Chemistry a molecular approach 4th edition tro solutions manual

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Chapter 7. The Quantum-Mechanical Model of the Atom Student Objectives 7.1 Schrödinger’s Cat • •

Know that the behavior of macroscopic objects like baseballs is strikingly different from the behavior of microscopic objects like electrons.

Know that the quantum-mechanical model provides the basis for the organization of the periodic table and our understanding of chemical bonding.

7.2 The Nature of Light • • • • • • •

Define and understand electromagnetic radiation.

Define and understand amplitude, wavelength, and frequency.

Use the speed of light to convert between wavelength and frequency.

Know the electromagnetic spectrum and its different forms of radiation.

Know and understand interference and diffraction and how they demonstrate the wave nature of light.

Know and explain the photoelectric effect and how it demonstrates the particle nature of light.

Use equations to interconvert energy, wavelength, and frequency of electromagnetic radiation.

7.3 Atomic Spectroscopy and the Bohr Model • •

Define and understand atomic spectroscopy and emission spectrum.

Understand how the Bohr model explains the emission spectrum of hydrogen.

7.4 The Wave Nature of Matter: The de Broglie Wavelength, the Uncertainty Principle, and Indeterminacy • • • • • •

Know that electrons and photons behave in similar ways: both can act as particles and as waves.

Know that photons and electrons, even when viewed as streams of particles, still display diffraction and interference patterns in a double-slit experiment.

Use de Broglie’s relation to interconvert wavelength, mass, and velocity.

Know the complementarity of position and velocity through Heisenberg’s uncertainty principle.

Know the similarities and differences in classical and quantum-mechanical concepts of trajectory.

Differentiate between deterministic and indeterminacy.

7.5 Quantum Mechanics and the Atom • • • •

Define orbital and wave function.

Know that the Schrödinger equation is the ultimate source of energies and orbitals for electrons in atoms.

Know the properties and allowed values of the principal quantum number, n.

Know the properties, allowed values, and letter designations of the angular momentum quantum number, l.

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Chapter 7. The Quantum-Mechanical Model of the Atom • • •

Know the properties and allowed values of the magnetic quantum number, ml.

Know and understand how atomic spectroscopy defines the energy levels of electrons in the hydrogen atom.

Calculate the energies and wavelengths of emitted and absorbed photons for hydrogen.

7.6 The Shapes of Atomic Orbitals • • • • • •

Define probability density and radial distribution function.

Define and understand node.

Identify the number of nodes in a radial distribution function for an s orbital.

Know the shapes of s, p, d, and f orbitals and the relationships to quantum numbers.

Know that the shape of an atom is dictated by the combined shapes of the collection of orbitals for that atom.

Define and understand phase.

Section Summaries Lecture Outline • •

Terms, Concepts, Relationships, Skills Figures, Tables, and Solved Examples

Teaching Tips • •

Suggestions and Examples Misconceptions and Pitfalls

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Chapter 7. The Quantum-Mechanical Model of the Atom Lecture Outline Terms, Concepts, Relationships, Skills

Figures, Tables, and Solved Examples

7.1 Schrödinger’s Cat • The strangeness of the quantum world does not transfer to the macroscopic world

•

Intro figure: Artistic representation of Schrödinger’s cat

7.2 The Nature of Light • Electromagnetic radiation o waves o amplitude o wavelength o frequency o speed  c = λν • Electromagnetic spectrum o radiowaves o microwaves o infrared o visible o ultraviolet o X-ray o gamma • Wave behavior o interference  constructive  destructive o diffraction • Particle behavior o photoelectric effect o photon or quantum of light  E = hν = hc/λ

