educators


Problem 1

The electron in a hydrogen atom spends most of its time $0.53 \times 10^{-10} \mathrm{m}$ from the nucleus, whose radius is about $0.88 \times 10^{-15} \mathrm{m} .$ If each dimension of this atom was increased by the same factor and the radius of the nucleus was increased to the size of a tennis ball, how from the nucleus would the electron be?

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Problem 2

A single layer of gold atoms lies on a table. The radius of each gold atom is about $1.5 \times 10^{-10} \mathrm{m},$ and the radius of each gold nucleus is about $7 \times 10^{-15} \mathrm{m} .$ A particle much smaller than the nucleus is shot at the layer of gold atoms. Roughly, what is its chance of hitting a nucleus and being scattered? (The electrons around the atom have no effect.)

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Problem 3

(a) Determine the mass of a gold foil that is 0.010 $\mathrm{cm}$ thick and whose area is 1 $\mathrm{cm} \times 1 \mathrm{cm} .$ The density of gold is $19,300 \mathrm{kg} / \mathrm{m}^{3} .$ (b) Determine the number of gold atoms in the foil if the mass of each atom is $3.27 \times 10^{-25} \mathrm{kg} .(\mathrm{c})$ The radius of a gold nucleus is $7 \times 10^{-15} \mathrm{m} .$ Determine the area of a circle with this radius. (d) Determine the chance that an alpha particle passing through the gold foil will hit a gold nucleus. Ignore the alpha particle's size and assume that all gold nuclei are exposed to it; that is, no gold nuclei are hidden behind other nuclei.

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Problem 4

An object of mass $M$ moving at speed $v_{0}$ has a direct elastic collision with a second object of mass $m$ that is at rest. Using the energy and momentum conservation principles (Chapters 5 and 6 ), show that the final velocity of the object of mass $M$ is $v=(M-m) v_{0} /(M+m) .$ Using this result, determine the final velocity of an alpha particle following a head-on collision with (a) an electron at rest and (b) a gold nucleus at rest. The alpha particle's velocity before the collision is 0.010$c$ $m_{\mathrm{alpha}}=6.6 \times 10^{-27} \mathrm{kg} ; \quad m_{\mathrm{electron}}=9.11 \times 10^{-31} \mathrm{kg} ; \quad$ and $m_{\mathrm{gold} \text { nucleus }}=3.3 \times 10^{-25} \mathrm{kg} .(\mathrm{c})$ Based on your answers, could an alpha particle be deflected backward by hitting an electron in a gold atom?

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Problem 5

Describe what happens to the energy of the atom and represent your reasoning with an energy bar chart when (a) a hydrogen atom emits a photon and (b) a hydrogen atom absorbs a photon.

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Problem 6

How do we know that the energy of the hydrogen atom in the ground state is -13.6 eV?

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Problem 7

Determine the wavelength, frequency, and photon energies of the line with n = 5 in the Balmer series.

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Problem 8

Determine the wavelengths, frequencies, and photon energies (in electron volts) of the first two lines in the Balmer series. In what part of the electromagnetic spectrum do the lines appear?

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Problem 9

Invent an equation An imaginary atom is observed to emit electromagnetic radiation at the following wavelengths: 250 nm, 2250 nm, 6250 nm, 12,250 nm, . . . . Invent an empirical equation for calculating these wavelengths; that is, determine $\lambda=f(n),$ where $f$ is an unknown function of an integer
$n,$ which can have the values $1,2,3,4, \ldots$

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Problem 10

Write three basic equations to derive the expressions for the allowed radii and the allowed energy of electron states in the Bohr model of the atom.

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Problem 11

If we know the value of n for the orbit of an electron in a hydrogen atom, we can determine the values of three other quantities related to either the electron’s motion or the atom as a whole. Briefly describe these quantities.

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Problem 12

Is it possible for a hydrogen atom to emit an X-ray? If so, describe the process and estimate the n value for the energy state. If not, indicate why not.

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Problem 13

A gas of hydrogen atoms in a tube is excited by collisions with free electrons. If the maximum excitation energy gained by an atom is 12.5 eV, determine all of the wavelengths of light emitted from the tube as atoms return to the ground state.

