CHAPTER 7 QUANTUM THEORY AND

ATOMIC STRUCTURE

The value for the speed of light will be 3.00x108 m/s except when more significant figures are necessary, in which cases,

2.9979x108 m/s will be used.

FOLLOW–UP PROBLEMS

7.1A

Plan: Given the frequency of the light, use the equation c = to solve for wavelength.

Solution:

c

3.00x108 m/s 1 nm

=

=

= 414.938 = 415 nm

ν

7.23x1014 s 1 109 m

=

7.1B

c

3.00x108 m/s 1 Å

=

= 4149.38 = 4150 Å

ν

7.23x1014 s 1 1010 m

Plan: Given the wavelength of the light, use the equation c = to solve for frequency. Remember that

wavelength must be in units of m in this equation.

Solution:

=

c

=

3.00x108 m/s

109 nm

940 nm

1m

= 3.1915x1014 = 3.2x1014 s–1

This is infrared radiation.

7.2A

Plan: Use the formula E = hto solve for the frequencyThen use the equation c = to solve for the wavelength.

Solution:

=

8.2x10–19 J

E

=

h

6.626x10–34 J•s

= 1.2375x1015 = 1.2x1015 s–1

(using the unrounded number in the next calculation to avoid rounding errors)

=

7.2B

3.00x108 m/s

c

=

1.2375x1015 s–1

ν

109 nm

1m

= 240 nm

Plan: To calculate the energy for each wavelength we use the formula E = hc/

Solution:

6.626x1034 J • s)(3.00x108 m/s

hc

=

= 1.9878x10–17 = 2x10–17 J

λ

1x108 m

6.626x1034 J • s)(3.00x108 m/s

hc

E=

=

= 3.9756x10–19 = 4x10–19 J

λ

5x107 m

6.626x1034 J • s)(3.00x108 m/s

hc

E=

=

= 1.9878x10–21 = 2x10–21 J

4

λ

1x10 m

As the wavelength of light increases from ultraviolet to visible to infrared, the energy of the light decreases.

E=

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7-1

7.3A

1

1

2

Plan: Use the equation relating ΔE = – 2.18x10– 18 J 2

to find the energy change; a photon in

n

n initial

final

the IR (infrared) region is emitted when has n has a final value of 3. Then use E = hc/to find the wavelength of

the photon.

Solution:

1

1

2

a) ΔE = – 2.18x10– 18 J 2

n

n initial

final

1

1

ΔE = – 2.18x10– 18 J 2 2

6

3

ΔE = –1.8166667x10– 19 = –1.82x10– 19 J

b) E =

7.3B

hc

λ

6.626x1034 J • s 3.00x108 m/s 1 Å

hc

4

4

10 1.094202x10 = 1.09x10 Å

19

E

1.8166667x10 J

10 m

Plan: Use the equation E = hc/ to find the energy change for this reaction. Then use the equation

1

1

2

ΔE = – 2.18x10– 18 J 2

to find the final energy level to which the electron moved.

n

n initial

final

Solution:

a) ΔE = hc/

(6.626x10–34 J•s) (3.00x108 m/s)

109 nm

410. nm

1m

= ΔE = 4.8483x10–19 = 4.85x10–19 J

Because the photon is emitted, energy is being given off, so the sign of ΔE should be negative. Therefore,

ΔE = –4.85x10–19 J

ΔE (kJ/mol) =

–4.85x10–19 J

6.022x1023 H atoms

1 kJ

1 H atom

1 mol H

1000 J

number)

1

1

2

b) ΔE = – 2.18x10– 18 J 2

n

n initial

final

1

n2final

1

n2final

1

n2final

=

=

ΔE

–18

J

–4.85x10

–19

J

–2.18x10

–18

–2.18x10

+

1

n2initial

1

62

= 0.25025

n2final = 3.9960 = 4

nfinal = 2

7.4A

J

+

= –292 kJ/mol (number of atoms is a positive

(The final energy level is an integer, so its square is also an integer.)

Plan: With the equation for the de Broglie wavelength, = h/mu and the given de Broglie wavelength, calculate

the electron speed. The wavelength must be expressed in meters. Use the same formulas to calculate the speed of

the golf ball. The mass of both the electron and the golf ball must be expressed in kg in their respective

calculations.

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7-2

Solution:

a) = h/mu

u=

h

=

mλ

kg • m2 /s2

6.626 x1034 J • s

J

109 m

31

9.11x10 kg 100.nm

1 nm

b) Mass (kg) of the golf ball = (45.9 g)

u=

7.4B

h

=

mλ

6.626x10–34 J•s

0.0459 kg

100. nm

1 kg

1000 g

kg•m2 /s2

10–9

J

= 7273.3 = 7.27x103 m/s

= 0.0459 kg

= 1.4436x10–25 = 1.44x10–25 m/s

1 nm

Plan: Use the equation for the de Broglie wavelength, = h/mu with the given mass and speed to calculate the de

Broglie wavelength of the racquetball. The mass of the racquetball must be expressed in kg, and the speed must be

expressed in m/s in the equation for the de Broglie wavelength.

Solution:

Mass (kg) of the racquetball = (39.7 g)

Speed (m/s) of the racquetball =

1 kg

1000 g

55 mi

1 hr

hr

3600 s

= 0.0397 kg

1.609 km

1000 m

mi

1 km

= h/mu

6.626x10–34 J•s

h

=

=

= 6.67607x10–34 = 6.7x10–34 m

0.0397 kg (25m/s)

mu

7.5A

Plan: Use the equation x • mu

= 24.5819 = 25 m/s

h

to solve for the uncertainty (Δx) in position of the baseball.

4π

Solution:

h

4π

Δu = 1% of u = 0.0100(44.7 m/s) = 0.447 m/s

h

6.626x1034 J • s

x

=

= 8.3070285x10– 34

4 0.142 kg 0.447 m/s

4πmu

Δx ≥ 8.31x10– 34 m

x • mu

7.5B

Plan: Use the equation x • mu

h

to solve for the uncertainty (Δx) in position of the neutron.

4π

Solution:

h

4π

Δu = 1% of u = 0.0100(8x107 m/s) = 8x105 m/s

x • mu

6.626x10–34 J•s

h

=

= 3.9467x10– 14 m

4πmu 4π 1.67x10–27 kg (8x105 m/s)

Δx ≥ 4x10– 14 m

x

7.6A

Plan: Following the rules for l (integer from 0 to n – 1) and ml (integer from –l to +l), write quantum numbers for

n = 4.

Solution:

For n = 4

l = 0, 1, 2, 3

For l = 0, ml = 0

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7-3

For l = 1, ml = –1, 0, 1

For l = 2, ml = –2, –1, 0, 1, 2

For l = 3, ml = –3, –2, –1, 0, 1, 2, 3

7.6B

Plan: Following the rules for l (integer from 0 to n – 1) and ml (integer from –l to +l), determine which value of

the principal quantum number has five allowed levels of l.

Solution:

The number of possible l values is equal to n, so the n = 5 principal quantum number has five allowed values of l.

l = 0, 1, 2, 3, 4

For n = 5

For l = 0, ml = 0

For l = 1, ml = –1, 0, 1

For l = 2, ml = –2, –1, 0, 1, 2

For l = 3, ml = –3, –2, –1, 0, 1, 2, 3

For l = 4, ml = –4, –3, –2, –1, 0, 1, 2, 3, 4

7.7A

Plan: Identify n and l from the sublevel designation. n is the integer in front of the sublevel letter. The sublevels

are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2, f represents l = 3.

Knowing the value for l, find the ml values (integer from –l to +l).

Solution:

Sublevel name

n value

l value

ml values

2p

2

1

–1, 0, 1

5f

5

3

–3, –2, –1, 0, 1, 2, 3

7.7B

Plan: Identify n and l from the sublevel designation. n is the integer in front of the sublevel letter. The sublevels

are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2, f represents l = 3.

Knowing the value for l, find the ml values (integer from –l to +l).

Solution:

Sublevel name

n value

l value

ml values

4d

4

2

–2, –1, 0, 1, 2

6s

6

0

0

The number of orbitals for each sublevel equals 2l + 1. Sublevel 4d should have 5 orbitals and sublevel 6s should

have 1 orbital. Both of these agree with the number of ml values for the sublevel.

7.8A

Plan: Use the rules for designating quantum numbers to fill in the blanks.

For a given n, l can be any integer from 0 to n–1.

For a given l, ml can be any integer from – l to + l.

The sublevels are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2,

f represents l = 3.

Solution:

The completed table is:

n

l

ml

Name

a)

4

1

0

4p

b)

2

1

0

2p

c)

3

2

–2

3d

d)

2

0

0

2s

7.8B

Plan: Use the rules for designating quantum numbers to determine what is wrong with the quantum number

designations provided in the problem.

For a given n, l can be any integer from 0 to n–1.

For a given l, ml can be any integer from – l to + l.

The sublevels are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2,

f represents l = 3.

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7-4

Solution:

The provided table is:

n

l

ml

a)

5

3

4

b)

2

2

1

c)

6

1

–1

a) For l = 3, the allowed values for ml are –3, –2, –1, 0, 1, 2, 3, not 4

b) For n = 2, l = 0 or 1 only, not 2; the sublevel is 2p, since ml = 1.

c) The value l = 1 indicates the p sublevel, not the s; the sublevel name is 6p.

Name

5f

2d

6s

TOOLS OF THE LABORATORY BOXED READING PROBLEMS

Plan: Plot absorbance on the y-axis and concentration on the x-axis. Since this is a linear plot, the graph is of the

type y = mx + b, with m = slope and b = intercept. Any two points may be used to find the slope, and the slope is

used to find the intercept. Once the equation for the line is known, the absorbance of the solution in part b) is

used to find the concentration of the diluted solution, after which the dilution equation is used to find the

molarity of the original solution.

Solution:

a) Absorbance vs. Concentration:

Absorbance

B7.1

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.00001

0.00002

0.00003

0.00004

Concentration (M)

This is a linear plot, thus, using the first and last points given:

0.396 0.131

y y1

m= 2

=

= 13,250 = 1.3x104/M

5

5

x2 x1

3.0x10 1.0x10 M

Using the slope just calculated and any of the data points, the value of the intercept may be found.

b = y – mx = 0.396 – (13,250/M)(3.0x10–5 M) = –0.0015 = 0.00 (absorbance has no units)

b) Use the equation just determined: y = (1.3x104/M) x + 0.00.

x = (y – 0.00)/(1.3x104/M) = (0.236/1.3x104) M = 1.81538x10–5 M = 1.8x10–5 M

This value is Mf in a dilution problem (MiVi)= (MfVf) with Vi = 20.0 mL and Vf = 150. mL.

Mi =

B7.2

M f Vf

Vi

=

1.81538x10

5

M 150. mL

20.0 mL

= 1.361538x10–4 = 1.4x10–4 M

Plan: The color of light associated with each wavelength can be found from Figure 7.3. The

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7-5

frequency of each wavelength can be determined from the relationship c = or =

c

. The

wavelength in nm must be converted to meters.

Solution:

3.00x108 m/s 1 nm

14

14 –1

a) red

=

= 4.4709x10 = 4.47x10 s

671 nm 109 m

3.00x108 m/s 1 nm

14

14 –1

b) blue

=

= 6.6225x10 = 6.62x10 s

453 nm 109 m

3.00x108 m/s 1 nm

14

14 –1

c) yellow-orange =

= 5.0933786x10 = 5.09x10 s

589 nm 109 m

END–OF–CHAPTER PROBLEMS

7.1

All types of electromagnetic radiation travel as waves at the same speed. They differ in both their frequency,

wavelength, and energy.

7.2

Plan: Recall that the shorter the wavelength, the higher the frequency and the greater the energy. Figure 7.3

describes the electromagnetic spectrum by wavelength and frequency.

Solution:

a) Wavelength increases from left (10–2 nm) to right (1012 nm) in Figure 7.3. The trend in increasing wavelength

is: x-ray < ultraviolet < visible < infrared < microwave < radio wave.

b) Frequency is inversely proportional to wavelength according to the equation c = λν, so frequency has the

opposite trend: radio wave < microwave < infrared < visible < ultraviolet < x-ray.

c) Energy is directly proportional to frequency according to the equation E = hν. Therefore, the trend in increasing

energy matches the trend in increasing frequency: radio wave < microwave < infrared < visible < ultraviolet <

x-ray.

7.3

a) Refraction is the bending of light waves at the boundary of two media, as when light travels from air into water.

b) Diffraction is the bending of light waves around an object, as when a wave passes through a slit about as wide as

its wavelength.

c) Dispersion is the separation of light into its component colors (wavelengths), as when light passes through a

prism.

d) Interference is the bending of light through a series of parallel slits to produce a diffraction pattern of

brighter and darker spots.

Note: Refraction leads to a dispersion effect and diffraction leads to an interference effect.

7.4

Evidence for the wave model is seen in the phenomena of diffraction and refraction. Evidence for the particle model

includes the photoelectric effect and blackbody radiation.

7.5

a) Frequency: C < B < A

b) Energy: C < B < A

c) Amplitude: B < C < A

d) Since wave A has a higher energy and frequency than B, wave A is more likely to cause a current.

e) Wave C is more likely to be infrared radiation since wave C has a longer wavelength than B.

7.6

Radiation (light energy) occurs as quanta of electromagnetic radiation, where each packet of energy is called a

photon. The energy associated with this photon is fixed by its frequency, E = hν. Since energy depends on frequency,

a threshold (minimum) frequency is to be expected. A current will flow as soon as a photon of sufficient energy

reaches the metal plate, so there is no time lag.

7.7

Plan: Wavelength is related to frequency through the equation c = ν. Recall that a Hz is a reciprocal second, or

1/s = s–1. Assume that the number “950” has three significant figures.

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7-6

Solution:

c = ν

c

(m) = =

3.00x108 m/s

= 315.789 = 316 m

103 Hz s1

950. kHz

1 kHz Hz

1 nm

11

11

= 315.789 m 9

= 3.15789x10 = 3.16x10 nm

10

m

c

(nm) =

(Å) =

7.8

1Å

= 315.789 m 10 = 3.158x1012 = 3.16x1012 Å

10 m

c

Wavelength and frequency relate through the equation c = . Recall that a Hz is a reciprocal second, or 1/s = s–1.

(m) =

c

(nm) =

(Å) =

7.9

=

3.00x108 m/s

= 3.208556 = 3.21 m

106 Hz s1

93.5 MHz

1 MHz Hz

1 nm

9

9

= 3.208556 m 9

= 3.208556x10 = 3.21x10 nm

10 m

c

1Å

= 3.208556 m 10 = 3.208556x1010 = 3.21x1010 Å

10 m

c

Plan: Frequency is related to energy through the equation E = h. Note that 1 Hz = 1 s–1.

