Chemistry

Raymond Chang, Kenneth A. Goldsby

Chapter 7

Quantum Theory and the Electronic Structure of Atoms - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

Problem 1

What is a wave? Explain the following terms associated with waves: wavelength, frequency, amplitude.

Ted Gray

Ted Gray

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Problem 2

What are the units for wavelength and frequency of electromagnetic waves? What is the speed of light in meters per second and miles per hour?

Tim Blackstad

Numerade Educator

Problem 3

List the types of electromagnetic radiation, starting with the radiation having the longest wavelength and ending with the radiation having the shortest wavelength.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 4

Give the high and low wavelength values that define the visible region of the electromagnetic spectrum.

Tim Blackstad

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Problem 5

Briefly explain Planck's quantum theory and explain what a quantum is. What are the units for Planck's constant?

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 6

Give two everyday examples that illustrate the concept of quantization.

Tim Blackstad

Numerade Educator

Problem 7

(a) What is the wavelength (in nanometers) of light having a frequency of $8.6 \times 10^{13} \mathrm{Hz} ?$ (b) What is the frequency (in $\mathrm{Hz}$ ) of light having a wavelength of $566 \mathrm{nm} ?$

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 8

(a) What is the frequency of light having a wavelength of $456 \mathrm{nm} ?$ (b) What is the wavelength (in nanometers) of radiation having a frequency of $2.45 \times 10^{9} \mathrm{Hz} ?$ (This is the type of radiation used in microwave ovens.)

Tim Blackstad

Numerade Educator

Problem 9

The average distance between Mars and Earth is about $1.3 \times 10^{8}$ miles. How long would it take $\mathrm{TV}$ pictures transmitted from the Viking space vehicle on Mars' surface to reach Earth? (1 mile = 1.61 km.)

Stephanie L

Numerade Educator

Problem 10

How many minutes would it take a radio wave to travel from the planet Venus to Earth? (Average distance from Venus to Earth is 28 million miles.)

Tim Blackstad

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Problem 11

The SI unit of time is the second, which is defined as 9,192,631,770 cycles of radiation associated with a certain emission process in the cesium atom. Calculate the wavelength of this radiation (to three significant figures). In which region of the electromagnetic spectrum is this wavelength found?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 12

The SI unit of length is the meter, which is defined as the length equal to 1,650,763.73 wavelengths of the light emitted by a particular energy transition in krypton atoms. Calculate the frequency of the light to three significant figures.

Tim Blackstad

Numerade Educator

Problem 13

What are photons? What role did Einstein's explanation of the photoelectric effect play in the development of the particle-wave interpretation of the nature of electromagnetic radiation?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 14

Consider the plots shown here for the photoelectric effect of two different metals A (green line) and B (red line). (a) Which metal has a greater work function? (b) What does the slope of the lines tell us?

Tim Blackstad

Numerade Educator

Problem 15

A photon has a wavelength of $624 \mathrm{nm}$. Calculate the energy of the photon in joules.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 16

The blue color of the sky results from the scattering of sunlight by air molecules. The blue light has a frequency of about $7.5 \times 10^{14} \mathrm{Hz}$. (a) Calculate the wavelength, in $\mathrm{nm}$, associated with this radiation, and (b) calculate the energy, in joules, of a single photon associated with this frequency.

Tim Blackstad

Numerade Educator

Problem 17

A photon has a frequency of $6.0 \times 10^{4} \mathrm{Hz}$. (a) Convert this frequency into wavelength (nm). Does this frequency fall in the visible region? (b) Calculate the energy (in joules) of this photon. (c) Calculate the energy (in joules) of 1 mole of photons all with this frequency.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 18

What is the wavelength, in $\mathrm{nm}$, of radiation that has an energy content of $1.0 \times 10^{3} \mathrm{kJ} / \mathrm{mol} ?$ In which region of the electromagnetic spectrum is this radiation found?

Tim Blackstad

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Problem 19

When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is $0.154 \mathrm{nm}$.

Stephanie L

Numerade Educator

Problem 20

A particular form of electromagnetic radiation has a frequency of $8.11 \times 10^{14} \mathrm{Hz}$. (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

Tim Blackstad

Numerade Educator

Problem 21

The work function of potassium is $3.68 \times 10^{-19} \mathrm{J}$ (a) What is the minimum frequency of light needed to eject electrons from the metal? (b) Calculate the kinetic energy of the ejected electrons when light of frequency equal to $8.62 \times 10^{14} \mathrm{s}^{-1}$ is used for irradiation.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 22

When light of frequency equal to $2.11 \times 10^{15} \mathrm{s}^{-1}$ shines on the surface of gold metal, the kinetic energy of ejected electrons is found to be $5.83 \times 10^{-19} \mathrm{J}$ What is the work function of gold?

Tim Blackstad

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Problem 23

(a) What is an energy level? Explain the difference between ground state and excited state. (b) What are emission spectra? How do line spectra differ from continuous spectra?

Ronald Prasad

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Problem 24

(a) Briefly describe Bohr's theory of the hydrogen atom and how it explains the appearance of an emission spectrum. How does Bohr's theory differ from concepts of classical physics? (b) Explain the meaning of the negative sign in Equation (7.5)

Tim Blackstad

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Problem 25

Explain why elements produce their own characteristic colors when they emit photons?

Ma Ednelyn Lim

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Problem 26

Some copper compounds emit green light when they are heated in a flame. How would you determine whether the light is of one wavelength or a mixture of two or more wavelengths?

Tim Blackstad

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Problem 27

Is it possible for a fluorescent material to emit radiation in the ultraviolet region after absorbing visible light? Explain your answer.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 28

Explain how astronomers are able to tell which elements are present in distant stars by analyzing the electromagnetic radiation emitted by the stars.

