CHEMISTRY: The Molecular Nature of Matter and Change 2016

Martin S. Silberberg, Patricia G. Amateis

Chapter 7

Quantum Theory and Atomic Structure

Educators

EB
MT
DR

Problem 1

In what ways are microwave and ultraviolet radiation the same? In what ways are they different?

EB
Emma B.
Numerade Educator

Problem 2

Consider the following types of electromagnetic radiation:
(1) Microwave (2) Ultraviolet (3) Radio waves
(4) Infrared (5) X-ray (6) Visible
(a) Arrange them in order of increasing wavelength.
(b) Arrange them in order of increasing frequency.
(c) Arrange them in order of increasing energy.

MT
Monica T.
Numerade Educator

Problem 3

Define each of the following wave phenomena, and give an example of where each occurs: (a) refraction; (b) diffraction; (c) dispersion; (d) interference.

Sam L.
Numerade Educator

Problem 4

In the $17^{\text { th }}$ century, Newton proposed that light was a stream of particles. The wave-particle debate continued for over 250 years until Planck and Einstein presented their ideas. Give two pieces of evidence for the wave model and two for the particle model.

Ashley M.
Numerade Educator

Problem 5

Portions of electromagnetic waves A, B, and C are represented by the following (not drawn to scale): Rank them in order of (a) increasing frequency; (b) increasing energy; (c) increasing amplitude. (d) If wave B just barely fails to cause a current when shining on a metal, is wave A or C more likely to do so? (e) If wave B represents visible radiation, is wave A or C more likely to be IR radiation?

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Problem 6

What new idea about light did Einstein use to explain the photoelectric effect? Why does the photoelectric effect exhibit a threshold frequency but not a time lag?

Ashley M.
Numerade Educator

Problem 7

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Problem 8

An FM station broadcasts music at 93.5 MHz (megahertz, or $10^{6} \mathrm{Hz}$ ). Find the wavelength (in $\mathrm{m}, \mathrm{nm},$ and $\mathrm{A} )$ of these waves.

MT
Monica T.
Numerade Educator

Problem 9

A radio wave has a frequency of $3.8 \times 10^{10} \mathrm{Hz}$ . What is the energy (in J) of one photon of this radiation?

Sam L.
Numerade Educator

Problem 10

An $x$ -ray has a wavelength of 1.3 A. Calculate the energy (in J) of one photon of this radiation.

MT
Monica T.
Numerade Educator

Problem 11

Rank these photons in terms of increasing energy: (a) blue $(\lambda=453 \mathrm{nm}) ;(b)$ red $(\lambda=660 \mathrm{nm}) ;(c)$ yellow $(\lambda=595 \mathrm{nm})$

Sam L.
Numerade Educator

Problem 12

Rank these photons in terms of decreasing energy: (a) IR $\left(v=6.5 \times 10^{13} \mathrm{s}^{-1}\right) ;$ (b) microwave $\left(v=9.8 \times 10^{11} \mathrm{s}^{-1}\right);$ (c) $\mathrm{UV}\left(v=8.0 \times 10^{15} \mathrm{s}^{-1}\right)$

MT
Monica T.
Numerade Educator

Problem 13

Police often monitor traffic with "K-band" radar guns, which operate in the microwave region at 22.235 $\mathrm{GHz}\left(1 \mathrm{GHz}=10^{9} \mathrm{Hz}\right)$ Find the wavelength (in $\mathrm{nm}$ and $\hat{\mathrm{A}} )$ of this radiation.

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Problem 14

Covalent bonds in a molecule absorb radiation in the IR region and vibrate at characteristic frequencies.
(a) The $C-O$ bond absorbs radiation of wavelength 9.6$\mu \mathrm{m}$ . What frequency (in $\mathrm{s}^{-1} )$ corresponds to that wavelength? (b) The $\mathrm{H}-\mathrm{Cl}$ bond has a frequency of vibration of $8.652 \times 10^{13}$ Hz. What wavelength (in $\mu \mathrm{m} )$ corresponds to that frequency?