Solution:

E = h

E = (6.626x10–34 J•s)(3.8x1010 s–1) = 2.51788x10–23 = 2.5x10–23 J

hc

6.626x10

=

34

J•s 3.00x108 m/s 1 Å

–15

–15

10 = 1.5291x10 = 1.5x10 J

1.3 Å

10

m

7.10

E=

7.11

Plan: Energy is inversely proportional to wavelength ( E =

hc

). As wavelength decreases, energy increases.

Solution:

In terms of increasing energy the order is red < yellow < blue.

7.12

Since energy is directly proportional to frequency (E = h): UV ( = 8.0x1015 s–1) > IR ( = 6.5x1013 s–1) >

microwave ( = 9.8x1011 s–1) or UV > IR > microwave.

7.13

Plan: Wavelength is related to frequency through the equation c = ν. Recall that a Hz is a reciprocal second,

or 1/s = s–1.

Solution:

109 Hz s 1

10 –1

= (s–1) = 22.235 GHz

= 2.2235x10 s

1

GHz

Hz

(nm) =

c

=

2.9979x108 m/s 1 nm

7

7

= 1.3482797x10 = 1.3483x10 nm

9

10 1

2.2235x10 s 10 m

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7-7

c

(Å) =

7.14

=

2.9979x108 m/s 1 Å

8

8

= 1.3482797x10 = 1.3483x10 Å

2.2235x1010 s1 1010 m

Frequency and wavelength can be calculated using the speed of light: c = ν.

3.00x108 m/s 1 m

c

13

13 –1

a) = =

= 3.125x10 = 3.1x10 s

9.6 m 106 m

1 m

c

2.9979x108 m/s

b) (m) =

=

= 3.464979 = 3.465 m

1

s 106 m

13

8.652x10 Hz

Hz

7.15

Frequency and energy are related by E = h, and wavelength and energy are related by E = hc/.

106 eV 1.602x1019 J

1.33 MeV

1 MeV

1 eV

E

Hz = 3.2156x1020 = 3.22x1020 Hz

(Hz) =

=

1

34

h

6.626x10 J•s

s

hc

(m) =

=

E

6.626x10

34

J•s 3.00x108 m/s

= 9.32950x10–13 = 9.33x10–13 m

10 eV 1.602x10 J

1.33 MeV

1 eV

1 MeV

The wavelength can also be found using the frequency calculated in the equation c =

7.16

19

6

Plan: The least energetic photon in part a) has the longest wavelength (242 nm). The most energetic photon in

part b) has the shortest wavelength (2200 Å). Use the relationship c = to find the frequency of the photons and

hc

to find the energy.

relationship E =

Solution:

a) c =

=

E=

c

hc

b) =

E=

=

c

hc

3.00x108 m/s 1 nm

15

15 –1

= 1.239669x10 = 1.24x10 s

242 nm 109 m

=

=

=

6.626x10

34

J•s 3.00x108 m/s 1 nm

–19

–19

9 = 8.2140x10 = 8.21x10 J

242 nm

10 m

3.00x108 m/s 1 Å

15

15 –1

= 1.3636x10 = 1.4x10 s

2200 Å 1010 m

6.626x10

34

J•s 3.00x108 m/s 1 Å

–19

–19

10 = 9.03545x10 = 9.0x10 J

2200 Å

10

m

7.17

“n” in the Rydberg equation is equal to a Bohr orbit of quantum number “n” where n = 1, 2, 3, ...

7.18

Bohr’s key assumption was that the electron in an atom does not radiate energy while in a stationary state, and the

electron can move to a different orbit by absorbing or emitting a photon whose energy is equal to the difference in

energy between two states. These differences in energy correspond to the wavelengths in the known spectra for

the hydrogen atoms. A Solar System model does not allow for the movement of electrons between levels.

7.19

An absorption spectrum is produced when atoms absorb certain wavelengths of incoming light as electrons move

from lower to higher energy levels and results in dark lines against a bright background. An emission spectrum is

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7-8

produced when atoms that have been excited to higher energy emit photons as their electrons return to lower

energy levels and results in colored lines against a dark background. Bohr worked with emission spectra.

7.20

Plan: The quantum number n is related to the energy level of the electron. An electron absorbs energy to

change from lower energy (lower n) to higher energy (higher n), giving an absorption spectrum. An electron emits

energy as it drops from a higher energy level (higher n) to a lower one (lower n), giving an emission spectrum.

Solution:

a) The electron is moving from a lower value of n (2) to a higher value of n (4): absorption

b) The electron is moving from a higher value of n (3) to a lower value of n (1): emission

c) The electron is moving from a higher value of n (5) to a lower value of n (2):emission

d) The electron is moving from a lower value of n (3) to a higher value of n (4): absorption

7.21

The Bohr model works only for a one-electron system. The additional attractions and repulsions in many-electron

systems make it impossible to predict accurately the spectral lines.

7.22

The Bohr model has successfully predicted the line spectra for the H atom and Be3+ ion since both are oneelectron species. The energies could be predicted from En =

Z 2 2.18x1018 J

where Z is the atomic number

n

for the atom or ion. The line spectra for H would not match the line spectra for Be3+ since the H nucleus contains

one proton while the Be3+ nucleus contains 4 protons (the Z values in the equation do not match); the force of

attraction of the nucleus for the electron would be greater in the beryllium ion than in the hydrogen atom. This

means that the pattern of lines would be similar, but at different wavelengths.

7.23

Plan: Calculate wavelength by substituting the given values into Equation 7.3, where n1 = 2 and n2 = 5 because

n2 > n1. Although more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

R = 1.096776x107 m–1

R 2 2

n

n

2

1

n1 = 2 n2 = 5

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 2,303,229.6 m–1

5

n2

2

n1

1 nm

1

(nm) =

= 434.1729544 = 434.17 nm

1

9

2,303, 229.6 m 10 m

7.24

Calculate wavelength by substituting the given values into the Rydberg equation, where n1 = 1 and n2 = 3 because

n2 > n1. Although more significant figures could be used, five significant figures are adequate for this calculation.

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 9,749,120 m–1

n

1

3

n

2

1

1Å

1

(Å) =

= 1025.7336 = 1025.7 Å

1

10

9,749,120 m 10 m

7.25

2

Plan: The Rydberg equation is needed. For the infrared series of the H atom, n1 equals 3. The least energetic

spectral line in this series would represent an electron moving from the next highest energy level, n2 = 4. Although

more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 533,155 m–1

4

n2

3

n1

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7-9

1 nm

1

(nm) =

= 1875.627 = 1875.6 nm

1

9

533,155 m 10 m

7.26

Plan: The Rydberg equation is needed. For the visible series of the H atom, n1 equals 2. The least energetic

spectral line in this series would represent an electron moving from the next highest energy level, n = 3. Although

more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 1,523,300 m–1

2

3

n

n

2

1

1 nm

1

(nm) =

= 656.4695 = 656.47 nm

1

9

1,523,300 m 10 m

7.27

7.28

7.29

Plan: To find the transition energy, use the equation for the energy of an electron transition and multiply by

Avogadro’s number to convert to energy per mole.

Solution:

1

1

E = 2.18x10 18 J 2

2

n

ninitial

final

1

1

E = 2.18x10 18 J 2 2 = –4.578x10–19 J/photon

5

2

19

4.578x10

J 6.022x1023 photons

5

5

E =

= –2.75687x10 = –2.76x10 J/mol

photon

1

mol

The energy has a negative value since this electron transition to a lower n value is an emission of energy.

To find the transition energy, use the equation for the energy of an electron transition and multiply by Avogadro’s

number.

1

1

E = 2.18x10 18 J 2

2

n

ninitial

final

1

1

E = 2.18x10 18 J 2 2 = 1.93778x10–18 J/photon

1

3

1.93778x10 18 J 6.022x1023 photons

6

6

E =

= 1.1669x10 = 1.17x10 J/mol

photon

1

mol

Plan: Determine the relative energy of the electron transitions. Remember that energy is directly proportional

to frequency (E = h).

Solution:

Looking at an energy chart will help answer this question.

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7-10

n=5 n=4

n=3

(d)

(a)

(c)

n=2

(b)

n=1

7.30

7.31

Frequency is proportional to energy so the smallest frequency will be d) n = 4 to n = 3; levels 3 and 4 have a

smaller E than the levels in the other transitions. The largest frequency is b) n = 2 to n = 1 since levels 1 and

2 have a larger E than the levels in the other transitions. Transition a) n = 2 to n = 4 will be smaller than

transition c) n = 2 to n = 5 since level 5 is a higher energy than level 4. In order of increasing frequency the

transitions are d < a < c < b.

b>c>a>d

Plan: Use the Rydberg equation. Since the electron is in the ground state (lowest energy level), n1 = 1. Convert

the wavelength from nm to units of meters.

Solution:

10 9 m

–8

= 97.20 nm

ground state: n1 = 1; n2 = ?

= 9.720x10 m

1

nm

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

= 1.096776x107 m 1

9.72x108 m

1

1

0.93803 = 2 2

1

n2

1

n22

11

2

1

n22

= 1 – 0.93803 = 0.06197

n 22 = 16.14

n2 = 4

7.32

109 m

–6

= 1.281x10 m

1

nm

= 1281 nm

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

1

1

= 1.096776x107 m 1 2 2

6

1.281x10 m

5

n1

1

1

0.07118 = 2 2

n

5

1

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7-11

1

n12

= 0.07118 + 0.04000 = 0.11118

n12 = 8.9944

n1 = 3

hc

6.626x10

34

J•s 3.00x108 m/s 1 nm

–19

–19

9 = 4.55917x10 = 4.56x10 J

436 nm

10

m

7.33

E=

7.34

a) Absorptions: A, C, D; Emissions: B, E, F

b) Energy of emissions: E < F < B

c) Wavelength of absorption: D < A < C

7.35

If an electron occupies a circular orbit, only integral numbers of wavelengths (= 2nr) are allowed for acceptable

standing waves. A wave with a fractional number of wavelengths is forbidden due to destructive interference with

itself. In a musical analogy to electron waves, the only acceptable guitar string wavelengths are those that are an

integral multiple of twice the guitar string length (2 L).

7.36

De Broglie’s concept is supported by the diffraction properties of electrons demonstrated in an electron

microscope.

7.37

Macroscopic objects have significant mass. A large m in the denominator of = h/mu will result in a very small

wavelength. Macroscopic objects do exhibit a wavelike motion, but the wavelength is too small for humans to see

it.

7.38

The Heisenberg uncertainty principle states that there is fundamental limit to the accuracy of measurements. This

limit is not dependent on the precision of the measuring instruments, but is inherent in nature.

7.39

Plan: Use the de Broglie equation. Mass in lb must be converted to kg and velocity in mi/h must

be converted to m/s because a joule is equivalent to kg•m2/s2.

Solution:

1 kg

a) Mass (kg) = 232 lb

= 105.2154 kg

2.205 lb

=

3

19.8 mi 1 km 10 m 1 h

Velocity (m/s) =

= 8.87097 m/s

h 0.62 mi 1 km 3600 s

=

h

=

mu

6.626x10

34

J•s

kg•m2 /s2

–37

–37

= 7.099063x10 = 7.10x10 m

m

J

105.2154 kg 8.87097

s

3

0.1 mi 1 km 10 m 1 h

b) Uncertainty in velocity (m/s) =

= 0.0448029 m/s

h 0.62 mi 1 km 3600 s

x•mv

x

h

4

h

4 mv

6.626x10

34

J•s

kg•m2 /s2

–35

–35

1.11855x10 1x10 m

J

0.0448029 m

4 105.2154 kg

s

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7-12

7.40

a) =

h

=

mu

6.626x10 J•s kg•m /s

6.6x10 g 3.4x10 mih J

34

2

24

2

7

103 g 0.62 mi 1 km 3600 s

3

1 kg 1 km 10 m 1 h

= 6.59057x10–15 = 6.6x10–15 m

h

b) x•mv

4

6.626x10 J•s

0.1x10

4 6.6x10 g

h

34

kg•m2 /s2 103 g 0.62 mi 1 km 3600 s

3

7

J

mi

24

1 kg 1 km 10 m 1 h

–14

–14

1.783166x10 2x10 m

h

x

4 mv

7.41

Plan: Use the de Broglie equation. Mass in g must be converted to kg and wavelength in Ǻ must be converted to

m because a joule is equivalent to kg•m2/s2.

Solution:

1 kg

Mass (kg) = 56.5 g 3 = 0.0565 kg

10 g

10 10 m

= 5.4x10–7 m

Wavelength (m) = 5400 Å

1 Å

h

=

mu

6.626x10 34 J•s

h

u=

=

m

0.0565 kg 5.4x107 m

7.42

=

kg•m 2 /s 2

J

h

mu

6.626x1034 J•s kg•m 2 /s2

h

u=

=

m

142 g 100. pm J

7.43

–26

–26

= 2.1717x10 = 2.2x10 m/s

103 g 1 pm

–23

–23

12 = 4.666197x10 = 4.67x10 m/s

1

kg

10 m

Plan: The de Broglie wavelength equation will give the mass equivalent of a photon with known wavelength and

velocity. The term “mass equivalent” is used instead of “mass of photon” because photons are quanta of

electromagnetic energy that have no mass. A light photon’s velocity is the speed of light, 3.00x108 m/s.

Wavelength in nm must be converted to m.

Solution:

109 m

Wavelength (m) = 589 nm

= 5.89x10–7 m

1 nm

h

=

mu

6.626x10 34 J•s

h

m=

=

u

5.89x10 7 m 3.00 x108 m /s

kg•m 2 /s 2

J

–36

–36

= 3.7499x10 = 3.75x10 kg/photon

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7-13

7.44

=

h

mu

m=

6.626x10 34 J•s

kg•m 2 /s 2 1 nm

h

–36

=

9 = 3.2916x10 kg/photon

8

u

J

10

m

671 nm 3.00x10 m/s

3.2916x1036 kg 6.022x1023 photons

–12

–12

= 1.9822x10 = 1.98x10 kg/mol

photon

mol

7.45

The quantity 2 expresses the probability of finding an electron within a specified tiny region of space.

7.46

Since 2 is the probability of finding an electron within a small region or volume, electron density would

represent a probability per unit volume and would more accurately be called electron probability density.

7.47

A peak in the radial probability distribution at a certain distance means that the total probability of finding the

electron is greatest within a thin spherical volume having a radius very close to that distance. Since principal

quantum number (n) correlates with distance from the nucleus, the peak for n = 2 would occur at a greater

distance from the nucleus than 0.529 Å. Thus, the probability of finding an electron at 0.529 Å is much greater for

the 1s orbital than for the 2s.