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Problem 14

Some of the energy states of a hypothetical atom, in units of electron volts, are $E_{1}=-31.50 E_{2}=-12.10, E_{3}=-5.20$ and $E_{4}=-3.60 .(\text { a })$ Draw an energy diagram for this atom. (b) Determine the energy and wavelength of the least energetic photon that can be absorbed by these atoms when initially in their ground state.

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Problem 15

An atom in an excited state usually remains in that state only about $10^{-8}$ s before transitioning to a lower energy state. How many times will an electron in the $n=3$ state of hydrogen move around the $n=3$ orbit before the atom transitions to the $n=2$ or $n=1$ state?

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Problem 16

Determine the speed and frequency of an electron moving around the first Bohr orbit in hydrogen. According to classical physics, the atom should emit electromagnetic radiation at this frequency. In what portion of the electromagnetic spectrum is this frequency?

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Problem 17

Show that the frequency of revolution of an electron around the nucleus of a hydrogen atom is $f=\left(4 \pi^{2} k^{2} e^{4} m / h^{3}\right)\left(1 / n^{3}\right)$

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Problem 18

Are we justified in using nonrelativistic energy equations in the Bohr theory for hydrogen? (That is, is the electron’s speed smaller than 0.1c?)

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Problem 19

Determine the ratio of the electric force between the nucleus and an electron in the ground state of the hydrogen atom and the gravitational force between the two particles. Based on your answer, are we justified in ignoring the gravitational force in the Bohr theory?

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Problem 20

A group of hydrogen atoms in a discharge tube emit violet light of wavelength 410 nm. Determine the quantum numbers of the atom’s initial and final states when undergoing this transition.

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Problem 21

Draw an energy state diagram for a hydrogen atom and explain (a) how an emission spectrum is formed and (b) how an absorption spectrum is formed.

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Problem 22

Explain what spectral lines could be emitted by hydrogen gas in a gas discharge tube with an 11.5-V potential difference across it. What assumptions did you make?

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Problem 23

The fractional population of an excited state of energy $E_{n}$ compared to the population of the ground of energy $E_{o}$ is $N_{n} / N_{o}=e^{-\left(E_{n}-E_{0}\right) / k T} .$ At approximately what temperature $T$ are 20 percent of hydrogen atoms in the first excited state?

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Problem 24

Draw an energy bar chart that describes the ionization process for a hydrogen atom (a) due to collisions with other atoms and (b) when placed in an external electric field.

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Problem 25

(a) A laser pulse emits 2.0 $\mathrm{J}$ of energy during $1.0 \times 10^{-9} \mathrm{s} .$ Determine the average power emitted during that short time interval. (b) Determine the average light intensity (power per unit area) in the laser beam if its cross-sectional area is $8.0 \times 10^{-9} \mathrm{m}^{2}$

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Problem 26

A pulsed laser used for welding produces 100 $\mathrm{W}$ of power during 10 $\mathrm{ms}$ . Determine the energy delivered to the weld.

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Problem 27

Welding the retina A pulsed argon laser of 476.5 -nm wave- length emits $3.0 \times 10^{-3} \mathrm{J}$ of energy to produce a tiny weld to repair a detached retina. How many photons are in the laser pulse?

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Problem 28

More welding the retina A laser used to weld the damaged retina of an eye emits 20 mW of power for 100 ms. The light is focused on a spot 0.10 mm in diameter. Assume that the laser’s energy is deposited in a small sheet of water of 0.10-mm diameter and 0.30-mm thickness. (a) Determine the energy deposited. (b) Determine the mass of this water. (c) Determine the increase in temperature of the water (assume that it does not boil and that its heat capacity is 4180 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{C}^{\circ} )$

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Problem 29

(a) An electron moves counterclockwise around a nucleus in a horizontal plane. Assuming that it behaves as a classical particle, what is the direction of the magnetic field produced by the electron? (b) The same one-electron atom is placed in an external magnetic field whose $\vec{B}$ field points horizontally from right to left. Draw a picture showing what happens to the atom.

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Problem 30

Estimate the wavelength of a tennis ball after a good serve.

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Problem 31

What is the wavelength of the electron in a hydrogen atom at a distance of one Bohr radius from the nucleus?

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Problem 32

Estimate the average wavelength of hydrogen atoms at room temperature.