MT
Monica T.
Numerade Educator

Problem 15

Cobalt-60 is a radioactive isotope used to treat cancers. A gamma ray emitted by this isotope has an energy of 1.33 $\mathrm{MeV}$ (million electron volts; $1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{J} ) .$ What is the frequency (in $\mathrm{Hz}$ ) and the wavelength (in $\mathrm{m} )$ of this gamma ray?

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Problem 16

(a) Ozone formation in the upper atmosphere starts when oxygen molecules absorb UV radiation of wavelengths $\leq 242 \mathrm{nm}$ . Find the frequency and energy of the least energetic of these photons. (b) Ozone absorbs radiation of wavelengths $2200-2900 \mathrm{A}$ thus protecting organisms from this radiation. Find the frequency and energy of the most energetic of these photons.

Ashley M.
Numerade Educator

Problem 17

How is $n_{1}$ in the Rydberg equation (Equation 7.4$)$ related to the quantum number $n$ in the Bohr model?

Sam L.
Numerade Educator

Problem 18

What key assumption of Bohr’s model would a “Solar System” model of the atom violate? What was the theoretical basis for this assumption?

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Problem 19

Distinguish between an absorption spectrum and an emission spectrum. With which did Bohr work?

Sam L.
Numerade Educator

Problem 20

Which of these electron transitions correspond to absorption of energy and which to emission?
$\begin{array}{ll}{\text { (a) } n=2 \text { to } n=4} & {\text { (b) } n=3 \text { to } n=1} \\ {\text { (c) } n=5 \text { to } n=2} & {\text { (d) } n=3 \text { to } n=4}\end{array}$

MT
Monica T.
Numerade Educator

Problem 21

Why couldn’t the Bohr model predict spectra for atoms other than hydrogen?

Sam L.
Numerade Educator

Problem 22

The $\mathrm{H}$ atom and the $\mathrm{Be}^{3+}$ ion each have electron. Would you expect the Bohr model to predict their spectra accurately? Would you expect their spectra to be identical? Explain.

Ashley M.
Numerade Educator

Problem 23

Use the Rydberg equation to find the wavelength (in $\mathrm{nm} )$ of the photon emitted when an electron in an $\mathrm{H}$ atom undergoes a transition from $n=5$ to $n=2$

Sam L.
Numerade Educator

Problem 24

Use the Rydberg equation to find the wavelength (in $\hat{\mathrm{A}} )$ of the photon absorbed when an electron in an H atom undergoes a transition from $n=1$ to $n=3$

MT
Monica T.
Numerade Educator

Problem 25

What is the wavelength (in nm) of the least energetic spectral line in the infrared series of the H atom?

Sam L.
Numerade Educator

Problem 26

What is the wavelength (in nm) of the least energetic spectral line in the visible series of the H atom?

Ashley M.
Numerade Educator

Problem 27

Calculate the energy difference $(\Delta E)$ for the transition in Problem 7.23 for 1 mol of $\mathrm{H}$ atoms.

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Problem 28

Calculate the energy difference $(\Delta E)$ for the transition in Problem 7.24 for 1 mol of $\mathrm{H}$ atoms.

Ashley M.
Numerade Educator

Problem 29

Arrange the following H atom electron transitions in order of increasing frequency of the photon absorbed or emitted:
$\begin{array}{ll}{\text { (a) } n=2 \text { to } n=4} & {\text { (b) } n=2 \text { to } n=1} \\ {\text { (c) } n=2 \text { to } n=5} & {\text { (d) } n=4 \text { to } n=3}\end{array}$

Sam L.
Numerade Educator

Problem 30

Arrange the following H atom electron transitions in order of decreasing wavelength of the photon absorbed or emitted:
$\begin{array}{ll}{\text { (a) } n=2 \text { to } n=\infty} & {\text { (b) } n=4 \text { to } n=20} \\ {\text { (c) } n=3 \text { to } n=10} & {\text { (d) } n=2 \text { to } n=1}\end{array}$

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Problem 31

The electron in a ground-state H atom absorbs a photon of wavelength 97.20 nm. To what energy level does it move?

Sam L.
Numerade Educator

Problem 32

An electron in the $n=5$ level of an $\mathrm{H}$ atom emits a photon of wavelength 97.20 $\mathrm{nm}$ . To what energy level does it move?