7.48

a) Principal quantum number, n, relates to the size of the orbital. More specifically, it relates to the distance from

the nucleus at which the probability of finding an electron is greatest. This distance is determined by the energy of

the electron.

b) Angular momentum quantum number, l, relates to the shape of the orbital. It is also called the azimuthal

quantum number.

c) Magnetic quantum number, ml, relates to the orientation of the orbital in space in three-dimensional space.

7.49

Plan: The following letter designations correlate with the following l quantum numbers:

l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are – l to + l.

The number of orbitals of a particular type is given by the number of possible ml values.

Solution:

a) There is only a single s orbital in any shell. l = 1 and ml = 0: one value of ml = one s orbital.

b) There are five d orbitals in any shell. l = 2 and ml = –2, –1, 0, +1, +2. Five values of ml = five d orbitals.

c) There are three p orbitals in any shell. l = 1 and ml = –1, 0, +1. Three values of ml = three p orbitals.

d) If n = 3, l = 0(s), 1(p), and 2(d). There is a 3s (1 orbital), a 3p set (3 orbitals), and a 3d set (5 orbitals) for a

total of nine orbitals (1 + 3 + 5 = 9).

7.50

a) All f orbitals consist of sets of seven (l = 3 and ml = –3, –2, –1, 0, +1, +2, +3).

b) All p orbitals consist of sets of three (l = 1 and ml = –1, 0, +1).

c) All d orbitals consist of sets of five (l = 2 and ml = –2, –1, 0, +1, +2).

d) If n = 2, then there is a 2s (1 orbital) and a 2p set (3 orbitals) for a total of four orbitals (1 + 3 = 4).

7.51

Plan: Magnetic quantum numbers (ml) can have integer values from –l to + l. The l quantum number can

have integer values from 0 to n – 1.

Solution:

a) l = 2 so ml = –2, –1, 0, +1, +2

b) n = 1 so l = 1 – 1 = 0 and ml = 0

c) l = 3 so ml = –3, –2, –1, 0, +1, +2, +3

7.52

Magnetic quantum numbers can have integer values from –l to +l. The l quantum number can

have integer values from 0 to n – 1.

a) l = 3 so ml = –3, –2, –1, 0, +1, +2, +3

b) n = 2 so l = 0 or 1; for l = 0, ml = 0; for l = l, ml = –1,0,+1

c) l = 1 so ml = –1, 0, +1

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7-14

7.53

Plan: The s orbital is spherical; p orbitals have two lobes; the subscript x indicates that this orbital lies along

the x-axis.

Solution:

a) s: spherical

b) px: 2 lobes along the x-axis

z

z

x

x

y

y

The variations in coloring of the p orbital are a consequence of the quantum mechanical derivation of atomic

orbitals that are beyond the scope of this course.

7.54

a) pz: 2 lobes along the z-axis

z

y

b) dxy: 4 lobes

y

x

x

The variations in coloring of the p and d orbitals are a consequence of the quantum mechanical derivation of

atomic orbitals that are beyond the scope of this course.

7.55

Plan: The following letter designations for the various sublevels (orbitals) correlate with the following l quantum

numbers: l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are

– l to + l. The number of orbitals of a particular type is given by the number of possible ml values.

Solution:

sublevel

allowable ml

# of possible orbitals

a) d (l = 2)

–2, –1, 0, +1, +2

5

b) p (l = 1)

–1, 0, +1

3

c) f (l = 3)

–3, –2, –1, 0, +1, +2, +3

7

7.56

sublevel

a) s (l = 0)

b) d (l = 2)

c) p (l = 1)

7.57

Plan: The integer in front of the letter represents the n value. The letter designates the l value:

l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are – l to + l.

Solution:

a) For the 5s subshell, n = 5 and l = 0. Since ml = 0, there is one orbital.

allowable ml

0

–2, –1, 0, +1, +2

–1, 0, +1

# of possible orbitals

1

5

3

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7-15

b) For the 3p subshell, n = 3 and l = 1. Since ml = –1, 0, +1, there are three orbitals.

c) For the 4f subshell, n = 4 and l = 3. Since ml = –3, –2, –1, 0, +1, +2, +3, there are seven orbitals.

7.58

a) n = 6; l = 4; 9 orbitals (ml = –4, –3, –2, –1, 0, +1, +2, +3, +4)

b) n = 4; l = 0; 1 orbital (ml = 0)

c) n = 3; l = 2; 5 orbitals (ml = –2, –1, 0, +1, +2)

7.59

Plan: Allowed values of quantum numbers: n = positive integers; l = integers from 0 to n – 1;

ml = integers from – l through 0 to + l.

Solution:

a) n = 2; l = 0; ml = –1: With n = 2, l can be 0 or 1; with l = 0, the only allowable ml value is 0. This

combination is not allowed. To correct, either change the l or ml value.

Correct: n = 2; l = 1; ml = –1 or n = 2; l = 0; ml = 0.

b) n = 4; l = 3; ml = –1: With n = 4, l can be 0, 1, 2, or 3; with l = 3, the allowable ml values are –3, –2, –1, 0, +1,

+2, +3. Combination is allowed.

c) n = 3; l = 1; ml = 0: With n = 3, l can be 0, 1, or 2; with l = 1, the allowable ml values are –1, 0, +1.

Combination is allowed.

d) n = 5; l = 2; ml = +3: With n = 5, l can be 0, 1, 2, 3, or 4; with l = 2, the allowable ml values are –2, –1, 0, +1,

+2. +3 is not an allowable ml value. To correct, either change l or ml value.

Correct: n = 5; l = 3; ml = +3 or n = 5; l = 2; ml = 0.

7.60

a) Combination is allowed.

b) No; n = 2; l = 1; ml = +1 or n = 2; l = 1; ml = 0

c) No; n = 7; l = 1; ml = +1 or n = 7; l = 3; ml = 0

d) No; n = 3; l = 1; ml = –1 or n = 3; l = 2; ml = –2

7.61

Determine the max for -carotene by measuring its absorbance in the 610-640 nm region of the visible spectrum.

Prepare a series of solutions of -carotene of accurately known concentration (using benzene or chloroform as

solvent), and measure the absorbance for each solution. Prepare a graph of absorbance versus concentration for these

solutions and determine its slope (assuming that this material obeys Beer’s law). Measure the absorbance of the

oil expressed from orange peel (diluting with solvent if necessary). The -carotene concentration can then either

be read directly from the calibration curve or calculated from the slope (A = kC, where k = slope of the line and

C = concentration).

7.62

Plan: For Ppart a, use the values of the constants h, π, me, and a0 to find the overall constant in the equation.

hc

Use the resulting equation to calculate E in part b). Use the relationship E =

to calculate the wavelength in

part c). Remember that a joule is equivalent to kg•m2/s2.

Solution:

a) h = 6.626x10–34 J•s; me = 9.1094x10–31 kg; a0 = 52.92x10–12 m

E=

E=

h2

8

2

me a02 n2

=

h2

8

2

me a02

6.626x10

34

1

2

n

J•s

2

8 2 9.1094x10 31 kg 52.92x10 12 m

2

kg•m 2 /s 2

J

1

2

n

1

1

= –(2.17963x10–18 J) 2 = –(2.180x10–18 J) 2

n

n

This is identical with the result from Bohr’s theory. For the H atom, Z = 1 and Bohr’s constant = –2.18x10–18 J.

For the hydrogen atom, derivation using classical principles or quantum-mechanical principles yields the same

constant.

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7-16

b) The n = 3 energy level is higher in energy than the n = 2 level. Because the zero point of the atom’s energy is

defined as an electron’s infinite distance from the nucleus, a larger negative number describes a lower energy

level. Although this may be confusing, it makes sense that an energy change would be a positive number.

1

1

E = –(2.180x10–18 J) 2 2 = –3.027778x10–19 = 3.028x10–19 J

3

2

hc

c) E =

(m) =

6.626x10 34 J•s 2.9979x108 m/s

hc

=

= 6.56061x10–7 = 6.561x10–7 m

E

3.027778x10 19 J

1 nm

= 656.061 = 656.1 nm

9

10 m

This is the wavelength for the observed red line in the hydrogen spectrum.

(nm) = 6.56061x107 m

7.63

Plan: When light of sufficient frequency (energy) shines on metal, electrons in the metal break free and a current

flows.

Solution:

a) The lines do not begin at the origin because an electron must absorb a minimum amount of energy before it has

enough energy to overcome the attraction of the nucleus and leave the atom. This minimum energy is the energy

of photons of light at the threshold frequency.

b) The lines for K and Ag do not begin at the same point. The amount of energy that an electron must absorb to

leave the K atom is less than the amount of energy that an electron must absorb to leave the Ag atom, where the

attraction between the nucleus and outer electron is stronger than in a K atom.

c) Wavelength is inversely proportional to energy. Thus, the metal that requires a larger amount of energy to be

absorbed before electrons are emitted will require a shorter wavelength of light. Electrons in Ag atoms require

more energy to leave, so Ag requires a shorter wavelength of light than K to eject an electron.

d) The slopes of the line show an increase in kinetic energy as the frequency (or energy) of light is increased.

Since the slopes are the same, this means that for an increase of one unit of frequency (or energy) of light, the

increase in kinetic energy of an electron ejected from K is the same as the increase in the kinetic energy of an

electron ejected from Ag. After an electron is ejected, the energy that it absorbs above the threshold energy

becomes the kinetic energy of the electron. For the same increase in energy above the threshold energy, for either

K or Ag, the kinetic energy of the ejected electron will be the same.

7.64

a) E =

6.626x10

34

J•s)(3.00x108 m/s 1 nm

–19

9 = 2.8397x10 J

700. nm

10

m

This is the value for each photon, that is, J/photon.

1 photon

= 70.430 = 70. photons

Number of photons = 2.0x10 17 J

19

J

2.8397x10

hc

=

6.626x10

=

34

J•s)(3.00x108 m/s 1nm

–19

b) E =

9 = 4.18484x10 J

475. nm

10 m

This is the value for each photon, that is, J/photon.

hc

1 photon

Number of photons = 2.0x10 17 J

= 47.7916 = 48 photons

19

J

4.18484x10

7.65

Determine the wavelength:

= 1/(1953 cm–1) = 5.1203277x10–4 cm

10 2 m 1 nm

3

(nm) = 5.1203277x104 cm

9 = 5120.3277 = 5.120x10 nm

1

cm

10

m

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7-17

10 2 m 1 Å

4

10 = 51203.277 = 5.120x10 Å

1

cm

10 m

(Å) = 5.1203277x104 cm

= c/=

7.66

2.9979x108 m/s 1 cm 1 Hz

13

13

= 5.8548987x10 = 5.855x10 Hz

5.1203277x104 cm 102 m 1 s1

Plan: The Bohr model has been successfully applied to predict the spectral lines for one-electron species other

than H. Common one-electron species are small cations with all but one electron removed. Since the problem

specifies a metal ion, assume that the possible choices are Li2+ or Be3+. Use the relationship E = h to convert the

Z2

frequency to energy and then solve Bohr’s equation E = 2.18x1018 J 2 to verify if a whole number for Z

n

can be calculated. Recall that the negative sign is a convention based on the zero point of the atom’s energy; it is

deleted in this calculation to avoid taking the square root of a negative number.

Solution:

The highest energy line corresponds to the transition from n = 1 to n = .

E = h = (6.626x10–34 J•s) (2.961x1016 Hz) (s–1/Hz) = 1.9619586x10–17 J

Z2

E = 2.18x1018 J 2

Z = charge of the nucleus

n

1.9619586x1017 (12 )

En2

=

= 8.99998

2.18x1018 J

2.18x10 18 J

Then Z2 = 9 and Z = 3.

Therefore, the ion is Li2+ with an atomic number of 3.

Z2 =

7.67

6.626 x10 J•s kg•m /s = 2.139214x10 = 2.1x10

9.11x10 kg 3.4x10 ms J

6.626x10 J•s kg•m /s = 1.16696x10 = 1.2x10 m

=

1.67x10 kg 3.4x10 ms J

34

h

=

a) Electron: =

mu

h

Proton: =

mu

b) E = 1/2mu2

u=

2

31

2

–10

–10

m

6

34

2

27

2

–13

–13

6

therefore u2 = 2E/m

2E

m

Electron: u =

2 2.7x1015 J kg•m 2 /s2

J

9.11x1031 kg

7

= 7.6991x10 m/s

2 2.7x10 J kg•m /s

Proton: u =

= 1.7982x10 m/s

1.67x10 kg J

kg•m /s

6.626x10 J•s

h

=

Electron: =

= 9.44698x10

m

J

mu

9.11x10

kg

7.6991x10

s

kg•m /s

6.626x10 J•s

h

Proton: =

=

= 2.20646 x 10

m

J

mu

1.67x10 kg 1.7982x10 s

15

2

2

6

27

34

2

31

–12

= 9.4x10–12 m

7

34

27

2

2

2

–13

= 2.2 x 10–13 m

6

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7-18

7.68

Plan: The electromagnetic spectrum shows that the visible region goes from 400 to 750 nm (4000 Å to 7500 Å).

Thus, wavelengths b, c, and d are for the three transitions in the visible series with nfinal = 2. Wavelength a is in the

ultraviolet region of the spectrum and the ultraviolet series has nfinal = 1. Wavelength e is in the infrared region of

the spectrum and the infrared series has nfinal = 3. Use the Rydberg equation to find the ninitial for each line. Convert

the wavelengths from Å to units of m.

Solution:

n = ? n = 1; = 1212.7 Å (shortest corresponds to the largest E)

10 10 m

–7

(m) = 1212.7

= 1.2127x10 m

1

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

7 1

= 1.096776x10 m

7

1.2127x10 m

1

1

0.7518456 = 2 2

1

n2

11

2

1

n22

1

2 = 1 – 0.7518456

n2

1

2 = 0.2481544

n2

n2 = 2 for line (a) (n = 2 n = 1)

n = ? n = 3; = 10,938 Å (longest corresponds to the smallest E)

10 10 m

–6

(m) = 10,938

= 1.0938x10 m

1

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

7 1

= 1.096776x10 m

6

1.0938x10 m

1

1

0.083357397 = 2 2

3

n2

31

2

1

n22

0.083357397 = 0.111111111

1

n22

1

2

n2

= 0.11111111 – 0.083357396

1

2

n2

= 0.0277537151

n 22 =36.03121

n2 = 6 for line (e) (n = 6 n = 3)

For the other three lines, n1 = 2.

For line (d), n2 = 3 (largest smallest E).

For line (b), n2 = 5 (smallest largest E).

For line (c), n2 = 4.