• •

Figure 7.1 Electromagnetic Radiation unnumbered figure: photo of lightning showing the speed of sound vs. light Figure 7.2 Wavelength and Amplitude Figure 7.3 Components of White Light Figure 7.4 The Color of an Object Example 7.1 Wavelength and Frequency Figure 7.5 The Electromagnetic Spectrum unnumbered figure: photos of medical X-ray Chemistry and Medicine: Radiation Treatment for Cancer unnumbered figure: thermal image unnumbered figures: constructive and destructive interference unnumbered figure: photo of water wave interference Figure 7.6 Diffraction Figure 7.7 Interference from Two Slits Figure 7.8 The Photoelectric Effect Figure 7.9 The Photoelectric Effect Example 7.2 Photon Energy Example 7.3 Wavelength, Energy, and Frequency

7.3 Atomic Spectroscopy and the Bohr Model • Atomic spectroscopy o emission spectrum o Bohr model

• • • •

• • • • • • • • • • • • • • • •

• • • • •

unnumbered figure: photo of neon sign Figure 7.10 Mercury, Helium, and Hydrogen Figure 7.11 Emission Spectra Figure 7.12 The Bohr Model and Emission Spectra Chemistry in Your Day: Atomic Spectroscopy, a Bar Code for Atoms unnumbered figure: photo of fireworks Figure 7.13 Emission Spectra of Oxygen and Neon Figure 7.14 Flame Tests for Sodium, Potassium, Lithium, and Barium Figure 7.15 Emission and Absorption Spectrum of Mercury

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Chapter 7. The Quantum-Mechanical Model of the Atom Teaching Tips Suggestions and Examples

Misconceptions and Pitfalls

7.1 Schrödinger’s Cat

7.2 The Nature of Light • Water waves provide a practical example of some of the properties of waves, especially wavelength and amplitude.

• Standing waves is a useful demonstration.

• Sound waves are compared to the light from an exploding firework.

• The use of color (wavelength) and brightness (amplitude) in Figure 7.2 can be compared to water and air (sound) analogies.

• The radiation treatment for cancer is a positive use of radiation. Its use is possible because delivery can be controlled and focused.

• Diffraction patterns are very interesting; a demonstration with white and laser light can use masks and patterns that can be printed onto slides or overheads. [Lisensky, George C.; Kelly, Thomas F.; Neu, Donald R.; Ellis, Arthur B. J. Chem. Educ. 1991, 68, 91.]

• Albert Einstein won the Nobel Prize in Physics for the photoelectric effect and not for relativity.

• Point out to students that we calculate the energy of a particle (photon) using a wave property (wavelength or frequency).

• Conceptual Connection 7.2 The Photoelectric Effect

•

7.3 Atomic Spectroscopy and the Bohr Model • Demonstrations of emission spectra are relatively simple. Lamps often show single colors (sodium vapor lamps). Mercury lamps (low pressure) or even mercury arc lamps used for lighting (medium and high pressure) have many lines and colors. Red and green laser pointers are effective.

• Emission spectra are used in astronomy to detect elemental makeup of stars, planets, and other glowing bodies.

• The colors in flame tests can be demonstrated easily.

•

•

•

•

Wavelengths of electromagnetic radiation like the signals for a radio or cell phone have very little energy.

Gamma rays and X-rays are much more likely to damage cells and tissue.

Radiation may have a negative connotation, but radiation keeps the Earth warm enough to inhabit, and ionizing radiation from gamma rays and X-rays can be useful.

The analogy with water waves can go too far, especially with respect to the significance of amplitude.

The Bohr model predicts emission spectra for H and for other singleelectron atoms, but it is not a correct model.

The most common misconception about electrons comes from the Bohr model: that electrons move in orbits as planets do around the sun.

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Chapter 7. The Quantum-Mechanical Model of the Atom Lecture Outline Terms, Concepts, Relationships, Skills

Figures, Tables, and Solved Examples

7.4 The Wave Nature of Matter: The de Broglie Wavelength, the Uncertainty Principle, and Indeterminacy • Interference o electron diffraction o particle beam o de Broglie wavelength: λ = h/mv • Complementary properties and uncertainty o Heisenberg’s uncertainty principle: ∆x × m∆v ≥ h/4π • Determinacy, indeterminacy, and probability o classical concept of trajectory o quantum mechanical probability

