Chapter 7

The Quantum-Mechanical Model of the Atom

Educators

ES
RK
XW

Problem 1

Why is the quantum-mechanical model of the atom important for understanding chemistry?

Lizabeth T.
Numerade Educator

Problem 2

What is light? How fast does it travel in a vacuum?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 3

Define the wavelength and amplitude of a wave. How are these related to the energy of the wave?

Lizabeth T.
Numerade Educator

Problem 4

Define the frequency of electromagnetic radiation. How is frequency related to wavelength?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 5

What determines the color of light? Describe the difference between red light and blue light.

Lizabeth T.
Numerade Educator

Problem 6

What determines the color of a colored object? Explain why grass appears green.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 7

Give an approximate range of wavelengths for each type of electromagnetic radiation and summarize the characteristics and or the uses of each.
$\begin{array}{ll}{\text { a. gamma rays }} & {\text { b. X-rays }} \\ {\text { c. ultraviolet radiation }} & {\text { d. visible light }} \\ {\text { e. infrared radiation }} & {\text { f. microwave radiation }} \\ {g . \text { radio waves }}\end{array}$

Lizabeth T.
Numerade Educator

Problem 8

Explain the wave behavior known as interference. Explain the difference between constructive and destructive interference.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 9

Explain the wave behavior known as diffraction. Draw the diffraction pattern that occurs when light travels through two slits comparable in size and separation to the light's wavelength.

Lizabeth T.
Numerade Educator

Problem 10

Describe the photoelectric effect. How did experimental observations of this phenomenon differ from the predictions of classical electromagnetic theory?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 11

How did the photoelectric effect lead Einstein to propose that light is quantized?

Lizabeth T.
Numerade Educator

Problem 12

What is a photon? How is the energy of a photon related to its wavelength? Its frequency?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 13

What is an emission spectrum? How does an emission spectrum of a gas in a discharge tube differ from a white light spectrum?

Lizabeth T.
Numerade Educator

Problem 14

Describe the Bohr model for the atom. How did the Bohr model account for the emission spectra of atoms?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 15

Explain electron diffraction.

Cathy G.
Numerade Educator

Problem 16

What is the de Broglie wavelength of an electron? What determines the value of the de Broglie wavelength for an electron?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 17

What are complementary properties? How does electron diffraction demonstrate the complementarity of the wave nature and particle nature of the electron?

Cathy G.
Numerade Educator

Problem 18

Explain Heisenberg's uncertainty principle. What paradox is at least partially solved by the uncertainty principle?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 19

What is a trajectory? What kind of information do you need to predict the trajectory of a particle?

Cathy G.
Numerade Educator

Problem 20

Why does the uncertainty principle make it impossible to predict a trajectory for the electron?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 21

Newton's laws of motion are deterministic. Explain this statement.

Cathy G.
Numerade Educator

Problem 22

An electron behaves in ways that are at least partially indeterminate. Explain this statement.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 23

What is a probability distribution map?

Cathy G.
Numerade Educator

Problem 24

For each solution to the Schrodinger equation, what can be precisely specified: the electron's energy or its position? Explain.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 25

What is a quantum-mechanical orbital?

Cathy G.
Numerade Educator

Problem 26

What is the Schrodinger equation? What is a wave function? How is a waye function related to an orbital?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 27

What are the possible values of the principal quantum number n? What does the principal quantum number determine?

Cathy G.
Numerade Educator

Problem 28

What are the possible values of the angular momentum quantum number $R$ . What does the angular momentum quantum number determine?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 29

What are the possible values of the magnetic quantum number $m_{l} ?$ What does the magnetic quantum number determine?

Cathy G.
Numerade Educator

Problem 30

List all the orbitals in each principal level. Specify the three guantum numbers for each orbital.
$$\text { a. }n=1 \quad \text { b. } n=2 \quad \text { c. } \quad n=3 \quad \text { d. } n=4$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 31

Explain the difference between a plot showing the probability density for an orbital and one showing the radial distribution function.

