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<strong>atom</strong>, (b) <strong>the</strong> <strong>volume</strong>- <strong>of</strong> <strong>the</strong> <strong>nucleus</strong>, <strong>and'</strong> (c) <strong>the</strong> <strong>percentage</strong> <strong>of</strong> <strong>the</strong><br />
<strong>volume</strong> <strong>of</strong> <strong>the</strong> <strong>atom</strong> that is occupied by <strong>the</strong> <strong>nucleus</strong>.<br />
2. In a Ru<strong>the</strong>rford scattering experiment a target <strong>nucleus</strong> has a diameter<br />
<strong>of</strong> lA X 10-14 m. The incoming Cl particle has a mass <strong>of</strong><br />
6.64 X 10-27 kg. What is <strong>the</strong> kinetic energy <strong>of</strong> an Cl particle that has<br />
a de Broglie wavelength equal to <strong>the</strong> diameter <strong>of</strong> <strong>the</strong> target <strong>nucleus</strong>?<br />
Ignore relativistic effects.<br />
3. The <strong>nucleus</strong> <strong>of</strong> a hydrogen <strong>atom</strong> is a single proton, which has a<br />
radius <strong>of</strong> about 1.0 X IO-IY m. The single electron in a hydrogen<br />
<strong>atom</strong> normally orbits <strong>the</strong> <strong>nucleus</strong> at a distance <strong>of</strong> 5.3 X 10-11 m.<br />
What is <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> density <strong>of</strong> <strong>the</strong> hydrogen <strong>nucleus</strong> to <strong>the</strong> density<br />
<strong>of</strong> <strong>the</strong> complete hydrogen <strong>atom</strong>?<br />
4. Review Conceptual Example I and use <strong>the</strong> information <strong>the</strong>rein as<br />
an aid in working this problem. Suppose you're building a scale<br />
model <strong>of</strong> <strong>the</strong> hydrogen <strong>atom</strong>, and <strong>the</strong> <strong>nucleus</strong> is represented by a ball<br />
<strong>of</strong> radius 3.2 cm (somewhat smaller than a baseball). How many<br />
miles away (I mi = 1.61 X 105 cm) should <strong>the</strong> electron be placed?<br />
* 5. ssm The <strong>nucleus</strong> <strong>of</strong> a copper <strong>atom</strong> contains 29 protons and has a<br />
radius <strong>of</strong> 4.8 X 10-15 m. How much work (in electron volts) is done<br />
by <strong>the</strong> electric force as a proton is brought from infinity, where it is<br />
at rest, to <strong>the</strong> "surface" <strong>of</strong> a copper <strong>nucleus</strong>?<br />
* 6. There are Z protons in <strong>the</strong> <strong>nucleus</strong> <strong>of</strong> an <strong>atom</strong>, where Z is <strong>the</strong><br />
<strong>atom</strong>ic number <strong>of</strong> <strong>the</strong> element. An Cl particle carries a charge <strong>of</strong><br />
+2e. In a scattering experiment, an Cl particle, heading directly toward<br />
a <strong>nucleus</strong> in a metal foil, will come to a halt when all <strong>the</strong> particle's<br />
kinetic energy is converted to electric potential energy. In such<br />
a situation, how close will an Cl particle with a kinetic energy <strong>of</strong><br />
5.0 X 10-13 J come to a gold <strong>nucleus</strong> (Z = 79)?<br />
Section 30.2 Line Spectra<br />
Section 30.3 The Bohr Model <strong>of</strong> <strong>the</strong> Hydrogen Atom<br />
7. ssm www Concept Simulation 30.1 at www.wiley.comlcollege/<br />
cutnell reviews <strong>the</strong> concepts on which <strong>the</strong> solution to this problem<br />
depends. The electron in a hydrogen <strong>atom</strong> is in <strong>the</strong> first excited state,<br />
when <strong>the</strong> electron acquires an additional 2.86 eV <strong>of</strong> energy. What is<br />
<strong>the</strong> quantum number n <strong>of</strong> <strong>the</strong> state'into which <strong>the</strong> electron moves?<br />
8. Using <strong>the</strong> Bohr model, determine <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> energy <strong>of</strong> <strong>the</strong> nth<br />
orbit <strong>of</strong> a triply ionized beryllium <strong>atom</strong> (BeH, Z = 4) to <strong>the</strong> energy<br />
<strong>of</strong> <strong>the</strong> nth orbit <strong>of</strong> a hydrogen <strong>atom</strong> (H).<br />
9. A singly ionized helium <strong>atom</strong> (He+) has only one electron in orbit<br />
about <strong>the</strong> <strong>nucleus</strong>. What is <strong>the</strong> radius <strong>of</strong> <strong>the</strong> ion when it is in <strong>the</strong><br />
~ond excited state?<br />
(!!J: (a) What is <strong>the</strong> minimum energy (in electron volts) that is required<br />
to remove <strong>the</strong> electron from <strong>the</strong> ground state <strong>of</strong> a singly ionized<br />
helium <strong>atom</strong> (He+, Z = 2)? (b) What is <strong>the</strong> ionization energy<br />
for He+?<br />
11. ssm Find <strong>the</strong> energy (in joules) <strong>of</strong> <strong>the</strong> photon that is emitted<br />
when <strong>the</strong> electron in a hydrogen <strong>atom</strong> undergoes a transition from<br />
<strong>the</strong> n = 7 energy level to produce a line in <strong>the</strong> Paschen series .<br />
.-@ A hydrogen <strong>atom</strong> is in <strong>the</strong> ground state. It absorbs energy and<br />
makes a transition to <strong>the</strong> n = 3 excited state. The <strong>atom</strong> returns to <strong>the</strong><br />
ground state by emitting two photons. What are <strong>the</strong>ir wavelengths?<br />
13. Consider <strong>the</strong> Bohr energy expression (Equation 30.13) as it applies<br />
to singly ionized helium He+ (Z = 2) and doubly ionized<br />
lithium Li2+ (Z = 3). This expression predicts equal electron ener<br />
gies for <strong>the</strong>se two species for certain values <strong>of</strong> <strong>the</strong> quantum number<br />
n (<strong>the</strong> quantum number is different for each species). For quantum<br />
numbers less than or equal to 9, what are <strong>the</strong> lowest three energies<br />
