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The Quantum
Mechanical Atom
CHAPTER 8
Chemistry: The Molecular Nature of Matter, 6th edition
By Jesperson, Brady, & Hyslop
CHAPTER 8: Quantum Mechanical Atom
Learning Objectives
q  Light as Waves, Wavelength and Frequency
q  The Photoelectric Effect, Light as Particles and the Relationship between
Energy and Frequency
q  Atomic Emission and Energy Levels
q  The Bohr Model and its Failures
q  Electron Diffraction and Electrons as Waves
q  Quantum Numbers, Shells, Subshells, and Orbitals
q  Electron Configuration, Noble Gas Configuration and Orbital Diagrams
q  Aufbau Principle, Hund’s Rule, and Pauli Exclusion Principle, Heisenberg
Uncertainty Principle
q  Valence vs Inner Core Electrons
q  Nuclear Charge vs Electron Repulsion
q  Periodic Trends: Atomic Radius, Ionization Energy, and Electron Affinity
2 Particle-Wave
Duality
Light Exhibits Interference
Constructive interference
–  Waves in-phase lead to greater amplitude
–  They add together
Destructive interference
–  Waves out-of-phase lead to lower amplitude
–  They cancel out
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 3 Particle-Wave
Duality
Are Electrons Waves or Particles?
Light behaves like both a particle and a wave:
–  Exhibits interference
–  Has particle-like nature
When studying behavior of electrons:
–  Known to be particles
–  Also demonstrate interference
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 4 Particle-Wave
Duality
Standing vs Traveling Waves
Traveling wave
–  Produced by wind on surfaces
of lakes and oceans
Standing wave
–  Produced when guitar string
is plucked
–  Center of string vibrates
–  Ends remain fixed
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 5 Particle-Wave
Duality
Standing Wave on a Wire
•  Integer number (n) of peaks and troughs is required •  Wavelength is quanMzed: •  L is the length of the string 2L
!=
n
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 6 Particle-Wave
Duality
Standing Wave on a Wire
•  Has both wave-­‐like and parMcle-­‐like properMes •  Energy of moving electron on a wire is E =½ mv 2 •  Wavelength is related to the quantum number, n, and the wire length: 2L
!=
n
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 7 Particle-Wave
Duality
Electron on a Wire
Standing wave •  Half-­‐wavelength must occur integer number of Mmes along wire s length 2L
!=
n
de Broglie s equa3on relates the mass and speed of the parMcle to its wavelength h
h
!
=
v
=
mv
!m
•  m = mass of parMcle •  v = velocity of parMcle Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 8 Particle-Wave
Duality
Electron on a Wire
StarMng with the equaMon of the standing wave and the de Broglie equaMon 2L
v= h
!=
n
!m
Combining with E = ½mv 2, subsMtuMng for v and then λ, we get !
$
"
h#
" h#
#
# #
!E = m # # =
#
&
"
h
n
h
# !m
# m! #
!E = #
&=
#
# # m #L / n & $mL#
"
%
Combining gives: (
2
)
2
nh
E =
8mL2
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 9 Particle-Wave
Duality
De Broglie & Quantized Energy
•  Electron energy quanMzed –  Depends on integer n •  Energy level spacing changes when posiMve charge in nucleus changes –  Line spectra different for each element •  Lowest energy allowed is for n =1 •  Energy cannot be zero, hence atom cannot collapse Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 10 Particle-Wave
Duality
Ex: Wavelength of an Electron
What is the de Broglie wavelength associated with an electron of mass 9.11 × 10 –31 kg traveling at a velocity of 1.0 × 107 m/s? 6.626 × 10 J s
1 kg m /s
λ=
×
(1.0 × 10 m/s)(9.11 × 10 kg)
1J
−34
7
2
2
−31
λ = 7.27 × 10–11 m
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 11 Particle-Wave
Duality
Ex: Wavelength of an Electron
Calculate the de Broglie wavelength of a baseball with a mass of 0.10 kg and traveling at a velocity of 35 m/s. A.  1.9 × 10–35 m B.  6.6 × 10–33 m # !.!"! ! #$"%& '() & # #(-.(+" ,)" &
C.  1.9 × 10–34 m (( ! %%
((
! = %%
D.  2.3 × 10–33 m %*(+,) ! $.#$(-. ' $
#('
$
'
–31
E.  2.3 × 10 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 12 Particle-Wave
Duality
Wave Functions
Schrödinger s equa3on –  SoluMons give wave funcMons and energy levels of electrons Wave func3on –  Wave that corresponds to electron –  Called orbitals for electrons in atoms Amplitude of wave funcMon squared –  Can be related to probability of finding electron at that given point Nodes –  Regions where electrons will not be found Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 13 Quantum
Numbers
Orbitals Characterized by 3 Quantum #’s
Quantum Numbers: –  Shorthand –  Describes characterisMcs of electron s posiMon –  Predicts its behavior n = principal quantum number –  All orbitals with same n are in same shell ℓ = secondary quantum number –  Divides shells into smaller groups called subshells mℓ = magne3c quantum number –  Divides subshells into individual orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 14 Quantum
Numbers
Principal Quantum Number (n)
•  Allowed values: posiMve integers from 1 to ∞ –  n = 1, 2, 3, 4, 5, … ∞ •  Determines: –  Size of orbital E =!
