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The Quantum
Mechanical Atom
CHAPTER 7
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 Electromagnetic
Radiation
Light Energy is a Wave
Electromagne,c Spectrum Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 3 Electromagnetic
Radiation
Light Energy is a Wave
Waves travel through space at speed of light in vacuum
c = speed of light = 2.9979 × 108 m/s
Can define waves as systematic fluctuations in
intensities of electrical and magnetic forces that vary
regularly with time and exhibit a wide range of energy.
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 4 Electromagnetic
Radiation
Light Energy is a Wave
Wavelength (λ)
–  Distance between two successive peaks or troughs
–  Units are in meters, centimeters, nanometers
Frequency (ν)
–  Number of waves per second that pass a given
point in space
–  Units are in Hertz (Hz = cycles/sec = 1/sec = s–1)
Related by λ × ν = c
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 5 Electromagnetic
Radiation
Light Energy is a Wave
Amplitude –  Maximum and minimum height –  Intensity of wave, or brightness –  Varies with Kme as travels through space Nodes –  Points of zero amplitude –  Place where wave goes though axis –  Distance between nodes is constant Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 6 Electromagnetic
Radiation
Ex: Converting between
Wavelengths and Frequency
Example: The bright red color in fireworks is due to emission of light when Sr(NO3)2 is heated. If the wavelength is ~650 nm, what is the frequency of this light? c 3.00 ! 108 m/s
!= =
"
650 ! 10"9 m
ν = 4.61 × 1014 s–1 = 4.6 × 1014 Hz
Example: WCBS broadcasts at a frequency of 880 kHz. What is the wavelength of their signal? c !."" ! #"$ %&'(
!= =
!
!
$$" ! #" / (
= 341 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 7 Electromagnetic
Radiation
Electromagnetic Spectrum
low energy, long waves
high energy, short waves
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 8 Electromagnetic
Radiation
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E Electromagnetic Spectrum
9 Electromagnetic
Radiation
Electromagnetic Spectrum
Visible light •  Band of wavelengths that human eyes can see •  400 to 700 nm make up spectrum of colors •  White light is a combinaKon of all these colors and can be separated into individual colors with a prism. Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 10 Electromagnetic
Radiation
Particle Theory of Light
Max Planck and Albert Einstein (1905) •  ElectromagneKc radiaKon is stream of small packets of energy •  Quanta of energy or photons •  Each photon travels with velocity = c •  Waves with frequency = ν Energy of photon of electromagneKc radiaKon is proporKonal to its frequency •  Energy of photon E = h ν •  h = Planck’s constant = 6.626 × 10–34 J s Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 11 Electromagnetic
Radiation
Ex: Determining Energy from
Frequency
Example: A microwave oven uses radiaKon with a frequency of 2450 MHz (megahertz, 106 s–1) to warm up food. What is the energy of such photons in joules? E = h!
# 1 ! 106 s "1 &
((
E = 6.626 ! 10"34 J s ! 2450 MHz ! %%
$ MHz '
(
) (
)
= 1.62 × 10–24 J Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 12 Electromagnetic
Radiation
Photoelectric Effect
If shine light on a metal surface: •  Below a certain frequency nothing happens •  Above a certain frequency electrons are ejected •  Increasing intensity increases # of electrons ejected •  Increasing frequency increases KE of electrons KE = hν – BE hν = energy of light shining on surface BE = binding energy of electron Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E h>p://hyperphysics.phy-­‐astr.gsu.edu/hbase/mod1.html 13 Electromagnetic
Radiation
Photoelectric Effect
Therefore Energy is Quan,zed •  Can occur only in discrete units of size hν •  1 photon = 1 quantum of energy •  Energy gained or lost in whole number mulKples of hν E = nhν •  If n = NA, then one mole of photons gained or lost E = 6.02 × 1023 hν If light is required to start reacKon •  Must have light above certain frequency to start reacKon •  Below minimum threshold energy, intensity is NOT important Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 14 Electromagnetic
Radiation
Ex: Energy, Frequency & Moles
Example: If a mole of photons has an energy of 1.60 × 10–3 J/mol, what is the frequency of each photon? Assume all photons have the same frequency. E
!=
N Ah
!=
!."# ! !#"$ %&'()*
+$
(".#+ ! !# %()* )("."+" ! !#
"!
"$,
%&%-)
= 4.01 × 106 Hz Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 15 Atomic
Spectra
Electronic Structure of the Atom
excited state
Because energy is quanKzed we can study the electronic structure of an atom the frequency of light it absorbs or emits: 1. Study of light absorpKon +hν
ground state
•  Electron absorbs energy •  Moves to higher energy excited state excited state
−hν
2. Study of light emission •  Electron loses photon of light •  Drops back down to lower energy ground state Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E ground state
16 Atomic
Spectra
Spectrum of Light
A con,nuous spectrum of light is an unbroken spectrum of all colors •  i.e., visible light through a prism; sunlight; incandescent light bulb; or a very hot metal rod An atomic spectrum or the light emi>ed by an atom is a discon,nuous (or line) spectrum of light •  A disconKnuous spectrum has only a few discrete lines •  Each element has a unique emission spectrum Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 17 Atomic
Spectra
Spectrum of Light
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 18 Atomic
Spectra
Electronic Structure of the Atom
Hydrogen is the simplest atomic spectra with only one electron Emission: (Hydrogen, Mercury, Neon) AbsorpKon: (Hydrogen) Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E h>p://facstaff.cbu.edu/~jvarrian/252/emspex.html 19 Atomic
Spectra
Rydberg Equation
⎛ 1
⎞
1
1
⎟
= R H ⎜
−
⎜ n 2 n 2 ⎟
λ
2 ⎠
⎝ 1
RH = 109,678 cm–1 = Rydberg constant
λ = wavelength of light emitted
n1 and n2 = whole numbers (integers) from 1 to ∞
where n2 > n1
If n1 = 1, then n2 = 2, 3, 4, …
•  Can be used to calculate all spectral lines of hydrogen •  The values for n correspond to allowed energy levels for atom Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 20 Atomic
Spectra
Ex: Rydberg Equation
Example: Consider the Balmer series where n1 = 2 Calculate λ (in nm) for the transiKon from n2 = 6 down to n1 = 2. "1
"
%
1
1%
1
1
= RH $$ 2 ! 2 '' = 109,678 cm!1 $$ ! '' = 24,373 cm–1
!