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Problem 32

Make sketches of the general shapes of the $s, p,$ and $d$ orbitals.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 33

List the four different sublevels. Given that only a maximum of two electrons can occupy an orbital, determine the maximum number of electrons that can exist in each sublevel.

Cathy G.
Numerade Educator

Problem 34

Why are atoms usually portrayed as spheres when most orbitals are not spherically shaped?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 35

The distance from the sun to Earth is $1.496 \times 10^{8} \mathrm{km}$ . How long does it take light to travel from the sun to Earth?

RK
Rachel K.
Numerade Educator

Problem 36

The nearest star to our sun is Proxima Centauri, at a distance of 4.3 light-years from the sun. A light-year is the distance that light travels in one year (365 days). How far away, in $\mathrm{km}$ , is Proxima Centauri from the sun?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 37

List these types of electromagnetic radiation in order of (i) increasing wavelength and (ii) increasing energy per photon.
$$\begin{array}{l}{\text { a. radio waves }} \\ {\text { b. microwaves }} \\ {\text { c. infrared radiation }} \\ {\text { d. ultraviolet radiation }}\end{array}$$

Cathy G.
Numerade Educator

Problem 38

List these types of electromagnetic radiation in order of (i) increasing frequency and (ii) decreasing energy per photon.
$$\begin{array}{l}{\text { a. gamma rays }} \\ {\text { b. radio waves }} \\ {\text { c. microwaves }} \\ {\text { d. yisible light }}\end{array}$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 39

Calculate the frequency of each wavelength of electromagnetic radiation.
$$\begin{array}{l}{\text { a. } 632.8 \mathrm{nm} \text { (wavelength of red light from helium-neon laser) }} \\ {\text { b. } 503 \mathrm{nm} \text { (wavelength of maximum solar radiation) }} \\ {\text { c. } 0.052 \mathrm{nm} \text { (a wavelength contained in medical X-rays) }}\end{array}$$

Cathy G.
Numerade Educator

Problem 40

Calculate the wavelength of each frequency of electromagnetic radiation.
a. 100.2 $\mathrm{MHz}$ (typical frequency for FM radio broadcasting)
b. 1070 $\mathrm{kHz}$ (typical frequency for AM radio broadcasting) (assume four significant figures)
c. 835.6 $\mathrm{MHz}$ (common frequency used for cell phone communication)

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 41

Calculate the energy of a photon of electromagnetic radiation at each of the wavelengths indicated in Problem 39.

Cathy G.
Numerade Educator

Problem 42

Calculate the energy of a photon of electromagnetic radiation at each of the frequencies indicated in Problem $40 .$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 43

A laser pulse with wavelength 532 nm contains 3.85 $\mathrm{mJ}$ of energy. How many photons are in the laser pulse?

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Problem 44

A heat lamp produces 32.8 watts of power at a wavelength of 6.5$\mu \mathrm{m} .$ How many photons are emitted per second? $(1$ watt $=1 \mathrm{J} / \mathrm{s})$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 45

Determine the energy of 1 mol of photons for each kind of light. (Assume three significant figures.)
$$\begin{array}{l}{\text { a. infrared radiation }(1500 \mathrm{nm})} \\ {\text { b. visible light }(500 \mathrm{nm})} \\ {\text { c. ultraviolet radiation }(150 \mathrm{nm})}\end{array}$$

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Problem 46

How much energy is contained in 1 mol of each?
a. X-ray photons with a wavelength of 0.135 $\mathrm{nm}$
b. $\gamma$ -ray photons with a wavelength of $2.15 \times 10^{-5} \mathrm{nm}$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 47

Sketch the interference pattern that results from the diffraction of electrons passing through two closely spaced slits.

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Problem 48

What happens to the interference pattern described in Problem 47 if the rate of electrons going through the slits is decreased to one electron per hour? What happens to the pattern if we try to determine which slit the electron goes through by using a laser placed directly behind the slits?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 49

The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.20 $\mathrm{nm}$ in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 $\mathrm{nm}$ ?

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Problem 50

The smallest atoms can themselves exhibit quantum-mechanical behavior. Calculate the de Broglie wavelength (in pm) of a hydrogen atom traveling at 475 $\mathrm{m} / \mathrm{s}$ .