Tim Blackstad

Numerade Educator

Problem 29

Consider the following energy levels of a hypothetical atom:
$E_{4}$_______$-1.0 \times 10^{-19} \mathrm{J}$
$E_{3}$ ______$-5.0 \times 10^{-19} \mathrm{J}$
$E_{2}$_______$-10 \times 10^{-19} \mathrm{J}$
$E_{1}$_______$-15 \times 10^{-19} \mathrm{J}$
(a) What is the wavelength of the photon needed to excite an electron from $E_{1}$ to $E_{4} ?$ (b) What is the energy (in joules) a photon must have in order to excite an electron from $E_{2}$ to $E_{3} ?$ (c) When an electron drops from the $E_{3}$ level to the $E_{1}$ level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 30

The first line of the Balmer series occurs at a wavelength of $656.3 \mathrm{nm} .$ What is the energy difference between the two energy levels involved in the emission that results in this spectral line?

Tim Blackstad

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Problem 31

Calculate the wavelength (in nanometers) of a photon emitted by a hydrogen atom when its electron drops from the $n=5$ state to the $n=3$ state.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 32

Calculate the frequency (Hz) and wavelength (nm) of the emitted photon when an electron drops from the $n=4$ to the $n=2$ level in a hydrogen atom.

Tim Blackstad

Numerade Educator

Problem 33

Careful spectral analysis shows that the familiar yellow light of sodium lamps (such as street lamps) is made up of photons of two wavelengths, $589.0 \mathrm{nm}$ and $589.6 \mathrm{nm} .$ What is the difference in energy (in joules) between photons with these wavelengths?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 34

An electron in the hydrogen atom makes a transition from an energy state of principal quantum numbers
$n_{i}$ to the $n=2$ state. If the photon emitted has a wavelength of $434 \mathrm{nm},$ what is the value of $n_{\mathrm{i}} ?$

Tim Blackstad

Numerade Educator

Problem 35

Explain the statement, Matter and radiation have a "dual nature."

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 36

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Tim Blackstad

Numerade Educator

Problem 37

Why is Equation (7.8) meaningful only for submicroscopic particles, such as electrons and atoms, and not for macroscopic objects?

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 38

(a) If a $\mathrm{H}$ atom and a He atom are traveling at the same speed, what will be the relative wavelengths of the two atoms? (b) If a $\mathrm{H}$ atom and a He atom have the same kinetic energy, what will be the relative wavelengths of the two atoms?

Tim Blackstad

Numerade Educator

Problem 39

Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among $^{235} \mathrm{U}$ isotopes. Calculate the wavelength (in $\mathrm{nm}$ ) associated with a beam of neutrons moving at $7.00 \times 10^{2} \mathrm{m} / \mathrm{s}$. (Mass of a neutron $\left.=1.675 \times 10^{-27} \mathrm{kg} .\right)$

Stephanie L

Numerade Educator

Problem 40

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at $2.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ (Mass of a proton $\left.=1.673 \times 10^{-27} \mathrm{kg} .\right)$

Tim Blackstad

Numerade Educator

Problem 41

What is the de Broglie wavelength, in $\mathrm{cm},$ of a $12.4-\mathrm{g}$ hummingbird flying at $1.20 \times 10^{2} \mathrm{mph} ?(1 \text { mile }=$ $1.61 \mathrm{km} .)$

Stephanie L

Numerade Educator

Problem 42

What is the de Broglie wavelength (in $\mathrm{nm}$ ) associated with a $2.5-\mathrm{g}$ Ping-Pong ball traveling $35 \mathrm{mph} ?$

Tim Blackstad

Numerade Educator

Problem 43

What are the inadequacies of Bohr's theory?

Ronald Prasad

Numerade Educator

Problem 44

What is the Heisenberg uncertainty principle? What is the Schrödinger equation?

Tim Blackstad

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Problem 45

What is the physical significance of the wave function?

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 46

How is the concept of electron density used to describe the position of an electron in the quantum mechanical treatment of an atom?

Tim Blackstad

Numerade Educator

Problem 47

What is an atomic orbital? How does an atomic orbital differ from an orbit?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 48

Describe the shapes of $s, p,$ and $d$ orbitals. How are these orbitals related to the quantum numbers $n, \ell$ and $m_{\ell} ?$

Tim Blackstad

Numerade Educator

Problem 49

List the hydrogen orbitals in increasing order of energy.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 50

Describe the characteristics of an $s$ orbital, a $p$ orbital, and a $d$ orbital. Which of the following orbitals do not exist: $1 p, 2 s, 2 d, 3 p, 3 d, 3 f, 4 g ?$

Tim Blackstad

Numerade Educator

Problem 51

Why is a boundary surface diagram useful in representing an atomic orbital?

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 52

Describe the four quantum numbers used to characterize an electron in an atom.

Tim Blackstad

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Problem 53

Which quantum number defines a shell? Which quantum numbers define a subshell?

Ronald Prasad

Numerade Educator

Problem 54

Which of the four quantum numbers $\left(n, \ell, m_{\ell}, m_{s}\right)$ determine (a) the energy of an electron in a hydrogen atom and in a many-electron atom, (b) the size of an orbital, (c) the shape of an orbital, (d) the orientation of an orbital in space?

Tim Blackstad

Numerade Educator

Problem 55

An electron in a certain atom is in the $n=2$ quantum level. List the possible values of $\ell$ and $m_{\ell}$ that it can have.

Stephanie L

Numerade Educator

Problem 56

An electron in an atom is in the $n=3$ quantum level. List the possible values of $\ell$ and $m_{\ell}$ that it can have.