MT
Monica T.
Numerade Educator

Problem 33

In addition to continuous radiation, fluorescent lamps emit some visible lines from mercury. A prominent line has a wavelength of 436 nm. What is the energy (in J) of one photon of it?

Sam L.
Numerade Educator

Problem 34

A Bohr-model representation of the H atom is shown below with several electron transitions depicted by arrows: (a) Which transitions are absorptions and which are emissions? (b) Rank the emissions in terms of increasing energy. (c) Rank the absorptions in terms of increasing wavelength of light absorbed.

DR
David R.
Numerade Educator

Problem 35

In what sense is the wave motion of a guitar string analogous to the motion of an electron in an atom?

Sam L.
Numerade Educator

Problem 36

What experimental support did de Broglie’s concept receive?

Ashley M.
Numerade Educator

Problem 37

If particles have wavelike motion, why don’t we observe that motion in the macroscopic world?

Sam L.
Numerade Educator

Problem 38

Why can't we overcome the uncertainty predicted by Heisenberg's principle by building more precise instruments to reduce the error in measurements below the $h / 4 \pi$ limit?

Ashley M.
Numerade Educator

Problem 39

$\mathrm{A} 232-\mathrm{lb}$ fullback runs 40 yd at $19.8 \pm 0.1 \mathrm{mi} / \mathrm{h}$
(a) What is his de Broglie wavelength (in meters)?
(b) What is the uncertainty in his position?

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Problem 40

An alpha particle (mass $=6.6 \times 10^{-24} \mathrm{g} )$ emitted by a radium isotope travels at $3.4 \times 10^{7} \pm 0.1 \times 10^{7} \mathrm{mi} / \mathrm{h}$ .
(a) What is its de Broglie wavelength (in meters)?
(b) What is the uncertainty in its position?

Ashley M.
Numerade Educator

Problem 41

How fast must a 56.5 -g tennis ball travel to have a de Broglie wavelength equal to that of a photon of green light $(5400 \text { A)? }$

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Problem 42

How fast must a 142 -g baseball travel to have a de Broglie wavelength equal to that of an x-ray photon with $\lambda=100 . \mathrm{pm}$ ?

Ashley M.
Numerade Educator

Problem 43

A sodium flame has a characteristic yellow color due to emissions of wavelength 589 $\mathrm{nm}$ . What is the mass equivalence of one photon with this wavelength $\left(1 \mathrm{J}=1 \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\right) ?$

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Problem 44

A lithium flame has a characteristic red color due to emissions of wavelength 671 $\mathrm{nm}$ . What is the mass equivalence of 1 mol of photons with this wavelength $\left(1 \mathrm{J}=1 \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\right) ?$

Ashley M.
Numerade Educator

Problem 45

What physical meaning is attributed to $\psi^{2} ?$

Sam L.
Numerade Educator

Problem 46

What does "electron density in a tiny volume of space" mean?

Ashley M.
Numerade Educator

Problem 47

Explain what it means for the peak in the radial probability distribution plot for the $n=1$ level of an $\mathrm{H}$ atom to be at 0.529 $\mathrm{A} .$ Is the probability of finding an electron at 0.529 $\mathrm{A}$ from the nucleus greater for the 1$s$ or the 2$s$ orbital?

Sam L.
Numerade Educator

Problem 48

What feature of an orbital is related to each of the following?
(a) Principal quantum number (n)
(b) Angular momentum quantum number $(l)$
(c) Magnetic quantum number $\left(m_{l}\right)$

MT
Monica T.
Numerade Educator

Problem 49

How many orbitals in an atom can have each of the following designations: (a) $1 s ;(\text { b) } 4 d ;(\mathrm{c}) 3 p ;(\mathrm{d}) n=3 ?$

Sam L.
Numerade Educator

Problem 50

How many orbitals in an atom can have each of the following designations: ( a ) $5 f ;(\text { b) } 4 p ;(c) 5 d ;(d) n=2 ?$

MT
Monica T.
Numerade Educator

Problem 51

Give all possible $m_{l}$ values for orbitals that have each of the following: (a) $l=2 ;(\mathrm{b}) n=1 ;(\mathrm{c}) n=4, l=3$