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 51

What is the de Broglie wavelength of an electron traveling at $1.35 \times 10^{5} \mathrm{m} / \mathrm{s}$ ?

Cathy G.
Numerade Educator

Problem 52

A proton in a linear accelerator has a de Broglie wavelength of 122 $\mathrm{pm}$ . What is the speed of the proton?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 53

Calculate the de Broglie wavelength of a $143-$ baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?

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Problem 54

A 2.22 -caliber handgun fires a 1.9 -g- bullet at a velocity of 765 $\mathrm{m} / \mathrm{s}$ . Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 55

An electron has an uncertainty in its position of 552 $\mathrm{pm} .$ What is the uncertainty in its velocity?

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Problem 56

An electron traveling at $3.7 \times 10^{5} \mathrm{m} / \mathrm{s}$ has an uncertainty its velocity of $1.88 \times 10^{5} \mathrm{m} / \mathrm{s}$ . What is the uncertainty in its position?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 57

Which electron is, on average, closer to the nucleus: an electron in a 2 orbital or an electron in a 3 s orbital?

Cathy G.
Numerade Educator

Problem 58

Which electron is, on average, farther from the nucleus: an electron in a 3$p$ orbital or an electron in a 4$p$ orbital?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 59

What are the possible values of $I$ for each value of $n ?$
$$\text { a. }1 \quad \text { b. } 2 \quad \text { c. } 3 \quad \text { d. } 4$$

Cathy G.
Numerade Educator

Problem 60

What are the possible values of $m_{l}$ for each value of $R$
$$\text { b. }0 \quad \text { b. } 1 \quad \text { c. } 2 \quad \text { d. } 3$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 61

Which set of quantum numbers cannot specify an orbital?
$$\begin{array}{ll}{\text { a. } n=2, l=1, m_{l}=-1} & {\text { b. } n=3, l=2, m_{l}=0} \\ {\text { c. } n=3, l=3, m_{l}=2} & {\text { d. } n=4, l=3, m_{l}=0}\end{array}$$

Cathy G.
Numerade Educator

Problem 62

Which combinations of $n$ and $t$ represent real orbitals, and which do not exist?
$$\text { a. }1 \text { s } \quad \text { b. } 2 p \quad \text { c. } 4 \mathrm{s} \quad \text { d. } 2 d$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 63

Sketch the 15 and 2$p$ orbitals. How do the 2 s and 3p orbitals differ from the 1$s$ and 2$p$ orbitals?

Cathy G.
Numerade Educator

Problem 64

Sketch the 3$d$ orbitals. How do the 4$d$ orbitals differ from the 3$d$ orbitals?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 65

An electron in a hydrogen atom is excited with electrical energy to an excited state with $n=2 .$ The atom then emits a photon. What is the value of $n$ for the electron following the emission?

Cathy G.
Numerade Educator

Problem 66

Determine whether each transition in the hydrogen atom corresponds to absorption or emission of energy.
$$\begin{array}{l}{\text { a. } n=3 \longrightarrow n=1} \\ {\text { b. } n=2 \longrightarrow n=4} \\ {\text { c. } n=4 \longrightarrow n=3}\end{array}$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 67

According to the quantum-mechanical model for the hydrogen atom, which electron transitions produces light with the longer wavelength: 2$p \longrightarrow 1 s$ or 3$p \longrightarrow 1 s ?$

Cathy G.
Numerade Educator

Problem 68

According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength: 3$p \longrightarrow 2 s$ or 4$p \longrightarrow 3 p ?$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 69

Calculate the wavelength of the light emitted when an electron in a hydrogen atom makes each transition and indicate the region of the electromagnetic spectrum (infrared, visible, ultraviolet, etc. $.$ where the light is found.
$$\begin{array}{ll}{\text { a. } n=2 \longrightarrow n=1} & {\text { b. } n=3 \longrightarrow n=1} \\ {\text { c. } n=4 \longrightarrow n=2} & {\text { d. } n=5 \longrightarrow n=2}\end{array}$$

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Problem 70

Calculate the frequency of the light emitted when an electron in a hydrogen atom makes each transition.
$$\begin{array}{ll}{\text { a. } n=4 \longrightarrow n=3} & {\text { h. } n=5 \longrightarrow n=1} \\ {\text { c. } n=5 \longrightarrow n=4} & {\text { d. } n=6 \longrightarrow n=5}\end{array}$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 71

An electron in the $n=7$ level of the hydrogen atom relaxes to a lower-energy level, emitting light of 397 $\mathrm{nm} .$ What is the value of $n$ for the level to which the electron relaxed?

