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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 28

Atomic Physics

Educators


Problem 1

The wavelengths of the Lyman series for hydrogen are given by
$$\frac{1}{\lambda}=R_{\mathrm{H}}\left(1-\frac{1}{n^{2}}\right) \qquad n=2,3,4, \ldots$$
(a) Calculate the wavelengths of the first three lines in this
series. (b) Identify the region of the electromagnetic spectrum in which these lines appear.

Guilherme B.
Numerade Educator

Problem 2

The wavelengths of the Paschen series for hydrogen are given by\
$$\frac{1}{\lambda}=R_{\mathrm{H}}\left(\frac{1}{3^{2}}-\frac{1}{n^{2}}\right) \qquad n=4,5,6, \ldots$$
(a) Calculate the wavelengths of the first three lines in this series. (b) Identify the region of the electromagnetic spectrum in which these lines appear.

Guilherme B.
Numerade Educator

Problem 3

The “size” of the atom in Rutherford’s model is about $1.0 \times 10^{-10} \mathrm{m} .$ (a) Determine the attractive electrostatic force between an electron and a proton separated by this distance. (b) Determine (in eV) the electrostatic potential energy of the atom.

Guilherme B.
Numerade Educator

Problem 4

An isolated atom of a certain element emits light of wavelength 520. nm when the atom falls from its fifth excited state into its second excited state. The atom emits a photon of wavelength 410. nm when it drops from its sixth excited state into its second excited state. Find the wavelength of the light radiated when the atom makes a transition from its sixth to its fifth excited state.

Guilherme B.
Numerade Educator

Problem 5

The “size” of the atom in Rutherford’s model is about $1.0 \times 10^{-10} \mathrm{m} .$ (a) Determine the speed of an electron moving about the proton using the attractive electrostatic force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered in studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton. (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must be considered in studying the atom?

Guilherme B.
Numerade Educator

Problem 6

In a Rutherford scattering experiment, an a - particle (charge $=+2 e )$ heads directly toward a gold nucleus (charge $=+79 e ) .$ The $\alpha$ -particle had a kinetic energy of 5.0 $\mathrm{MeV}$ when very far $(r \rightarrow \infty)$ from the nucleus. Assuming the gold
nucleus to be fixed in space, determine the distance of closest approach. Hint: Use conservation of energy with $P E=k_{e} q_{1} q_{2} / r$.

Guilherme B.
Numerade Educator

Problem 7

The so-called Lyman $-\alpha$ photon is the lowest energy photon in the Lyman series of hydrogen and results from an electron transitioning from the n 5 2 to the n 5 1 energy level. Determine (a) the energy in eV and (b) the wavelength in nm of a Lyman - a photon.

Guilherme B.
Numerade Educator

Problem 8

Determine the energies in eV of the (a) second and (b) third energy levels of the hydrogen atom. Calculate the orbital radius in nm of an electron in hydrogen’s (c) second and (d) third energy levels.

Guilherme B.
Numerade Educator

Problem 9

Singly ionized helium (He $^{+} )$ is a hydrogen-like atom. Determine the energy in eV required to raise a $\mathrm{He}^{+}$ electron from the $n=1$ to the $n=2$ energy level.

Guilherme B.
Numerade Educator

Problem 10

What is the (a) energy in eV and (b) wavelength in $\mu \mathrm{m}$ of a photon that, when absorbed by a hydrogen atom, could cause a transition from the $n=3$ to the $n=6$ energy level?

Guilherme B.
Numerade Educator

Problem 11

A hydrogen atom is in its first excited state $(n=2)$. Using the Bohr theory of the atom, calculate (a) the radius of the orbit, (b) the linear momentum of the electron, (c) the angular momentum of the electron, (d) the kinetic energy, (e) the potential energy, and (f) the total energy.

Guilherme B.
Numerade Educator

Problem 12

For a hydrogen atom in its ground state, use the Bohr model to compute (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electrical potential energy of the atom.