(in electron volts) for which <strong>the</strong> helium energy level is equal to <strong>the</strong><br />
lithium energy level?<br />
PROBLEMS I 971<br />
rilf.)In <strong>the</strong> hydrogen <strong>atom</strong> <strong>the</strong> radius <strong>of</strong> orbit B is sixteen times greater<br />
~ <strong>the</strong> radius <strong>of</strong> orbit A. The total energy <strong>of</strong> <strong>the</strong> electron in orbit A is<br />
- 3AO eV. What is <strong>the</strong> total energy <strong>of</strong> <strong>the</strong> electron in orbit B?<br />
* 15. ssm A wavelength <strong>of</strong> 410.2 nm is emitted by <strong>the</strong> hydrogen<br />
<strong>atom</strong>s in a high-voltage discharge tube. What are <strong>the</strong> initial and final<br />
values <strong>of</strong> <strong>the</strong> quantum number n for <strong>the</strong> energy level transition that<br />
produces this wavelength?<br />
*@ The energy <strong>of</strong> <strong>the</strong> n = 2 Bohr orbit is - 30.6 eV for an unidentified<br />
ionized <strong>atom</strong> in which only one electron moves about <strong>the</strong> <strong>nucleus</strong>.<br />
What is <strong>the</strong> radius <strong>of</strong> <strong>the</strong> n = 5 orbit for this species?<br />
* 17. ssm For <strong>atom</strong>ic hydrogen, <strong>the</strong> Paschen series <strong>of</strong> lines occurs<br />
when nf = 3, whereas <strong>the</strong> Brackett series occurs when nf = 4 in<br />
Equation 30.14. Using this equation, show that <strong>the</strong> ranges <strong>of</strong> wavelengths<br />
in <strong>the</strong>se two series overlap.<br />
*@ Interactive LearningWare 30.1 at www.wiley.comlcolIege/cutnell<br />
reviews <strong>the</strong> concepts that play roles in this problem. A hydrogen<br />
<strong>atom</strong> emits a photon that has momentum with a magnitude <strong>of</strong><br />
5A52 X 10-27 kg' m/so This photon is emitted because <strong>the</strong> electron<br />
in <strong>the</strong> <strong>atom</strong> falls from a higher energy level into <strong>the</strong> n = I leveL<br />
What is <strong>the</strong> quantum number <strong>of</strong> <strong>the</strong> level from which <strong>the</strong> electron<br />
falls? Use a value <strong>of</strong> 6.626 X 10-34 J. s for Planck's constant.<br />
** 19. A diffraction grating is used in <strong>the</strong> first order to separate <strong>the</strong><br />
wavelengths in <strong>the</strong> Balmer series <strong>of</strong> <strong>atom</strong>ic hydrogen. (Section 27.7<br />
discusses diffraction gratings.) The grating and an observation<br />
screen (see Figure 27.32) are separated by a distance <strong>of</strong> 81.0 cm.<br />
You may assume that ()is small, so sin ()= ()when radian measure<br />
is used for (). How many lines per centimeter should <strong>the</strong> grating<br />
have so that <strong>the</strong> longest and <strong>the</strong> next-to-<strong>the</strong>-Iongest wavelengths in<br />
<strong>the</strong> series will be separated by 3.00 cm on <strong>the</strong> screen?<br />
** 20. A certain species <strong>of</strong> ionized <strong>atom</strong>s produces an emission line<br />
spectrum according to <strong>the</strong> Bohr model, but <strong>the</strong> number <strong>of</strong> protons Z<br />
in <strong>the</strong> <strong>nucleus</strong> is unknown. A group <strong>of</strong> lines in <strong>the</strong> spectrum forms a<br />
series in which <strong>the</strong> shortest wavelength is 22.79 nm and <strong>the</strong> longest<br />
wavelength is 41.02 nm. Find <strong>the</strong> next-to-<strong>the</strong>-Iongest wavelength in<br />
<strong>the</strong> series <strong>of</strong> lines.<br />
"@ The<br />
Section 30.5 The Quantum Mechanical Picture<br />
<strong>of</strong> <strong>the</strong> Hydrogen Atom<br />
orbital quantum number for <strong>the</strong> electron in a hydrogen <strong>atom</strong><br />
IS £ = 5. What is <strong>the</strong> smallest possible value (algebraically) for <strong>the</strong><br />
total energy <strong>of</strong> this electron? Give your answer in electron volts.<br />
22. A hydrogen <strong>atom</strong> is in its second excited state. Determine, according<br />
to quantum mechanics, (a) <strong>the</strong> total energy (in eV) <strong>of</strong> <strong>the</strong><br />
<strong>atom</strong>, (b) <strong>the</strong> magnitude <strong>of</strong> <strong>the</strong> maximum angular momentum <strong>the</strong><br />
electron can have in this state, and (c) <strong>the</strong> maximum value that <strong>the</strong> z<br />
, 23. ssm www The principal quantum number for an electron in an<br />
~mponent a om is n = Lz6, <strong>of</strong>and <strong>the</strong> <strong>the</strong> angular magnetic momentum quantumcannumber have. is me = 2. What<br />
possible values for <strong>the</strong> orbital quantum<br />
have?<br />
number £ could this electron<br />
24. The maximum value for <strong>the</strong> magnetic quantum number in state<br />
A is me = 2, while in state B it is me = I. What is <strong>the</strong> ratio LA/LB <strong>of</strong><br />
<strong>the</strong> magnitudes<br />
<strong>the</strong>se two states?<br />
<strong>of</strong> <strong>the</strong> orbital angular momenta <strong>of</strong> an electron in<br />
* 25. Interactive Solution 30.25 at www.wiley.comlcollege/cutnell<br />
<strong>of</strong>fers one approach to problems <strong>of</strong> this type. For an electron in a hydrogen<br />
<strong>atom</strong>, <strong>the</strong> z component <strong>of</strong> <strong>the</strong> angular momentum has a maximum<br />
value <strong>of</strong> L, = 4.22 X 10-34 J . s. Find <strong>the</strong> three smallest possible<br />
values (algebraically)<br />
this <strong>atom</strong> could have.<br />
for <strong>the</strong> total energy (in electron volts) that
972 I CHAPTER 30 I THE NATURE OF THE ATOM<br />
ff6J Review Conceptual Example 6 as backg;ound for this problem.<br />