Sam L.
Numerade Educator

Problem 52

Give all possible $m_{l}$ values for orbitals that have each of the following: $(a) l=3 ;(b) n=2 ;(c) n=6, l=1$

MT
Monica T.
Numerade Educator

Problem 53

Draw 90$\%$ probability contours (with axes) for each of the following orbitals: (a) $s ;(\mathrm{b}) p_{x}$

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Problem 54

Draw 90$\%$ probability contours (with axes) for each of the following orbitals: (a) $p_{z} ;(\mathrm{b}) d_{x y^{*}}$

Ashley M.
Numerade Educator

Problem 55

For each of the following, give the sublevel designation, the allowable $m_{l}$ values, and the number of orbitals:
(a) $n=4, l=2 \quad$ (b) $n=5, l=1 \quad$ (c) $n=6, l=3$

Sam L.
Numerade Educator

Problem 56

For each of the following, give the sublevel designation, the allowable $m_{l}$ values, and the number of orbitals:
(a) $n=2, l=0 \quad$ (b) $n=3, l=2 \quad$ (c) $n=5, l=1$

Ashley M.
Numerade Educator

Problem 57

For each of the following sublevels, give the $n$ and $l$ values and the number of orbitals: (a) $5 s ;(\mathrm{b}) 3 p ;(\mathrm{c}) 4 f$

Sam L.
Numerade Educator

Problem 58

For each of the following sublevels, give the $n$ and $l$ values and the number of orbitals: (a) $6 g ;(b) 4 s ;(c) 3 d$

MT
Monica T.
Numerade Educator

Problem 59

Are the following combinations allowed? If not, show two ways to correct them:
$\begin{array}{ll}{\text { (a) } n=2 ; l=0 ; m_{l}=-1} & {\text { (b) } n=4 ; l=3 ; m_{l}=-1} \\ {\text { (c) } n=3 ; l=1 ; m_{l}=0} & {\text { (d) } n=5 ; l=2 ; m_{l}=+3}\end{array}$

Sam L.
Numerade Educator

Problem 60

Are the following combinations allowed? If not, show two ways to correct them:
$\begin{array}{ll}{\text { (a) } n=1 ; l=0 ; m_{l}=0} & {\text { (b) } n=2 ; l=2 ; m_{l}=+1} \\ {\text { (c) } n=7 ; l=1 ; m_{l}=+2} & {\text { (d) } n=3 ; l=1 ; m_{l}=-2}\end{array}$

Ashley M.
Numerade Educator

Problem 61

The orange color of carrots and orange peel is due mostly to $\beta$ -carotene, an organic compound insoluble in water but soluble in benzene and chloroform. Describe an experiment to determine the concentration of $\beta$ -carotene in the oil from orange peel.

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Problem 62

The quantum-mechanical treatment of the H atom gives the energy, E, of the electron as a function of n:
$$
E=-\frac{h^{2}}{8 \pi^{2} m_{\mathrm{e}} a_{0}^{2} n^{2}} \quad(n=1,2,3, \ldots)
$$
where $h$ is Planck's constant, $m_{\mathrm{e}}$ is the electron mass, and $a_{0}$ is $52.92 \times 10^{-12} \mathrm{m} .$
(a) Write the expression in the form $E=-(\text { constant })\left(1 / n^{2}\right)$ evaluate the constant (in J), and compare it with the corresponding expression from Bohr's theory.
(b) Use the expression from part (a) to find $\Delta E$ between $n=2$ and $n=3$
(c) Calculate the wavelength of the photon that corresponds to this energy change.

Ashley M.
Numerade Educator

Problem 63

Sam L.
Numerade Educator

Problem 64

The optic nerve needs a minimum of $2.0 \times 10^{-17} \mathrm{J}$ of energy to trigger a series of impulses that eventually reach the brain. (a) How many photons of red light $(700 . \mathrm{nm})$ are needed? (b) How many photons of blue light $(475 \mathrm{nm}) ?$

Ashley M.
Numerade Educator

Problem 65

One reason carbon monoxide (CO) is toxic is that it binds to the blood protein hemoglobin more strongly than oxygen does. The bond between hemoglobin and $\mathrm{CO}$ absorbs radiation of 1953 $\mathrm{cm}^{-1} .$ (The units are the reciprocal of the wavelength in centimeters.) Calculate the wavelength (in nm and A) and the frequency (in $\mathrm{Hz} )$ of the absorbed radiation.