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Problem 33

Describe how you will determine the wavelength of an electron in a cathode ray tube if you know the potential difference between the electrodes.

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Problem 34

An electron first has an infinite wavelength and then after it travels through a potential difference has a de Broglie wave-length of $1.0 \times 10^{-10} \mathrm{m} .$ What is the potential difference that it traversed? Draw a picture of the situation and an energy bar chart, and explain why the electron's wavelength decreased.

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Problem 35

Describe two experiments whose outcomes can be explained using the concept of the de Broglie wavelength.

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Problem 36

How does the wavelength of an electron relate to the radius of its orbit in a hydrogen atom? Why?

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Problem 37

Discuss the similarities and differences in the way a hydrogen atom is pictured in Bohr’s model and in quantum mechanics.

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Problem 38

A high-energy particle scattered from the nucleus of an atom helps determine the size and shape of the nucleus. For best results, the de Broglie wavelength of the particle should be the same size as the nucleus (approximately $10^{-14} \mathrm{m} )$ or smaller. If the mass of the particle is $6.6 \times 10^{-27} \mathrm{kg},$ at what speed must it travel to produce a wavelength of $10^{-14} \mathrm{m} ?$

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Problem 39

(a) Use de Broglie's hypothesis to determine the speed of the electron in a hydrogen atom when in the $n=1$ orbit. The radius of the orbit is $0.53 \times 10^{-10} \mathrm{m} .$ (b) Determine the electron's de Broglie wavelength. (c) Confirm that the circumference of the orbit equals one de Broglie wavelength.

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Problem 40

Repeat Problem 39 for the $n=3$ orbit, whose radius is $4.77 \times 10^{-10} \mathrm{m} .$ Three de Broglie wavelengths should fit around the $n=3$ orbit.

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Problem 41

A beam of electrons accelerated in an electric field is passing through two slits separated by a very small distance d and then hits a screen that glows when an electron hits it. What do you need to know about the electrons to be able to predict where on the screen, which is L meters from the slits, you will see the brightest and the darkest spots?

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Problem 42

(a) An electron is in an $n=4$ state. List the possible values of its $l$ quantum number. (b) If the electron happens to be in an $l=3$ state, list the possible values of its $m_{l}$ quantum number. (c) List the possible values of its $m_{s}$ quantum number.

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Problem 43

(a) An electron in an atom is in a state with $m_{l}=3$ and $m_{s}=+1 / 2 .$ What are the smallest possible values of $l$ and $n$ for that state? (b) Repeat for an $m_{l}=2$ and $m_{s}=-1 / 2$ state.

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Problem 44

(a) An atom is in the $n=7$ state. List the possible values of the electron's $l$ quantum number. (b) Of these different states, the electron occupies the $l=4$ state. List the possible values of the $m_{l}$ quantum number.

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Problem 45

Draw schematic orbits and arrows representing the different $m_{l}$ quantum states for an $l=2$ atomic electron in a magnetic field.

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Problem 46

List the $n, l, m_{b}$ and $m_{s}$ states available for an electron in a 4$p$ subshell.

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Problem 47

List the $n, l, m_{b}$ and $m_{s}$ states available for an electron in a 4$f$ subshell.

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Problem 48

Identify the values of $n$ and $l$ for each of the following subshell designations: $3 s, 2 p, 4 d, 5 f,$ and 6$s .$

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Problem 49

(a) Determine the electron configuration of sulfur (its atomic number is 16 ). (b) Why are sulfur and oxygen (atomic number 8 ) in the same group on the periodic table?

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Problem 50

(a) Determine the electron configuration of silicon (atomic number 14$) .$ (b) Why are carbon and silicon in the same group of the periodic table?

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Problem 51

Determine the electron configuration for iron (atomic number 26$) .$

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Problem 52

Manganese (atomic number 25$)$ has two 4 s electrons. How many 3$d$ electrons does it have? Explain your answer.

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Problem 53

Determine the electron configurations of four elements of group I in the periodic table. Explain why these elements are likely to have similar properties. Note that a higher $s$ shell fills before the next lower $d$ shell- the electron in the $s$ shell spends more time on average closer to the nucleus.

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Problem 54

Determine the electron configurations of three elements in group VI of the periodic table. Explain why these elements are likely to have similar properties.

