General Chemistry

Darrell Ebbing, Steven D. Gammon

Chapter 7

Quantum Theory of the Atom - all with Video Answers

Educators

Chapter Questions

Problem 1

Give a brief wave description of light. What are two characteristics of light waves?

Nicole Smina

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Problem 2

What is the mathematical relationship among the different characteristics of light waves? State the meaning of each of the terms in the equation.

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Problem 3

Briefly describe the portions of the electromagnetic spectrum, starting with shortest wavelengths and going to longer wavelengths.

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Problem 4

Planck originated the idea that energies can be quantized. What does the term quantized mean? What was Planck trying to explain when he was led to the concept of quantization of energy? Give the formula he arrived at and explain each of the terms in the formula.

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Problem 5

In your own words, explain the photoelectric effect. How does the photon concept explain this effect?

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Problem 6

Describe the wave-particle picture of light.

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Problem 7

Give the equation that relates particle properties of light. Explain the meaning of each symbol in the equation.

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Problem 8

Physical theory at the time Rutherford proposed his nuclear model of the atom was not able to explain how this model could give a stable atom. Explain the nature of this difficulty.

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Problem 9

Explain the main features of Bohr's theory. Do these features solve the difficulty alluded to in Question $7.8 ?$

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Problem 10

Explain the process of emission of light by an atom.

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Problem 11

Explain the process of absorption of light by an atom.

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Problem 12

What is the evidence for electron waves? Give a practical application.

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Problem 13

What kind of information does a wave function give about an electron in an atom?

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Problem 14

The atom is sometimes said to be similar to a miniature planetary system, with electrons orbiting the nucleus. What does the uncertainty principle have to say about this view of the atom?

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Problem 15

Bohr described the hydrogen atom as an electron orbiting a hydrogen nucleus. Although certain aspects of his theory are still valid, his theory agreed quantitatively with experiment only in the case of the hydrogen atom. In what way does quantum mechanics change Bohr's original picture of the hydrogen atom?

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Problem 16

Give the possible values of (a) the principal quantum number, (b) the angular momentum quantum number,
(c) the magnetic quantum number, and (d) the spin quantum number.

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Problem 17

What is the notation for the subshell in which $n=4$ and $l=3$ ? How many orbitals are in this subshell?

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Problem 18

What is the general shape of an $s$ orbital? of a $p$ orbital?

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Problem 19

Which of the following statements about a hydrogen atom is false?
a. An electron in the $n=1$ level of the hydrogen atom is in its ground state.
b. On average, an electron in the $n=3$ level is farther from the nucleus than an electron in the $n=2$ state.
c. The wavelength of light emitted when the electron goes from the $n=3$ level to the $n=1$ level is the same as the wavelength of light absorbed when the electron goes from the $n=1$ level to $n=3$ level.
d. An electron in the $n=1$ level is higher in energy than an electron in the $n=4$ level.
e. Light of greater frequency is required for a transition from the $n=1$ level to $n=3$ level than is required for a transition from the $n=2$ level to $n=3$ level.

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Problem 20

Which of the following statements is (are) true?
I. The product of wavelength and frequency of light is a constant.
II. As the energy of electromagnetic radiation increases, its frequency decreases.
III. As the wavelength of light increases, its frequency increases.
a. I only
b. II only
c. III only
d. I and III only
e. II and III only

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Problem 21

Of the following possible transitions of an electron in a hydrogen atom, which emits light of the highest energy?
a. Transition from the $n=1$ to the $n=3$ level
b. Transition from the $n=1$ to the $n=2$ level
c. Transition from the $n=3$ to the $n=1$ level
d. Transition from the $n=2$ to the $n=1$ level
e. Transition from the $n=5$ to the $n=4$ level

Nicole Smina

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Problem 22

What wavelength of electromagnetic radiation corresponds to a frequency of $3.46 \times 10^{13} \mathrm{~s}^{-1}$ ?
a. $8.66 \times 10^{-6} \mathrm{~m}$
b. $1.15 \times 10^{5} \mathrm{~m}$
c. $7.65 \times 10^{-29} \mathrm{~m}$
d. $9.10 \times 10^{-6} \mathrm{~m}$
e. $8.99 \times 10^{-6} \mathrm{~m}$