XW
Xinran W.
Numerade Educator

Problem 72

An electron in a hydrogen atom relaxes to the $n=4$ level, emitting light of 114 THz What is the value of $n$ for the level in which the electron originated?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 73

Ultraviolet radiation and radiation of shorter wavelengths can damage biological molecules because these kinds of radiation carry enough energy to break bonds within the molecules. A typical carbon-carbon bond requires 348 $\mathrm{kJ} / \mathrm{mol}$ to break. What is the longest wavelength of radiation with enough energy to break carbon-carbon bonds?

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Problem 74

The human eye contains a molecule called lil-cis-retinal that changes shape when struck with light of sufficient energy. The change in shape triggers a series of events that results in an electrical signal being sent to the brain that results in vision. The minimum energy required to change the conformation of
11-cis-retinal within the eye is about 164 kJ/mol. Calculate the longest wavelength visible to the human eye.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 75

An argon ion laser puts out 5.0 $\mathrm{W}$ of continuous power at a wavelength of 532 $\mathrm{nm}$ . The diameter of the laser beam is 5.5 mm. If the laser is pointed toward a pinhole with a diameter of 1.2 $\mathrm{mm}$ , how many photons travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})$

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Problem 76

A green leaf has a surface area of 2.50 $\mathrm{cm}^{2} .$ If solar radiation is $1000 \mathrm{W} / \mathrm{m}^{2},$ how many photons strike the leaf every second? Assume three significant figures and an average wavelength of 504 $\mathrm{nm}$ for solar radiation.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 77

In a technique used for surface analysis called Auger electron spectroscopy (AES), electrons are accelerated toward a metal surface. These electrons cause the emissions of secondary electrons called Auger electrons-from the metal surface. The kinetic energy of the Auger electrons depends on the composition of the surface. The presence of oxygen atoms on the surface results in Auger electrons with a kinetic energy of approximately 506 eV. What is the de Broglie wavelength of one of these electrons?
$$\left[\mathrm{KE}=\frac{1}{2} m v^{2} ; 1 \text { electron volt }(\mathrm{eV})=1.602 \times 10^{-19} \mathrm{J}\right]$$

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Problem 78

An $\mathrm{X}$ -ray photon of wavelength 0.989 $\mathrm{nm}$ strikes a surface. The emitted electron has a kinetic energy of 969 $\mathrm{eV} .$ What is the binding energy of the electron in $\mathrm{kJ} / \mathrm{mol} ?$
$$\left[\mathrm{KE}=\frac{1}{2} m v^{2} ; 1 \text { electron volt }(\mathrm{eV})=1.602 \times 10^{-19} \mathrm{J}\right]$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 79

Ionization involves completely removing an electron from an atom. How much energy is required to ionize a hydrogen atom in its ground (or lowest energy) state? What wavelength of light contains enough energy in a single photon to ionize a hydrogen atom?

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Problem 80

The energy required to ionize sodium is 496 $\mathrm{kJ} / \mathrm{mol} .$ What minimum frequency of light is required to ionize sodium?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 81

Suppose that in an alternate universe, the possible values of $/$ are the integer values from 0 to $n$ (instead of 0 to $n-1 )$ . Assuming no other differences berween this imaginary untverse and ours, how many orbitals exist in each level in the alternate universe?
$$\text { a. }n=1 \quad \text { b. } n=2 \quad \text { c. } n=3$$

Cathy G.
Numerade Educator

Problem 82

Suppose that, in an alternate universe, the possible values of $m_{1}$ are the integer values including 0 ranging from $-l-1$ to $l+1$ (instead of simply $-1$ to $+l )$ . How many orbitals exist in each sublevel in the alternate universe?
$$a. ssublevel \quad b. p sublevel \quad c. d sublevel$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 83

An atomic emission spectrum of hydrogen shows three wavelengths: $1875 \mathrm{nm}, 1282 \mathrm{nm},$ and 1093 $\mathrm{nm} .$ Assign these wavelengths to transitions in the hydrogen atom.