Guilherme B.
Numerade Educator

Problem 13

Show that the speed of the electron in the nth Bohr orbit in hydrogen is given by
$$v_{n}=\frac{k_{e} e^{2}}{n \hbar}$$

Guilherme B.
Numerade Educator

Problem 14

A photon is emitted when a hydrogen atom undergoes a transition from the $n=5$ state to the $n=3$ state. Calculate (a) the wavelength, (b) the frequency, and (c) the energy (in eV) of the emitted photon.

Guilherme B.
Numerade Educator

Problem 15

A hydrogen atom emits a photon of wavelength 656 nm. From what energy orbit to what lower - energy orbit did the electron jump?

Guilherme B.
Numerade Educator

Problem 16

Following are four possible transitions for a hydrogen atom
$$\begin{array}{ll}{\text { I. } n_{i}=2 ; n_{f}=5} & {\text { II. } n_{i}=5 ; n_{f}=3} \\ {\text { III. } n_{i}=7 ; n_{f}=4} & {\text { IV. } n_{i}=4 ; n_{f}=7}\end{array}$$
(a) Which transition will emit the shortest - wavelength photon? (b) For which transition will the atom gain the most energy? (c) For which transition(s) does the atom lose energy?

Guilherme B.
Numerade Educator

Problem 17

What is the energy of a photon that, when absorbed by a hydrogen atom, could cause an electronic transition from (a) the $n=2$ state to the $n=5$ state and (b) the $n=4$ state to
the $n=6$ state?

Guilherme B.
Numerade Educator

Problem 18

A hydrogen atom initially in its ground state $(n=1)$ absorbs a photon and ends up in the state for which $n=3$. (a) What is the energy of the absorbed photon? (b) If the atom eventually returns to the ground state, what photon energies could the atom emit?

Guilherme B.
Numerade Educator

Problem 19

The Balmer series for the hydrogen atom corresponds to electronic transitions that terminate in the state with quantum number $n=2$ as shown in Figure P28.19. Consider the photon of longest wavelength corresponding to a transition shown in the figure. Determine (a) its energy and (b) its wavelength. Consider the spectral line of shortest wavelength corresponding to a transition shown in the figure. Find (c) its photon energy and (d) its wavelength. (e) What is the shortest possible wavelength in the Balmer series?

Guilherme B.
Numerade Educator

Problem 20

A particle of charge $q$ and mass $m$, moving with a constant speed $v$, perpendicular to a constant magnetic field $B$, follows a circular path. If in this case the angular momentum about the center of this circle is quantized so that $m v r=2 n \hbar,$ show that the allowed radii for the particle are
$$r_{n}=\sqrt{\frac{2 n \hbar}{q B}}$$

Guilherme B.
Numerade Educator

Problem 21

(a) If an electron makes a transition from the $n=4$ Bohr orbit to the $n=2$ orbit, determine the wavelength of the photon created in the process. (b) Assuming that the atom was initially at rest, determine the recoil speed of the hydrogen atom when this photon is emitted.

Guilherme B.
Numerade Educator

Problem 22

Consider a large number of hydrogen atoms, with electrons all initially in the $n=4$ .(a) How many different wavelengths would be observed in the emission spectrum of these atoms? (b) What is the longest wavelength that could be observed? (c) To which series does the wavelength found in (b) belong?

Guilherme B.
Numerade Educator

Problem 23

A photon with energy 2.28 $\mathrm{eV}$ is absorbed by a hydrogen atom. Find (a) the minimum $n$ for a hydrogen atom that can be ionized by such a photon and (b) the speed of the electron released from the state in part (a) when it is far from the nucleus.