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Problem 23

Light, Energy, and the Hydrogen Atom
a. Which has the greater wavelength, blue light or red light?
b. How do the frequencies of blue light and red light compare?
C. How does the energy of blue light compare with that of red light?
d. Does blue light have a greater speed than red light?
e. How does the energy of three photons from a blue light source compare with the energy of one photon of blue light from the same source? How does the energy of two photons corresponding to a wavelength of $451 \mathrm{~nm}$ (blue light) compare with the energy of three photons corresponding to a wavelength of $704 \mathrm{~nm}$ (red light)?
f. A hydrogen atom with an electron in its ground state interacts with a photon of light with a wavelength of $1.22 \times$ $10^{-6} \mathrm{~m} .$ Could the electron make a transition from the ground state to a higher energy level? If it does make a transition, indicate which one. If no transition can occur, explain.
g. If you have one mole of hydrogen atoms with their electrons in the $n=1$ level, what is the minimum number of photons you would need to interact with these atoms in order to have all of their electrons promoted to the $n=3$ level? What wavelength of light would you need to perform this experiment?

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Problem 24

Investigating Energy Levels Consider the hypothetical atom $\mathrm{X}$ that has one electron like the $\mathrm{H}$ atom but has different energy levels. The energies of an electron in an $\mathrm{X}$ atom are described by the equation
$$
E=-\frac{R_{\mathrm{H}}}{n^{3}}
$$
where $R_{\mathrm{H}}$ is the same as for hydrogen $\left(2.179 \times 10^{-18} \mathrm{~J}\right)$. Answer the following questions, without calculating energy values.
a. How would the ground-state energy levels of $\mathrm{X}$ and $\mathrm{H}$ compare?
b. Would the energy of an electron in the $n=2$ level of $\mathrm{H}$ be higher or lower than that of an electron in the $n=2$ level of X? Explain your answer.
c. How do the spacings of the energy levels of $X$ and $H$ compare?
d. Which would involve the emission of a higher frequency of light, the transition of an electron in an $\mathrm{H}$ atom from the $n=$ 5 to the $n=3$ level or a similar transition in an $\mathrm{X}$ atom?
e. Which atom, $\mathrm{X}$ or $\mathrm{H}$, would require more energy to completely remove its electron?
f. A photon corresponding to a particular frequency of blue light produces a transition from the $n=2$ to the $n=5$ level of a hydrogen atom. Could this photon produce the same transition $(n=2$ to $n=5$ ) in an atom of $\mathrm{X} ?$ Explain.

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Problem 25

Consider two beams of the same yellow light. Imagine that one beam has its wavelength doubled; the other has its frequency doubled. Which of these two beams is then in the ultraviolet region?

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Problem 26

Some infrared radiation has a wavelength that is 1000 times larger than that of a certain visible light. This visible light has a frequency that is 1000 times smaller than that of some $X$ radiation. How many times more energy is there in a photon of this X radiation than there is in a photon of the infrared radiation?

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Problem 27

One photon of green light has less than twice the energy of two photons of red light. Consider two hypothetical experiments. In one experiment, potassium metal is exposed to one photon of green light; in another experiment, potassium metal is exposed to two photons of red light. In one of these experiments, no electrons are ejected by the photoelectric effect (no matter how many times this experiment is repeated). In the other experiment, at least one electron was observed to be ejected. What is the maximum number of electrons that could be ejected during this other experiment, one or two?

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Problem 28

An atom in its ground state absorbs a photon (photon 1 ), then quickly emits another photon (photon 2). One of these photons corresponds to ultraviolet radiation, whereas the other one corresponds to red light. Explain what is happening. Which electromagnetic radiation, ultraviolet or red light, is associated with the emitted photon (photon 2)?

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Problem 29

Three emission lines involving three energy levels in an atom occur at wavelengths $x, 1.5 x$, and $3.0 x$ nanometers. Which wavelength corresponds to the transition from the highest to the lowest of the three energy levels?

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Problem 30

An atom emits yellow light when an electron makes the transition from the $n=5$ to the $n=1$ level. In separate experiments, suppose you bombarded the $n=1$ level of this atom with red light, yellow light (obtained from the previous emission), and blue light. In which experiment or experiments would the electron be promoted to the $n=5$ level?

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Problem 31

Which of the following particles has the longest wavelength?
a. an electron traveling at $x$ meters per second
b. a proton traveling at $x$ meters per second
C. a proton traveling at $2 x$ meters per second

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Problem 32

Imagine a world in which the rule for the $l$ quantum number is that values start with 1 and go up to $n$. The rules for the $n$ and $m_{l}$ quantum numbers are unchanged from those of our world. Write the quantum numbers for the first two shells (i.e., $n=1$ and $n=2$ ).