Tim Blackstad

Numerade Educator

Problem 57

Give the values of the quantum numbers associated with the following orbitals: (a) $2 p,$ (b) $3 s,$ (c) $5 d$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 58

Give the values of the four quantum numbers of an electron in the following orbitals: (a) $3 s,$ (b) $4 p$
(c) $3 d$.

Tim Blackstad

Numerade Educator

Problem 59

Discuss the similarities and differences between a $1 s$ and a $2 s$ orbital.

Ronald Prasad

Numerade Educator

Problem 60

What is the difference between a $2 p_{x}$ and a $2 p_{y}$ orbital?

Tim Blackstad

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Problem 61

List all the possible subshells and orbitals associated with the principal quantum number $n,$ if $n=5$.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 62

List all the possible subshells and orbitals associated with the principal quantum number $n,$ if $n=6$.

Tim Blackstad

Numerade Educator

Problem 63

Calculate the total number of electrons that can occupy (a) one $s$ orbital, (b) three $p$ orbitals, (c) five $d$ orbitals, (d) seven $f$ orbitals.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 64

What is the total number of electrons that can be held in all orbitals having the same principal quantum number $n ?$

Tim Blackstad

Numerade Educator

Problem 65

Determine the maximum number of electrons that can be found in each of the following subshells: $3 s$
$3 d, 4 p, 4 f, 5 f$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 66

Indicate the total number of (a) $p$ electrons in $\mathrm{N}(Z=7) ;$ (b) $s$ electrons in $\mathrm{Si}(Z=14) ;$ and $(\mathrm{c}) 3 d$ electrons in $\mathrm{S}(Z=16)$.

Tim Blackstad

Numerade Educator

Problem 67

Make a chart of all allowable orbitals in the first four principal energy levels of the hydrogen atom. Designate each by type (for example, $s, p$ ) and indicate how many orbitals of each type there are.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 68

Why do the $3 s, 3 p,$ and $3 d$ orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

Tim Blackstad

Numerade Educator

Problem 69

For each of the following pairs of hydrogen orbitals, indicate which is higher in energy: (a) $1 s, 2 s ;$ (b) $2 p$ $3 p ;(\mathrm{c}) 3 d_{x y}, 3 d_{y z} ;(\mathrm{d}) 3 s, 3 d ;(\mathrm{e}) 4 f, 5 s$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 70

Which orbital in each of the following pairs is lower in energy in a many-electron atom? (a) $2 s, 2 p ;$ (b) $3 p$
$3 \bar{d} ;(\mathrm{c}) 3 s, 4 s ;(\mathrm{d}) 4 d, 5 f$.

Tim Blackstad

Numerade Educator

Problem 71

What is electron configuration? Describe the roles that the Pauli exclusion principle and Hund's rule play in writing the electron configuration of elements.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 72

Explain the meaning of the symbol $4 d^{6}$.

Tim Blackstad

Numerade Educator

Problem 73

Explain the meaning of diamagnetic and paramagnetic. Give an example of an element that is diamagnetic and one that is paramagnetic. What does it mean when we say that electrons are paired?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 74

What is meant by the term "shielding of electrons" in an atom? Using the Li atom as an example, describe the effect of shielding on the energy of electrons in $=7$ an atom.

Tim Blackstad

Numerade Educator

Problem 75

Indicate which of the following sets of quantum numbers in an atom are unacceptable and explain why:
(a) $\left(1,0, \frac{1}{2}, \frac{1}{2}\right),(\mathrm{b})\left(3,0,0,+\frac{1}{2}\right),(\mathrm{c})\left(2,2,1,+\frac{1}{2}\right)$ (d) $\left(4,3,-2,+\frac{1}{2}\right),(\text { e })(3,2,1,1)$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 76

The ground-state electron configurations listed here are incorrect. Explain what mistakes have been made in each and write the correct electron configurations.
Al: $1 s^{2} 2 s^{2} 2 p^{4} 3 s^{2} 3 p^{3}$
$\mathrm{B}: 1 s^{2} 2 s^{2} 2 p^{5}$
$\mathrm{F}: 1 s^{2} 2 s^{2} 2 p^{6}$

Tim Blackstad

Numerade Educator

Problem 77

The atomic number of an element is $73 .$ Is this element diamagnetic or paramagnetic?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 78

Indicate the number of unpaired electrons present in each of the following atoms: $\mathrm{B}, \mathrm{Ne}, \mathrm{P}, \mathrm{Sc}, \overline{\mathrm{Mn}}, \mathrm{Se}$ $\mathrm{Kr}, \mathrm{Fe}, \mathrm{Cd}, \mathrm{I}, \mathrm{Pb}$.

Tim Blackstad

Numerade Educator

Problem 79

State the Aufbau principle and explain the role it plays in classifying the elements in the periodic table.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 80

Describe the characteristics of the following groups of elements: transition metals, lanthanides, actinides.

Tim Blackstad

Numerade Educator

Problem 81

What is the noble gas core? How does it simplify the writing of electron configurations?

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 82

What are the group and period of the element osmium?

Tim Blackstad

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Problem 83

Define the following terms and give an example of each: transition metals, lanthanides, actinides.

Kim Trang Nguyen

Kim Trang Nguyen

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Problem 84

Explain why the ground-state electron configurations of Cr and Cu are different from what we might expect.

Tim Blackstad

Numerade Educator

Problem 85

Explain what is meant by a noble gas core. Write the electron configuration of a xenon core.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 86

Comment on the correctness of the following statement: The probability of finding two electrons with the same four quantum numbers in an atom is zero.

Tim Blackstad

Numerade Educator

Problem 87

Use the Aufbau principle to obtain the ground-state electron configuration of selenium.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 88

Use the Aufbau principle to obtain the ground-state electron configuration of technetium.