Guilherme B.
Numerade Educator

Problem 24

(a) Calculate the angular momentum of the Moon due to its orbital motion about Earth. In your calculation use $3.84 \times$ $10^{8} \mathrm{m}$ as the average Earth-Moon distance and $2.36 \times 10^{6} \mathrm{s}$ as the period of the Moon in its orbit. (b) If the angular
momentum of the Moon obeys Bohr's quantization rule $(L=n \hbar),$ determine the value of the quantum number $n .$ (c) By what fraction would the Earth-Moon radius have to be increased to increase the quantum number by 1?

Guilherme B.
Numerade Educator

Problem 25

An electron is in the second excited orbit of hydrogen, corresponding to $n=3 .$ Find (a) the radius of the orbit and (b) the wavelength of the electron in this orbit.

Guilherme B.
Numerade Educator

Problem 26

(a) Write an expression relating the kinetic energy $K E$ of the electron and the potential energy $P E$ in the Bohr model of the hydrogen atom. (b) Suppose a hydrogen atom absorbs a photon of energy $E,$ resulting in the transfer of the electron to a higher-energy level. Express the resulting change in the potential energy of the system in terms of $E$ . (c) What is the change in the electron's kinetic energy during this process?

Guilherme B.
Numerade Educator

Problem 27

The orbital radii of a hydrogen - like atom is given by the equation
$$r_{n}=\frac{n^{2} \hbar^{2}}{Z m_{e} k_{e} e^{2}}$$
What is the radius of the first Bohr orbit in (a) $\mathrm{He}^{+},(\mathrm{b}) \mathrm{Li}^{2+}$
and $(\mathrm{c}) \mathrm{Be}^{3+} ?$

Guilherme B.
Numerade Educator

Problem 28

Consider a Bohr model of doubly ionized lithium. (a) Write an expression similar to Equation 28.14 for the energy levels of the sole remaining electron. (b) Find the energy corresponding to $n=4 .$ (c) Find the energy corresponding to $n=2 .$ (d) Calculate the energy of the photon
emitted when the electron transits from the fourth energy level to the second energy level. Express the answer both in electron volts and in joules. (e) Find the frequency and wavelength of the emitted photon. (f) In what part of the spectrum is the emitted light?

Guilherme B.
Numerade Educator

Problem 29

A general expression for the energy levels of one - electron atoms and ions is
$$E_{n}=-\frac{\mu k_{e}^{2} q_{1}^{2} q_{2}^{2}}{2 \hbar^{2} n^{2}}$$
Here $\mu$ is the reduced mass of the atom, given by $\mu=m_{1} m_{2} /$ $\left(m_{1}+m_{2}\right),$ where $m_{1}$ is the mass of the electron and $m_{2}$ is the mass of the nucleus; $k_{e}$ is the Coulomb constant; and $q_{1}$ and $q_{2}$ are the charges of the electron and the nucleus, respectively. The wavelength for the $n=3$ to $n=2$ transition of the hydrogen atom is 656.3 $\mathrm{nm}$ (visible red light). What are the wavelengths for this same transition in (a) positronium, which consists of an electron and a positron, and (b) singly ionized helium? Note: A positron is a positively charged electron.

Guilherme B.
Numerade Educator

Problem 30

Using the concept of standing waves, de Broglie was able to derive Bohr’s stationary orbit postulate. He assumed a confined electron could exist only in states where its de Broglie waves form standing wave patterns, as in Figure 28.6. Consider a particle confined in a box of length $L$ to be equivalent to a string of length $L$ and fixed at both ends. Apply de Broglie's
concept to show that (a) the linear momentum of this particle is quantized with $p=m v=n h / 2 L$ and $(\mathrm{b})$ the allowed states correspond to particle energies of $E_{n}=n^{2} E_{0},$ where $E_{0}=h^{2} /\left(8 m L^{2}\right)$.

Guilherme B.
Numerade Educator

Problem 31

Hydrogen's single electron can occupy any of the atom's distinct quantum states. Determine the number of distinct quantum states in the (a) $n=1,(\mathrm{b}) n=2,$ and $(\mathrm{c}) n=3$ energy levels.