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Problem 33

Given the following energy level diagram for an atom that contains an electron in the $n=3$ level, answer the following questions.
a. Which transition of the electron will emit light of the lowest frequency?
b. Using only those levels depicted in the diagram, which transition of the electron would require the highestfrequency light?
c. If the transition from the $n=3$ level to the $n=1$ level emits green light, what color light is absorbed when an electron makes the transition from the $n=1$ to $n=3$ level?

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Problem 34

The following shapes each represent an orbital of an atom in a hypothetical universe. The small circle is the location of the nucleus in each orbital.
a. If you placed an electron in each orbital, which one would be higher in energy?
b. When an electron makes a transition from the orbital represented on the right to the orbital on the left, would you expect energy to be absorbed or released?
c. Draw a sketch of an orbital of the same type that would be higher in energy than either of the two pictured orbitals.

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Problem 35

Radio waves in the AM region have frequencies in the range 530 to 1700 kilocycles per second $(530$ to $1700 \mathrm{kHz})$. Calculate the wavelength corresponding to a radio wave of frequency $1.365 \times 10^{6} / \mathrm{s}$ (that is, $1365 \mathrm{kHz}$ ).

Nicole Smina

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Problem 36

Microwaves have frequencies in the range $10^{9}$ to $10^{12} / \mathrm{s}$ (cycles per second), equivalent to between 1 gigahertz and 1 terahertz. What is the wavelength of microwave radiation whose frequency is $1.258 \times 10^{10} / \mathrm{s}$ ?

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Problem 37

Light with a wavelength of $478 \mathrm{~nm}$ lies in the blue region of the visible spectrum. Calculate the frequency of this light.

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Problem 38

Calculate the frequency associated with light of wavelength $656 \mathrm{~nm}$. (This corresponds to one of the wavelengths of light emitted by the hydrogen atom.)

Jennifer Hudspeth

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Problem 39

At its closest approach, Mars is 56 million $\mathrm{km}$ from Earth. How long would it take to send a radio message from a space probe of Mars to Earth when the planets are at this closest distance?

Cheryl Glor

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Problem 40

The space probe Pioneer 11 was launched April 5,1973 , and reached Jupiter in December 1974, traveling a distance of 998 million $\mathrm{km}$. How long did it take an electromagnetic signal to travel to Earth from Pioneer 11 when it was near Jupiter?

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Problem 41

The meter was defined in 1963 as the length equal to $1,650,763.73$ wavelengths of the orange-red radiation emitted by the krypton-86 atom (the meter has since been redefined). What is the wavelength of this transition? What is the frequency?

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Problem 42

The second is defined as the time it takes for $9,192,631,770$ wavelengths of a certain transition of the cesium- 133 atom to pass a fixed point. What is the frequency of this electromagnetic radiation? What is the wavelength?

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Problem 43

What is the energy of a photon corresponding to radio waves of frequency $1.365 \times 10^{6} / \mathrm{s}$ ?

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Problem 44

What is the energy of a photon corresponding to microwave radiation of frequency $1.258 \times 10^{10} / \mathrm{s}$ ?

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Problem 45

The green line in the atomic spectrum of thallium has a wavelength of $535 \mathrm{~nm}$. Calculate the energy of a photon of this light.

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Problem 46

Indium compounds give a blue-violet flame test. The atomic emission responsible for this blue-violet color has a wavelength of $451 \mathrm{~nm}$. Obtain the energy of a single photon of this wavelength.

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Problem 47

A particular transition of the rubidium atom emits light whose frequency is $3.84 \times 10^{14} \mathrm{~Hz} .(\mathrm{Hz}$ is the abbreviation for hertz, which is equivalent to the unit $/ \mathrm{s}$, or $\mathrm{s}^{-1}$.) Is this light in the visible spectrum? If so, what is the color of the light? (See Figure 7.5.)

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Problem 48

Barium atoms have a particular transition that emits light of frequency $5.41 \times 10^{14} \mathrm{~Hz} .(\mathrm{Hz}$ is the abbreviation for hertz, which is equivalent to the unit $/ \mathrm{s}$, or $\mathrm{s}^{-1}$.) Is this light in the visible spectrum? If so, what is the color of the light? (See Figure 7.5.)