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7-19

7.69

E=

hc

a) (nm) =

thus =

hc

E

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

=

9 = 432.130 = 432 nm

E

4.60x10 19 J

10 m

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

b) (nm) =

=

9 = 286.4265 = 286 nm

E

6.94x1019 J

10 m

c) (nm) =

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

=

9 = 450.748 = 451 nm

E

4.41x1019 J

10 m

7.70

Index of refraction = c/v; v = c/(index of refraction)

a) Water

v = c/(index of refraction) = (3.00x108 m/s)/(1.33) = 2.2556x108 = 2.26x108 m/s

b) Diamond

v = c/(index of refraction) = (3.00x108 m/s)/(2.42) = 1.239669x108 = 1.24x108 m/s

7.71

Extra significant figures are necessary because of the data presented in the problem.

He–Ne = 632.8 nm

Ar

= 6.148x1014 s–1

Ar–Kr E = 3.499x10–19 J

= 663.7 nm

Dye

Calculating missing values:

Ar

= c/ = (2.9979x108 m/s)/(6.148x1014 s–1) = 4.8762199x 10–7 = 4.876x10–7 m

Ar–Kr = hc/E = (6.626x10–34 J•s) (2.9979x108 m/s)/(3.499x10–19 J) = 5.67707x10–7 = 5.677x10–7 m

Calculating missing values:

He–Ne = c/ = (2.9979x108 m/s)/[632.8 nm (10–9 m/nm)] = 4.7375 x 1014 = 4.738 x 1014 s–1

Ar–Kr = E/h = (3.499x10–19 J)/(6.626x10–34 J•s) = 5.28071x1014 = 5.281x1014 s–1

Dye

= c/ = (2.9979x108 m/s)/[663.7 nm (10–9 m/nm)] = 4.51695x1014 = 4.517x1014 s–1

Calculating missing E values:

He–Ne E = hc/ = [(6.626x10–34 J•s)(2.9979x108 m/s)]/[632.8 nm (10–9 m/nm)]

= 3.13907797x10–19 = 3.139x10–19 J

E = h = (6.626x10–34 J•s)(6.148x1014 s–1) = 4.0736648x10–19 = 4.074x10–19 J

Ar

Dye

E = hc/ = [(6.626x10–34 J•s)(2.9979x108 m/s)]/[663.7 nm (10–9 m/nm)]

= 2.99293x10–19 = 2.993x10–19 J

The colors may be predicted from Figure 7.3 and the frequencies.

Orange

He–Ne = 4.738x1014 s–1

Ar

= 6.148x1014 s–1

Green

Ar–Kr = 5.281x1014 s–1

Yellow

Dye

= 4.517x1014 s–1

Red

7.72

Plan: Allowed values of quantum numbers: n = positive integers; l = integers from 0 to n – 1;

ml = integers from – l through 0 to + l.

Solution:

a) The l value must be at least 1 for ml to be – 1, but cannot be greater than n – 1 = 3 –1 = 2. Increase the l value to

1 or 2 to create an allowable combination.

b) The l value must be at least 1 for ml to be +1, but cannot be greater than n – 1 = 3 – 1 = 2. Decrease the l value

to 1 or 2 to create an allowable combination.

c) The l value must be at least 3 for ml to be +3, but cannot be greater than n – 1 = 7 – 1 = 6. Increase the l value to

3, 4, 5, or 6 to create an allowable combination.

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7-20

d) The l value must be at least 2 for ml to be –2, but cannot be greater than n – 1 = 4 – 1 = 3. Increase the l value to

2 or 3 to create an allowable combination.

7.73

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

a)

1

1 nm

= 1.096776x107 m 1

94.91 nm 109 m

1

1

0.9606608 = 2 2

1

n2

n2 = 5

1 nm

1

7 1

b)

= 1.096776x10 m

1281 nm 10 9 m

11

2

n1

2

1

1

n22

1

52

1

1

0.071175894 = 2 2

n

5

1

n1 = 3

1

1 1

c)

= 1.096776x107 m 1 2 2

1 3

1

= 9.74912x106 m–1

10

1 nm

= 1.02573x107 m 9 = 102.573 = 102.6 nm

7.74

m

Plan: Ionization occurs when the electron is completely removed from the atom, or when nfinal = . We can

use the equation for the energy of an electron transition to find the quantity of energy needed to remove

completely the electron, called the ionization energy (IE). To obtain the ionization energy per mole of species,

multiply by Avogadro’s number. The charge on the nucleus must affect the IE because a larger nucleus would

exert a greater pull on the escaping electron. The Bohr equation applies to H and other one-electron species. Use

the expression to determine the ionization energy of B4+. Then use the expression to find the energies of the

hc

transitions listed and use E =

to convert energy to wavelength.

Solution:

Z2

a) E = 2.18x1018 J 2

n

Z = atomic number

1

1

E = 2.18x1018 J 2

2

n

ninitial

final

2

Z

1

1 2 6.022x1023

E = 2.18x1018 J 2 2

Z

ninitial 1 mol

= (1.312796x106) Z2 for n = 1

b) In the ground state n = 1, the initial energy level for the single electron in B4+. Once ionized, n = is the final

energy level.

Z = 5 for B4+.

E = IE = (1.312796x106) Z2 = (1.312796x106 J/mol)(52) = 3.28199x107 = 3.28x107 J/mol

c) nfinal = , ninitial = 3, and Z = 2 for He+.

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7-21

1

1

E = 2.18x10 18 J 2 2

n

final ninitial

hc

E=

2

18

Z = 2.18x10 J

1

2

1

32

2

–19

2 = 9.68889x10 J

6.626x1034 J•s 3.00x108 m/s

hc

=

= 2.051628x10–7 m

E

9.68889x1019 J

1 nm

(nm) = 2.051628x107 m 9 = 205.1628 = 205 nm

10 m

d) nfinal = , ninitial = 2, and Z = 4 for Be3+.

1

1

1 2

1

E = 2.18x10 18 J 2 2

Z = 2.18x10 18 J 2 2 4 2 = 8.72x10–18 J

2

ninitial

(m) =

6.626x10

=

hc

(m) =

E

34

J•s 3.00x108 m/s

8.72x10

18

J

= 2.279587x10

–8

m

1 nm

(nm) = 2.279587x108 m 9 = 22.79587 = 22.8 nm

10 m

7.75

a) Orbital D has the largest value of n, given that it is the largest orbital.

b) l = 1 indicates a p orbital. Orbitals A and C are p orbitals. l = 2 indicates a d orbital. Orbitals B and D are d

orbitals.

c) In an atom, there would be four other orbitals with the same value of n and the same shape as orbital B.

There would be two other orbitals with the same value of n and the same shape as orbital C.

d) Orbital D has the highest energy and orbital C has the lowest energy.

7.76

Plan: Use the values and the equation given in the problem to calculate the appropriate values.

Solution:

n2 h2 0

a) rn =

me e2

2

C2

12 6.626x1034 J•s 8.854x1012

J•m kg•m 2 /s2

r1 =

2

J

9.109x1031 kg 1.602x1019 C

–11

–11

= 5.2929377x10 = 5.293x10 m

2

C2

102 6.626x1034 J•s 8.854x1012

J•m kg•m 2 /s2

–9

–9

b) r10 =

= 5.2929377x10 = 5.293x10 m

2

31

19

J

9.109x10 kg 1.602x10 C

7.77

a) rn =

n2 h2 0

me e2

2

C2

32 6.626x1034 J•s 8.854x1012

J•m kg•m2 /s2

–10

–10

r3 =

= 4.76364x10 = 4.764x10 m

2

31

19

J

9.109x10 kg 1.602x10 C

b) Z = 1 for an H atom

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7-22

Z2

En = 2.18x1018 J 2

n

c) Z = 3 for a Li atom

18

= 2.18x10 J

2

13

2

–19

–19

= – 2.42222x10 = – 2.42x10 J

Z2

32

En = 2.18x1018 J 2 = 2.18x1018 J 2 = – 2.18x10–1

n

3

d) The greater number of protons in the Li nucleus results in a greater interaction between the Li nucleus and its

electrons. Thus, the energy of an electron in a particular orbital becomes more negative with increasing atomic

number.

7.78

Plan: Refer to Chapter 6 for the calculation of the amount of heat energy absorbed by a substance from its

specific heat capacity and temperature change (q = c x mass x T). Using this equation, calculate the energy

absorbed by the water. This energy equals the energy from the microwave photons. The energy of each photon

can be calculated from its wavelength: E = hc/. Dividing the total energy by the energy of each photon gives the

number of photons absorbed by the water.

Solution:

q = c x mass x T

q = (4.184 J/g°C)(252 g)(98 – 20.)°C = 8.22407x104 J

E=

hc

6.626x10

=

34

J•s 3.00x108 m/s

1.55x10

2

m

7.79

= 1.28245x10

–23

J/photon

1 photon

= 6.41278x1027 = 6.4x1027 photons

Number of photons = 8.22407x104 J

23

1.28245x10 J

One sample calculation will be done using the equation in the book:

1 1

=

a0

For r = 50 pm:

r

3

2

e

r

a0

1

1

=

52.92 pm

3

2

e

r

a0

= 1.465532x10–3 e

r

a0

50

= 1.465532x10–3 e a0 = 1.465532x10–3 e 52.92 = 5.69724x10–4

2 = (5.69724x10–4)2 = 3.24585x10–7

4r22 = 4(50)2(3.24585x10–7) = 1.0197x10–2

r (pm)

(pm–3/2)

2 (pm–3)

4r22 (pm–1)

–3

–6

0

1.47x10

2.15x10

0

50

0.570x10–3

0.325x10–6

1.02x10–2

–3

–6

100

0.221x10

0.0491x10

0.616x10–2

–3

–6

200

0.0335x10

0.00112x10

0.0563x10–2

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7-23

The plots are similar to Figure 7.17A in the text.

7.80

Plan: In general, to test for overlap of the two series, compare the longest wavelength in the “n” series with the

shortest wavelength in the “n+1” series. The longest wavelength in any series corresponds to the transition

between the n1 level and the next level above it; the shortest wavelength corresponds to the transition between the n1

1

1

1

level and the n = level. Use the relationship = R 2 2 to calculate the wavelengths.

n

n2

1

Solution:

1

1

1

1

1

= R 2 2 = 1.096776x107 m 1 2 2

n2

n2

n1

n1

a) The overlap between the n1 = 1 series and the n1 = 2 series would occur between the longest wavelengths for

n1 = 1 and the shortest wavelengths for n1 = 2.

Longest wavelength in n1 = 1 series has n2 equal to 2.

1

1

1

= 1.096776x107 m 1 2 2 = 8,225,820 m–1

2

1

1

=

= 1.215684272x10–7 = 1.215684x10–7 m

1

8,225,820 m

Shortest wavelength in the n1 = 2 series:

1

1

1

–1

= 1.096776x107 m 1 2

= 2,741,940 m

2

2

1

=

= 3.647052817x10–7 = 3.647053x10–7 m

2,741,940 m 1

Since the longest wavelength for n1 = 1 series is shorter than shortest wavelength for n1 = 2 series, there

is no overlap between the two series.

b) The overlap between the n1 = 3 series and the n1 = 4 series would occur between the longest wavelengths for

n1 = 3 and the shortest wavelengths for n1 = 4.

Longest wavelength in n1 = 3 series has n2 equal to 4.

1

1

1

= 1.096776x107 m 1 2 2 = 533,155 m–1

4

3

1

=

= 1.875627163x10–6 = 1.875627x10–6 m

533,155 m 1

Shortest wavelength in n1 = 4 series has n2 = .

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7-24

1

1

1

–1

= 1.096776x107 m 1 2

= 685,485 m

2

4

1

=

= 1.458821127x10–6 = 1.458821x10–6 m

685,485 m 1

Since the n1 = 4 series shortest wavelength is shorter than the n1 = 3 series longest wavelength, the series do

overlap.

c) Shortest wavelength in n1 = 5 series has n2 = .

1

1

1

–1

= 1.096776x107 m 1 2

= 438,710.4 m

2

5

1

=

= 2.27940801x10–6 = 2.279408x10–6 m

438,710.4 m 1

Calculate the first few longest lines in the n1 = 4 series to determine if any overlap with the shortest wavelength

in the n1 = 5 series:

For n1 = 4, n2 = 5:

1

1

1

= 1.096776x107 m 1 2 2 = 246,774.6 m–1

4

5

1

–6

=

= 4.052281x10 m

246,774.6 m 1

For n1 = 4, n2 = 6:

1

1

1

= 1.096776x107 m 1 2 2 = 380,825 m–1

6

4

1

=

= = 2.625878x10–6 m

1

380,825 m

For n1 = 4, n2 = 7:

1

1

1

= 1.096776x107 m 1 2 2 = 461,653.2 m–1

7

4

1

=

= 2.166128x10–6 m

461,653.2 m 1

The wavelengths of the first two lines of the n1 = 4 series are longer than the shortest wavelength in the n1 = 5

series. Therefore, only the first two lines of the n1 = 4 series overlap the n1 = 5 series.

d) At longer wavelengths (i.e., lower energies), there is increasing overlap between the lines from different series

(i.e., with different n1 values). The hydrogen spectrum becomes more complex, since the lines begin to merge into a

more-or-less continuous band, and much more care is needed to interpret the information.

7.81

a) The highest frequency would correspond to the greatest energy difference. In this case, the greatest energy

difference would be between E3 and E1.

E = E3 – E1 = h = (– 15x10–19 J) – (– 20x10–19 J) = 5x10–19 J

= E/h = (5x10–19 J)/(6.626x10–34 J•s) = 7.54603x1014 = 8x1014 s–1

= c/ = (3.00x108 m/s)/(7.54603x1014 s–1) = 3.97560x10–7 = 4x10–7 m

b) The ionization energy (IE) is the same as the reverse of E1. Thus, the value of the IE is 20x10–19 J/atom.

IE = (20x10–19 J/atom)(1kJ/103 J)(6.022x1023 atoms/mol) = 1204.4 = 1.2x103 kJ/mol

c) The shortest wavelength would correspond to an electron moving from the n = 4 level to the highest level

available in the problem (n = 6).

E = E6 – E4 = hc/ = (– 2x10–19 J) – (– 11x10–19 J) = 9x10–19 J

= hc/E =

7.82

6.626x10

34

J•s 3.00x108 m /s 1 nm

2

9 = 220.867 = 2x10 nm

19

9x10 J

10 m

Plan: The energy differences sought may be determined by looking at the energy changes in steps. The

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7-25

ATOMIC STRUCTURE

The value for the speed of light will be 3.00x108 m/s except when more significant figures are necessary, in which cases,

2.9979x108 m/s will be used.