Tim Blackstad

Numerade Educator

Problem 89

Write the ground-state electron configurations for the following elements: $\mathrm{B}, \mathrm{V}, \mathrm{Ni}, \mathrm{As}, \mathrm{I}, \mathrm{Au}$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 90

Write the ground-state electron configurations for the following elements: Ge, Fe, $\mathrm{Zn}, \mathrm{Ni}, \mathrm{W}, \mathrm{Tl}$.

Tim Blackstad

Numerade Educator

Problem 91

The electron configuration of a neutral atom is $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} .$ Write a complete set of quantum numbers for each of the electrons. Name the element.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 92

Which of the following species has the most unpaired electrons? $\mathrm{S}^{+}, \mathrm{S},$ or $\mathrm{S}^{-} .$ Explain how you arrive at your answer.

Tim Blackstad

Numerade Educator

Problem 93

A sample tube consisted of atomic hydrogens in their ground state. A student illuminated the atoms with monochromatic light, that is, light of a single wavelength. If only two spectral emission lines in the visible region are observed, what is the wavelength (or wavelengths) of the incident radiation?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 94

A laser produces a beam of light with a wavelength of $532 \mathrm{nm} .$ If the power output is $25.0 \m$6.68 \times 10^{16}$ photons are emitted per secondathrm{mW}$ how many photons does the laser emit per second? $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s} .)$

David Collins

Numerade Educator

Problem 95

When a compound containing cesium ion is heated in a Bunsen burner flame, photons with an energy of $4.30 \times 10^{-19} \mathrm{J}$ are emitted. What color is the cesium flame?

Stephanie L

Numerade Educator

Problem 96

Discuss the current view of the correctness of the following statements. (a) The electron in the hydrogen atom is in an orbit that never brings it closer than $100 \mathrm{pm}$ to the nucleus. (b) Atomic absorption spectra result from transitions of electrons from lower to higher energy levels. (c) A many-electron atom behaves somewhat like a solar system that has a number of planets.

Tim Blackstad

Numerade Educator

Problem 97

What is the basis for thinking that atoms are spherical in shape even though the atomic orbitals $p, d, \ldots$ have distinctly nonspherical shapes?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 98

What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) $n=2, m_{s}=+\frac{1}{2} ;$ (b) $n=4, m_{\ell}=+1$.
(c) $n=3, \ell=2 ;$ (d) $n=2, \ell=0, m_{s}=-\frac{1}{2}.$
(e) $n=4, \ell=3, m_{\ell}=-2$.

Tim Blackstad

Numerade Educator

Problem 99

Identify the following individuals and their contributions to the development of quantum theory: Bohr, de Broglie, Einstein, Planck, Heisenberg, Schrödinger.

Ronald Prasad

Numerade Educator

Problem 100

What properties of electrons are used in the operation of an electron microscope?

Tim Blackstad

Numerade Educator

Problem 101

In a photoelectric experiment a student uses a light source whose frequency is greater than that needed to eject electrons from a certain metal. However, after continuously shining the light on the same area of the metal for a long period of time the student notices that the maximum kinetic energy of ejected electrons begins to decrease, even though the frequency of the light is held constant. How would you account for this behavior?

Ronald Prasad

Numerade Educator

Problem 102

A certain pitcher's fastballs have been clocked at about $100 \mathrm{mph} .$ (a) Calculate the wavelength of a 0.141-kg baseball (in nm) at this speed. (b) What is the wavelength of a hydrogen atom at the same speed? (1 mile $=1609 \mathrm{m} .)$

Tim Blackstad

Numerade Educator

Problem 103

A student carried out a photoelectric experiment by shining visible light on a clean piece of cesium metal. The table here shows the kinetic energies (KE) of the ejected electrons as a function of wavelengths ( $\lambda$ ). Determine graphically the work function and the Planck constant.$$\begin{array}{c|c|c|c|c|c}\lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\\hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\& 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19}\end{array}$$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 104

(a) What is the lowest possible value of the principal quantum number ( $n$ ) when the angular momentum quantum number $(\ell)$ is $1 ?$ (b) What are the possible values of the angular momentum quantum number (\ell) when the magnetic quantum number $\left(m_{\ell}\right)$ is 0 given than $n \leq 4 ?$

Tim Blackstad

Numerade Educator

Problem 105

Considering only the ground-state electron configuration, are there more diamagnetic or paramagnetic elements? Explain.

Ronald Prasad

Numerade Educator

Problem 106

A ruby laser produces radiation of wavelength $633 \mathrm{nm}$ in pulses whose duration is $1.00 \times 10^{-9}$ s. (a) If the laser produces 0.376 J of energy per pulse, how many photons are produced in each pulse? (b) Calculate the power (in watts) delivered by the laser per pulse. $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s} .)$

Tim Blackstad

Numerade Educator

Problem 107

A $368-\mathrm{g}$ sample of water absorbs infrared radiation at $1.06 \times 10^{4} \mathrm{nm}$ from a carbon dioxide laser. Suppose all the absorbed radiation is converted to heat. Calculate the number of photons at this wavelength required to raise the temperature of the water by $5.00^{\circ} \mathrm{C}$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 108

Photodissociation of water $$\mathrm{H}_{2} \mathrm{O}(l)+h v \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)$$ has been suggested as a source of hydrogen. The $\Delta H_{\mathrm{rxn}}^{\circ}$ for the reaction, calculated from thermochemical data, is $285.8 \mathrm{kJ}$ per mole of water decomposed. Calculate the maximum wavelength (in nm) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

Tim Blackstad

Numerade Educator

Problem 109

Spectral lines of the Lyman and Balmer series do not overlap. Verify this statement by calculating the longest wavelength associated with the Lyman series and the shortest wavelength associated with the Balmer series (in $\mathrm{nm}$ ).