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Problem 55

Describe the experimental evidence that supports the concept of the uncertainty principle.

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Problem 56

If you assume that the uncertainty in our knowledge of Bohr’s radius of the atom is 1% of the value of the radius, then what is the minimum uncertainty in our knowledge of the electron’s momentum? What component of momentum is it—radial or tangential? What does this uncertainty mean for our interpretation of the atomic orbits?

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Problem 57

The lifetime of the hydrogen atom in the $n=3$ second excited state is $10^{-9}$ s. What is the uncertainty of that energy state of the atom? Compare this uncertainty with the magnitude of the $-1.51$ eV energy of the atom in that state.

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Problem 58

Use the uncertainty principle to discuss whether lasers can emit 100$\%$ monochromatic light.

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Problem 59

(a) Determine the radii and energies of the $n=1,2,$ and 3 states in the $\mathrm{He}^{+}$ ion (it has two protons in its nucleus and one electron). (b) Construct an energy state diagram for this ion. Indicate any assumptions that you made.

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Problem 60

(a) Determine the radii and energies of the $n=1,2,$ and 3 states of a sodium ion in which 10 of its electrons have been removed. (b) Construct an energy state diagram for the ion. Indicate any assumptions that you made.

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Problem 61

A uranium atom with $Z=92$ has 92 protons in its nucleus. It has two electrons in an $n=1$ orbit. Estimate the radius of this orbit. Indicate any assumptions that you made.

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Problem 62

Estimate the energy needed to remove an electron from (a) the $n=1$ state of iron $(Z=26 \text { is the number of protons }$ in the nucleus) and (b) the $n=1$ state of hydrogen $(Z=1)$ .

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Problem 63

An electron in a hydrogen atom changes from the $n=4$ to the $n=3$ state. Determine the wavelength of the emitted photon.

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Problem 64

An electron in a He^ $^{+}$ ion changes its energy from the $n=3$ to the $n=1$ state. Determine the wavelength, frequency, and energy of the emitted photon.

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Problem 65

Determine the energy, frequency, and wavelength of a photon whose absorption changes a He^ $^{+}$ ion from the $n=1$ to the $n=6$ state.

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Problem 66

A helium ion He^ $^{+}$ emits an ultraviolet photon of wave-length 164 $\mathrm{nm}$ . Determine the quantum numbers of the ion's initial and final states.

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Problem 67

The average thermal energy due to the random translational motion of a hydrogen atom at room temperature is $(3 / 2) k T$ Here $k$ is the Boltzmann constant. Would a typical collision between two hydrogen atoms be likely to transfer enough energy to one of the atoms to raise its energy from the $n=1$ to the $n=2$ energy state? Explain your answer. [Note: Earth's free hydrogen is in the molecular form $\mathrm{H}_{2}$ . However, the above reasoning is similar for atomic and molecular hydrogen.]

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Problem 68

Which answer below is closest to the speed of an electron accelerated from rest across a 100-V potential difference?
$$
\begin{array}{ll}{\text { (a) } 600 \mathrm{m} / \mathrm{s}} & {\text { (b) } 6 \times 10^{4} \mathrm{m} / \mathrm{s}} \\ {\text { (c) } 6 \times 10^{6} \mathrm{m} / \mathrm{s}} & {\text { (d) } 6 \times 10^{7} \mathrm{m} / \mathrm{s}}\end{array}
$$

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Problem 69

Which answer below is closest to the wavelength of an electron accelerated from rest across a $100-\mathrm{V}$ potential difference?
$$
\begin{array}{ll}{\text { (a) } 1 \times 10^{-11} \mathrm{m}} & {\text { (b) } 1 \times 10^{-10} \mathrm{m}}\end{array}
$$
$$
\text { (c) } 1 \times 10^{-9} \mathrm{m} \quad \text { (d) } 1 \times 10^{-8} \mathrm{m}
$$

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Problem 70

Compared to the electron in Problem 69, the wavelength of an electron accelerated from rest across a 10,000-V potential difference would be
(a) the same length (b) 10 times longer
(c) 1000 times longer (d) 1>10 times as long
(e) 1/1000 times as long