6 &
#2
# 4 36 &
!=
1
24,372.9 cm
!1
= 4.1029 " 10
!5
1m
1 nm
cm "
"
100 cm 1 " 10!9 m
λ = 410.3 nm Violet line in spectrum Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 21 Atomic
Spectra
Ex: Rydberg Equation
Example: A photon undergoes a transiKon from nhigher down to n = 2 and the emi>ed light has a wavelength of 650.5 nm? 1 ! 10"7 cm
! = 650.5 nm !
= 650.5 ! 10"7 cm
1 nm
"1 1
1
1 )
=
109,
678
cm
(
"
650.5 ! 10"7 cm
22 (n )2
2
1
=(1 ! 12 )
7.13455
4 (n2 )
( )
n2
2
=
1
= 9.10
0.110
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 1 =1!
1
= 0.110
2
(n2 ) 4 7.13455
n2 = 3
22 Atomic
Spectra
Understanding Atomic Structure
Atomic line spectra tells us when excited atom loses energy •  Only fixed amounts of energy can be lost •  Only certain energy photons are emi>ed •  Electron restricted to certain fixed energy levels in atoms Atomic line spectra tells us Energy of electron is quan,zed and is the simple extension of Planck's Theory Therefore any theory of atomic structure must account for •  Atomic spectra •  QuanKzaKon of energy levels in atom Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>exr, 6E 23 “Quantum”
What do we mean by “Quantized”
•  Energy is quanKzed if only certain discrete values are allowed •  Presence of disconKnuiKes makes atomic emission quanKzed Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 24 Bohr Model
Bohr Model of an Atom
First theoreKcal model of atom to successfully account for Rydberg equaKon •  QuanKzaKon of energy in hydrogen atom •  Correctly explained atomic line spectra Proposed that electrons moved around nucleus like planets move around sun •  Move in fixed paths or orbits •  Each orbit has fixed energy Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 25 Bohr Model
Energy Level Diagram for a Hydrogen Atom
•  AbsorpKon of photon –  Electron raised to higher energy level •  Emission of photon –  Electron falls to lower energy level Energy levels are quantized
• 
• 
• 
Every time an electron drops from one
energy level to a lower energy level
Same frequency photon is emitted
Yields line spectra
RH hc
b
E =! 2 =! 2
n
n
Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 26 Bohr Model
Bohr model of the Hydrogen Atom
•  n = 1 First Bohr orbit –  Most stable energy state equals the ground state which is the lowest energy state –  Electron remains in lowest energy state unless disturbed How to change the energy of the atom? –  Add energy in the form of light: E = hν –  Electron raised to higher n orbit n = 2, 3, 4, … ∞ –  Higher n orbits = excited states = less stable –  So electron quickly drops to lower energy orbit and emits photon of energy equal to ΔE between levels ΔE = Eh – El h = higher l = lower Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 27 Bohr Model
Bohr’s Model Fails
•  Theory could not explain spectra of mulK-­‐electron atoms •  Theory doesn’t explain collapsing atom paradox •  If electron doesn’t move, atom collapses •  PosiKve nucleus should easily capture electron •  VibraKng charge should radiate and lose energy Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 28 Bohr Model
Ex: Bohr’s Model of Energy Levels
Example: In Bohr's atomic theory, when an electron moves from one energy level to another energy level more distant from the nucleus, A.  energy is emi>ed B.  energy is absorbed C.  no change in energy occurs D.  light is emi>ed E.  none of these Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 29 Problem
Set A
1.  Which electromagneKc radiaKon has a higher energy? Radio waves or microwaves? UV light or X rays? 2.  How does thermal imaging work? (Use what you have learned about the electromagneKc spectrum to briefly explain). 3.  Blue, red, and green lasers have wavelengths of 445 nm, 635 nm, and 532 nm respecKvely what are their frequencies, and what is the energy in Joules of a photon from each laser? 4.  In Neon there is a line with the frequency of 4.546 x1014 Hz. What is its wavelength and color of the line? And what is the energy of each of its photons? 5.  What is the wavelength of light (in nm) that is emi>ed when an excited electron in the hydrogen atom falls from n = 5 to n = 3? Would you expect to be able to see the light emi>ed? 6.  How many grams of water could have its temperature raised by 7°C by a mole of photons that have a wavelength of 450 nm?