FOLLOW–UP PROBLEMS

7.1A

Plan: Given the frequency of the light, use the equation c = to solve for wavelength.

Solution:

c

3.00x108 m/s 1 nm

=

=

= 414.938 = 415 nm

ν

7.23x1014 s 1 109 m

=

7.1B

c

3.00x108 m/s 1 Å

=

= 4149.38 = 4150 Å

ν

7.23x1014 s 1 1010 m

Plan: Given the wavelength of the light, use the equation c = to solve for frequency. Remember that

wavelength must be in units of m in this equation.

Solution:

=

c

=

3.00x108 m/s

109 nm

940 nm

1m

= 3.1915x1014 = 3.2x1014 s–1

This is infrared radiation.

7.2A

Plan: Use the formula E = hto solve for the frequencyThen use the equation c = to solve for the wavelength.

Solution:

=

8.2x10–19 J

E

=

h

6.626x10–34 J•s

= 1.2375x1015 = 1.2x1015 s–1

(using the unrounded number in the next calculation to avoid rounding errors)

=

7.2B

3.00x108 m/s

c

=

1.2375x1015 s–1

ν

109 nm

1m

= 240 nm

Plan: To calculate the energy for each wavelength we use the formula E = hc/

Solution:

6.626x1034 J • s)(3.00x108 m/s

hc

=

= 1.9878x10–17 = 2x10–17 J

λ

1x108 m

6.626x1034 J • s)(3.00x108 m/s

hc

E=

=

= 3.9756x10–19 = 4x10–19 J

λ

5x107 m

6.626x1034 J • s)(3.00x108 m/s

hc

E=

=

= 1.9878x10–21 = 2x10–21 J

4

λ

1x10 m

As the wavelength of light increases from ultraviolet to visible to infrared, the energy of the light decreases.

E=

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7-1

7.3A

1

1

2

Plan: Use the equation relating ΔE = – 2.18x10– 18 J 2

to find the energy change; a photon in

n

n initial

final

the IR (infrared) region is emitted when has n has a final value of 3. Then use E = hc/to find the wavelength of

the photon.

Solution:

1

1

2

a) ΔE = – 2.18x10– 18 J 2

n

n initial

final

1

1

ΔE = – 2.18x10– 18 J 2 2

6

3

ΔE = –1.8166667x10– 19 = –1.82x10– 19 J

b) E =

7.3B

hc

λ

6.626x1034 J • s 3.00x108 m/s 1 Å

hc

4

4

10 1.094202x10 = 1.09x10 Å

19

E

1.8166667x10 J

10 m

Plan: Use the equation E = hc/ to find the energy change for this reaction. Then use the equation

1

1

2

ΔE = – 2.18x10– 18 J 2

to find the final energy level to which the electron moved.

n

n initial

final

Solution:

a) ΔE = hc/

(6.626x10–34 J•s) (3.00x108 m/s)

109 nm

410. nm

1m

= ΔE = 4.8483x10–19 = 4.85x10–19 J

Because the photon is emitted, energy is being given off, so the sign of ΔE should be negative. Therefore,

ΔE = –4.85x10–19 J

ΔE (kJ/mol) =

–4.85x10–19 J

6.022x1023 H atoms

1 kJ

1 H atom

1 mol H

1000 J

number)

1

1

2

b) ΔE = – 2.18x10– 18 J 2

n

n initial

final

1

n2final

1

n2final

1

n2final

=

=

ΔE

–18

J

–4.85x10

–19

J

–2.18x10

–18

–2.18x10

+

1

n2initial

1

62

= 0.25025

n2final = 3.9960 = 4

nfinal = 2

7.4A

J

+

= –292 kJ/mol (number of atoms is a positive

(The final energy level is an integer, so its square is also an integer.)

Plan: With the equation for the de Broglie wavelength, = h/mu and the given de Broglie wavelength, calculate

the electron speed. The wavelength must be expressed in meters. Use the same formulas to calculate the speed of

the golf ball. The mass of both the electron and the golf ball must be expressed in kg in their respective

calculations.

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7-2

Solution:

a) = h/mu

u=

h

=

mλ

kg • m2 /s2

6.626 x1034 J • s

J

109 m

31

9.11x10 kg 100.nm

1 nm

b) Mass (kg) of the golf ball = (45.9 g)

u=

7.4B

h

=

mλ

6.626x10–34 J•s

0.0459 kg

100. nm

1 kg

1000 g

kg•m2 /s2

10–9

J

= 7273.3 = 7.27x103 m/s

= 0.0459 kg

= 1.4436x10–25 = 1.44x10–25 m/s

1 nm

Plan: Use the equation for the de Broglie wavelength, = h/mu with the given mass and speed to calculate the de

Broglie wavelength of the racquetball. The mass of the racquetball must be expressed in kg, and the speed must be

expressed in m/s in the equation for the de Broglie wavelength.

Solution:

Mass (kg) of the racquetball = (39.7 g)

Speed (m/s) of the racquetball =

1 kg

1000 g

55 mi

1 hr

hr

3600 s

= 0.0397 kg

1.609 km

1000 m

mi

1 km

= h/mu

6.626x10–34 J•s

h

=

=

= 6.67607x10–34 = 6.7x10–34 m

0.0397 kg (25m/s)

mu

7.5A

Plan: Use the equation x • mu

= 24.5819 = 25 m/s

h

to solve for the uncertainty (Δx) in position of the baseball.

4π

Solution:

h

4π

Δu = 1% of u = 0.0100(44.7 m/s) = 0.447 m/s

h

6.626x1034 J • s

x

=

= 8.3070285x10– 34

4 0.142 kg 0.447 m/s

4πmu

Δx ≥ 8.31x10– 34 m

x • mu

7.5B

Plan: Use the equation x • mu

h

to solve for the uncertainty (Δx) in position of the neutron.

4π

Solution:

h

4π

Δu = 1% of u = 0.0100(8x107 m/s) = 8x105 m/s

x • mu

6.626x10–34 J•s

h

=

= 3.9467x10– 14 m

4πmu 4π 1.67x10–27 kg (8x105 m/s)

Δx ≥ 4x10– 14 m

x

7.6A

Plan: Following the rules for l (integer from 0 to n – 1) and ml (integer from –l to +l), write quantum numbers for

n = 4.

Solution:

For n = 4

l = 0, 1, 2, 3

For l = 0, ml = 0

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7-3

For l = 1, ml = –1, 0, 1

For l = 2, ml = –2, –1, 0, 1, 2

For l = 3, ml = –3, –2, –1, 0, 1, 2, 3

7.6B

Plan: Following the rules for l (integer from 0 to n – 1) and ml (integer from –l to +l), determine which value of

the principal quantum number has five allowed levels of l.

Solution:

The number of possible l values is equal to n, so the n = 5 principal quantum number has five allowed values of l.

l = 0, 1, 2, 3, 4

For n = 5

For l = 0, ml = 0

For l = 1, ml = –1, 0, 1

For l = 2, ml = –2, –1, 0, 1, 2

For l = 3, ml = –3, –2, –1, 0, 1, 2, 3

For l = 4, ml = –4, –3, –2, –1, 0, 1, 2, 3, 4

7.7A

Plan: Identify n and l from the sublevel designation. n is the integer in front of the sublevel letter. The sublevels

are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2, f represents l = 3.

Knowing the value for l, find the ml values (integer from –l to +l).

Solution:

Sublevel name

n value

l value

ml values

2p

2

1

–1, 0, 1

5f

5

3

–3, –2, –1, 0, 1, 2, 3

7.7B

Plan: Identify n and l from the sublevel designation. n is the integer in front of the sublevel letter. The sublevels

are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2, f represents l = 3.

Knowing the value for l, find the ml values (integer from –l to +l).

Solution:

Sublevel name

n value

l value

ml values

4d

4

2

–2, –1, 0, 1, 2

6s

6

0

0

The number of orbitals for each sublevel equals 2l + 1. Sublevel 4d should have 5 orbitals and sublevel 6s should

have 1 orbital. Both of these agree with the number of ml values for the sublevel.

7.8A

Plan: Use the rules for designating quantum numbers to fill in the blanks.

For a given n, l can be any integer from 0 to n–1.

For a given l, ml can be any integer from – l to + l.

The sublevels are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2,

f represents l = 3.

Solution:

The completed table is:

n

l

ml

Name

a)

4

1

0

4p

b)

2

1

0

2p

c)

3

2

–2

3d

d)

2

0

0

2s

7.8B

Plan: Use the rules for designating quantum numbers to determine what is wrong with the quantum number

designations provided in the problem.

For a given n, l can be any integer from 0 to n–1.

For a given l, ml can be any integer from – l to + l.

The sublevels are given a letter designation, in which s represents l = 0, p represents l = 1, d represents l = 2,

f represents l = 3.

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7-4

Solution:

The provided table is:

n

l

ml

a)

5

3

4

b)

2

2

1

c)

6

1

–1

a) For l = 3, the allowed values for ml are –3, –2, –1, 0, 1, 2, 3, not 4

b) For n = 2, l = 0 or 1 only, not 2; the sublevel is 2p, since ml = 1.

c) The value l = 1 indicates the p sublevel, not the s; the sublevel name is 6p.

Name

5f

2d

6s

TOOLS OF THE LABORATORY BOXED READING PROBLEMS

Plan: Plot absorbance on the y-axis and concentration on the x-axis. Since this is a linear plot, the graph is of the

type y = mx + b, with m = slope and b = intercept. Any two points may be used to find the slope, and the slope is

used to find the intercept. Once the equation for the line is known, the absorbance of the solution in part b) is

used to find the concentration of the diluted solution, after which the dilution equation is used to find the

molarity of the original solution.

Solution:

a) Absorbance vs. Concentration:

Absorbance

B7.1

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.00001

0.00002

0.00003

0.00004

Concentration (M)

This is a linear plot, thus, using the first and last points given:

0.396 0.131

y y1

m= 2

=

= 13,250 = 1.3x104/M

5

5

x2 x1

3.0x10 1.0x10 M

Using the slope just calculated and any of the data points, the value of the intercept may be found.

b = y – mx = 0.396 – (13,250/M)(3.0x10–5 M) = –0.0015 = 0.00 (absorbance has no units)

b) Use the equation just determined: y = (1.3x104/M) x + 0.00.

x = (y – 0.00)/(1.3x104/M) = (0.236/1.3x104) M = 1.81538x10–5 M = 1.8x10–5 M

This value is Mf in a dilution problem (MiVi)= (MfVf) with Vi = 20.0 mL and Vf = 150. mL.

Mi =

B7.2

M f Vf

Vi

=

1.81538x10

5

M 150. mL

20.0 mL

= 1.361538x10–4 = 1.4x10–4 M

Plan: The color of light associated with each wavelength can be found from Figure 7.3. The

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7-5

frequency of each wavelength can be determined from the relationship c = or =

c

. The

wavelength in nm must be converted to meters.

Solution:

3.00x108 m/s 1 nm

14

14 –1

a) red

=

= 4.4709x10 = 4.47x10 s

671 nm 109 m

3.00x108 m/s 1 nm

14

14 –1

b) blue

=

= 6.6225x10 = 6.62x10 s

453 nm 109 m

3.00x108 m/s 1 nm

14

14 –1

c) yellow-orange =

= 5.0933786x10 = 5.09x10 s

589 nm 109 m

END–OF–CHAPTER PROBLEMS

7.1

All types of electromagnetic radiation travel as waves at the same speed. They differ in both their frequency,

wavelength, and energy.

7.2

Plan: Recall that the shorter the wavelength, the higher the frequency and the greater the energy. Figure 7.3

describes the electromagnetic spectrum by wavelength and frequency.

Solution:

a) Wavelength increases from left (10–2 nm) to right (1012 nm) in Figure 7.3. The trend in increasing wavelength

is: x-ray < ultraviolet < visible < infrared < microwave < radio wave.

b) Frequency is inversely proportional to wavelength according to the equation c = λν, so frequency has the

opposite trend: radio wave < microwave < infrared < visible < ultraviolet < x-ray.

c) Energy is directly proportional to frequency according to the equation E = hν. Therefore, the trend in increasing

energy matches the trend in increasing frequency: radio wave < microwave < infrared < visible < ultraviolet <

x-ray.

7.3

a) Refraction is the bending of light waves at the boundary of two media, as when light travels from air into water.

b) Diffraction is the bending of light waves around an object, as when a wave passes through a slit about as wide as

its wavelength.

c) Dispersion is the separation of light into its component colors (wavelengths), as when light passes through a

prism.

d) Interference is the bending of light through a series of parallel slits to produce a diffraction pattern of

brighter and darker spots.

Note: Refraction leads to a dispersion effect and diffraction leads to an interference effect.

7.4

Evidence for the wave model is seen in the phenomena of diffraction and refraction. Evidence for the particle model

includes the photoelectric effect and blackbody radiation.

7.5

a) Frequency: C < B < A

b) Energy: C < B < A

c) Amplitude: B < C < A

d) Since wave A has a higher energy and frequency than B, wave A is more likely to cause a current.

e) Wave C is more likely to be infrared radiation since wave C has a longer wavelength than B.

7.6

Radiation (light energy) occurs as quanta of electromagnetic radiation, where each packet of energy is called a

photon. The energy associated with this photon is fixed by its frequency, E = hν. Since energy depends on frequency,

a threshold (minimum) frequency is to be expected. A current will flow as soon as a photon of sufficient energy

reaches the metal plate, so there is no time lag.

7.7

Plan: Wavelength is related to frequency through the equation c = ν. Recall that a Hz is a reciprocal second, or

1/s = s–1. Assume that the number “950” has three significant figures.

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7-6

Solution:

c = ν

c

(m) = =

3.00x108 m/s

= 315.789 = 316 m

103 Hz s1

950. kHz

1 kHz Hz

1 nm

11

11

= 315.789 m 9

= 3.15789x10 = 3.16x10 nm

10

m

c

(nm) =

(Å) =

7.8

1Å

= 315.789 m 10 = 3.158x1012 = 3.16x1012 Å

10 m

c

Wavelength and frequency relate through the equation c = . Recall that a Hz is a reciprocal second, or 1/s = s–1.

(m) =

c

(nm) =

(Å) =

7.9

=

3.00x108 m/s

= 3.208556 = 3.21 m

106 Hz s1

93.5 MHz

1 MHz Hz

1 nm

9

9

= 3.208556 m 9

= 3.208556x10 = 3.21x10 nm

10 m

c

1Å

= 3.208556 m 10 = 3.208556x1010 = 3.21x1010 Å

10 m

c

Plan: Frequency is related to energy through the equation E = h. Note that 1 Hz = 1 s–1.