Guilherme B.
Numerade Educator

Problem 32

For an electron in a 3$d$ state, determine (a) the principle quantum number and (b) the orbital quantum number. (c) How many different magnetic quantum numbers are possible for an electron in that state?

Guilherme B.
Numerade Educator

Problem 33

List the possible sets of quantum numbers for electrons in the 3$d$ subshell.

Guilherme B.
Numerade Educator

Problem 34

When the principal quantum number is $n=4,$ how many different values of $(a) \ell$ and $(b) m_{\ell}$ are possible?

Guilherme B.
Numerade Educator

Problem 35

The $\rho$ -meson has a charge of $-e,$ a spin quantum number of $1,$ and a mass 1507 times that of the electron. If the electrons in atoms were replaced by $\rho$ -mesons, list the possible sets of quantum numbers for $\rho$ -mesons in the 3$d$ subshell.

Guilherme B.
Numerade Educator

Problem 36

A hydrogen atom is immersed in a magnetic field so that its energy levels split according to the Zeeman effect. Neglecting any effects due to electron spin, how many unique energy levels are available to an electron in the 4$f$ subshell?

Guilherme B.
Numerade Educator

Problem 37

Apply the Pauli exclusion principle to determine the number of electrons that could occupy the quantum states described by $(\mathrm{a}) n=3, \ell=2, m_{\ell}=-1$ and (b) $n=3, \ell=1,$ and
(c) $n=4$.

Guilherme B.
Numerade Educator

Problem 38

(a) Write out the electronic configuration of the ground state for nitrogen $(Z=7) .$ (b) Write out the values for the possible set of quantum numbers $n, \ell, m_{\ell},$ and $m_{s}$ for the electrons in nitrogen.

Guilherme B.
Numerade Educator

Problem 39

A certain element has its outermost electron in a 3$p$ subshell. It has valence $+3$ because it has three more electrons than a certain noble gas. What element is it?

Guilherme B.
Numerade Educator

Problem 40

Two electrons in the same atom have $n=3$ and $\ell=1 .$ (a) List the quantum numbers for the possible states of the atom. (b) How many states would be possible if the exclusion principle did not apply to the atom?

Guilherme B.
Numerade Educator

Problem 41

Zirconium $(Z=40)$ has two electrons in an incomplete $d$ subshell. (a) What are the values of $n$ and $\ell$ for each electron? (b) What are all possible values of $m_{\ell}$ and $m_{s}^{7}(\mathrm{c})$ What is the electron configuration in the ground state of zirconium?

Guilherme B.
Numerade Educator

Problem 42

A tungsten target is struck by electrons that have been accelerated from rest through a 40.0 -kV potential difference. Find the shortest wavelength of the radiation emitted.

Guilherme B.
Numerade Educator

Problem 43

A bismuth target is struck by electrons, and x-rays are emitted. Estimate (a) the M-to L-shell transitional energy for bismuth and (b) the wavelength of the x-ray emitted when an electron falls from the M shell to the L shell.

Guilherme B.
Numerade Educator

Problem 44

When an electron drops from the M shell $(n=3)$ to a vacancy in the $\mathrm{K}$ shell $(n=1),$ the measured wavelength of the emitted $\mathrm{x}$ -ray is found to be 0.101 $\mathrm{nm} .$ Identify the element.

Guilherme B.
Numerade Educator

Problem 45

The $\mathrm{K}$ series of the discrete spectrum of tungsten contains wavelengths of $0.0185 \mathrm{nm}, 0.0209 \mathrm{nm},$ and 0.0215 $\mathrm{nm}$ . The K-shell ionization energy is 69.5 $\mathrm{keV}$ . Determine the ionization energies of the $\mathrm{L}, \mathrm{M},$ and $\mathrm{N}$ shells.