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Problem 49

An electron in a hydrogen atom in the level $n=5$ undergoes a transition to level $n=3 .$ What is the frequency of the emitted radiation?

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Problem 50

Calculate the frequency of electromagnetic radiation emitted by the hydrogen atom in the electron transition from $n=4$ to $n=3$.

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Problem 51

The first line of the Lyman series of the hydrogen atom emission results from a transition from the $n=2$ level to the $n=1$ level. What is the wavelength of the emitted photon? Using Figure $7.5$, describe the region of the electromagnetic spectrum in which this emission lies.

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Problem 52

What is the wavelength of the electromagnetic radiation emitted from a hydrogen atom when the electron undergoes the transition $n=5$ to $n=4 ?$ In what region of the spectrum does this line occur? (See Figure $7.5 .)$

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Problem 53

Calculate the shortest wavelength of the electromagnetic radiation emitted by the hydrogen atom in undergoing a transition from the $n=6$ level.

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Problem 54

Calculate the longest wavelength of the electromagnetic radiation emitted by the hydrogen atom in undergoing a transition from the $n=7$ level.

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Problem 55

What is the difference in energy between the two levels responsible for the violet emission line of the calcium atom at $422.7 \mathrm{~nm} ?$

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Problem 56

What is the difference in energy between the two levels responsible for the ultraviolet emission line of the magnesium atom at $285.2 \mathrm{~nm}$ ?

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Problem 57

What is the wavelength of a neutron traveling at a speed of $4.15 \mathrm{~km} / \mathrm{s} ?$ (Neutrons of these speeds are obtained from a nuclear pile.)

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Problem 58

What is the wavelength of a proton traveling at a speed of $6.58 \mathrm{~km} / \mathrm{s} ?$ What would be the region of the spectrum for electromagnetic radiation of this wavelength?

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Problem 59

At what speed must an electron travel to have a wavelength of $10.0 \mathrm{pm} ?$

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Problem 60

At what speed must a neutron travel to have a wavelength of $12.0 \mathrm{pm} ?$

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Problem 61

What is the de Broglie wavelength of a $145-\mathrm{g}$ baseball traveling at $30.0 \mathrm{~m} / \mathrm{s}(67.1 \mathrm{mph}) ?$ Is the wavelength much smaller or much larger than the diameter of an atom (on the order of $100 \mathrm{pm}$ )?

Nicole Smina

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Problem 62

What is the de Broglie wavelength of an oxygen molecule, $\mathrm{O}_{2}$, traveling at $521 \mathrm{~m} / \mathrm{s} ?$ Is the wavelength much smaller or much larger than the diameter of an atom (on the order of $100 \mathrm{pm}$ )?

Ronald Prasad

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Problem 63

If the $n$ quantum number of an atomic orbital is 4, what are the possible values of $l$ ? If the $l$ quantum number is 3, what are the possible values of $m_{l}$ ?

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Problem 64

The $n$ quantum number of an atomic orbital is 6 . What are the possible values of $l ?$ What are the possible values of $m_{l}$ if the $l$ quantum number is $5 ?$

Nicole Smina

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Problem 65

How many subshells are there in the $M$ shell? How many orbitals are there in the $f$ subshell?

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Problem 66

How many subshells are there in the $N$ shell? How many orbitals are there in the $g$ subshell?

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Problem 67

Give the notation (using letter designations for $l$ ) for the subshells denoted by the following quantum numbers.
a. $n=6, l=2$
c. $n=4, l=3$
b. $n=5, l=4$
d. $n=6, l=1$

Nicole Smina

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Problem 68

Give the notation (using letter designations for $l$ ) for the subshells denoted by the following quantum numbers.
a. $n=3, l=2$
b. $n=4, l=0$
c. $n=4, l=1$
d. $n=5, l=3$

Nicole Smina

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Problem 69

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers.
a. $n=1, l=0, m_{l}=0, m_{s}=+1$
b. $n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}$
c. $n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}$
d. $n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}$
e. $n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}$

Nicole Smina

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Problem 70

State which of the following sets of quantum numbers would be possible and which impossible for an electron in an atom.
a. $n=2, l=0, m_{l}=0, m_{s}=+\frac{1}{2}$,
b. $n=1, l=1, m_{l}=0, m_{s}=+\frac{1}{2}$
c. $n=0, l=0, m_{l}=0, m_{s}=-\frac{1}{2}$
d. $n=2, l=1, m_{l}=-1, m_{s}=+\frac{1}{2}$
e. $n=2, l=1, m_{l}=-2, m_{s}=+\frac{1}{2}^{2}$

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Problem 71

The blue line of the strontium atom emission has a wavelength of $461 \mathrm{~nm}$. What is the frequency of this light? What is the energy of a photon of this light?