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Problem 84

An atomic emission spectrum of hydrogen shows three wavelengths: $121.5 \mathrm{nm}, 102.6 \mathrm{nm}$ , and 97.23 $\mathrm{nm}$ . Assign these wavelengths to transitions in the hydrogen atom.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 85

The binding energy of electrons in a metal is 193 $\mathrm{kJ} / \mathrm{mol} .$ Find
the threshold frequency of the metal.

Cathy G.
Numerade Educator

Problem 86

In order for a thermonuclear fusion reaction of two deuterons $\left(2 \mathrm{H}^{4}\right)$ to take place, the deuterons must collide and each must have a velocity of about $1 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find the wavelength of such a deuteron.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 87

The speed of sound in air is 344 $\mathrm{m} / \mathrm{s}$ at room temperature. The lowest frequency of a large organ pipe is 30 $\mathrm{s}^{-1}$ and the highest frequency of a piccolo is $1.5 \times 10^{4} \mathrm{s}^{-1} .$ Find the difference in wavelength between these two sounds.

Cathy G.
Numerade Educator

Problem 88

The distance from Earth to the sun is $1.5 \times 10^{8} \mathrm{km} .$ Find the number of crests in a light wave of frequency $1.0 \times 10^{14} \mathrm{s}^{-1}$ traveling from the sun to Earth.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 89

The iodine molecule can be photodissociated (broken apart with light) into iodine atoms in the gas phase with light of wavelengths shorter than about 792 $\mathrm{nm}$ . A 100.0 -mL glass tube contains 55.7 mtorr of gaseous iodine at $25.0^{\circ} \mathrm{C}$ . What minimum amount of light energy must be absorbed by the iodine in the tube to dissociate 15. 0$\%$ of the molecules?

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Problem 90

A 5.00 -mL ampule of a $0.100-M$ solution of naphthalene in hexane is excited with a flash of light. The naphthalene emits 15.5 $\mathrm{J}$ of energy at an average wavelength of 349 $\mathrm{nm}$ . What percentage of the naphthalene molecules emitted a photon?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 91

A laser produces 20.0 $\mathrm{mW}$ of red light. In 1.00 $\mathrm{hr}$ , the laser emits $2.29 \times 10^{20}$ photons. What is the wavelength of the laser?

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Problem 92

A particular laser consumes 150.0 watts of electrical power and produces a stream of $1.33 \times 10^{19} 1064$ -nm photons per second. What is the percent efficiency of the laser in converting electrical power to light?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 93

The quantum yield of light-induced chemical reactions (called photochemical reactions) measures the efficiency of the process. The quantum yield, $\phi, \quad$ is defined as:
$$\phi=\frac{\text { number of reaction events }}{\text { number of photons absorbed }}Suppose\quad the\quad quantum\quad yield$$
for the reaction $\mathrm{CH}_{3} \mathrm{X} \longrightarrow \mathrm{CH}_{3}+\mathrm{X}$ is $\phi=0.24 .$ A cuvette containing a solution of $\mathrm{CH}_{3} \mathrm{X}$ is irradiated with 280 -nm light with a power of 885 $\mathrm{mW}$ for 10.0 minutes. Assuming total absorption of the light by the sample, what is the maximum amount (in moles) of $\mathrm{CH}_{3} \mathrm{X}$ that breaks apart?