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 110

An atom moving at its root-mean-square speed at $20^{\circ} \mathrm{C}$ has a wavelength of $3.28 \times 10^{-11} \mathrm{m} .$ Identify the atom.

Tim Blackstad

Numerade Educator

Problem 111

Certain sunglasses have small crystals of silver chloride (AgCl) incorporated in the lenses. When the lenses are exposed to light of the appropriate wavelength, the following reaction occurs: $$\mathrm{AgCl} \longrightarrow \mathrm{Ag}+\mathrm{Cl}$$ The Ag atoms formed produce a uniform gray color that reduces the glare. If $\Delta H$ for the preceding reaction is $248 \mathrm{kJ} / \mathrm{mol}$, calculate the maximum wavelength of light that can induce this process.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 112

The $\mathrm{He}^{+}$ ion contains only one electron and is therefore a hydrogenlike ion. Calculate the wavelengths, in increasing order, of the first four transitions in the Balmer series of the $\mathrm{He}^{+}$ ion. Compare these wavelengths with the same transitions in a H atom. Comment on the differences. (The Rydberg constant for $\left.\mathrm{He}^{+} \text {is } 8.72 \times 10^{-18} \mathrm{J} .\right)$

Crystal Wang

Numerade Educator

Problem 113

Ozone $\left(\mathrm{O}_{3}\right)$ in the stratosphere absorbs the harmful radiation from the sun by undergoing decomposition: $\mathrm{O}_{3} \longrightarrow \mathrm{O}+\mathrm{O}_{2} .$ (a) Referring to Table 6.4 calculate the $\Delta H^{\circ}$ for this process. (b) Calculate the maximum wavelength of photons (in $\mathrm{nm}$ ) that possess this energy to cause the decomposition of ozone photochemically.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 114

The retina of a human eye can detect light when radiant energy incident on it is at least $4.0 \times 10^{-17} \mathrm{J}$ For light of 600 -nm wavelength, how many photons does this correspond to?

Tim Blackstad

Numerade Educator

Problem 115

A helium atom and a xenon atom have the same kinetic energy. Calculate the ratio of the de Broglie wavelength of the helium atom to that of the xenon atom.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 116

A laser is used in treating retina detachment. The wavelength of the laser beam is $514 \mathrm{nm}$ and the power is $1.6 \mathrm{W}$. If the laser is turned on for $0.060 \mathrm{s}$ during surgery, calculate the number of photons emitted by the laser. ( $1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}$.)

Tim Blackstad

Numerade Educator

Problem 117

An electron in an excited state in a hydrogen atom can return to the ground state in two different ways:
(a) via a direct transition in which a photon of wavelength $\lambda_{1}$ is emitted and (b) via an intermediate excited state reached by the emission of a photon of wavelength $\lambda_{2} .$ This intermediate excited state then decays to the ground state by emitting another photon of wavelength $\lambda_{3}$. Derive an equation that relates $\lambda_{1}$ to $\lambda_{2}$ and $\lambda_{3}$.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 118

A photoelectric experiment was performed by separately shining a laser at $450 \mathrm{nm}$ (blue light) and a laser at $560 \mathrm{nm}$ (yellow light) on a clean metal surface and measuring the number and kinetic energy of the ejected electrons. Which light would generate more electrons? Which light would eject electrons with greater kinetic energy? Assume that the same amount of energy is delivered to the metal surface by each laser and that the frequencies of the laser lights exceed the threshold frequency.

Tim Blackstad

Numerade Educator

Problem 119

Draw the shapes (boundary surfaces) of the following orbitals: (a) $2 p_{y},$ (b) $3 d_{z^{2}},$ (c) $3 d_{x^{2}-y^{2}}$. (Show coordinate axes in your sketches.)

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 120

The electron configurations described in this chapter all refer to gaseous atoms in their ground states. An atom may absorb a quantum of energy and promote one of its electrons to a higher-energy orbital. When this happens, we say that the atom is in an excited state. The electron configurations of some excited atoms are given. Identify these atoms and write their ground-state configurations:
(a) $1 s^{1} 2 s^{1}$
(b) $1 s^{2} 2 s^{2} 2 p^{2} 3 d^{1}$
(c) $1 s^{2} 2 s^{2} 2 p^{6} 4 s^{1}$
(d) $[\mathrm{Ar}] 4 s^{1} 3 d^{10} 4 p^{4}$
(e) $[\mathrm{Ne}] 3 s^{2} 3 p^{4} 3 d^{1}$

Tim Blackstad

Numerade Educator

Problem 121

Draw orbital diagrams for atoms with the following electron configurations:
(a) $1 s^{2} 2 s^{2} 2 p^{5}$
(b) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{3}$
(c) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}$

Stephanie L

Numerade Educator

Problem 122

If Rutherford and his coworkers had used electrons instead of alpha particles to probe the structure of the nucleus as described in Section $2.2,$ what might they have discovered?

Tim Blackstad

Numerade Educator

Problem 123

Scientists have found interstellar hydrogen atoms with quantum number $n$ in the hundreds. Calculate the wavelength of light emitted when a hydrogen atom undergoes a transition from $n=236$ to $n=235$ In what region of the electromagnetic spectrum does this wavelength fall?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 124

Calculate the wavelength of a helium atom whose speed is equal to the root-mean-square speed at $20^{\circ} \mathrm{C}$.