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Problem 71

An electron in the SEM electron beam is moving parallel to the paper and downward toward the bottom of the page. In which direction should a magnetic field point to deflect the electron toward the right as seen while looking at the page?
(a) Right
(b) Left
(c) Toward the top of the page
(d) Toward the bottom of the page
(e) Into the paper (f) Out of the paper

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Problem 72

The SEM can also detect the types of atoms in the sample by measuring which of the following?
(a) Energy of the secondary electrons
(b) The wavelengths of X-rays emitted from the sample
(c) Both (a) and (b)

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Problem 73

The secondary electrons detected by the SEM detector are
(a) outer electrons knocked out of atoms on or near the surface of the sample.
(b) X-rays produced when outer electrons fall into vacant inner electron orbits.
(c) electrons in the electron beam slowed by passing through the top layer of the sample.
(d) none of the above. (e) (a)–(c) are correct.

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Problem 74

If all of the electron resting sites were in a line along the length of an ETC, what would be the approximate distance from one site to the next?
(a) 400 nm
(b) 20 nm
(c) 1 nm
(d) 11>202 nm

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Problem 75

Electron tunneling involves electrons
(a) jumping to a conduction band and then falling down at a different place.
(b) diffusing with a small protein from one place to another.
(c) converting to a photon and moving at light speed to another site.
(d) passing through a potential barrier to a different location.
(e) having an uncertain energy for a short time interval.
(f) (d) and (e)

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Problem 76

An electron with mass $9.1 \times 10^{-31} \mathrm{kg}$ and kinetic energy 20 $\mathrm{eV}$ is in a potential well that is 0.15 $\mathrm{nm}$ wide. Which answer is closest to the electron's speed?
$$
\begin{array}{ll}{\text { (a) } 3 \times 10^{3} \mathrm{m} / \mathrm{s}} & {\text { (b) } 3 \times 10^{4} \mathrm{m} / \mathrm{s}} \\ {\text { (c) } 3 \times 10^{5} \mathrm{m} / \mathrm{s}} & {\text { (d) } 3 \times 10^{6} \mathrm{m} / \mathrm{s}}\end{array}
$$
$$
\text { (e) } 9 \times 10^{12} \mathrm{s}^{-1}
$$

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Problem 77

The electron from Problem 76 is in a potential well that is 0.15 nm wide. Which answer is closest to the frequency with which the electron hits one side of the well?
$$
\begin{array}{ll}{\text { (a) } 9 \times 10^{16} \mathrm{s}^{-1}} & {\text { (b) } 9 \times 10^{15} \mathrm{s}^{-1}} \\ {\text { (c) } 9 \times 10^{14} \mathrm{s}^{-1}} & {\text { (d) } 9 \times 10^{13} \mathrm{s}^{-1}} \\ {\text { (e) } 9} { \times 10^{12} \mathrm{s}^{-1}}\end{array}
$$

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Problem 78

Suppose the barrier height above the electron energy $\left(U_{\text { barrier }}-E_{\text { electron }}\right)$ is 1 eV and that the barrier is 2 nm wide. According to the uncertainty principle, what is the minimum time interval that the electron's energy is in this classically forbidden region?
$$
\begin{array}{ll}{\text { (a) } 3 \times 10^{-16} \mathrm{s}} & {\text { (b) } 3 \times 10^{-15} \mathrm{s}} \\ {\text { (c) } 3 \times 10^{-14} \mathrm{s}} & {\text { (d) } 3 \times 10^{-13} \mathrm{s}}\end{array}
$$
$$
\text { (e) } 3 \times 10^{-12} \mathrm{s}
$$

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Problem 79

Electron transport chains are fundamental parts of:
(a) The conversion of glucose into energetic molecules used in the body
(b) Molecules used for fuel in trains and other vehicles
(c) The photosynthetic conversion of light into stable chemical compounds
(d) The passing of electric current in nerve cells
(e) (a) and (c) (f) (a), (c), and (d)

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Problem 80

The conduction band model of electron transport in electron transport chains may not work because:
(a) The electron transport chain is not entirely metal ions.
(b) The conduction band is too localized.
(c) The electron has little chance of being “promoted” into the conduction band.
(d) The visible spectrum of the resting site is not a broad band as is expected.
(e) (c) and (d) (f) (a), (b), (c), and (d)

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