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Problem 66

A metal ion $\mathrm{M}^{n+}$ has a single electron. The highest energy line in its emission spectrum has a frequency of $2.961 \times 10^{16} \mathrm{Hz}$ . Identify the ion.

Ashley M.
Numerade Educator

Problem 67

Compare the wavelengths of an electron (mass $9.11 \times 10^{-31} \mathrm{kg} )$ and a proton (mass $=1.67 \times 10^{-27} \mathrm{kg},$ each having (a) a speed of $3.4 \times 10^{6} \mathrm{m} / \mathrm{s} ;(\mathrm{b})$ a kinetic energy of $2.7 \times 10^{-15} \mathrm{J} .$

Sam L.
Numerade Educator

Problem 68

Five lines in the $\mathrm{H}$ atom spectrum have these wavelengths $(\text { in } A) :(\text { a }) 1212.7 ;(\text { b ) } 4340.5$ (c) $4861.3 ;(\mathrm{d}) 6562.8 ;(\mathrm{e}) 10,938$ Three lines result from transitions to $n_{\text { final }}=2$ (visible series). The other two result from transitions in different series, one with $n_{\text { final }}=1$ and the other with $n_{\text { final }}=3 .$ Identify $n_{\text { initial }}$ for each line.

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Problem 69

In his explanation of the threshold frequency in the photo-electric effect, Einstein reasoned that the absorbed photon must have a minimum energy to dislodge an electron from the metal surface. This energy is called the work function (\phi) of the metal. What is the longest wavelength of radiation (in $\mathrm{nm}$ ) that could cause the photoelectric effect in each of these metals: (a) calcium,
$\phi=4.60 \times 10^{-19} \mathrm{J} ;(\mathrm{b})$ titanium, $\phi=6.94 \times 10^{-19} \mathrm{J} ;(\mathrm{c})$ sodium, $\phi=4.41 \times 10^{-19} \mathrm{J} ?$

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Problem 70

Refractometry is based on the difference in the speed of light through a substance $(v)$ and through a vacuum $(c) .$ In the procedure, light of known wavelength passes through a fixed thickness of the substance at a known temperature. The index of refraction equals $c / v$ . Using yellow light $(\lambda=589 \mathrm{nm})$ at $20^{\circ} \mathrm{C},$ for example, the index of refraction of water is 1.33 and that of diamond is 2.42 Calculate the speed of light in (a) water and (b) diamond.

Ashley M.
Numerade Educator

Problem 71

A laser (light amplification by stimulated emission of radiation) provides nearly monochromatic high-intensity light. Lasers are used in eye surgery, CD/DVD players, basic research, and many other areas. Some dye lasers can be "tuned" to emit a desired wavelength. Fill in the blanks in the following table of the properties of some common types of lasers:

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Problem 72

The following combinations are not allowed. If $n$ and $m_{l}$ are correct, change the $l$ value to create an allowable combination:
$\begin{array}{ll}{\text { (a) } n=3 ; l=0 ; m_{l}=-1} & {\text { (b) } n=3 ; l=3 ; m_{l}=+1} \\ {\text { (c) } n=7 ; l=2 ; m_{l}=+3} & {\text { (d) } n=4 ; l=1 ; m_{l}=-2}\end{array}$

Ashley M.
Numerade Educator

Problem 73

A ground-state H atom absorbs a photon of wavelength 94.91 nm, and its electron attains a higher energy level. The atom then emits two photons: one of wavelength 1281 nm to reach an intermediate energy level, and a second to return to the ground state.
(a) What higher level did the electron reach?
(b) What intermediate level did the electron reach?
(c) What was the wavelength of the second photon emitted?