Tim Blackstad

Numerade Educator

Problem 125

Ionization energy is the minimum energy required to remove an electron from an atom. It is usually expressed in units of $\mathrm{kJ} / \mathrm{mol}$, that is, the energy in kilojoules required to remove one mole of electrons from one mole of atoms. (a) Calculate the ionization energy for the hydrogen atom. (b) Repeat the calculation, assuming in this second case that the electrons are removed from the $n=2$ state.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 126

An electron in a hydrogen atom is excited from the ground state to the $n=4$ state. Comment on the correctness of the following statements (true or false).
(a) $n=4$ is the first excited state.
(b) It takes more energy to ionize (remove) the electron from $n=4$ than from the ground state.
(c) The electron is farther from the nucleus (on average) in $n=4$ than in the ground state.
(d) The wavelength of light emitted when the electron drops from $n=4$ to $n=1$ is longer than that from $n=4$ to $n=2$.
(e) The wavelength the atom absorbs in going from $n=1$ to $n=4$ is the same as that emitted as it goes from $n=4$ to $n=1$.

Tim Blackstad

Numerade Educator

Problem 127

The ionization energy of a certain element is 412 kJ/mol (see Problem 7.125). However, when the atoms of this element are in the first excited state, the ionization energy is only $126 \mathrm{kJ} / \mathrm{mol}$. Based on this information, calculate the wavelength of light emitted in a transition from the first excited state to the ground state.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 128

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136 ) whose average diameter is $5.0 \times 10^{-5} \mathrm{m}$ Consider an oxygen molecule $\left(5.3 \times 10^{-26} \mathrm{kg}\right)$ trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

Tim Blackstad

Numerade Educator

Problem 129

How many photons at 660 nm must be absorbed to melt $5.0 \times 10^{2} \mathrm{g}$ of ice? On average, how many $\mathrm{H}_{2} \mathrm{O}$ molecules does one photon convert from ice to water? (Hint: It takes $334 \mathrm{J}$ to melt $1 \mathrm{g}$ of ice at $\left.0^{\circ} \mathrm{C} .\right)$

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 130

Shown are portions of orbital diagrams representing the ground-state electron configurations of certain elements. Which of them violate the Pauli exclusion principle? Hund's rule?

Ronald Prasad

Numerade Educator

Problem 131

The UV light that is responsible for tanning the skin falls in the 320 - to 400 -nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for $2.0 \mathrm{h}$, given that there are $2.0 \times 10^{16} \mathrm{pho}-$ tons hitting Earth's surface per square centimeter per second over a $80-\mathrm{nm}(320 \mathrm{nm} \text { to } 400 \mathrm{nm})$ range and that the exposed body area is $0.45 \mathrm{m}^{2}$. Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Hint: Use an average wavelength of $360 \mathrm{nm}$ in calculating the energy of a photon.)

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 132

The sun is surrounded by a white circle of gaseous material called the corona, which becomes visible during a total eclipse of the sun. The temperature of the corona is in the millions of degrees Celsius, which is high enough to break up molecules and remove some or all of the electrons from atoms. One way astronomers have been able to estimate the temperature of the corona is by studying the emission lines of ions of certain elements. For example, the emission spectrum of $\mathrm{Fe}^{14+}$ ions has been recorded and analyzed. Knowing that it takes $3.5 \times 10^{4} \mathrm{kJ} / \mathrm{mol}$ to convert $\mathrm{Fe}^{13+}$ to $\mathrm{Fe}^{14+}$ estimate the temperature of the sun's corona. (Hint: The average kinetic energy of one mole of a gas is $\frac{3}{2} R T$.)

Tim Blackstad

Numerade Educator

Problem 133

In 1996 physicists created an anti-atom of hydrogen. In such an atom, which is the antimatter equivalent of an ordinary atom, the electrical charges of all the component particles are reversed. Thus, the nucleus of an anti-atom is made of an anti-proton, which has the same mass as a proton but bears a negative charge, while the electron is replaced by an antielectron (also called positron) with the same mass as an electron, but bearing a positive charge. Would you expect the energy levels, emission spectra, and atomic orbitals of an antihydrogen atom to be different from those of a hydrogen atom? What would happen if an anti-atom of hydrogen collided with a hydrogen atom?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 134

Use Equation (5.16) to calculate the de Broglie wavelength of a $\mathrm{N}_{2}$ molecule at $300 \mathrm{K}$.

Tim Blackstad

Numerade Educator

Problem 135

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number $n .$ However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum $\ell .$ This is the selection rule,which states that $\Delta \ell=\pm 1 ;$ that is, in a transition, the value of $\ell$ can only increase or decrease by one. According to this rule, which of the following transitions are allowed: $(\mathrm{a}) 2 \mathrm{s} \longrightarrow 1 s,$ (b) $3 p \longrightarrow 1 s$ (c) $3 d \longrightarrow 4 f,$ (d) $4 d \longrightarrow 3 s ?$ In view of this selection rule, explain why it is possible to observe the various emission series shown in Figure 7.11.

Crystal Wang

Numerade Educator

Problem 136

In an electron microscope, electrons are accelerated by passing them through a voltage difference. The kinetic energy thus acquired by the electrons is equal to the voltage times the charge on the electron. Thus, a voltage difference of $1 \mathrm{V}$ imparts a kinetic energy of $1.602 \times 10^{-19} \mathrm{C} \times \mathrm{V}$ or $1.602 \times 10^{-19} \mathrm{J}$.Calculate the wavelength associated with electrons accelerated by $5.00 \times 10^{3} \mathrm{V}$.

Tim Blackstad

Numerade Educator

Problem 137

A microwave oven operating at $1.22 \times 10^{8} \mathrm{nm}$ is used to heat $150 \mathrm{mL}$ of water (roughly the volume of a tea cup) from $20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$. Calculate the number of photons needed if 92.0 percent of microwave energy is converted to the thermal energy of water.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 138

The radioactive $\mathrm{Co}-60$ isotope is used in nuclear medicine to treat certain types of cancer. Calculate the wavelength and frequency of an emitted gamma photon having the energy of $1.29 \times$ $10^{11} \mathrm{J} / \mathrm{mol}$.