Lottie Adams

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Problem 72

The barium atom has an emission with wavelength 554 $\mathrm{nm}$ (green). Calculate the frequency of this light and the energy of a photon of this light.

Lottie Adams

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Problem 73

The energy of a photon is $4.10 \times 10^{-19} \mathrm{~J}$. What is the wavelength of the corresponding light? What is the color of this light?

Ronald Prasad

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Problem 74

The energy of a photon is $3.34 \times 10^{-19} \mathrm{~J} .$ What is the wavelength of the corresponding light? What is the color of this light?

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Problem 75

The photoelectric work function of a metal is the minimum energy needed to eject an electron by irradiating the metal with light. For calcium, this work function equals $4.34 \times 10^{-19} \mathrm{~J}$. What is the minimum frequency of light for the photoelectric effect in calcium?

Lottie Adams

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Problem 76

The photoelectric work function for magnesium is $5.90 \times$ $10^{-19} \mathrm{~J}$. (The work function is the minimum energy needed to eject an electron from the metal by irradiating it with light.) Calculate the minimum frequency of light required to eject electrons from magnesium.

Ronald Prasad

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Problem 77

Light of wavelength $345 \mathrm{~nm}$ shines on a piece of calcium metal. What is the speed of the ejected electron? (Light energy greater than that of the work function of calcium ends up as kinetic energy of the ejected electron. See Problem $7.75$ for the definition of work function and its value for calcium.)

Ronald Prasad

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Problem 78

Light of wavelength $276 \mathrm{~nm}$ shines on a piece of magnesium metal. What is the speed of the ejected electron? (Light energy greater than that of the work function of magnesium ends up as kinetic energy of the ejected electron. See Problem $7.76$ for the definition of work function and its value for magnesium.)

Ronald Prasad

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Problem 79

Calculate the wavelength of the Balmer line of the hydrogen spectrum in which the initial $n$ quantum number is 5 and the final $n$ quantum number is $2 .$

Lottie Adams

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Problem 80

Calculate the wavelength of the Balmer line of the hydrogen spectrum in which the initial $n$ quantum number is 6 and the final $n$ quantum number is $2 .$

Ronald Prasad

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Problem 81

One of the lines in the Balmer series of the hydrogen atom emission spectrum is at $397 \mathrm{~nm}$. It results from a transition from an upper energy level to $n=2 .$ What is the principal quantum number of the upper level?

Lottie Adams

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Problem 82

A line of the Lyman series of the hydrogen atom spectrum has the wavelength $9.50 \times 10^{-8} \mathrm{~m} .$ It results from a transition from an upper energy level to $n=1$. What is the principal quantum number of the upper level?

Lottie Adams

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Problem 83

A hydrogen-like ion has a nucleus of charge $+Z e$ and a single electron outside this nucleus. The energy levels of these ions are $-Z^{2} R_{\mathrm{H}} / n^{2}$ (where $Z=$ atomic number). Calculate the wavelength of the transition from $n=3$ to $n=2$ for $\mathrm{He}^{+}$, a hydrogen-like ion. In what region of the spectrum does this emission occur?

Lottie Adams

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Problem 84

What is the wavelength of the transition from $n=5$ to $n=3$ for $\mathrm{Li}^{2+} ?$ In what region of the spectrum does this emission occur? $\mathrm{Li}^{2+}$ is a hydrogen-like ion. Such an ion has a nucleus of charge $+Z e$ and a single electron outside this nucleus. The energy levels of the ion are $-Z^{2} R_{\mathrm{H}} / n^{2}$, where $Z$ is the atomic number.