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Problem 74

Ground-state ionization energies of some one-electron species are
$$
\begin{array}{rl}{\mathrm{H}=1.31 \times 10^{3} \mathrm{kJ} / \mathrm{mol}} & {\mathrm{He}^{+}=5.24 \times 10^{3} \mathrm{kJ} / \mathrm{mol}} \\ {\mathrm{Li}^{2+}=1.18 \times 10^{4} \mathrm{kJ} / \mathrm{mol}}\end{array}
$$
(a) Write a general expression for the ionization energy of any one-electron species. (b) Use your expression to calculate the ionization energy of $\mathrm{B}^{4+},(\mathrm{c})$ What is the minimum wavelength required to remove the electron from the $n=3$ level of $\mathrm{He}^{+} ?$ (d) What is the minimum wavelength required to remove the electron from the $n=2$ level of $\mathrm{Be}^{3+} ?$

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Problem 75

Use the relative size of the 3$s$ orbital below to answer the following questions about orbitals $\mathrm{A}-\mathrm{D}$ .
(a) Which orbital has the highest value of $n ?(b)$ Which orbital(s) have a value of $l=1 ? l=2 ?(\mathrm{c})$ How many other orbitals with the same value of $n$ have the same shape as orbital B? Orbital C? (d) Which orbital has the highest energy? Lowest energy?

Sam L.
Numerade Educator

Problem 76

In the course of developing his model, Bohr arrived at the following formula for the radius of the electron's orbit: $r_{n}=n^{2} h^{2} \epsilon_{0} / \pi m_{\mathrm{c}} e^{2},$ where $m_{\mathrm{c}}$ is the electron mass, $e$ is its charge, and $\epsilon_{0}$ is a constant related to charge attraction in a vacuum. Given that $m_{e}=9.109 \times 10^{-31} \mathrm{kg}, e=1.602 \times 10^{-19} \mathrm{C},$ and $\epsilon_{0}=8.854 \times 10^{-12} \mathrm{C}^{2} \mathrm{J} \cdot \mathrm{m},$ calculate the following:
(a) The radius of the first $(n=1)$ orbit in the $\mathrm{H}$ atom
(b) The radius of the tenth $(n=10)$ orbit in the $\mathrm{H}$ atom

Ashley M.
Numerade Educator

Problem 77

(a) Find the Bohr radius of an electron in the $n=3$ orbit of an H atom (see Problem 7.76$) .$ (b) What is the energy (in J) of the atom in $(\mathrm{a}) ?(\mathrm{c})$ What is the energy of an $\mathrm{Li}^{2+}$ ion with its electron in the $n=3$ orbit? (d) Why are the answers to (b) and (c) different?

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Problem 78

Enormous numbers of microwave photons are needed to warm macroscopic samples of matter. A portion of soup containing 252 g of water is heated in a microwave oven from $20 .^{\circ} \mathrm{C}$ to $98^{\circ} \mathrm{C}$ , with radiation of wavelength $1.55 \times 10^{-2} \mathrm{m} .$ How many photons are absorbed by the water in the soup?

Ashley M.
Numerade Educator

Problem 79

The quantum-mechanical treatment of the hydrogen atom gives this expression for the wave function, $\psi,$ of the 1$s$ orbital:
$$
\psi=\frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} e^{-r / a_{0}}
$$
where $r$ is the distance from the nucleus and $a_{0}$ is 52.92 pm. The probability of finding the electron in a tiny volume at distance $r$ from the nucleus is proportional to $\psi^{2} .$ The total probability of finding the electron at all points at distance $r$ from the nucleus is proportional to 4$\pi r^{2} \psi^{2}$ . Calculate the values (to three significant figures) of $\psi, \psi^{2},$ and 4$\pi r^{2} \psi^{2}$ to fill in the following table and sketch a plot of each set of values versus $r$ . Compare the latter two plots with those in Figure $7.17 \mathrm{A}, \mathrm{p} .311 :$

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Problem 80

Lines in one spectral series can overlap lines in another. (a) Does the range of wavelengths in the $n_{1}=1$ series for the $\mathrm{H}$ atom overlap the range in the $n_{1}=2$ series? (b) Does the range in the $n_{1}=3$ series overlap the range in the $n_{1}=4$ series? (c) How many lines in the $n_{1}=4$ series lie in the range of the $n_{1}=5$ series? (d) What does this overlap imply about the $\mathrm{H}$ atom line spectrum at longer wavelengths?