~ <strong>the</strong> hydrogen <strong>atom</strong>, <strong>the</strong> Bohr model and quantum mechanics<br />
both give <strong>the</strong> same value for <strong>the</strong> energy <strong>of</strong> <strong>the</strong> nth state. However,<br />
<strong>the</strong>y do not give <strong>the</strong> same value for <strong>the</strong> orbital angular momentum L.<br />
(a) For n = I, determine <strong>the</strong> values <strong>of</strong> L [in units <strong>of</strong> h/(27T)] predicted<br />
by <strong>the</strong> Bohr model and quantum mechanics. (b) Repeat part<br />
(a) for n = 3, noting that quantum mechanics permits more than one<br />
value <strong>of</strong> £ when <strong>the</strong> electron is in <strong>the</strong> n = 3 state.<br />
** 27. ssm www An electron is in <strong>the</strong> n = 5 state. What is <strong>the</strong> small<br />
est possible value for <strong>the</strong> angle between <strong>the</strong> z component <strong>of</strong> <strong>the</strong> orbital<br />
angular momentum and <strong>the</strong> orbital angular momentum?<br />
Section 30.6 The Pauli Exclusion Principle and <strong>the</strong> Periodic<br />
Table <strong>of</strong> <strong>the</strong> Elements<br />
28. Two <strong>of</strong> <strong>the</strong> three electrons in a lithium <strong>atom</strong> have quantum numbers<br />
<strong>of</strong> n = 1, £ = 0, me = 0, m, = +! and n = 1, £ = 0, me = 0,<br />
m, = -!. What quantum numbers can <strong>the</strong> third electron have if <strong>the</strong><br />
<strong>atom</strong> is in (a) its ground state and (b) its first excited state?<br />
29. In <strong>the</strong> style shown in Table 30.3, write down <strong>the</strong> ground-state<br />
electronic configuration for arsenic As (2 = 33). Refer to Figure<br />
30.17 for <strong>the</strong> order in which <strong>the</strong> sub shells fill.<br />
30. Figure 30.17 was constructed using <strong>the</strong> Pauli exclusion principle<br />
and indicates that <strong>the</strong> n = 1 shell holds 2 electrons, <strong>the</strong> n = 2 shell<br />
holds 8 electrons, and <strong>the</strong> n = 3 shell holds 18 electrons. These num<br />
bers can be obtained by adding <strong>the</strong> numbers given in'<strong>the</strong> figure for<br />
<strong>the</strong> subshells contained within a given shell. How many electrons can<br />
be put into <strong>the</strong> n = 5 shell, which is only partly shown in <strong>the</strong> figure?<br />
31. Write down <strong>the</strong> fourteen sets <strong>of</strong> <strong>the</strong> four quantum numbers that<br />
correspond to <strong>the</strong> electrons in a completely filled 4f subshell.<br />
* 32. What is <strong>the</strong> <strong>atom</strong> with <strong>the</strong> smallest <strong>atom</strong>ic number that contains<br />
<strong>the</strong> same number <strong>of</strong> electrons in its s subshells as it does in its d sub<br />
shell? Refer to Figure 30.17 for <strong>the</strong> order in which <strong>the</strong> subshells fill.<br />
Section 30.7 X-Rays<br />
33. ssm Molybdenum has an <strong>atom</strong>ic number <strong>of</strong> 2 = 42. Using <strong>the</strong><br />
Bohr model, estimate <strong>the</strong> wavelength <strong>of</strong> <strong>the</strong> Kcx X-ray.<br />
34. Interactive LearningWare 30.2 at www.wiley.comlcollege/cutnell<br />
reviews <strong>the</strong> concepts that are pertinent to this problem. By using <strong>the</strong><br />
Bohr model, decide which element is likely to emit a Kcx X-ray with<br />
a wavelength <strong>of</strong> 4.5 X 10-9 m.<br />
35. Interactive Solution 30.35 at www.wiley.comlcollege/cutnell<br />
provides one model for solving problems such as this one. An X-ray<br />
tube is being operated at a potential difference <strong>of</strong> 52.0 kV. What is<br />
<strong>the</strong> Bremsstrahlung wavelength that corresponds to 35.0% <strong>of</strong> <strong>the</strong> kinetic<br />
energy with which an electron collides with <strong>the</strong> metal target in<br />
<strong>the</strong> tube?<br />
IADDITIONAL PROBLEMS<br />
44. Referring to Figure 30.17 for <strong>the</strong> order in which <strong>the</strong> sub shells<br />
fill and following <strong>the</strong> style used in Table 30.3, determine <strong>the</strong> groundstate<br />
electronic configuration for cadmium Cd (2 = 48).<br />
45. ssm www In <strong>the</strong> line spectrum <strong>of</strong> <strong>atom</strong>ic hydrogen <strong>the</strong>re is<br />
also a group <strong>of</strong> lines known as <strong>the</strong> Pfund series. These lines are produced<br />
when electrons, excited to high energy levels, make transitions<br />
to <strong>the</strong> n = 5 level. Determine (a) <strong>the</strong> longest wavelength and<br />
36. What is <strong>the</strong> minimum potential difference that must be applied<br />
to an X-ray tube to knock a K-shell electron completely out <strong>of</strong> an<br />
<strong>atom</strong> in a copper (2 = 29) target? Use <strong>the</strong> Bohr model as needed.<br />
* 37. ssm An X-ray tube contains a silver (2 = 47) target. The high<br />
voltage in this tube is increased from zero. Using <strong>the</strong> Bohr model,<br />
find <strong>the</strong> value <strong>of</strong> <strong>the</strong> voltage at which <strong>the</strong> Kcx X-ray just appears in<br />
<strong>the</strong> X-ray spectrum.<br />
* 38. Multiple-Concept Example 9 reviews <strong>the</strong> concepts that are<br />
important in this problem. An electron, traveling at a speed <strong>of</strong><br />
6.00 X 107 m/s, strikes <strong>the</strong> target <strong>of</strong> an X-ray tube. Upon impact, <strong>the</strong><br />
electron decelerates to one-quarter <strong>of</strong> its original speed, emitting an<br />
X-ray in <strong>the</strong> process. What is <strong>the</strong> wavelength <strong>of</strong> <strong>the</strong> X-ray photon?<br />
39.r.'<br />
Section 30.8 The Laser<br />