Ronald Prasad

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Problem 85

An electron microscope employs a beam of electrons to obtain an image of an object. What energy must be imparted to each electron of the beam to obtain a wavelength of $10.0 \mathrm{pm} ?$ Obtain the energy in electron volts $(\mathrm{eV})(1 \mathrm{eV}=1.602 \times$ $\left.10^{-19} \mathrm{~J}\right)$

Lottie Adams

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Problem 86

Neutrons are used to obtain images of the hydrogen atoms in molecules. What energy must be imparted to each neutron in a neutron beam to obtain a wavelength of $10.0 \mathrm{pm}$ ? Obtain the energy in electron volts $(\mathrm{eV})\left(1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}\right)$

Lottie Adams

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Problem 87

What is the number of different orbitals in each of the following subshells?
a. $3 d$
b. $4 f \quad$ c. $4 p$
d. $5 s$

Nicole Smina

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Problem 88

What is the number of different orbitals in each of the following subshells?
a. $6 g$
b. $4 f \quad$ c. $6 s$
d. $5 p$

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Problem 89

List the possible subshells for the $n=6$ shell.

Lottie Adams

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Problem 90

List the possible subshells for the $n=7$ shell.

Nicole Smina

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Problem 91

What are gamma rays? How does the gamma radiation of foods improve their shelf life?

Nicole Smina

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Problem 92

How can gamma rays that are used in food irradiation be produced? Does such irradiated food show any radioactivity?

Nicole Smina

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Problem 93

The word laser is an acronym meaning light $a$ mplification
by stimulated $e$ mission of radiation. What is the stimulated emission of radiation?

Ronald Prasad

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Problem 94

Explain how lasers are used to "read" a compact disc.

Lottie Adams

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Problem 95

Explain the concept of quantum mechanical tunneling.

Lottie Adams

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Problem 96

Explain how the probe in a scanning tunneling microscope scans a sample on the surface of a metal.

Lottie Adams

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Problem 97

What wavelength of electromagnetic radiation corresponds to a frequency of $7.76 \times 10^{9} \mathrm{~s}^{-1}$ ? Note that Planck's constant is $6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}$, and the speed of light is $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$.

Lottie Adams

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Problem 98

AM radio stations broadcast at frequencies between $530 \mathrm{kHz}$ and $1700 \mathrm{kHz}$. (1 $\mathrm{kHz}=10^{3} \mathrm{~s}^{-1}$.) For a station broadcasting at $1.69 \times 10^{3} \mathrm{kHz}$, what is the energy of this radio wave? Note that Planck's constant is $6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}$, and the speed of light is $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$.

Lottie Adams

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Problem 99

The photoelectric work function of a metal is the minimum energy required to eject an electron by shining light on the metal. The work function of calcium is $4.60 \times 10^{-19} \mathrm{~J}$. What is the longest wavelength of light (in nanometers) that can cause an electron to be ejected from calcium metal.

Lottie Adams

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Problem 100

Calculate the shortest wavelength of visible light (in nanometers) seen in the spectrum of the hydrogen atom. What are the principal quantum numbers for the levels in this transition? Does Figure $7.11$ include all visible lines?

Lottie Adams

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Problem 101

Light of wavelength $1.03 \times 10^{-7} \mathrm{~m}$ is emitted when an electron in an excited level of a hydrogen atom undergoes a transition to the $n=1$ level. What is the region of the spectrum of this light? What is the principal quantum number of the excited level?

Lottie Adams

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Problem 102

It requires $799 \mathrm{~kJ}$ of energy to break one mole of carbonoxygen double bonds in carbon dioxide. What wavelength of light does this correspond to per bond? Is there any transition in the hydrogen atom that has at least this quantity of energy in one photon?

Ronald Prasad

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Problem 103

The root-mean-square speed of an oxygen molecule, $\mathrm{O}_{2}$, at $21^{\circ} \mathrm{C}$ is $479 \mathrm{~m} / \mathrm{s}$. Calculate the de Broglie wavelength for an $\mathrm{O}_{2}$ molecule traveling at this speed. How does this wavelength compare with the approximate length of this molecule, which is about $242 \mathrm{pm}$ ? (For this comparison, state the wavelength as a percentage of the molecular length).

Lottie Adams

Numerade Educator

Problem 104

A particular microwave oven delivers 800 watts. (A watt is a unit of power, which is the joules of energy delivered, or used, per second.) If the oven uses microwave radiation of wavelength $12.2 \mathrm{~cm}$, how many photons of this radiation are required to heat $1.00 \mathrm{~g}$ of water $1.00^{\circ} \mathrm{C}$, assuming that all of the photons are absorbed?