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Problem 81

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Problem 82

Photoelectron spectroscopy applies the principle of the photoelectric effect to study orbital energies of atoms and molecules. High-energy radiation (usually UV or x-ray) is absorbed by a sample and an electron is ejected. The orbital energy can be calculated from the known energy of the radiation and the measured energy of the electron lost. The following energy differences were determined for several electron transitions:
$$
\begin{array}{ll}{\Delta E_{2} \longrightarrow_{1}=4.098 \times 10^{-17} \mathrm{J}} & {\Delta E_{3} \longrightarrow_{1}=4.854 \times 10^{-17} \mathrm{J}} \\ {\Delta E_{5} \longrightarrow_{1}=5.242 \times 10^{-17} \mathrm{J}} & {\Delta E_{4} \longrightarrow_{2}=1.024 \times 10^{-17} \mathrm{J}}\end{array}
$$
Calculate $\Delta E$ and $\lambda$ of a photon emitted in the following transitions: (a) level $3 \longrightarrow 2 ;(b)$ level $4 \longrightarrow 1 ;(c)$ level 5$\longrightarrow 4$ .

Ashley M.
Numerade Educator

Problem 83

Horticulturists know that, for many plants, dark green leaves are associated with low light levels and pale green with high levels. (a) Use the photon theory to explain this behavior. (b) What change in leaf composition might account for this difference?

Sam L.
Numerade Educator

Problem 84

In order to comply with the requirement that energy be conserved, Einstein showed in the photoelectric effect that the energy of a photon $(h v)$ absorbed by a metal is the sum of the work function ( $\phi$ ), the minimum energy needed to dislodge an electron from the metal's surface, and the kinetic energy $\left(E_{\mathrm{k}}\right)$ of the electron: $h v=\phi+E_{\mathrm{k}} .$ When light of wavelength 358.1 $\mathrm{nm}$ falls on the surface of potassium metal, the speed (u) of the dislodged electron
is $6.40 \times 10^{5} \mathrm{m} / \mathrm{s}$ . (a) What is $E_{\mathrm{k}}\left(\frac{1}{2} m u^{2}\right)$ of the dislodged electron? (b) What is $\phi(\text { in } \mathrm{J})$ of potassium?

Ashley M.
Numerade Educator

Problem 85

For any microscope, the size of the smallest observable object is one-half the wavelength of the radiation used. For example, the smallest object observable with 400 -nm light is $2 \times 10^{-7} \mathrm{m}$ . (a) What is the smallest observable object for an electron microscope using electrons moving at $5.5 \times 10^{4} \mathrm{m} / \mathrm{s} ?$ (b) At $3.0 \times 10^{7} \mathrm{m} / \mathrm{s} ?$

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Problem 86

In fireworks, the heat of the reaction of an oxidizing agent, such as $\mathrm{KClO}_{4},$ with an organic compound excites certain salts, which emit specific colors. Strontium salts have an intense emission at $641 \mathrm{nm},$ and barium salts have one at 493 $\mathrm{nm} .$
(a) What colors do these emissions produce?
(b) What is the energy (in $\mathrm{kJ}$ of these emissions for 5.00 $\mathrm{g}$ each of the chloride salts of $\mathrm{Sr}$ and Ba? (Assume that all the heat produced is converted to emitted light.)

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Problem 87

Atomic hydrogen produces several series of spectral lines. Each series fits the Rydberg equation with its own particular $n_{1}$ value. Calculate the value of $n_{1}$ (by trial and error if necessary) that would produce a series of lines in which:
(a) The highest energy line has a wavelength of 3282 $\mathrm{nm}$ .
(b) The lowest energy line has a wavelength of 7460 $\mathrm{nm}$ .

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Problem 88

Fish-liver oil is a good source of vitamin $\mathrm{A}$ , whose concentration is measured spectrometrically at a wavelength of 329 nm.
(a) Suggest a reason for using this wavelength.
(b) In what region of the spectrum does this wavelength lie?
(c) When 0.1232 g of fish-liver oil is dissolved in 500 . mL of solvent, the absorbance is 0.724 units. When $1.67 \times 10^{-3}$ g of vitamin A is dissolved in $250 .$ mL of solvent, the absorbance is 1.018 units. Calculate the vitamin A concentration in the fish-liver oil.