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Problem 94

A student is studying the photodissociation (dissociation with light) of $\mathrm{I}_{2} \longrightarrow 2 \mathrm{I}$ . When a sample of $\mathrm{I}_{2}$ is irradiated with a power of 255 $\mathrm{mW}$ at 590 $\mathrm{nm}$ for 35 seconds, 0.0256 $\mathrm{mmol}$ of $\mathrm{I}$ forms. Assuming complete absorption of the incident radiation, what is the quantum yield, $\phi$ , of the reaction? (See Problem 93 for definition of quantum yield.) where $n$ is a quantum number with possible values of $1,2, \ldots$ and $v$ is the frequency of vibration. The vibration frequency of HCl is approximately $8.85 \times 10^{13} \mathrm{s}^{-1} .$ What minimum energy is required to excite a vibration in HCl? What wavelength of light is required to excite this vibration?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 95

An electron confined to a one-dimensional box has energy levels given by the equation:
$$E_{n}=n^{2} h^{2} / 8 m L^{2}$$
where $n$ is a quantum number with possible values of $1,2,3, \ldots, m$ is the mass of the particle, and $L$ is the length of the box.
a. Calculate the energies of the $n=1, n=2,$ and $n=3$ levels for an electron in a box with a length of 155 $\mathrm{pm} .$
b. Calculate the wavelength of light required to make a transition from $n=1 \longrightarrow n=2$ and from $n=2 \longrightarrow n=3 .$ In what region of the electromagnetic spectrum do these wavelengths lie?

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Problem 96

The energy of a vibrating molecule is quantized much like the energy of an electron in the hydrogen atom. The energy levels of a vibrating molecule are given by the equation:
$$E_{n}=\left(n+\frac{1}{2}\right) h v$$
where $n$ is a quantum number with possible values of $1,2, \ldots,$ and $v$ is the frequency of vibration. The vibration frequency of HCl is approximately $8.85 \times 10^{13} \mathrm{s}^{-1} .$ What minimum energy is required to excite a vibration in HCl? What wavelength of light is required to excite this vibration?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 97

The wave functions for the 1 s and 2 s orbitals are as follows:
$$\begin{array}{l}{\text { 1s } \psi=(1 / \pi)^{1 / 2}\left(1 / a_{0}^{3 / 2}\right) \exp \left(-r / a_{0}\right)} \\ {\text { 2s } \psi=(1 / 32 \pi)^{1 / 2}\left(1 / a_{0}^{3 / 2}\right)\left(-2 r / a_{0}\right) \exp \left(-r / a_{0}\right)}\end{array}$$
where $a_{0}$ is a constant $\left(a_{0}=53 \mathrm{pm}\right)$ and $r$ is the distance from the nucleus. Use a spreadsheet to make a plot of each of these wave functions for values of $r$ ranging from 0 $\mathrm{pm}$ to 200 $\mathrm{pm}$ . Describe the differences in the plots and identify the node in the 2s wave function.

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Problem 98

Before quantum mechanics was developed, Johannes Rydberg developed an equation that predicted the wavelengths $(\lambda)$ in the atomic spectrum of hydrogen:
$$1 / \lambda=R\left(1 / m^{2}-1 / n^{2}\right)$$
In this equation $R$ is a constant and $m$ and $n$ are integers. Use the quantum-mechanical model for the hydrogen atom to derive the Rydberg equation.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 99

Find the velocity of an electron emitted by a metal with a threshold frequency of $2.25 \times 10^{14} \mathrm{s}^{-1}$ when it is exposed to visible light of wavelength $5.00 \times 10^{-7} \mathrm{m}$ .

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Problem 100

Water is exposed to infrared radiation of wavelength $2.8 \times 10^{4} \mathrm{cm}$ . Assume that all the radiation is absorbed and converted to heat. How many photons are required to raise the temperature of 2.0 $\mathrm{g}$ of water by 2.0 $\mathrm{K}$ ?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 101

The 2005 Nobel Prize in Physics was given, in part, to scientists who had made ultrashort pulses of light. These pulses are important in making measurements involving very short time periods. One challenge in making such pulses is the uncertainty principle, which can be stated with respect to energy and time as $\Delta E \cdot \Delta t \geqslant h / 4 \pi .$ What is the energy uncertainty $(\Delta E)$ associated with a short pulse of laser light that lasts for only 5.0 femtoseconds (fs)? Suppose the low-energy end of the pulse had a wavelength of 722 $\mathrm{nm}$ . What is the wavelength of the high-energy end of the pulse that is limited only by the uncertainty principle?