R tached retina back into place. The wavelength <strong>of</strong> <strong>the</strong><br />
laser ssm wwwAlaserisusedineyesurgerytoweldade<br />
beam is 514 nm, and <strong>the</strong> power is 1.5 W. During<br />
surgery, <strong>the</strong> laser beam is turned on for 0.050 s. During this time,<br />
how many photons are emitted by <strong>the</strong> laser?<br />
40. ~ The dye laser used in <strong>the</strong> treatment <strong>of</strong> <strong>the</strong> port-wine stain<br />
8~ in 585Figure nm. A30.30 carbon (seedioxide Sectionlaser 30.9) produces has a wavelength a wavelength<strong>of</strong><br />
<strong>of</strong> 1.06 X 10-5 m. What is <strong>the</strong> minimum number <strong>of</strong> photons that <strong>the</strong><br />
carbon dioxide laser must produce to deliver at least as much or<br />
more energy to a target as does a single photon from <strong>the</strong> dye laser?<br />
41. A pulsed laser emits light in a series <strong>of</strong> short pulses, each having<br />
a duration <strong>of</strong> 25.0 ms. The average power <strong>of</strong> each pulse is<br />
5.00 mW, and <strong>the</strong> wavelength <strong>of</strong> <strong>the</strong> light is 633 nm. Find (a) <strong>the</strong><br />
energy <strong>of</strong> each pulse and (b) <strong>the</strong> number <strong>of</strong> photons in each pulse.<br />
. an eye condition known as narrow-angle glaucoma, in<br />
42. T'A • which laserpressure peripheral buildup iridotomy in <strong>the</strong> is eye a procedure can lead t<strong>of</strong>or loss treating <strong>of</strong> vision.<br />
A neodymium YAG laser (wavelength = 1064 nm) is used in<br />
<strong>the</strong> procedure to punch a tiny hole in <strong>the</strong> peripheral iris, <strong>the</strong>reby<br />
relieving <strong>the</strong> pressure buildup. In one application <strong>the</strong> laser delivers<br />
4.1 X 10-3 J <strong>of</strong> energy to <strong>the</strong> iris in creating <strong>the</strong> hole. How many<br />
photons does <strong>the</strong> laser deliver?<br />
* 43. Fusion is <strong>the</strong> process by which <strong>the</strong> sun produces energy. One<br />
experimental technique for creating controlled fusion utilizes<br />
a solid-state laser that emits a wavelength <strong>of</strong> 1060 nm and can produce<br />
a power <strong>of</strong> 1.0 X 1014 W for a pulse duration <strong>of</strong> 1.1 X 10-11 s.<br />
In contrast, <strong>the</strong> helium/neon laser used at <strong>the</strong> checkout counter<br />
in a bar-code scanner emits a wavelength <strong>of</strong> 633 nm and produces<br />
a power <strong>of</strong> about 1.0 X 10-3 W. How long (in days)<br />
would <strong>the</strong> helium/neon laser have to operate to produce <strong>the</strong><br />
same number <strong>of</strong> photons that <strong>the</strong> solid-state laser produces in<br />
1.1 X 1O-11 s?<br />
(b) <strong>the</strong> shortest wavelength in this series. (c) Refer to Figure 24.10<br />
and identify <strong>the</strong> region <strong>of</strong> <strong>the</strong> electromagnetic spectrum in which<br />
<strong>the</strong>se lines are found.<br />
46. The <strong>atom</strong>ic number <strong>of</strong> lead is 2 = 82. According to <strong>the</strong> Bohr<br />
model, what is <strong>the</strong> energy (in joules) <strong>of</strong> a Kcx X-ray photon?<br />
47. ssm When an electron makes a transition between energy levels<br />
<strong>of</strong> an <strong>atom</strong>, <strong>the</strong>re are no restrictions on <strong>the</strong> initial and final values <strong>of</strong>
Chapter 30 Problems 1515<br />
8. REASONING According to <strong>the</strong> Bohr model, <strong>the</strong> energy (in joules) <strong>of</strong> <strong>the</strong> nth orbit <strong>of</strong> an<br />
<strong>atom</strong> containing a single electron is<br />
En = -(2.18x10-18 J) Z:<br />
n<br />
(30.12)<br />
where Z is <strong>the</strong> <strong>atom</strong>ic number <strong>of</strong> <strong>the</strong> <strong>atom</strong>. The ratio <strong>of</strong> <strong>the</strong> energies <strong>of</strong> <strong>the</strong> two <strong>atom</strong>s can be<br />
obtained directly by using this relation.<br />
SOLUTION Taking <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> energy E B 3+ <strong>of</strong> <strong>the</strong> nth orbit <strong>of</strong> a beryllium <strong>atom</strong><br />
n, e<br />
(ZBe3+ = 4) to <strong>the</strong> energy En, H <strong>of</strong> <strong>the</strong> nth orbit <strong>of</strong> a hydrogen (ZH = I) <strong>atom</strong> gives<br />
En, B,;' -(2.18xlO-18 J) Z~,;.<br />
= 2 Z2<br />
En.H "2 =~= (4)2<br />
-(2.18x 10-18 J) ZH Z~ (1)2 =[lli<br />
n2<br />
9. REASONING The <strong>atom</strong>ic number for helium is Z = 2. The ground state is <strong>the</strong> n = 1 state,<br />
<strong>the</strong> first excited state is <strong>the</strong> n = 2 state, and <strong>the</strong> second excited state is <strong>the</strong> n = 3 state. With<br />
Z= 2 and n = 3, we can use Equation 30.10 to find <strong>the</strong> radius <strong>of</strong> <strong>the</strong> ion.<br />
SOLUTION The radius <strong>of</strong><strong>the</strong> second excited state is<br />
10. REASONING<br />
a. The total energy En for a single electron in <strong>the</strong> nth state is given by<br />
Z2<br />
En = -(13.6 eV)-2 n<br />
(30.10)<br />
(30.13)<br />
where Z = 2 for helium. The minimum amount <strong>of</strong> energy required to remove <strong>the</strong> electron<br />
from <strong>the</strong> ground state (n = 1) is that needed to move <strong>the</strong> electron into <strong>the</strong> state for which<br />
n = 2. This amount equals <strong>the</strong> difference between <strong>the</strong> two energy levels.<br />
b. The ionization energy defined as <strong>the</strong> minimum amount <strong>of</strong> energy required to remove <strong>the</strong><br />
electron from <strong>the</strong> n = 1 orbit to <strong>the</strong> highest possible excited state (n = 00) .