Tim Blackstad

Numerade Educator

Problem 139

(a) An electron in the ground state of the hydrogen atom moves at an average speed of $5 \times 10^{6} \mathrm{m} / \mathrm{s}$. If the speed is known to an uncertainty of 1 percent, what is the uncertainty in knowing its position? Given that the radius of the hydrogen atom in the ground state is $5.29 \times 10^{-11} \mathrm{m},$ comment on your result. The mass of an electron is $9.1094 \times 10^{-31} \mathrm{kg}$.(a) An electron in the ground state of the hydrogen atom moves at an average speed of $5 \times 10^{6} \mathrm{m} / \mathrm{s}$. If the speed is known to an uncertainty of 1 percent, what is the uncertainty in knowing its position? Given that the radius of the hydrogen atom in the ground state is $5.29 \times 10^{-11} \mathrm{m},$ comment on your result. The mass of an electron is $9.1094 \times 10^{-31} \mathrm{kg}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 140

One wavelength in the hydrogen emission spectrum is $1280 \mathrm{nm}$. What are the initial and final states of the transition responsible for this emission?

Tim Blackstad

Numerade Educator

Problem 141

Owls have good night vision because their eyes can detect a light intensity as low as $5.0 \times 10^{-13} \mathrm{W} / \mathrm{m}^{2}$ Calculate the number of photons per second that an owl's eye can detect if its pupil has a diameter of $9.0 \mathrm{mm}$ and the light has a wavelength of $500 \mathrm{nm}$
$(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s} .)$

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 142

For hydrogenlike ions, that is, ions containing only one electron, Equation (7.5) is modified as follows: $E_{n}=-R_{\mathrm{H}} Z^{2}\left(1 / n^{2}\right),$ where $Z$ is the atomic number of the parent atom. The figure here represents the emission spectrum of such a hydrogenlike ion in the gas phase. All the lines result from the electronic transitions from the excited states to the $n=2$ state. (a) What electronic transitions correspond to lines $B$ and $C ?$ (b) If the wavelength of line $C$ is $27.1 \mathrm{nm},$ calculate the wavelengths of lines A and B. (c) Calculate the energy needed to remove the electron from the ion in the $n=4$ state. (d) What is the physical significance of the continuum?

Crystal Wang

Numerade Educator

Problem 143

Calculate the energies needed to remove an electron from the $n=1$ state and the $n=5$ state in the $L i^{2+}$ ion. What is the wavelength (in $n m$ ) of the emitted photon in a transition from $n=5$ to $n=1 ?$ The Rydberg constant for hydrogenlike ions is $\left(2.18 \times 10^{-18} \mathrm{J}\right) Z^{2},$ where $Z$ is the atomic number.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 144

Calculate the energies needed to remove an electron from the $n=1$ state and the $n=5$ state in the $L i^{2+}$ ion. What is the wavelength (in $n m$ ) of the emitted photon in a transition from $n=5$ to $n=1 ?$ The Rydberg constant for hydrogenlike ions is $\left(2.18 \times 10^{-18} \mathrm{J}\right) Z^{2},$ where $Z$ is the atomic number.

Tim Blackstad

Numerade Educator

Problem 145

The de Broglie wavelength of an accelerating proton in the Large Hadron Collider is $2.5 \times 10^{-14} \mathrm{m}$. What is the kinetic energy (in joules) of the proton?

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 146

The minimum uncertainty in the position of a certain moving particle is equal to its de Broglie wavelength. If the speed of the particle is $1.2 \times 10^{5} \mathrm{m} / \mathrm{s}$ what is the minimum uncertainty in its speed?

Ronald Prasad

Numerade Educator

Problem 147

According to Einstein's special theory of relativity, the mass of a moving particle, $m_{\text {moving }},$ is related to its mass at rest, $m_{\text {rest }},$ by the following equation $m_{\text {moving }}=\frac{m_{\text {rest }}}{\sqrt{1-\left(\frac{u}{c}\right)^{2}}}$,where $u$ and $c$ are the speeds of the particle and light, respectively. (a) In particle accelerators, protons, electrons, and other charged particles are often accelerated to speeds close to the speed of light. Calculate the wavelength (in $\mathrm{nm}$ ) of a proton moving at
50.0 percent the speed of light. The mass of a proton is $1.673 \times 10^{-27} \mathrm{kg} .$ (b) Calculate the mass of a 6.0 $\times 10^{-2} \mathrm{kg}$ tennis ball moving at $63 \mathrm{m} / \mathrm{s} .$ Comment on your results.

Kim Trang Nguyen

Kim Trang Nguyen

Numerade Educator

Problem 148

The mathematical equation for studying the photoelectric effect is $$h v=W+\frac{1}{2} m_{e} u^{2}$$ where $v$ is the frequency of light shining on the metal, $W$ is the work function, and $m_{e}$ and $u$ are the mass and speed of the ejected electron. In an experiment, a student found that a maximum wavelength of $351 \mathrm{nm}$ is needed to just dislodge electrons from a zinc metal surface. Calculate the speed (in $\mathrm{m} / \mathrm{s}$ ) of an ejected electron when she employed light with a wavelength of $313 \mathrm{nm}$.

Tim Blackstad

Numerade Educator

Problem 149

In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. Use the Heisenberg uncertainty principle to show that an electron cannot be confined within a nucleus. Repeat the calculation for a proton. Comment on your results. Assume the radius of a nucleus to be $1.0 \times 10^{-15} \mathrm{m} .$ The masses of an electron and a proton are $9.109 \times 10^{-31} \mathrm{kg}$ and $1.673 \times 10^{-27} \mathrm{kg},$ respectively. (Hint: Treat the diameter of the nucleus as the uncertainty in position.)