Solution:

E = h

E = (6.626x10–34 J•s)(3.8x1010 s–1) = 2.51788x10–23 = 2.5x10–23 J

hc

6.626x10

=

34

J•s 3.00x108 m/s 1 Å

–15

–15

10 = 1.5291x10 = 1.5x10 J

1.3 Å

10

m

7.10

E=

7.11

Plan: Energy is inversely proportional to wavelength ( E =

hc

). As wavelength decreases, energy increases.

Solution:

In terms of increasing energy the order is red < yellow < blue.

7.12

Since energy is directly proportional to frequency (E = h): UV ( = 8.0x1015 s–1) > IR ( = 6.5x1013 s–1) >

microwave ( = 9.8x1011 s–1) or UV > IR > microwave.

7.13

Plan: Wavelength is related to frequency through the equation c = ν. Recall that a Hz is a reciprocal second,

or 1/s = s–1.

Solution:

109 Hz s 1

10 –1

= (s–1) = 22.235 GHz

= 2.2235x10 s

1

GHz

Hz

(nm) =

c

=

2.9979x108 m/s 1 nm

7

7

= 1.3482797x10 = 1.3483x10 nm

9

10 1

2.2235x10 s 10 m

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7-7

c

(Å) =

7.14

=

2.9979x108 m/s 1 Å

8

8

= 1.3482797x10 = 1.3483x10 Å

2.2235x1010 s1 1010 m

Frequency and wavelength can be calculated using the speed of light: c = ν.

3.00x108 m/s 1 m

c

13

13 –1

a) = =

= 3.125x10 = 3.1x10 s

9.6 m 106 m

1 m

c

2.9979x108 m/s

b) (m) =

=

= 3.464979 = 3.465 m

1

s 106 m

13

8.652x10 Hz

Hz

7.15

Frequency and energy are related by E = h, and wavelength and energy are related by E = hc/.

106 eV 1.602x1019 J

1.33 MeV

1 MeV

1 eV

E

Hz = 3.2156x1020 = 3.22x1020 Hz

(Hz) =

=

1

34

h

6.626x10 J•s

s

hc

(m) =

=

E

6.626x10

34

J•s 3.00x108 m/s

= 9.32950x10–13 = 9.33x10–13 m

10 eV 1.602x10 J

1.33 MeV

1 eV

1 MeV

The wavelength can also be found using the frequency calculated in the equation c =

7.16

19

6

Plan: The least energetic photon in part a) has the longest wavelength (242 nm). The most energetic photon in

part b) has the shortest wavelength (2200 Å). Use the relationship c = to find the frequency of the photons and

hc

to find the energy.

relationship E =

Solution:

a) c =

=

E=

c

hc

b) =

E=

=

c

hc

3.00x108 m/s 1 nm

15

15 –1

= 1.239669x10 = 1.24x10 s

242 nm 109 m

=

=

=

6.626x10

34

J•s 3.00x108 m/s 1 nm

–19

–19

9 = 8.2140x10 = 8.21x10 J

242 nm

10 m

3.00x108 m/s 1 Å

15

15 –1

= 1.3636x10 = 1.4x10 s

2200 Å 1010 m

6.626x10

34

J•s 3.00x108 m/s 1 Å

–19

–19

10 = 9.03545x10 = 9.0x10 J

2200 Å

10

m

7.17

“n” in the Rydberg equation is equal to a Bohr orbit of quantum number “n” where n = 1, 2, 3, ...

7.18

Bohr’s key assumption was that the electron in an atom does not radiate energy while in a stationary state, and the

electron can move to a different orbit by absorbing or emitting a photon whose energy is equal to the difference in

energy between two states. These differences in energy correspond to the wavelengths in the known spectra for

the hydrogen atoms. A Solar System model does not allow for the movement of electrons between levels.

7.19

An absorption spectrum is produced when atoms absorb certain wavelengths of incoming light as electrons move

from lower to higher energy levels and results in dark lines against a bright background. An emission spectrum is

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7-8

produced when atoms that have been excited to higher energy emit photons as their electrons return to lower

energy levels and results in colored lines against a dark background. Bohr worked with emission spectra.

7.20

Plan: The quantum number n is related to the energy level of the electron. An electron absorbs energy to

change from lower energy (lower n) to higher energy (higher n), giving an absorption spectrum. An electron emits

energy as it drops from a higher energy level (higher n) to a lower one (lower n), giving an emission spectrum.

Solution:

a) The electron is moving from a lower value of n (2) to a higher value of n (4): absorption

b) The electron is moving from a higher value of n (3) to a lower value of n (1): emission

c) The electron is moving from a higher value of n (5) to a lower value of n (2):emission

d) The electron is moving from a lower value of n (3) to a higher value of n (4): absorption

7.21

The Bohr model works only for a one-electron system. The additional attractions and repulsions in many-electron

systems make it impossible to predict accurately the spectral lines.

7.22

The Bohr model has successfully predicted the line spectra for the H atom and Be3+ ion since both are oneelectron species. The energies could be predicted from En =

Z 2 2.18x1018 J

where Z is the atomic number

n

for the atom or ion. The line spectra for H would not match the line spectra for Be3+ since the H nucleus contains

one proton while the Be3+ nucleus contains 4 protons (the Z values in the equation do not match); the force of

attraction of the nucleus for the electron would be greater in the beryllium ion than in the hydrogen atom. This

means that the pattern of lines would be similar, but at different wavelengths.

7.23

Plan: Calculate wavelength by substituting the given values into Equation 7.3, where n1 = 2 and n2 = 5 because

n2 > n1. Although more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

R = 1.096776x107 m–1

R 2 2

n

n

2

1

n1 = 2 n2 = 5

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 2,303,229.6 m–1

5

n2

2

n1

1 nm

1

(nm) =

= 434.1729544 = 434.17 nm

1

9

2,303, 229.6 m 10 m

7.24

Calculate wavelength by substituting the given values into the Rydberg equation, where n1 = 1 and n2 = 3 because

n2 > n1. Although more significant figures could be used, five significant figures are adequate for this calculation.

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 9,749,120 m–1

n

1

3

n

2

1

1Å

1

(Å) =

= 1025.7336 = 1025.7 Å

1

10

9,749,120 m 10 m

7.25

2

Plan: The Rydberg equation is needed. For the infrared series of the H atom, n1 equals 3. The least energetic

spectral line in this series would represent an electron moving from the next highest energy level, n2 = 4. Although

more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 533,155 m–1

4

n2

3

n1

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7-9

1 nm

1

(nm) =

= 1875.627 = 1875.6 nm

1

9

533,155 m 10 m

7.26

Plan: The Rydberg equation is needed. For the visible series of the H atom, n1 equals 2. The least energetic

spectral line in this series would represent an electron moving from the next highest energy level, n = 3. Although

more significant figures could be used, five significant figures are adequate for this calculation.

Solution:

1

1

1

1

1

R 2 2 = 1.096776x107 m 1 2 2 = 1,523,300 m–1

2

3

n

n

2

1

1 nm

1

(nm) =

= 656.4695 = 656.47 nm

1

9

1,523,300 m 10 m

7.27

7.28

7.29

Plan: To find the transition energy, use the equation for the energy of an electron transition and multiply by

Avogadro’s number to convert to energy per mole.

Solution:

1

1

E = 2.18x10 18 J 2

2

n

ninitial

final

1

1

E = 2.18x10 18 J 2 2 = –4.578x10–19 J/photon

5

2

19

4.578x10

J 6.022x1023 photons

5

5

E =

= –2.75687x10 = –2.76x10 J/mol

photon

1

mol

The energy has a negative value since this electron transition to a lower n value is an emission of energy.

To find the transition energy, use the equation for the energy of an electron transition and multiply by Avogadro’s

number.

1

1

E = 2.18x10 18 J 2

2

n

ninitial

final

1

1

E = 2.18x10 18 J 2 2 = 1.93778x10–18 J/photon

1

3

1.93778x10 18 J 6.022x1023 photons

6

6

E =

= 1.1669x10 = 1.17x10 J/mol

photon

1

mol

Plan: Determine the relative energy of the electron transitions. Remember that energy is directly proportional

to frequency (E = h).

Solution:

Looking at an energy chart will help answer this question.

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7-10

n=5 n=4

n=3

(d)

(a)

(c)

n=2

(b)

n=1

7.30

7.31

Frequency is proportional to energy so the smallest frequency will be d) n = 4 to n = 3; levels 3 and 4 have a

smaller E than the levels in the other transitions. The largest frequency is b) n = 2 to n = 1 since levels 1 and

2 have a larger E than the levels in the other transitions. Transition a) n = 2 to n = 4 will be smaller than

transition c) n = 2 to n = 5 since level 5 is a higher energy than level 4. In order of increasing frequency the

transitions are d < a < c < b.

b>c>a>d

Plan: Use the Rydberg equation. Since the electron is in the ground state (lowest energy level), n1 = 1. Convert

the wavelength from nm to units of meters.

Solution:

10 9 m

–8

= 97.20 nm

ground state: n1 = 1; n2 = ?

= 9.720x10 m

1

nm

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

= 1.096776x107 m 1

9.72x108 m

1

1

0.93803 = 2 2

1

n2

1

n22

11

2

1

n22

= 1 – 0.93803 = 0.06197

n 22 = 16.14

n2 = 4

7.32

109 m

–6

= 1.281x10 m

1

nm

= 1281 nm

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

1

1

= 1.096776x107 m 1 2 2

6

1.281x10 m

5

n1

1

1

0.07118 = 2 2

n

5

1

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7-11

1

n12

= 0.07118 + 0.04000 = 0.11118

n12 = 8.9944

n1 = 3

hc

6.626x10

34

J•s 3.00x108 m/s 1 nm

–19

–19

9 = 4.55917x10 = 4.56x10 J

436 nm

10

m

7.33

E=

7.34

a) Absorptions: A, C, D; Emissions: B, E, F

b) Energy of emissions: E < F < B

c) Wavelength of absorption: D < A < C

7.35

If an electron occupies a circular orbit, only integral numbers of wavelengths (= 2nr) are allowed for acceptable

standing waves. A wave with a fractional number of wavelengths is forbidden due to destructive interference with

itself. In a musical analogy to electron waves, the only acceptable guitar string wavelengths are those that are an

integral multiple of twice the guitar string length (2 L).

7.36

De Broglie’s concept is supported by the diffraction properties of electrons demonstrated in an electron

microscope.

7.37

Macroscopic objects have significant mass. A large m in the denominator of = h/mu will result in a very small

wavelength. Macroscopic objects do exhibit a wavelike motion, but the wavelength is too small for humans to see

it.

7.38

The Heisenberg uncertainty principle states that there is fundamental limit to the accuracy of measurements. This

limit is not dependent on the precision of the measuring instruments, but is inherent in nature.

7.39

Plan: Use the de Broglie equation. Mass in lb must be converted to kg and velocity in mi/h must

be converted to m/s because a joule is equivalent to kg•m2/s2.

Solution:

1 kg

a) Mass (kg) = 232 lb

= 105.2154 kg

2.205 lb

=

3

19.8 mi 1 km 10 m 1 h

Velocity (m/s) =

= 8.87097 m/s

h 0.62 mi 1 km 3600 s

=

h

=

mu

6.626x10

34

J•s

kg•m2 /s2

–37

–37

= 7.099063x10 = 7.10x10 m

m

J

105.2154 kg 8.87097

s

3

0.1 mi 1 km 10 m 1 h

b) Uncertainty in velocity (m/s) =

= 0.0448029 m/s

h 0.62 mi 1 km 3600 s

x•mv

x

h

4

h

4 mv

6.626x10

34

J•s

kg•m2 /s2

–35

–35

1.11855x10 1x10 m

J

0.0448029 m

4 105.2154 kg

s

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7-12

7.40

a) =

h

=

mu

6.626x10 J•s kg•m /s

6.6x10 g 3.4x10 mih J

34

2

24

2

7

103 g 0.62 mi 1 km 3600 s

3

1 kg 1 km 10 m 1 h

= 6.59057x10–15 = 6.6x10–15 m

h

b) x•mv

4

6.626x10 J•s

0.1x10

4 6.6x10 g

h

34

kg•m2 /s2 103 g 0.62 mi 1 km 3600 s

3

7

J

mi

24

1 kg 1 km 10 m 1 h

–14

–14

1.783166x10 2x10 m

h

x

4 mv

7.41

Plan: Use the de Broglie equation. Mass in g must be converted to kg and wavelength in Ǻ must be converted to

m because a joule is equivalent to kg•m2/s2.

Solution:

1 kg

Mass (kg) = 56.5 g 3 = 0.0565 kg

10 g

10 10 m

= 5.4x10–7 m

Wavelength (m) = 5400 Å

1 Å

h

=

mu

6.626x10 34 J•s

h

u=

=

m

0.0565 kg 5.4x107 m

7.42

=

kg•m 2 /s 2

J

h

mu

6.626x1034 J•s kg•m 2 /s2

h

u=

=

m

142 g 100. pm J

7.43

–26

–26

= 2.1717x10 = 2.2x10 m/s

103 g 1 pm

–23

–23

12 = 4.666197x10 = 4.67x10 m/s

1

kg

10 m

Plan: The de Broglie wavelength equation will give the mass equivalent of a photon with known wavelength and

velocity. The term “mass equivalent” is used instead of “mass of photon” because photons are quanta of

electromagnetic energy that have no mass. A light photon’s velocity is the speed of light, 3.00x108 m/s.

Wavelength in nm must be converted to m.

Solution:

109 m

Wavelength (m) = 589 nm

= 5.89x10–7 m

1 nm

h

=

mu

6.626x10 34 J•s

h

m=

=

u

5.89x10 7 m 3.00 x108 m /s

kg•m 2 /s 2

J

–36

–36

= 3.7499x10 = 3.75x10 kg/photon

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7-13

7.44

=

h

mu

m=

6.626x10 34 J•s

kg•m 2 /s 2 1 nm

h

–36

=

9 = 3.2916x10 kg/photon

8

u

J

10

m

671 nm 3.00x10 m/s

3.2916x1036 kg 6.022x1023 photons

–12

–12

= 1.9822x10 = 1.98x10 kg/mol

photon

mol

7.45

The quantity 2 expresses the probability of finding an electron within a specified tiny region of space.

7.46

Since 2 is the probability of finding an electron within a small region or volume, electron density would

represent a probability per unit volume and would more accurately be called electron probability density.

7.47

A peak in the radial probability distribution at a certain distance means that the total probability of finding the

electron is greatest within a thin spherical volume having a radius very close to that distance. Since principal

quantum number (n) correlates with distance from the nucleus, the peak for n = 2 would occur at a greater

distance from the nucleus than 0.529 Å. Thus, the probability of finding an electron at 0.529 Å is much greater for

the 1s orbital than for the 2s.