Guilherme B.
Numerade Educator

Problem 46

In a hydrogen atom, what is the principal quantum number of the electron orbit with a radius closest to 1.0$\mu \mathrm{m} ?$

Guilherme B.
Numerade Educator

Problem 47

(a) How much energy is required to cause an electron in hydrogen to move from the $n=1$ state to the $n=2$ state? $(\mathrm{b})$ If the electrons gain this energy by collision between hydrogen atoms in a high-temperature gas, find the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by $3 k_{\mathrm{B}} T / 2,$ where $k_{\mathrm{B}}$ is the Boltzmann constant.

Guilherme B.
Numerade Educator

Problem 48

A pulsed ruby laser emits light at 694.3 nm. For a 14.0 -ps pulse containing 3.00 $\mathrm{J}$ of energy, find (a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) If the beam has a circular cross section 0.600 $\mathrm{cm}$ in diameter, what is the number of photons per cubic millimeter?

Guilherme B.
Numerade Educator

Problem 49

An electron in chromium moves from the $n=2$ state to the $n=1$ state without emitting a photon. Instead, the excess energy is transferred to an outer electron (one in the $n=4$ state), which is then ejected by the atom. In this Auger (pronounced "ohjay") process, the ejected electron is referred to as an Auger electron. (a) Find the change in energy associated with the transition from $n=2$ into the vacant $n=1$ state using Bohr theory. Assume only one electron in the $\mathrm{K}$ shell is shielding part of the nuclear charge. (b) Find the energy needed to ionize an $n=4$ electron, assuming 22 electrons shield the nucleus. (c) Find the kinetic energy of the ejected (Auger) electron. (All answers should be in electron volts.)

Guilherme B.
Numerade Educator

Problem 50

As the Earth moves around the Sun, its orbits are quantized. (a) Follow the steps of Bohr’s analysis of the hydrogen atom to show that the allowed radii of the Earth’s orbit are given by
$$r_{n}=\frac{n^{2} \hbar^{2}}{G M_{S} M_{E}^{2}}$$
where $n$ is an integer quantum number, $M_{S}$ is the mass of the Sun, and $M_{E}$ is the mass of the Earth. (b) Calculate the numerical value of $n$ for the Sun-Earth system. (c) Find the distance between the orbit for quantum number $n$ and the next orbit out from the Sun corresponding to the quantum number $n+1 .$ (d) Discuss the significance of your results from parts (b) and (c).

Guilherme B.
Numerade Educator

Problem 51

A laser used in eye surgery emits a $3.00-\mathrm{m}$ J pulse in 1.00 ns, focused to a spot 30.0$\mu \mathrm{m}$ in diameter on the retina. (a) Find (in SI units) the power per unit area at the retina. (This quantity is called the irradiance.) (b) What energy is delivered per pulse to an area of molecular size (say, a circular area 0.600 $\mathrm{nm}$ in diameter)?

Guilherme B.
Numerade Educator

Problem 52

An electron has a de Broglie wavelength equal to the diameter of a hydrogen atom in its ground state. (a) What is the kinetic energy of the electron? (b) How does this energy compare with the magnitude of the ground - state energy of the hydrogen atom?

Guilherme B.
Numerade Educator

Problem 53

Use Bohr’s model of the hydrogen atom to show that when the atom makes a transition from the state n to the state $n-1,$ the frequency of the emitted light is given by
$$f=\frac{2 \pi^{2} m k_{e}^{2} e^{4}}{h^{3}}\left[\frac{2 n-1}{(n-1)^{2} n^{2}}\right]$$

Guilherme B.
Numerade Educator

Problem 54

Suppose the ionization energy of an atom is 4.100 $\mathrm{eV}$ . In this same atom, we observe emission lines that have wavelengths of $310.0 \mathrm{nm}, 400.0 \mathrm{nm},$ and 1378 $\mathrm{nm} .$ Use this information to construct the energy level diagram with the least number of levels. Assume that the higher energy levels are closer together.

Guilherme B.
Numerade Educator