Z 2RH hc
n2
–  Total energy of orbital •  RHhc = 2.18 × 10–18 J/atom •  For given atom, –  Lower n = Lower (more negaMve) E = More stable Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 15 Quantum
Numbers
Orbital Angular Momentum (ℓ)
–  Allowed values: 0, 1, 2, 3, 4, 5…(n – 1) –  Le>ers: s, p, d, f, g, h Orbital designa3on number nℓ le>er •  Possible values of ℓ depend on n –  n different values of ℓ for given n •  Determines •  Shape of orbital Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 16 Quantum
Numbers
Magnetic Quantum Number (mℓ)
•  Allowed values: from –ℓ to 0 to +ℓ –  Ex. when ℓ=2 then mℓ can be •  –2, –1, 0, +1, +2 •  Possible values of mℓ depend on ℓ –  There are 2ℓ+1 different values of mℓ for given ℓ •  Determines orientaMon of orbital in space •  To designate specific orbital, you need three quantum numbers –  n, ℓ, mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 17 Quantum
Numbers
n, ℓ, and mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 18 Quantum
Numbers
Multiple Electrons in Orbitals
Orbital Designation
§  Based on first two
quantum numbers
§  Number for n and
letter for ℓ
§  How many electrons
can go in each
orbital?
§  Two electrons
§  Need another
quantum number
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 19 Quantum
Numbers
Spin Quantum Number (ms)
•  Arises out of behavior of electron in magneMc field •  electron acts like a top •  Spinning charge is like a magnet –  Electron behave like Mny magnets •  Leads to two possible direcMons of electron spin –  Up and down –  North and south Possible Values:
+½
↑
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E -½
↓
20 Quantum
Numbers
Pauli Exclusion Principle
•  No two electrons in same atom can have same set of all four quantum numbers (n, ℓ, mℓ , ms) Can only have two electrons per orbital •  Two electrons in same orbital must have opposite spin –  Electrons are said to be paired Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 21 Quantum
Numbers
Number of Orbitals
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 22 Quantum
Numbers
Magnetic Properties
•  Two electrons in same orbital have different spins –  Spins paired—diamagne3c –  Sample not a>racted to magneMc field –  MagneMc effects tend to cancel each other •  Two electrons in different orbital with same spin –  Spins unpaired—paramagne3c –  Sample a>racted to a magneMc field –  MagneMc effects add •  Measure extent of a>racMon –  Gives number of unpaired spins Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 23 Quantum
Numbers
Ex: Number of Electrons
What is the maximum number of electrons allowed in a set of 4p orbitals? A.  14 B.  6 C.  0 D.  2 E.  10 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 24 Electron
Configurations
Ground State
Electron Configura3ons • 
DistribuMon of electrons among orbitals of atom 1.  List subshells that contain electrons 2.  Indicate their electron populaMon with superscript e.g. N is 1s 2 2s 2 2p 3 Orbital Diagrams • 
Way to represent electrons in orbitals 1.  Represent each orbital with circle (or line) 2.  Use arrows to indicate spin of each electron e.g. N is 1s
2s
2p
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 25 Electron
Configurations
6s
5s
Energy Level Diagram
4f
5p
4d
4p
3d
4s
Energy
3p
3s
2p
2s
§  How to put electrons into a diagram?
§  Need some rules
1s
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 26 Electron
Configurations
Aufbau Principle
•  Building-­‐up principle Pauli Exclusion Principle •  Two electrons per orbital •  Fill following the order suggested by the periodic table •  Spins must be paired Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 27 Electron
Configurations
Hund’s Rule
•  If you have more than one orbital all at the same energy –  Put one electron into each orbital with spins parallel (all up) unMl all are half filled –  Aler orbitals are half full, pair up electrons Why? •  Repulsion of electrons in same region of space •  Empirical observaMon based on magneMc properMes Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 28 Electron
Configurations
Orbital Diagram & Electron
Configurations: e.g. N, Z = 7
4p
3d
4s
Energy
3p
3s
2p
2s
Each arrow represents electron
1s 2 2s 2 2p 3
1s
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 29 Electron
Configurations
Orbital Diagram and Electron
Configurations: e.g. V, Z = 23
4p
3d
4s
Energy
3p
3s
2p
2s
Each arrow represents an electron
1s 2 2s 2 2p 2 3s 2 3p 2 4s 2 3d 3
1s
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 30 Electron
Configurations
Ex: Orbital Diagrams &
Electron Configurations
Give electron configurations and orbital diagrams for Na and As
6s
5s
5p
4d
4p
3d
4s
Energy
3p
3s
2p
2s
Na Z = 11
1s 2 2s 2 2p 2 3s 1
As Z = 33
1s
1s 22s 22p 63s 23p 64s 23d 104p 3
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 31 Electron
Configurations
Ex: Ground State Electron
Configurations
What is the correct ground state electron configuraMon for Si? A.  1s 22s 22p 63s 23p 6 B.  1s 22s 22p 63s 23p 4 C.  1s 22s 22p 62d 4 D.  1s 22s 22p 63s 23p 2 E.  1s 22s 22p 63s 13p 3 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 32 Problem
Set B