11,<br />
1516 THE NATURE OF THE ATOM<br />
SOLUTION<br />
a. The minimum amount <strong>of</strong> energy required to remove <strong>the</strong> electron from <strong>the</strong> ground state<br />
(n = 1) and move it into <strong>the</strong> state for which n = 2 is<br />
Minimum energy = E, _ El = -(13.6 22 eV)(2)' - [-(13.6 12 eV) (2)' ] = 140.8 eVI<br />
b. The ionization energy is <strong>the</strong> difference between <strong>the</strong> ground-state energy (n = 1) and <strong>the</strong><br />
energy in <strong>the</strong> highest possible excited state (n = 00) . Thus,<br />
Ionization energy ~ E__ El = -(13.6 e~) (2)' [-(13.6 eV)(2)']_<br />
(00) 12 -154.4 eVI<br />
11. I SSMI REASONING According to Equation 30.14, <strong>the</strong> wavelength A emitted by <strong>the</strong><br />
hydrogen <strong>atom</strong> when it makes a transition from <strong>the</strong> level with nj to <strong>the</strong> level with nf is given<br />
by<br />
-=---- (Z2) --- wIth nj,nf =1,2,3, ... and nj >nf<br />
A1 2tr2mk2e4 h3e ( nl 1 ni2 1 J .<br />
where 2tr2mk2e4 /(h3e) = 1.097 X107 m-I and Z= 1 for hydrogen. Once <strong>the</strong> wavelength<br />
for <strong>the</strong> particular transition in question is determined, Equation 29.2 (E = hf = he / A) can<br />
be used to find <strong>the</strong> energy <strong>of</strong> <strong>the</strong> emitted photon.<br />
SOLUTION In <strong>the</strong> Paschen series, nf= 3. Using <strong>the</strong> above expression with Z= 1, nj = 7<br />
and nf= 3, we find that<br />
~ = (1.097 X 107 m-I)(I') (3~- 71,) or A=1.005xlO-6 m<br />
The photon energy is<br />
he (6.63XlO-34 J.s)(3.00XI08 rnls) I I<br />
E=-=-----------= 1.98xl0-19 J<br />
A 1.005xl0-6 m '------
Chapter 30 Problems 1517<br />
12. REASONING Since <strong>the</strong> <strong>atom</strong> emits two photons as it returns to <strong>the</strong> ground state, one is<br />
emitted when <strong>the</strong> electron falls from n = 3 to n = 2, and <strong>the</strong> o<strong>the</strong>r is emitted when it<br />
subsequently drops from n = 2 to n = 1. The wavelengths <strong>of</strong> <strong>the</strong> photons emitted during<br />
<strong>the</strong>se transitions are given by Equation 30.14 with <strong>the</strong> appropriate values for <strong>the</strong> initial and<br />
final numbers, ni and ne<br />
SOLUTION The wavelengths <strong>of</strong> <strong>the</strong> photons are<br />
n =3 ton =2 ,.1,= 1 ( 1.097x10 7 m_1)()2(1 1 "22-"32 1) =1.524xlO 6 m-1<br />
n = 2 to n = 1<br />
,.1,= !6.56XlO-7ml<br />
~ =(1.097X107 m-1)(1)2(*_ 212)=8.228X106 m-I<br />
A=I1.22X10-7 m!<br />
(30.14)<br />
(30.14)<br />
13. REASONING The Bohr expression as it applies to anyone-electron species <strong>of</strong> <strong>atom</strong>ic<br />
number Z, is given by Equation 30.13: En = -(13.6 eV)(Z2 / n2). For certain values <strong>of</strong> <strong>the</strong><br />
quantum number n, this expression predicts equal electron energies for singly ionized<br />
helium He + (Z = 2) and doubly ionized lithium Li + (Z = 3). As stated in <strong>the</strong> problem, <strong>the</strong><br />
quantum number n is different for <strong>the</strong> equal energy states for each species.<br />
SOLUTION For, equal energies, we can write<br />
Simplifying, this becomes<br />
Thus,<br />
or<br />
Z2 Z2<br />
-(13.6 eV) ~e = -(13.6 eV) ii<br />
nHe nLi<br />
or<br />
4 9<br />
2 2<br />
nHe nLi<br />
Therefore, <strong>the</strong> value <strong>of</strong> <strong>the</strong> helium energy level is equal to <strong>the</strong> lithium energy level for any<br />
value <strong>of</strong> nHe that is two-thirds <strong>of</strong> nLi· For quantum numbers less than or equal to 9, an
I.<br />
-I' j.<br />
-3.40eV 9-1.S1eV<br />
3-13.6 6nLiEnergy<br />
eV<br />
--- --~- --- -- -- ----<br />
1518 THE NATURE OF THE ATOM<br />
equality in energy levels will occur for nHe = 2,4, 6 corresponding to nLi = 3, 6, 9. The<br />
results are summarized in <strong>the</strong> following table.<br />
24<br />
nHe<br />
14. REASONING In <strong>the</strong> Bohr model <strong>of</strong> <strong>the</strong> hydrogen <strong>atom</strong> <strong>the</strong> total energy En <strong>of</strong> <strong>the</strong> electron is<br />
given in electron volts by Equation 30.13 and <strong>the</strong> orbital radius rn is given in meters by<br />
Equation 30.10:<br />
Solving <strong>the</strong> radius equation for n2 and substituting <strong>the</strong> result into <strong>the</strong> energy equation gives<br />
En = -13.6 = (-13.6)(S.29X10-11)<br />
rnl(S.29X10-11) rn<br />
Thus, <strong>the</strong> energy is inversely proportional to <strong>the</strong> radius, and it is on this fact that we base our<br />