7.48

a) Principal quantum number, n, relates to the size of the orbital. More specifically, it relates to the distance from

the nucleus at which the probability of finding an electron is greatest. This distance is determined by the energy of

the electron.

b) Angular momentum quantum number, l, relates to the shape of the orbital. It is also called the azimuthal

quantum number.

c) Magnetic quantum number, ml, relates to the orientation of the orbital in space in three-dimensional space.

7.49

Plan: The following letter designations correlate with the following l quantum numbers:

l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are – l to + l.

The number of orbitals of a particular type is given by the number of possible ml values.

Solution:

a) There is only a single s orbital in any shell. l = 1 and ml = 0: one value of ml = one s orbital.

b) There are five d orbitals in any shell. l = 2 and ml = –2, –1, 0, +1, +2. Five values of ml = five d orbitals.

c) There are three p orbitals in any shell. l = 1 and ml = –1, 0, +1. Three values of ml = three p orbitals.

d) If n = 3, l = 0(s), 1(p), and 2(d). There is a 3s (1 orbital), a 3p set (3 orbitals), and a 3d set (5 orbitals) for a

total of nine orbitals (1 + 3 + 5 = 9).

7.50

a) All f orbitals consist of sets of seven (l = 3 and ml = –3, –2, –1, 0, +1, +2, +3).

b) All p orbitals consist of sets of three (l = 1 and ml = –1, 0, +1).

c) All d orbitals consist of sets of five (l = 2 and ml = –2, –1, 0, +1, +2).

d) If n = 2, then there is a 2s (1 orbital) and a 2p set (3 orbitals) for a total of four orbitals (1 + 3 = 4).

7.51

Plan: Magnetic quantum numbers (ml) can have integer values from –l to + l. The l quantum number can

have integer values from 0 to n – 1.

Solution:

a) l = 2 so ml = –2, –1, 0, +1, +2

b) n = 1 so l = 1 – 1 = 0 and ml = 0

c) l = 3 so ml = –3, –2, –1, 0, +1, +2, +3

7.52

Magnetic quantum numbers can have integer values from –l to +l. The l quantum number can

have integer values from 0 to n – 1.

a) l = 3 so ml = –3, –2, –1, 0, +1, +2, +3

b) n = 2 so l = 0 or 1; for l = 0, ml = 0; for l = l, ml = –1,0,+1

c) l = 1 so ml = –1, 0, +1

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7-14

7.53

Plan: The s orbital is spherical; p orbitals have two lobes; the subscript x indicates that this orbital lies along

the x-axis.

Solution:

a) s: spherical

b) px: 2 lobes along the x-axis

z

z

x

x

y

y

The variations in coloring of the p orbital are a consequence of the quantum mechanical derivation of atomic

orbitals that are beyond the scope of this course.

7.54

a) pz: 2 lobes along the z-axis

z

y

b) dxy: 4 lobes

y

x

x

The variations in coloring of the p and d orbitals are a consequence of the quantum mechanical derivation of

atomic orbitals that are beyond the scope of this course.

7.55

Plan: The following letter designations for the various sublevels (orbitals) correlate with the following l quantum

numbers: l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are

– l to + l. The number of orbitals of a particular type is given by the number of possible ml values.

Solution:

sublevel

allowable ml

# of possible orbitals

a) d (l = 2)

–2, –1, 0, +1, +2

5

b) p (l = 1)

–1, 0, +1

3

c) f (l = 3)

–3, –2, –1, 0, +1, +2, +3

7

7.56

sublevel

a) s (l = 0)

b) d (l = 2)

c) p (l = 1)

7.57

Plan: The integer in front of the letter represents the n value. The letter designates the l value:

l = 0 = s orbital; l = 1 = p orbital; l = 2 = d orbital; l = 3 = f orbital. Remember that allowed ml values are – l to + l.

Solution:

a) For the 5s subshell, n = 5 and l = 0. Since ml = 0, there is one orbital.

allowable ml

0

–2, –1, 0, +1, +2

–1, 0, +1

# of possible orbitals

1

5

3

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7-15

b) For the 3p subshell, n = 3 and l = 1. Since ml = –1, 0, +1, there are three orbitals.

c) For the 4f subshell, n = 4 and l = 3. Since ml = –3, –2, –1, 0, +1, +2, +3, there are seven orbitals.

7.58

a) n = 6; l = 4; 9 orbitals (ml = –4, –3, –2, –1, 0, +1, +2, +3, +4)

b) n = 4; l = 0; 1 orbital (ml = 0)

c) n = 3; l = 2; 5 orbitals (ml = –2, –1, 0, +1, +2)

7.59

Plan: Allowed values of quantum numbers: n = positive integers; l = integers from 0 to n – 1;

ml = integers from – l through 0 to + l.

Solution:

a) n = 2; l = 0; ml = –1: With n = 2, l can be 0 or 1; with l = 0, the only allowable ml value is 0. This

combination is not allowed. To correct, either change the l or ml value.

Correct: n = 2; l = 1; ml = –1 or n = 2; l = 0; ml = 0.

b) n = 4; l = 3; ml = –1: With n = 4, l can be 0, 1, 2, or 3; with l = 3, the allowable ml values are –3, –2, –1, 0, +1,

+2, +3. Combination is allowed.

c) n = 3; l = 1; ml = 0: With n = 3, l can be 0, 1, or 2; with l = 1, the allowable ml values are –1, 0, +1.

Combination is allowed.

d) n = 5; l = 2; ml = +3: With n = 5, l can be 0, 1, 2, 3, or 4; with l = 2, the allowable ml values are –2, –1, 0, +1,

+2. +3 is not an allowable ml value. To correct, either change l or ml value.

Correct: n = 5; l = 3; ml = +3 or n = 5; l = 2; ml = 0.

7.60

a) Combination is allowed.

b) No; n = 2; l = 1; ml = +1 or n = 2; l = 1; ml = 0

c) No; n = 7; l = 1; ml = +1 or n = 7; l = 3; ml = 0

d) No; n = 3; l = 1; ml = –1 or n = 3; l = 2; ml = –2

7.61

Determine the max for -carotene by measuring its absorbance in the 610-640 nm region of the visible spectrum.

Prepare a series of solutions of -carotene of accurately known concentration (using benzene or chloroform as

solvent), and measure the absorbance for each solution. Prepare a graph of absorbance versus concentration for these

solutions and determine its slope (assuming that this material obeys Beer’s law). Measure the absorbance of the

oil expressed from orange peel (diluting with solvent if necessary). The -carotene concentration can then either

be read directly from the calibration curve or calculated from the slope (A = kC, where k = slope of the line and

C = concentration).

7.62

Plan: For Ppart a, use the values of the constants h, π, me, and a0 to find the overall constant in the equation.

hc

Use the resulting equation to calculate E in part b). Use the relationship E =

to calculate the wavelength in

part c). Remember that a joule is equivalent to kg•m2/s2.

Solution:

a) h = 6.626x10–34 J•s; me = 9.1094x10–31 kg; a0 = 52.92x10–12 m

E=

E=

h2

8

2

me a02 n2

=

h2

8

2

me a02

6.626x10

34

1

2

n

J•s

2

8 2 9.1094x10 31 kg 52.92x10 12 m

2

kg•m 2 /s 2

J

1

2

n

1

1

= –(2.17963x10–18 J) 2 = –(2.180x10–18 J) 2

n

n

This is identical with the result from Bohr’s theory. For the H atom, Z = 1 and Bohr’s constant = –2.18x10–18 J.

For the hydrogen atom, derivation using classical principles or quantum-mechanical principles yields the same

constant.

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7-16

b) The n = 3 energy level is higher in energy than the n = 2 level. Because the zero point of the atom’s energy is

defined as an electron’s infinite distance from the nucleus, a larger negative number describes a lower energy

level. Although this may be confusing, it makes sense that an energy change would be a positive number.

1

1

E = –(2.180x10–18 J) 2 2 = –3.027778x10–19 = 3.028x10–19 J

3

2

hc

c) E =

(m) =

6.626x10 34 J•s 2.9979x108 m/s

hc

=

= 6.56061x10–7 = 6.561x10–7 m

E

3.027778x10 19 J

1 nm

= 656.061 = 656.1 nm

9

10 m

This is the wavelength for the observed red line in the hydrogen spectrum.

(nm) = 6.56061x107 m

7.63

Plan: When light of sufficient frequency (energy) shines on metal, electrons in the metal break free and a current

flows.

Solution:

a) The lines do not begin at the origin because an electron must absorb a minimum amount of energy before it has

enough energy to overcome the attraction of the nucleus and leave the atom. This minimum energy is the energy

of photons of light at the threshold frequency.

b) The lines for K and Ag do not begin at the same point. The amount of energy that an electron must absorb to

leave the K atom is less than the amount of energy that an electron must absorb to leave the Ag atom, where the

attraction between the nucleus and outer electron is stronger than in a K atom.

c) Wavelength is inversely proportional to energy. Thus, the metal that requires a larger amount of energy to be

absorbed before electrons are emitted will require a shorter wavelength of light. Electrons in Ag atoms require

more energy to leave, so Ag requires a shorter wavelength of light than K to eject an electron.

d) The slopes of the line show an increase in kinetic energy as the frequency (or energy) of light is increased.

Since the slopes are the same, this means that for an increase of one unit of frequency (or energy) of light, the

increase in kinetic energy of an electron ejected from K is the same as the increase in the kinetic energy of an

electron ejected from Ag. After an electron is ejected, the energy that it absorbs above the threshold energy

becomes the kinetic energy of the electron. For the same increase in energy above the threshold energy, for either

K or Ag, the kinetic energy of the ejected electron will be the same.

7.64

a) E =

6.626x10

34

J•s)(3.00x108 m/s 1 nm

–19

9 = 2.8397x10 J

700. nm

10

m

This is the value for each photon, that is, J/photon.

1 photon

= 70.430 = 70. photons

Number of photons = 2.0x10 17 J

19

J

2.8397x10

hc

=

6.626x10

=

34

J•s)(3.00x108 m/s 1nm

–19

b) E =

9 = 4.18484x10 J

475. nm

10 m

This is the value for each photon, that is, J/photon.

hc

1 photon

Number of photons = 2.0x10 17 J

= 47.7916 = 48 photons

19

J

4.18484x10

7.65

Determine the wavelength:

= 1/(1953 cm–1) = 5.1203277x10–4 cm

10 2 m 1 nm

3

(nm) = 5.1203277x104 cm

9 = 5120.3277 = 5.120x10 nm

1

cm

10

m

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7-17

10 2 m 1 Å

4

10 = 51203.277 = 5.120x10 Å

1

cm

10 m

(Å) = 5.1203277x104 cm

= c/=

7.66

2.9979x108 m/s 1 cm 1 Hz

13

13

= 5.8548987x10 = 5.855x10 Hz

5.1203277x104 cm 102 m 1 s1

Plan: The Bohr model has been successfully applied to predict the spectral lines for one-electron species other

than H. Common one-electron species are small cations with all but one electron removed. Since the problem

specifies a metal ion, assume that the possible choices are Li2+ or Be3+. Use the relationship E = h to convert the

Z2

frequency to energy and then solve Bohr’s equation E = 2.18x1018 J 2 to verify if a whole number for Z

n

can be calculated. Recall that the negative sign is a convention based on the zero point of the atom’s energy; it is

deleted in this calculation to avoid taking the square root of a negative number.

Solution:

The highest energy line corresponds to the transition from n = 1 to n = .

E = h = (6.626x10–34 J•s) (2.961x1016 Hz) (s–1/Hz) = 1.9619586x10–17 J

Z2

E = 2.18x1018 J 2

Z = charge of the nucleus

n

1.9619586x1017 (12 )

En2

=

= 8.99998

2.18x1018 J

2.18x10 18 J

Then Z2 = 9 and Z = 3.

Therefore, the ion is Li2+ with an atomic number of 3.

Z2 =

7.67

6.626 x10 J•s kg•m /s = 2.139214x10 = 2.1x10

9.11x10 kg 3.4x10 ms J

6.626x10 J•s kg•m /s = 1.16696x10 = 1.2x10 m

=

1.67x10 kg 3.4x10 ms J

34

h

=

a) Electron: =

mu

h

Proton: =

mu

b) E = 1/2mu2

u=

2

31

2

–10

–10

m

6

34

2

27

2

–13

–13

6

therefore u2 = 2E/m

2E

m

Electron: u =

2 2.7x1015 J kg•m 2 /s2

J

9.11x1031 kg

7

= 7.6991x10 m/s

2 2.7x10 J kg•m /s

Proton: u =

= 1.7982x10 m/s

1.67x10 kg J

kg•m /s

6.626x10 J•s

h

=

Electron: =

= 9.44698x10

m

J

mu

9.11x10

kg

7.6991x10

s

kg•m /s

6.626x10 J•s

h

Proton: =

=

= 2.20646 x 10

m

J

mu

1.67x10 kg 1.7982x10 s

15

2

2

6

27

34

2

31

–12

= 9.4x10–12 m

7

34

27

2

2

2

–13

= 2.2 x 10–13 m

6

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7-18

7.68

Plan: The electromagnetic spectrum shows that the visible region goes from 400 to 750 nm (4000 Å to 7500 Å).

Thus, wavelengths b, c, and d are for the three transitions in the visible series with nfinal = 2. Wavelength a is in the

ultraviolet region of the spectrum and the ultraviolet series has nfinal = 1. Wavelength e is in the infrared region of

the spectrum and the infrared series has nfinal = 3. Use the Rydberg equation to find the ninitial for each line. Convert

the wavelengths from Å to units of m.

Solution:

n = ? n = 1; = 1212.7 Å (shortest corresponds to the largest E)

10 10 m

–7

(m) = 1212.7

= 1.2127x10 m

1

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

7 1

= 1.096776x10 m

7

1.2127x10 m

1

1

0.7518456 = 2 2

1

n2

11

2

1

n22

1

2 = 1 – 0.7518456

n2

1

2 = 0.2481544

n2

n2 = 2 for line (a) (n = 2 n = 1)

n = ? n = 3; = 10,938 Å (longest corresponds to the smallest E)

10 10 m

–6

(m) = 10,938

= 1.0938x10 m

1

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

1

7 1

= 1.096776x10 m

6

1.0938x10 m

1

1

0.083357397 = 2 2

3

n2

31

2

1

n22

0.083357397 = 0.111111111

1

n22

1

2

n2

= 0.11111111 – 0.083357396

1

2

n2

= 0.0277537151

n 22 =36.03121

n2 = 6 for line (e) (n = 6 n = 3)

For the other three lines, n1 = 2.

For line (d), n2 = 3 (largest smallest E).

For line (b), n2 = 5 (smallest largest E).