Ronald Prasad

Numerade Educator

Problem 105

For each of the following combinations of quantum numbers, make changes that produce an allowed combination. Count 3 for each change of $n, 2$ for each change of $l$, and 1 for each change of $m_{l}$. What is the lowest possible count that you can obtain?
a. $n=3, l=0, m_{l}=-2$
c. $n=3, l=3, m_{l}=-3$
b. $n=5, l=5, m_{l}=4$
d. $n=5, l=6, m_{l}=3$

Nicole Smina

Numerade Educator

Problem 106

The term degeneracy means the number of different quantum states of an atom or molecule having the same energy. For example, the degeneracy of the $n=2$ level of the hydrogen atom is 4 (a $2 s$ quantum state, and three different $2 p$ states). What is the degeneracy of the $n=5$ level?

Lottie Adams

Numerade Educator

Problem 107

The energy required to dissociate the $\mathrm{Cl}_{2}$ molecule to $\mathrm{Cl}$ atoms is $239 \mathrm{~kJ} / \mathrm{mol} \mathrm{Cl}_{2}$. If the dissociation of a $\mathrm{Cl}_{2}$ molecule were accomplished by the absorption of a single photon whose energy was exactly the quantity required, what would be its wavelength (in meters)?

Lottie Adams

Numerade Educator

Problem 108

The energy required to dissociate the $\mathrm{H}_{2}$ molecule to $\mathrm{H}$ atoms is $432 \mathrm{~kJ} / \mathrm{mol} \mathrm{H}_{2}$. If the dissociation of an $\mathrm{H}_{2}$ molecule were accomplished by the absorption of a single photon whose energy was exactly the quantity required, what would be its wavelength (in meters)?

Lottie Adams

Numerade Educator

Problem 109

A microwave oven heats by radiating food with microwave radiation, which is absorbed by the food and converted to heat. Suppose an oven's radiation wavelength is $12.5 \mathrm{~cm} . \mathrm{A}$ container with $0.250 \mathrm{~L}$ of water was placed in the oven, and the temperature of the water rose from $20.0^{\circ} \mathrm{C}$ to $100.0^{\circ} \mathrm{C}$. How many photons of this microwave radiation were required? Assume that all the energy from the radiation was used to raise the temperature of the water.

Ronald Prasad

Numerade Educator

Problem 110

Warm objects emit electromagnetic radiation in the infrared region. Heat lamps employ this principle to generate infrared radiation. Water absorbs infrared radiation with wavelengths near $2.80$ $\mu \mathrm{m}$. Suppose this radiation is absorbed by the water and converted to heat. A 1.00-L sample of water absorbs infrared radiation, and its temperature increases from $20.0^{\circ} \mathrm{C}$ to $30.0^{\circ} \mathrm{C}$. How many photons of this radiation are used to heat the water?

Lottie Adams

Numerade Educator

Problem 111

Light with a wavelength of $425 \mathrm{~nm}$ fell on a potassium surface, and electrons were ejected at a speed of $4.88 \times 10^{5} \mathrm{~m} / \mathrm{s}$. What energy was expended in removing an electron from the metal? Express the answer in joules (per electron) and in kilojoules per mole (of electrons).

Lottie Adams

Numerade Educator

Problem 112

Light with a wavelength of $405 \mathrm{~nm}$ fell on a strontium surface, and electrons were ejected. If the speed of an ejected electron is $3.36 \times 10^{5} \mathrm{~m} / \mathrm{s}$, what energy was expended in removing the electron from the metal? Express the answer in joules (per electron) and in kilojoules per mole (of electrons).

Lottie Adams

Numerade Educator

Problem 113

When an electron is accelerated by a voltage difference, the kinetic energy acquired by the electron equals the voltage times the charge on the electron. Thus, one volt imparts a kinetic energy of $1.602 \times 10^{-19}$ volt-coulombs, which equals $1.602 \times$ $10^{-19} \mathrm{~J}$. What is the wavelength associated with electrons accelerated by $4.00 \times 10^{3}$ volts?

Lottie Adams

Numerade Educator

Problem 114

When an electron is accelerated by a voltage difference, the kinetic energy acquired by the electron equals the voltage times the charge on the electron. Thus, one volt imparts a kinetic energy of $1.602 \times 10^{-19}$ volt-coulombs, or $1.602 \times$ $10^{-19} \mathrm{~J}$. What is the wavelength for electrons accelerated by $1.00 \times 10^{4}$ volts?

Lottie Adams

Numerade Educator