solution.<br />
SOLUTION We know that <strong>the</strong> radius <strong>of</strong> orbit B is sixteen times greater than <strong>the</strong> radius <strong>of</strong><br />
orbit A. Since <strong>the</strong> total energy is inversely proportional to <strong>the</strong> radius, it follows that <strong>the</strong> total<br />
energy <strong>of</strong> <strong>the</strong> electron in orbit B is one-sixteenth <strong>of</strong> <strong>the</strong> total energy in orbit A:<br />
EA _ -3.40 eV = 1-0.213 eVI<br />
EB=16.0- 16.0<br />
IS. I SSM I REASONING A wavelength <strong>of</strong> 410.2 nm is emitted by <strong>the</strong> hydrogen <strong>atom</strong>s in a<br />
high-voltage discharge tube. This transition lies in <strong>the</strong> visible region (380-7S0 nm) <strong>of</strong> <strong>the</strong><br />
hydrogen spectrum. Thus, we can conclude that <strong>the</strong> transition is in <strong>the</strong> Balmer series and,<br />
<strong>the</strong>refore, that nf = 2. The value <strong>of</strong> ni can be found using Equation 30.14, according to which<br />
<strong>the</strong> Ba1mer series transitions are given by<br />
11 = 3, 4, S, ...
Chapter 30 Problems 1519<br />
This expression may be solved for ni for <strong>the</strong> energy transition that produces <strong>the</strong> given<br />
wavelength.<br />
SOLUTION Solving for ni' we find that<br />
Therefore, <strong>the</strong> initial and final states are identified by I ni = 6 and nf = 2 I·<br />
16. REASONING The energy levels and radii <strong>of</strong> a hydrogenic species <strong>of</strong> <strong>atom</strong>ic number Z are<br />
given by Equations30.13 and 30.10, respectively: En=-(13.6eV)(Z2/n2) and<br />
rn = (5.29 x 10-11 m)( n2 /Z) . We can use Equation 30.13 to find <strong>the</strong> value <strong>of</strong> Z for <strong>the</strong><br />
unidentified ionized <strong>atom</strong> and <strong>the</strong>n calculate <strong>the</strong> radius <strong>of</strong> <strong>the</strong> n = 5 orbit using<br />
Equation 30.10.<br />
SOLUTION Solving Equation 30.13 for <strong>atom</strong>ic number Z <strong>of</strong> <strong>the</strong> unknown species, we have<br />
Z=<br />
Therefore, <strong>the</strong> radius <strong>of</strong> <strong>the</strong> n = 5 orbit is<br />
-13.6 Enn2 eV = ~ {(-30.6 -13.6eV)(2)2 eV = 3<br />
17. I SSMI REASONING AND SOLUTION For <strong>the</strong> Paschen series, nf = 3. The range <strong>of</strong><br />
wavelengths occurs for values <strong>of</strong> nj = 4 to ni= 00. Using Equation 30.14, we find that <strong>the</strong><br />
shortest wavelength occurs for nj = 00 and is given by<br />
/L=8.204x10-7 m<br />
~---~v~---~<br />
Shortest wavelength in<br />
Paschen series
1520 THE NATURE OF THE ATOM<br />
The longest wavelength in <strong>the</strong> Paschen series occurs for ni= 4 and is given by<br />
A = 1.875 X 10-6 m<br />
~---~v·~---~<br />
Longest wavelength in<br />
Paschen series<br />
For <strong>the</strong> Brackett series, nf= 4. The range <strong>of</strong> wavelengths occurs for values <strong>of</strong> ni= 5 to<br />
ni= 00, Using Equation 30.14, we find that <strong>the</strong> shortest wavelength occurs for ni= 00 and is<br />
given by<br />
~ =(l.097Xl07 m-I)( 41,- 5; J<br />
or<br />
or<br />
A = 4.051 X 10-6 m<br />
v<br />
Longest wavelength in<br />
Brackett series<br />
A=1.459x10-6 m<br />
v<br />
Shortest wavelength in<br />
Brackett series<br />
The longest wavelength in <strong>the</strong> Brackett series occurs for ni= 5 and is given by<br />
Since <strong>the</strong> longest wavelength in <strong>the</strong> Paschen series falls within <strong>the</strong> Brackett series, <strong>the</strong><br />
wavelengths <strong>of</strong> <strong>the</strong> two series overlap.<br />
18. REASONING· To obtain <strong>the</strong> quantum number <strong>of</strong> <strong>the</strong> higher level from which <strong>the</strong> electron<br />
falls, we will use Equation 30.14 for <strong>the</strong> reciprocal <strong>of</strong> <strong>the</strong> wavelength A <strong>of</strong> <strong>the</strong> photon:<br />
where R is <strong>the</strong> Rydberg constant and nf and ni' respectively, are <strong>the</strong> quantum numbers <strong>of</strong> <strong>the</strong><br />
final and initial levels. Although we are not directly given <strong>the</strong> wavelength, we do have a<br />
value for <strong>the</strong> magnitude p <strong>of</strong> <strong>the</strong> photon's momentum, and <strong>the</strong> momentum and <strong>the</strong><br />
wavelength are related according to Equation 29.6:<br />
j~.<br />
'~<br />
',[i<br />
o i where his Planck's constant.<br />