For line (c), n2 = 4.

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7-19

7.69

E=

hc

a) (nm) =

thus =

hc

E

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

=

9 = 432.130 = 432 nm

E

4.60x10 19 J

10 m

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

b) (nm) =

=

9 = 286.4265 = 286 nm

E

6.94x1019 J

10 m

c) (nm) =

6.626x1034 J•s 3.00x108 m/s 1 nm

hc

=

9 = 450.748 = 451 nm

E

4.41x1019 J

10 m

7.70

Index of refraction = c/v; v = c/(index of refraction)

a) Water

v = c/(index of refraction) = (3.00x108 m/s)/(1.33) = 2.2556x108 = 2.26x108 m/s

b) Diamond

v = c/(index of refraction) = (3.00x108 m/s)/(2.42) = 1.239669x108 = 1.24x108 m/s

7.71

Extra significant figures are necessary because of the data presented in the problem.

He–Ne = 632.8 nm

Ar

= 6.148x1014 s–1

Ar–Kr E = 3.499x10–19 J

= 663.7 nm

Dye

Calculating missing values:

Ar

= c/ = (2.9979x108 m/s)/(6.148x1014 s–1) = 4.8762199x 10–7 = 4.876x10–7 m

Ar–Kr = hc/E = (6.626x10–34 J•s) (2.9979x108 m/s)/(3.499x10–19 J) = 5.67707x10–7 = 5.677x10–7 m

Calculating missing values:

He–Ne = c/ = (2.9979x108 m/s)/[632.8 nm (10–9 m/nm)] = 4.7375 x 1014 = 4.738 x 1014 s–1

Ar–Kr = E/h = (3.499x10–19 J)/(6.626x10–34 J•s) = 5.28071x1014 = 5.281x1014 s–1

Dye

= c/ = (2.9979x108 m/s)/[663.7 nm (10–9 m/nm)] = 4.51695x1014 = 4.517x1014 s–1

Calculating missing E values:

He–Ne E = hc/ = [(6.626x10–34 J•s)(2.9979x108 m/s)]/[632.8 nm (10–9 m/nm)]

= 3.13907797x10–19 = 3.139x10–19 J

E = h = (6.626x10–34 J•s)(6.148x1014 s–1) = 4.0736648x10–19 = 4.074x10–19 J

Ar

Dye

E = hc/ = [(6.626x10–34 J•s)(2.9979x108 m/s)]/[663.7 nm (10–9 m/nm)]

= 2.99293x10–19 = 2.993x10–19 J

The colors may be predicted from Figure 7.3 and the frequencies.

Orange

He–Ne = 4.738x1014 s–1

Ar

= 6.148x1014 s–1

Green

Ar–Kr = 5.281x1014 s–1

Yellow

Dye

= 4.517x1014 s–1

Red

7.72

Plan: Allowed values of quantum numbers: n = positive integers; l = integers from 0 to n – 1;

ml = integers from – l through 0 to + l.

Solution:

a) The l value must be at least 1 for ml to be – 1, but cannot be greater than n – 1 = 3 –1 = 2. Increase the l value to

1 or 2 to create an allowable combination.

b) The l value must be at least 1 for ml to be +1, but cannot be greater than n – 1 = 3 – 1 = 2. Decrease the l value

to 1 or 2 to create an allowable combination.

c) The l value must be at least 3 for ml to be +3, but cannot be greater than n – 1 = 7 – 1 = 6. Increase the l value to

3, 4, 5, or 6 to create an allowable combination.

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7-20

d) The l value must be at least 2 for ml to be –2, but cannot be greater than n – 1 = 4 – 1 = 3. Increase the l value to

2 or 3 to create an allowable combination.

7.73

1

1

= 1.096776x107 m 1 2 2

n2

n1

1

a)

1

1 nm

= 1.096776x107 m 1

94.91 nm 109 m

1

1

0.9606608 = 2 2

1

n2

n2 = 5

1 nm

1

7 1

b)

= 1.096776x10 m

1281 nm 10 9 m

11

2

n1

2

1

1

n22

1

52

1

1

0.071175894 = 2 2

n

5

1

n1 = 3

1

1 1

c)

= 1.096776x107 m 1 2 2

1 3

1

= 9.74912x106 m–1

10

1 nm

= 1.02573x107 m 9 = 102.573 = 102.6 nm

7.74

m

Plan: Ionization occurs when the electron is completely removed from the atom, or when nfinal = . We can

use the equation for the energy of an electron transition to find the quantity of energy needed to remove

completely the electron, called the ionization energy (IE). To obtain the ionization energy per mole of species,

multiply by Avogadro’s number. The charge on the nucleus must affect the IE because a larger nucleus would

exert a greater pull on the escaping electron. The Bohr equation applies to H and other one-electron species. Use

the expression to determine the ionization energy of B4+. Then use the expression to find the energies of the

hc

transitions listed and use E =

to convert energy to wavelength.

Solution:

Z2

a) E = 2.18x1018 J 2

n

Z = atomic number

1

1

E = 2.18x1018 J 2

2

n

ninitial

final

2

Z

1

1 2 6.022x1023

E = 2.18x1018 J 2 2

Z

ninitial 1 mol

= (1.312796x106) Z2 for n = 1

b) In the ground state n = 1, the initial energy level for the single electron in B4+. Once ionized, n = is the final

energy level.

Z = 5 for B4+.

E = IE = (1.312796x106) Z2 = (1.312796x106 J/mol)(52) = 3.28199x107 = 3.28x107 J/mol

c) nfinal = , ninitial = 3, and Z = 2 for He+.

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7-21

1

1

E = 2.18x10 18 J 2 2

n

final ninitial

hc

E=

2

18

Z = 2.18x10 J

1

2

1

32

2

–19

2 = 9.68889x10 J

6.626x1034 J•s 3.00x108 m/s

hc

=

= 2.051628x10–7 m

E

9.68889x1019 J

1 nm

(nm) = 2.051628x107 m 9 = 205.1628 = 205 nm

10 m

d) nfinal = , ninitial = 2, and Z = 4 for Be3+.

1

1

1 2

1

E = 2.18x10 18 J 2 2

Z = 2.18x10 18 J 2 2 4 2 = 8.72x10–18 J

2

ninitial

(m) =

6.626x10

=

hc

(m) =

E

34

J•s 3.00x108 m/s

8.72x10

18

J

= 2.279587x10

–8

m

1 nm

(nm) = 2.279587x108 m 9 = 22.79587 = 22.8 nm

10 m

7.75

a) Orbital D has the largest value of n, given that it is the largest orbital.

b) l = 1 indicates a p orbital. Orbitals A and C are p orbitals. l = 2 indicates a d orbital. Orbitals B and D are d

orbitals.

c) In an atom, there would be four other orbitals with the same value of n and the same shape as orbital B.

There would be two other orbitals with the same value of n and the same shape as orbital C.

d) Orbital D has the highest energy and orbital C has the lowest energy.

7.76

Plan: Use the values and the equation given in the problem to calculate the appropriate values.

Solution:

n2 h2 0

a) rn =

me e2

2

C2

12 6.626x1034 J•s 8.854x1012

J•m kg•m 2 /s2

r1 =

2

J

9.109x1031 kg 1.602x1019 C

–11

–11

= 5.2929377x10 = 5.293x10 m

2

C2

102 6.626x1034 J•s 8.854x1012

J•m kg•m 2 /s2

–9

–9

b) r10 =

= 5.2929377x10 = 5.293x10 m

2

31

19

J

9.109x10 kg 1.602x10 C

7.77

a) rn =

n2 h2 0

me e2

2

C2

32 6.626x1034 J•s 8.854x1012

J•m kg•m2 /s2

–10

–10

r3 =

= 4.76364x10 = 4.764x10 m

2

31

19

J

9.109x10 kg 1.602x10 C

b) Z = 1 for an H atom

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7-22

Z2

En = 2.18x1018 J 2

n

c) Z = 3 for a Li atom

18

= 2.18x10 J

2

13

2

–19

–19

= – 2.42222x10 = – 2.42x10 J

Z2

32

En = 2.18x1018 J 2 = 2.18x1018 J 2 = – 2.18x10–1

n

3

d) The greater number of protons in the Li nucleus results in a greater interaction between the Li nucleus and its

electrons. Thus, the energy of an electron in a particular orbital becomes more negative with increasing atomic

number.

7.78

Plan: Refer to Chapter 6 for the calculation of the amount of heat energy absorbed by a substance from its

specific heat capacity and temperature change (q = c x mass x T). Using this equation, calculate the energy

absorbed by the water. This energy equals the energy from the microwave photons. The energy of each photon

can be calculated from its wavelength: E = hc/. Dividing the total energy by the energy of each photon gives the

number of photons absorbed by the water.

Solution:

q = c x mass x T

q = (4.184 J/g°C)(252 g)(98 – 20.)°C = 8.22407x104 J

E=

hc

6.626x10

=

34

J•s 3.00x108 m/s

1.55x10

2

m

7.79

= 1.28245x10

–23

J/photon

1 photon

= 6.41278x1027 = 6.4x1027 photons

Number of photons = 8.22407x104 J

23

1.28245x10 J

One sample calculation will be done using the equation in the book:

1 1

=

a0

For r = 50 pm:

r

3

2

e

r

a0

1

1

=

52.92 pm

3

2

e

r

a0

= 1.465532x10–3 e

r

a0

50

= 1.465532x10–3 e a0 = 1.465532x10–3 e 52.92 = 5.69724x10–4

2 = (5.69724x10–4)2 = 3.24585x10–7

4r22 = 4(50)2(3.24585x10–7) = 1.0197x10–2

r (pm)

(pm–3/2)

2 (pm–3)

4r22 (pm–1)

–3

–6

0

1.47x10

2.15x10

0

50

0.570x10–3

0.325x10–6

1.02x10–2

–3

–6

100

0.221x10

0.0491x10

0.616x10–2

–3

–6

200

0.0335x10

0.00112x10

0.0563x10–2

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7-23

The plots are similar to Figure 7.17A in the text.

7.80

Plan: In general, to test for overlap of the two series, compare the longest wavelength in the “n” series with the

shortest wavelength in the “n+1” series. The longest wavelength in any series corresponds to the transition

between the n1 level and the next level above it; the shortest wavelength corresponds to the transition between the n1

1

1

1

level and the n = level. Use the relationship = R 2 2 to calculate the wavelengths.

n

n2

1

Solution:

1

1

1

1

1

= R 2 2 = 1.096776x107 m 1 2 2

n2

n2

n1

n1

a) The overlap between the n1 = 1 series and the n1 = 2 series would occur between the longest wavelengths for

n1 = 1 and the shortest wavelengths for n1 = 2.

Longest wavelength in n1 = 1 series has n2 equal to 2.

1

1

1

= 1.096776x107 m 1 2 2 = 8,225,820 m–1

2

1

1

=

= 1.215684272x10–7 = 1.215684x10–7 m

1

8,225,820 m

Shortest wavelength in the n1 = 2 series:

1

1

1

–1

= 1.096776x107 m 1 2

= 2,741,940 m

2

2

1

=

= 3.647052817x10–7 = 3.647053x10–7 m

2,741,940 m 1

Since the longest wavelength for n1 = 1 series is shorter than shortest wavelength for n1 = 2 series, there

is no overlap between the two series.

b) The overlap between the n1 = 3 series and the n1 = 4 series would occur between the longest wavelengths for

n1 = 3 and the shortest wavelengths for n1 = 4.

Longest wavelength in n1 = 3 series has n2 equal to 4.

1

1

1

= 1.096776x107 m 1 2 2 = 533,155 m–1

4

3

1

=

= 1.875627163x10–6 = 1.875627x10–6 m

533,155 m 1

Shortest wavelength in n1 = 4 series has n2 = .

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7-24

1

1

1

–1

= 1.096776x107 m 1 2

= 685,485 m

2

4

1

=

= 1.458821127x10–6 = 1.458821x10–6 m

685,485 m 1

Since the n1 = 4 series shortest wavelength is shorter than the n1 = 3 series longest wavelength, the series do

overlap.

c) Shortest wavelength in n1 = 5 series has n2 = .

1

1

1

–1

= 1.096776x107 m 1 2

= 438,710.4 m

2

5

1

=

= 2.27940801x10–6 = 2.279408x10–6 m

438,710.4 m 1

Calculate the first few longest lines in the n1 = 4 series to determine if any overlap with the shortest wavelength

in the n1 = 5 series:

For n1 = 4, n2 = 5:

1

1

1

= 1.096776x107 m 1 2 2 = 246,774.6 m–1

4

5

1

–6

=

= 4.052281x10 m

246,774.6 m 1

For n1 = 4, n2 = 6:

1

1

1

= 1.096776x107 m 1 2 2 = 380,825 m–1

6

4

1

=

= = 2.625878x10–6 m

1

380,825 m

For n1 = 4, n2 = 7:

1

1

1

= 1.096776x107 m 1 2 2 = 461,653.2 m–1

7

4

1

=

= 2.166128x10–6 m

461,653.2 m 1

The wavelengths of the first two lines of the n1 = 4 series are longer than the shortest wavelength in the n1 = 5

series. Therefore, only the first two lines of the n1 = 4 series overlap the n1 = 5 series.

d) At longer wavelengths (i.e., lower energies), there is increasing overlap between the lines from different series

(i.e., with different n1 values). The hydrogen spectrum becomes more complex, since the lines begin to merge into a

more-or-less continuous band, and much more care is needed to interpret the information.

7.81

a) The highest frequency would correspond to the greatest energy difference. In this case, the greatest energy

difference would be between E3 and E1.

E = E3 – E1 = h = (– 15x10–19 J) – (– 20x10–19 J) = 5x10–19 J

= E/h = (5x10–19 J)/(6.626x10–34 J•s) = 7.54603x1014 = 8x1014 s–1

= c/ = (3.00x108 m/s)/(7.54603x1014 s–1) = 3.97560x10–7 = 4x10–7 m

b) The ionization energy (IE) is the same as the reverse of E1. Thus, the value of the IE is 20x10–19 J/atom.

IE = (20x10–19 J/atom)(1kJ/103 J)(6.022x1023 atoms/mol) = 1204.4 = 1.2x103 kJ/mol

c) The shortest wavelength would correspond to an electron moving from the n = 4 level to the highest level

available in the problem (n = 6).

E = E6 – E4 = hc/ = (– 2x10–19 J) – (– 11x10–19 J) = 9x10–19 J

= hc/E =

7.82

6.626x10

34

J•s 3.00x108 m /s 1 nm

2

9 = 220.867 = 2x10 nm

19

9x10 J

10 m

Plan: The energy differences sought may be determined by looking at the energy changes in steps. The

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7-25

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