• • •

7.5 Quantum Mechanics and the Atom • Orbital, a probability map o Schrödinger equation o hydrogen energy levels

•

æ1 ö En = - 2.18´ 10- 18 J çç 2 ÷ ÷ çè n ÷ ø o principal quantum number, n o angular momentum quantum number, l o magnetic quantum number, ml Atomic spectroscopy o excitation and radiation o hydrogen atom



•



• • • • •

• • • • • • •

Figure 7.16 Electron Diffraction Example 7.4 De Broglie Wavelength unnumbered figure: illustration of double-slit experiment with electrons unnumbered figure: photo of Werner Heisenberg Figure 7.17 The Concept of Trajectory Figure 7.18 Trajectory and Probability Figure 7.19 Trajectory of a Macroscopic Object Figure 7.20 The Quantum-Mechanical Strike Zone

unnumbered figure: diagram of principal energy levels unnumbered figure: letter designations of l quantum number unnumbered figure: table of n, l, and ml quantum numbers for n = 1–3 Example 7.5 Quantum Numbers I Example 7.6 Quantum Numbers II Figure 7.21 Excitation and Radiation Figure 7.22 Hydrogen Energy Transitions and Radiation Example 7.7 Wavelength of Light for a Transition in the Hydrogen Atom

 1 1 ∆E = −2.18 × 10−18 J 2 − 2  n   f ni 

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Chapter 7. The Quantum-Mechanical Model of the Atom Teaching Tips Suggestions and Examples

Misconceptions and Pitfalls

7.4 The Wave Nature of Matter: The de Broglie Wavelength, the Uncertainty Principle, and Indeterminacy • An electron exhibits wave properties, but each electron has the same mass and the same charge regardless of wavelength.

• Conceptual Connection 7.3 The de Broglie Wavelength of Macroscopic Objects (illustrates the insignificance of the wavelengths of macroscopic objects) • Heisenberg’s uncertainty principle in particular challenges the centuries-old scientific tenet that two experiments arranged the same way should give identical results.

• Schrödinger’s cat is a popular illustration of the absurdity of quantum mechanics at the macroscopic level.

•

7.5 Quantum Mechanics and the Atom • The Schrödinger equation is a mathematical model beyond the scope of the course. Quantum numbers are results of the application of the Schrödinger equation. The names of the quantum numbers may be confusing, especially the angular momentum and magnetic quantum numbers, as most students will have no connection to the meanings of those terms.

• Conceptual Connection 7.4 The Relationship between n and l • Conceptual Connection 7.5 The Relationship Between l and ml • Conceptual Connection 7.6 Emission Spectra

•

• •

Electron interference patterns occur even when the electrons go through the double slits singly and cannot interact with each other.

Students have a hard time visualizing what the wavelength of a particle means.

Students are misled by the probabilistic nature of quantum mechanics in much the same way that Einstein was. They presume that probabilities must be invoked because of an incompletness in the theory.

The exact values of n, l, and ml are obtained only through tedious mathematics. The logic of the mathematics is the only guiding light in quantum theory since its basic principles are so counterintuitive in the macroscopic world.

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Chapter 7. The Quantum-Mechanical Model of the Atom Lecture Outline Terms, Concepts, Relationships, Skills

Figures, Tables, and Solved Examples

7.6 The Shapes of Atomic Orbitals • Orbital representations o probability density o radial distribution function o nodes o orbital surface shapes  s  p  d  f

• • • • • • • • • • •

Figure 7.23 The 1s Orbital: Two Representations Figure 7.24 The 1s Orbital Surface Figure 7.25 The Radial Distribution Function for the 1s Orbital Figure 7.26 Probability Densities and Radial Distribution Functions for the 2s and 3s Orbitals unnumbered figure: illustration of nodes on a vibrating string Figure 7.27 The 2p Orbitals and Their Radial Distribution Function Figure 7.28 The 3d Orbitals Figure 7.29 The 4f Orbitals unnumbered figure: illustration of waves and phase unnumbered figures: illustrations of phases of s and p orbitals Figure 7.30 Why Atoms Are Spherical

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Chapter 7. The Quantum-Mechanical Model of the Atom Teaching Tips Suggestions and Examples

Misconceptions and Pitfalls

7.6 The Shapes of Atomic Orbitals • The dot representation of probability in Figure 7.22 is much more realistic than the solid sphere in Figure 7.23.