h<br />
p=- A<br />
Using Equation 30.14 to substitute for ~, we obtain<br />
A
SOLUTION Rearranging Equation (1) gives<br />
Thus, we find<br />
Chapter 30 Problems 1521<br />
p=-=hR --- (1)<br />
Ah (1nf nf 1 J<br />
_1 __ 1 _ P<br />
nf nf - hR<br />
_1 __ 1 _L<br />
nf - nf hR<br />
1 1 P _ 1 5.452x10-27 kg·m/s -0.2499 or n =!lI<br />
nj2 = ni - hR -12- (6.626X10-34 J.s)(1.097X107 m-l) 1<br />
or<br />
19. REASONING AND SOLUTION We need to use Equation 30.14 to find <strong>the</strong> spacing<br />
between <strong>the</strong> longest and next-to-<strong>the</strong> longest wavelengths in <strong>the</strong> Ba1mer series. In order to do<br />
this, we need to first find <strong>the</strong>se two wavelengths.<br />
Longest:<br />
-=R --- = 1.097x10 m ---<br />
/41 ( nf 1 ni2 1 J ( 7 -1)( 22 1 32 1 )<br />
Next-to-Iongest:<br />
_I ~ =R( ~_~ nf nj J~(1.097XI07 m-I)(_l 22 __1 42 )<br />
or /4 = 656.3 nm<br />
or ~ = 486.2 nm<br />
Equation 27.7 states that sin B= mAid. Using <strong>the</strong> small angle approximation, we have<br />
sin B"'" tan B"'" B = Y<br />
L<br />
so that y/L = mAid. The position <strong>of</strong> <strong>the</strong> fringe due to <strong>the</strong> longest wavelength is Yl = mAlL/d.<br />
For <strong>the</strong> next-to-Iongest, Y2 = m~L/d. The difference in <strong>the</strong> positions on <strong>the</strong> screen is,<br />
<strong>the</strong>refore, Yl - Y2 = (mL/d)(Al -~) which gives<br />
d= mL(/4 -~)<br />
Yj- Y2
Chapter 30 Problems 1523<br />
21. REASONING The orbital quantum number .e has values <strong>of</strong> 0, 1, 2, ....., (n -1), according<br />
to <strong>the</strong> discussion in Section 30.5. Since.e = 5, we can conclude, <strong>the</strong>refore, that n ;? 6. This<br />
knowledge about <strong>the</strong> principal quantum number n can be used with Equation 30.13,<br />
E = -(13.6 eV)Z2/n2, to determine <strong>the</strong> smallest value for <strong>the</strong> total energy E .<br />
n n<br />
SOLUTION The smallest value <strong>of</strong> E (i.e., <strong>the</strong> most negative) occurs when n = 6. Thus,<br />
n<br />
using Z = 1 for hydrogen, we find<br />
Z2 12<br />
En = -(13.6 eV)~ = -(13.6 eV)"62 = 1-0.378 eV I<br />
22. REASONING<br />
a. The ground state is <strong>the</strong> n = 1 state, <strong>the</strong> first excited state is <strong>the</strong> n = 2 state, and <strong>the</strong> second<br />
excited state is <strong>the</strong> n = 3 state. The total energy (in eV) <strong>of</strong> a hydrogen <strong>atom</strong> in <strong>the</strong> n = 3<br />
state is given by Equation 30.13.<br />
b. According to quantum mechanics, <strong>the</strong> magnitude L <strong>of</strong> <strong>the</strong> angular momentum is given by<br />
Equation 30.15 as L=~.e(.e+l)(h/21r), where .e is <strong>the</strong> orbital quantum number. The<br />
discussion in Section 30.5 indicates that <strong>the</strong> maximum value that .e can have is one less<br />
than <strong>the</strong> principal quantum number, so that .e max = n - 1.<br />
c. Equation 30.16 gives <strong>the</strong> z-component Lz <strong>of</strong> <strong>the</strong> angular momentum as Lz = m£ (h / 21r) ,<br />
where me is <strong>the</strong> magnetic quantum number. According to <strong>the</strong> discussion in Section 30.5,<br />
<strong>the</strong> maximum value that m£ can attain is when it is equal to <strong>the</strong> orbital quantum number,<br />
which is .e max'<br />
SOLUTION<br />
a. The total energy <strong>of</strong> <strong>the</strong> hydrogen <strong>atom</strong> is given by Equation 30.13. Using n = 3, we have<br />
(13.6 eV)(1)2 = 1-1.51 eVI<br />
E3 = 32<br />
b. The maximum orbital quantum number is .e max = n - 1 = 3 - 1 = 2. The maximum<br />
angular momentum Lmax has a magnitude given by Equation 30.15:
1524 THE NATURE OF THE ATOM<br />
c. The maximum value for <strong>the</strong> z-component Lz <strong>of</strong> <strong>the</strong> angular momentum is<br />
(withmf =J!max=2)<br />
L =m ~=(2)6.63XlO-34 ],s=12.11X10-34 J.sl<br />
z £ 2tr 2tr .------<br />
23. ISSMII wwwl REASONING The values that I can have depend on <strong>the</strong> value <strong>of</strong> n, and<br />
only <strong>the</strong> following integers are allowed: I = 0, 1, 2, ... (n -1). The values that ml can<br />
have depend on <strong>the</strong> value <strong>of</strong> I , with only <strong>the</strong> following positive and negative integers being<br />
permitted: mf = -J!, ... -2, -1,0, +1, +2, .. .+e.<br />
SOLUTION Thus, when n = 6, <strong>the</strong> possible values <strong>of</strong> I are 0, 1,2,3, 4,5. Now when<br />
mf = 2 , <strong>the</strong> possible values <strong>of</strong> I are 2, 3, 4, 5, ... These two series <strong>of</strong> integers overlap for<br />