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 150

Blackbody radiation is the term used to describe the dependence of the radiation energy emitted by an object on wavelength at a certain temperature. Planck proposed the quantum theory to account for this dependence. Shown in the figure is a plot of the radiation energy emitted by our sun versus wavelength. This curve is characteristic of the temperature at the surface of the sun. At a higher temperature, the curve has a similar shape but the maximum will shift to a shorter wavelength. What does this curve reveal about two consequences of great biological significance on Earth?

Tim Blackstad

Numerade Educator

Problem 151

All molecules undergo vibrational motions. Quantum mechanical treatment shows that the vibrational energy, $E_{\text {vib }},$ of a diatomic molecule like HCl is given by $$E_{\mathrm{vib}}=\left(n+\frac{1}{2}\right) h v$$ where $n$ is a quantum number given by $n=0,1,2$ 3, .... and $v$ is the fundamental frequency of vibration. (a) Sketch the first three vibrational energy levels for HCl. (b) Calculate the energy required to excite a HCl molecule from the ground level to the first excited level. The fundamental frequency of vibration for $\mathrm{HCl}$ is $8.66 \times 10^{13} \mathrm{s}^{-1}$. (c) The fact that the lowest vibrational energy in the ground level is not zero but equal to $\frac{1}{2} h v$ means that molecules will vibrate at all temperatures, including the absolute zero. Use the Heisenberg uncertainty principle to justify this prediction. (Hint: Consider a nonvibrating molecule and predict the uncertainty in the momentum and hence the uncertainty in the position.)

Ronald Prasad

Numerade Educator

Problem 152

The wave function for the $2 s$ orbital in the hydrogen atom is $$\Psi_{2 s}=\frac{1}{\sqrt{2 a_{0}^{3}}}\left(1-\frac{\rho}{2}\right) e^{-\rho / 2}$$ where $a_{0}$ is the value of the radius of the first Bohr orbit, equal to $0.529 \mathrm{nm}, \rho$ is $Z\left(r / a_{0}\right),$ and $r$ is the distance from the nucleus in meters. Calculate the location of the node of the $2 s$ wave function from the nucleus.

Manik Pulyani

Numerade Educator

Problem 153

A student placed a large unwrapped chocolate bar in a microwave oven without a rotating glass plate. After turning the oven on for less than a minute, she noticed there were evenly spaced dents (due to melting) about $6 \mathrm{cm}$ apart. Based on her observations, calculate the speed of light given that the microwave frequency is $2.45 \mathrm{GHz} .$ (Hint: The energy of a wave is proportional to the square of its amplitude.)

Ronald Prasad

Numerade Educator

Problem 154

The wave properties of matter can generally be ignored for macroscopic objects such as tennis balls; however, wave properties have been measured at the fringe of detection for some very large molecules. For example, wave patterns were detected for $\mathrm{C}_{60}\left(\mathrm{C}_{12} \mathrm{F}_{25}\right)_{8}$ molecules moving at a velocity of $63 \mathrm{m} / \mathrm{s} .$ (a) Calculate the wavelength of $\mathrm{a} \mathrm{C}_{60}\left(\mathrm{C}_{12} \mathrm{F}_{25}\right)_{8}$
molecule moving at this velocity. (b) How does the wavelength compare to the size of the molecule given that its diameter is roughly $3000 \mathrm{pm} ?$

Tim Blackstad

Numerade Educator

Problem 155

Atoms of an element have only two accessible excited states. In an emission experiment, however, three spectral lines were observed. Explain. Write an equation relating the shortest wavelength to the other two wavelengths.

Nicole Smina

Numerade Educator

Problem 156

According to Wien's law, the wavelength of maximum intensity in blackbody radiation, $\lambda_{\max },$ is given by $$\lambda_{\max }=\frac{b}{T}$$ where $b$ is a constant $\left(2.898 \times 10^{6} \mathrm{nm} \cdot \mathrm{K}\right)$ and $T$ is the temperature of the radiating body in kelvins.
(a) Estimate the temperature at the surface of the sun. (b) How are astronomers able to determine the temperature of stars in general? (See Problem 7.150 for a definition of blackbody radiation.)

Tim Blackstad

Numerade Educator

Problem 157

Only a fraction of the electrical energy supplied to an incandescent-tungsten lightbulb is converted to visible light. The rest of the energy shows up as infrared radiation (that is, heat). A $60-\mathrm{W}$ lightbulb converts about 15.0 percent of the energy supplied to it into visible light. Roughly how many photons are emitted by the lightbulb per second? $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}$.)

Crystal Wang

Numerade Educator

Problem 158

Photosynthesis makes use of photons of visible light to bring about chemical changes. Explain why heat energy in the form of infrared photons is ineffective for photosynthesis. (Hint: Typical chemical bond energies are $200 \mathrm{kJ} / \mathrm{mol}$ or greater.)

Tim Blackstad

Numerade Educator

Problem 159

A typical red laser pointer has a power of $5 \mathrm{mW}$ How long would it take a red laser pointer to emit the same number of photons emitted by a $1-\mathrm{W}$ blue laser in $1 \mathrm{s} ?(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s} .)$

Nicole Smina

Numerade Educator

Problem 160

Referring to the Chemistry in Action essay on p. $312,$ estimate the wavelength of light that would be emitted by a cadmium selenide (CdSe) quantum dot with a diameter of $10 \mathrm{nm}$. Would the emitted light be visible to the human eye? The diameter and emission wavelength for a series of quantumdots are given here.$$\begin{array}{l|c|c|c|c|c|c}
\text { Diameter (nm) } & 2.2 & 2.5 & 3.3 & 4.2 & 4.9 & 6.3 \\
\hline \text { Wavelength (nm) } & 462 & 503 & 528 & 560 & 583 & 626
\end{array}$$

David Collins

Numerade Educator