• A standing wave version of a vibrating string can be demonstrated with a Slinky toy. Two students can be used to demonstrate zero, one, or two nodes without much difficulty.

•

Orbitals define the probability of finding an electron. The shape and graphical representation may suggest a physical container.

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Chapter 7. The Quantum-Mechanical Model of the Atom

Additional Problem for Photon Energy (Example 7.2)

A 1-second pulse of a red laser pointer with a wavelength of 635 nm contains 5.0 mJ of energy. How many photons does it contain?

Sort

Given Epulse = 5.0 mJ

You are given the wavelength and total energy of a light pulse and asked to find the number of photons it contains.

Find number of photons

Strategize

Conceptual Plan

In the first part of the conceptual plan, calculate the energy of an individual photon from its wavelength.

λ = 635 nm

λ

→ E =

Epulse

In the second part, divide the total energy of the pulse by the energy of each photon to get the number of photons in a pulse.

Ephoton

Ephoton hc λ

= number of photons

Relationships Used E = hc/λ (Equation 7.3)

Solve

Solution

Convert wavelength to meters and substitute the values into the energy equation.

λ = 635 nm ×

hc Ephoton = = λ

10−9 m = 6.35 × 10−7 m 1 nm

(6.626 × 10 −34 J ⋅ s )(3.00 × 108

m ) s

6.35 × 10 −7 m

= 3.13 × 10 −19 J

5.0 mJ ×

10 −3 J = 5.0 × 10 −3 J 1 mJ

number of photons =

Epulse Ephoton

=

5.0 × 10−3 J =1.6 × 1016 −19 3.13 × 10 J

Convert the energy of the pulse to joules J. Then divide by the energy of a single photon.

Check

The magnitude of the answer makes physical sense since the pulse was much larger than the individual photon energy.

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Chapter 7. The Quantum-Mechanical Model of the Atom

Additional Problem for Wavelength of Light for a Transition in the Hydrogen Atom (Example 7.7)

Determine the wavelength of light emitted when an electron in a hydrogen atom makes a transition from an orbital in n = 5 to n = 4.

Sort

Given n = 5 →

You are given the energy levels of an atomic transition and asked to find the wavelength of emitted light.

Find λ

Strategize

Conceptual Plan

Calculate the energy of the electron in the n = 5 and n = 4 orbitals using Equation 7.7 and subtract to find the difference.

n = 5, n = 4

→

∆Eatom

∆E = E4 – E5 Ephoton

→

λ

∆Eatom = −Ephoton

E =

hc λ

∆Eatom The negative value of the difference indicates that the energy is being emitted. Convert the energy value to a wavelength using Equation 7.3.

n=4

→

Relationships Used En = −2.18 × 10−18 J (1 / n2) (Equation 7.7) E = hc/λ (Equation 7.3)

Solve

Solution

Follow the conceptual plan to solve the problem. Round the answer to three significant figures to reflect the three significant figures in the least precisely known quantity (4750).

These conversion factors are all exact and therefore do not limit the number of significant figures.

∆Eatom = E4 − E5   1   1  = − 2.18 × 10−18 J  2  −  −2.18 × 10−18 J  2   4   5   1  1 = − 2.18 × 10−18 J  2 − 2  5  4 = − 4.90 × 10−20 J

Ephoton = −∆Eatom = +4.90 x 10−20 J E

λ

hc or λ hc = E (6.626 × 10−34 J ⋅ s )(3.00 × 108 m/ s ) = 4.90 × 10−20 J =

= 4.06 × 10−6 m

Check

The units of the answer are correct (m). A comparison with the values in Figure 7.21 indicates that the answer should be in the infrared region. Figure 7.5 confirms that the answer is within that region.

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