<strong>the</strong> integers 2, 3, 4, and 5. Therefore, <strong>the</strong> possible values for <strong>the</strong> orbital quantum number I<br />
that this electron could have are 11 =<br />
2, 3, 4, 51.<br />
24. REASONING The maximum value for <strong>the</strong> magnetic quantum number is ml = I ; thus, in<br />
state A, I = 2, while in state B, I = 1. According to <strong>the</strong> quantum mechanical <strong>the</strong>ory <strong>of</strong><br />
angular momentum, <strong>the</strong> magnitude <strong>of</strong> <strong>the</strong> orbital angular momentum for a state <strong>of</strong> given I is<br />
L = -JJ!(J! + 1) (h / 2tr) (Equation 30.15). This expression can be used to form <strong>the</strong> ratio<br />
LA / ~ <strong>of</strong> <strong>the</strong> magnitudes <strong>of</strong> <strong>the</strong> orbital angular momenta for <strong>the</strong> two states.<br />
SOLUTION Using Equation 30.15, we find that<br />
h<br />
LA = .j2(2+ 1) 2& h ~ V2 @" ~v'3~11.7321<br />
LB -J1(1 + 1) 2tr<br />
25. REASONING The total energy En for a hydrogen <strong>atom</strong> in <strong>the</strong> quantum mechanical picture<br />
is <strong>the</strong> same as in <strong>the</strong> Bohr model and is given by Equation 30.13:<br />
E/1 = -(13.6 eV)2 n<br />
Thus, we need to determine values for <strong>the</strong> principal quantum number n if we are to calculate<br />
<strong>the</strong> three smallest possible values for E. Since <strong>the</strong> maximum value <strong>of</strong> <strong>the</strong> orbital quantum<br />
number J! is n - 1, we can obtain a minimum value for n as nmin = J! + 1. But how to obtain<br />
1<br />
(30.13)
Chapter 30 Problems 1525<br />
£? It can be obtained, because <strong>the</strong> problem statement gives <strong>the</strong> maximum value <strong>of</strong> Lz, <strong>the</strong> z<br />
component <strong>of</strong> <strong>the</strong> angular momentum. According to Equation 30.16, Lz is<br />
where mf is <strong>the</strong> magnetic quantum number and his Planck's constant. For a given value <strong>of</strong> £<br />
<strong>the</strong> allowed values for mf are as follows: -£, .. " -2, -1, 0, +1, +2, ... , +£. Thus, <strong>the</strong><br />
maximum value <strong>of</strong>mf is £, and we can use Equation 30.16 to calculate <strong>the</strong> maximum value <strong>of</strong><br />
mf from <strong>the</strong> maximum value given for Lz·<br />
h<br />
Lz = mf 21C<br />
SOLUTION Solving Equation 30.16 for mf gives<br />
rn, ~ 21fL, ~ 21f(4.22X10-34 J,s)_4<br />
h 6.63xl0-34 J. S -<br />
(30.16)<br />
As explained in <strong>the</strong> REASONING, this maximum value for mf indicates that £ = 4.<br />
Therefore, a minimum value for n is<br />
n . =£+1=4+1=5 or n>5<br />
mm -<br />
This means that <strong>the</strong> three energies we seek correspond to n = 5, n = 6, and n = 7. Using<br />
Equation 30.13, we find <strong>the</strong>m to be<br />
[n = 5] E5 = -(13.6 eV)~ = 1-0.544 eVj<br />
5<br />
[n = 6] E6 = -(13.6 eV)~ = 1-0.378 eV[<br />
6<br />
[n=7] E7 = -(13.6 eV)~ = 1-0.278 eV[<br />
7<br />
26. REASONING AND SOLUTION<br />
a. For <strong>the</strong> angular momentum, Bohr's value is given by Equation 30.8, with n = 1,<br />
According to quantum <strong>the</strong>ory, <strong>the</strong> angular momentum is given by Equation 30.15. For<br />
n=I,£=O
1526 THE NATURE OF THE ATOM<br />
b. For n = 3; Bohr <strong>the</strong>ory gives<br />
while quantum mechanics gives<br />
[n = 3, .e = 0]<br />
[n=3,.e=1]<br />
[n = 3, .e = 2]<br />
27. I SSMII wwwl REASONING Let e denote <strong>the</strong> angle between<br />
<strong>the</strong> angular momentum L and its z-component Lz· We can see<br />
from <strong>the</strong> figure at <strong>the</strong> right that Lz = L cos e . Using<br />
Equation 30.16 for Lz and Equation 30.15 for L, we have<br />
The smallest value for e corresponds to <strong>the</strong> largest value <strong>of</strong> cos e. For a given value <strong>of</strong> I ,<br />
<strong>the</strong> largest value for cos e corresponds to <strong>the</strong> largest value for rn,. But <strong>the</strong> largest possible<br />
value for rnl is rnl = I . Therefore, we find that<br />
cos e =.e re<br />
~.e(.e+l) ~£+1<br />
SOLUTION The smallest value for e corresponds to <strong>the</strong> largest value for I. When <strong>the</strong><br />
electron is in <strong>the</strong> n = 5 state, <strong>the</strong> largest allowed value <strong>of</strong> I is I = 4 ; <strong>the</strong>refore, we see that<br />
or<br />
e = cos-1 (J4i5) = I 26.6° I<br />
i<br />
"1<br />
,j<br />
i;<br />
1<br />
!<br />
"