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Problem 102

A metal with a threshold frequency of $6.71 \times 10^{14} \mathrm{s}^{-1}$ emits an electron with a velocity of $6.95 \times 10^{5} \mathrm{m} / \mathrm{s}$ when radiation of $1.01 \times 10^{15} \mathrm{s}^{-1}$ strikes the metal. Calculate the mass of the electron.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 103

Find the longest wavelength of a wave that can travel around in a circular orbit of radius 1.8 $\mathrm{m}$ .

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Problem 104

The heat of fusion of ice is 6.00 $\mathrm{k} / \mathrm{mol}$ . Find the number of photons of wavelength $=6.42 \times 10^{-6} \mathrm{m}$ that must be absorbed to melt 1.00 $\mathrm{g}$ of ice.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 105

Explain the difference between the Bohr model for the hydrogen atom and the quantum-mechanical model. Is the Bohr model consistent with Heisenberg's uncertainty principle?

Cathy G.
Numerade Educator

Problem 106

The light emitted from one of the following electronic transitions $(n=4 \longrightarrow n=3$ or $n=3 \longrightarrow n=2)$ in the hydrogen atom causes the photoelectric effect in a particular metal, while
light from the other transition does not. Which transition causes the photoelectric effect and why?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 107

Determine whether an interference pattern is observed on the other side of the slits in each experiment.
a. An electron beam is aimed at two closely spaced slits. The beam produces only one electron per minute.
b. An electron beam is aimed at two closely spaced slits. A light beam is placed at each slit to determine when an electron goes through the slit.
c. A high-intensity light beam is aimed at two closely spaced slits.
d. A gun is fired at a solid wall containing two closely spaced slits. (Will the bullets that pass through the slits form an interference pattern on the other side of the solid wall?

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Problem 108

Which transition in the hydrogen atom produces emitted light with the longest wayelength?
$$\begin{array}{l}{\text { a. } n=4 \longrightarrow n=3} \\ {\text { b. } n=2 \longrightarrow n=1} \\ {\text { c. } n=3 \longrightarrow n=2}\end{array}$$

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 109

Discuss the nature of light with your group. Ask each member of your group to transcribe one complete sentence about the physical nature of light.

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Problem 110

How are electrons like baseballs? How are they unlike baseballs?

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 111

What are all the possible values of $m_{1}$ if $l=0$ (an s orbital)? If $l=1$ (a p orbital)? If $l=2$ (a d orbital)? How many possible values of $m_{I}$ would there be if $l=20 ?$ Write an equation to determine the number of possible values of $m_{1}$ from the value of $L$ .

Cathy G.
Numerade Educator

Problem 112

Have each group member choose a set of quantum numbers for an electron in a hydrogen atom. Calculate the wavelength of light produced if an electron moves from your state to each state of the other group members. Make a table comparing all possible combinations, and list all wavelengths in order of increasing energy.

ES
Eugene S.
University of Minnesota - Twin Cities

Problem 113

How many nodes are there in the $1 s, 2 p,$ and 3$d$ orbitals? How many nodes are in a 4f orbital?

Cathy G.
Numerade Educator

Problem 114

On average sunlight shines on the surface of Earth with an intensity of 910 Watts/m. . Some of this energy is in the form of ultraviolet (UV) radiation. The compounds in chemical sunscreens absorb ultraviolet rays and release the energy as lower-energy infrared (IR) rays, thereby preventing the skin-damaging UV rays from reaching the skin. You may have noticed references to UV-A $(320-400 \mathrm{nm})$ and $\mathrm{UV}-\mathrm{B}(280-320 \mathrm{nm})$ protection in sunscreen packaging and advertisements. Figure at lists the chemical formula and molar mass of two compounds, 2-ethylhexyl $p$ -methoxycinnamate $(2-\mathrm{EHMC}),$ a UV-B sunscreen, and terephthalylidenedicamphor sulfonic acid (TDSA), a UV-A sunscreen.

ES
Eugene S.
University of Minnesota - Twin Cities