Ashley M.
Numerade Educator

Problem 89

Many calculators use photocells as their energy source. Find the maximum wavelength needed to remove an electron from silver $\left(\phi=7.59 \times 10^{-19} \mathrm{J} \text { ). Is silver a good choice for a photocell }\right.$ that uses visible light? [The concept of the work function (\phi) is
explained in Problem $7.69 . ]$

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Problem 90

In a game of "Clue," Ms. White is killed in the conservatory. A spectrometer in each room records who is present to help find the murderer. For example, if someone wearing yellow is in a room, light at 580 $\mathrm{nm}$ is reflected. The suspects are Col. Mustard, Prof. Plum, Mr. Green, Ms. Peacock (blue), and Ms. Scarlet.At the time of the murder, the spectrometer in the dining room shows a reflection at 520 $\mathrm{nm}$ , those in the lounge and the study record lower frequencies, and the one in the library records the shortest possible wavelength. Who killed Ms. White? Explain.

Ashley M.
Numerade Educator

Problem 91

Technetium $(\mathrm{Tc} ; Z=43)$ is a synthetic element used as a radioactive tracer in medical studies. A Te atom emits a beta particle (electron) with a kinetic energy $\left(E_{\mathrm{k}}=\frac{1}{2} m v^{2}\right)$ of $4.71 \times 10^{-15} \mathrm{J}$ . What is the de Broglie wavelength of this electron?

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Problem 92

Electric power is measured in watts $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})$ . About 95$\%$ of the power output of an incandescent bulb is converted to heat and 5$\%$ to light. If 10$\%$ of that light shines on your chemistry textbook, how many photons per second shine on the book from $\mathrm{a} 75-\mathrm{W}$ bulb? (Assume that the photons have a wavelength of 550 $\mathrm{nm} .$ )

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Problem 93

The flame tests for sodium and potassium are based on the emissions at 589 $\mathrm{nm}$ and 404 $\mathrm{nm}$ , respectively. When both elements are present, the Na' emission is so strong that the $\mathrm{K}^{+}$ emission can be seen only by looking through a cobalt-glass filter.
(a) What are the colors of these Na' and $\mathrm{K}^{+}$ emissions?
(b) What does the cobalt-glass filter do?
(c) Why is $\mathrm{KClO}_{4}$ used as an oxidizing agent in fireworks rather than $\mathrm{NaClO}_{4} ?$

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Problem 94

The net change during photosynthesis involves $\mathrm{CO}_{2}$ and $\mathrm{H}_{2} \mathrm{O}$ forming glucose $\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)$ and $\mathrm{O}_{2}$ . Chlorophyll absorbs light in the $600-700 \mathrm{nm}$ region. (a) Write a balanced thermochemical equation for formation of 1.00 mol of glucose. (b) What is the minimum number of photons with $\lambda=680 . \mathrm{nm}$ needed to form 1.00 $\mathrm{mol}$ of glucose?

Ashley M.
Numerade Educator

Problem 95

Only certain transitions are allowed from one energy level to another. In one-electron species, the change in $l$ of an allowed transition is $\pm 1 .$ For example, a 3$p$ electron can move to a 2$s$ orbital but not to a 2$p .$ Thus, in the UV series, where $n_{\text { final }}=1$ allowed transitions can start in a $p$ orbital $(l=1)$ of $n=2$ or higher, not in an $s(l=0)$ or $d(l=2)$ orbital of $n=2$ or higher. From what orbital do each of the allowed transitions start for the first four emission lines in the visible series $\left(n_{\text { final }}=2\right) ?$

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Problem 96

The discharge of phosphate in detergents to the environment has led to imbalances in the life cycle of freshwater lakes. A chemist uses a spectrometric method to measure total phosphate and obtains the following data for known standards:
(a) Draw a curve of absorbance vs. phosphate concentration.
(b) If a sample of lake water has an absorbance of $0.55,$ what is its phosphate concentration?

Ashley M.
Numerade Educator