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Daniel L. Reger
Scott R. Goode
David W. Ball
www.cengage.com/chemistry/reger
Lecture 07 (Chapter 7)
Electronic Structure
Core Concepts
• Wave nature of light
• Relationship between wavelength and frequency
•
•
•
•
•
EM Spectrum
Plancks quantization of energy
Photoelectric effect
Atomic line spectra and energy levels (Bohr)
Quantum Mechanics
• Heisenberg’s Uncertainty Principle
• Quantum numbers of electrons
Constants, Variables, Equations
•
•
•
•
•
•
•
λ = lambda (wavelength)
υ = upsilon (speed, de Broglie)
ν = nu (frequency)
h = Planck’s constant (6.626 x 10-34 J*s)
RH = 2.179 x 10-18 J
Mass of electron m=9.1 x 10-31 kg
c = wave speed of light (3.0 x 108 m/s)
c  
E  h
RH
E 2
n
h

m
For n = 1, 2 ,3,…∞
(x)(px )  h 4
Energy and stability of the atom
– Ernest Rutherford nuclear model (1911): a nucleus with
mass surrounded by electrons with little mass.
– Problem: according to current theory at that time, the
atoms would essentially collapse as flying electrons lost
energy.
– How can the atom remain stable over time?
-
-
+
-
-
How do electrons in atom remain stable over
time?
To answer this question, we need to understand 2
important aspects associated with electrons and their
place within the atom
• Quantization of energy
• Wave-Particle duality
Waves and Wave Properties
• Waves are periodic disturbances (through matter or a
physical field) that repeat at regular intervals of time and
distance.
• Wavelength () is the distance between one peak and
the next.
• Frequency () is the Number of wavelengths that pass
through a fixed point in a given unit of time (e.g., 1s or
Hz).
Waves
• Waves are periodic disturbances (through
matter or a physical field) that repeat at regular
intervals of time and distance.
Wavelength and Frequency
1 Hz
2 Hz
4 Hz
8 Hz
16 Hz
Time = 1s
http://www.physicsforums.com/showthread.php?t=57843
Electromagnetic Radiation
• Light or electromagnetic radiation consists of
oscillating electric and magnetic fields.
Electromagnetic Spectrum
• Light is visible portion of a much larger energy
continuum…
Speed of Light
• All electromagnetic waves travel at the same speed
in a vacuum, 3.00×108 m/s.
• The speed of a wave is the product of its frequency
and wavelength, so for light:
c    3.00  10 m/s
8
• So, if either the wavelength or frequency is known,
the other can be calculated.
Example: Electromagnetic Radiation
• An FM radio station broadcasts at a frequency of
100.3 MHz (1 Hz = 1 s-1). Calculate the
wavelength of this electromagnetic radiation.
Example: Electromagnetic Radiation
• An FM radio station broadcasts at a frequency of
100.3 MHz (1 Hz = 1 s-1). Calculate the
wavelength of this electromagnetic radiation.
Quantization of Energy
• At T > 0 K, matter emits EMR across EM spectrum.
• For visible light, different colors indicate differences in energy.
• In 1900, Max Planck, working with heated solids, observed
color changes with temperature increases.
• Planck theorized that atoms of the solid oscillate with a definite
frequency (ν), but atoms could have only certain energies of
vibrations (i.e., they are quantized into discrete levels).
• He then proposed a smallest unit of energy, called a quantum,
and this energy is represented by Planck’s equation:
E  h
where h is Planck’s constant, 6.626×10-34 J·s, and ν is the
frequency.
The Photoelectric Effect
• Planck’s experiments described the photoelectric effect: electrons
are ejected from a metal when it is exposed to light.
• All metals have a threshold frequency, 0, so if the frequency of
light is less than the threshold, no matter how intense the light, no
electrons are ejected.
• If the light frequency exceeds the threshold frequency, 0,
electrons are ejected, and increasing the light intensity increases
the number of electrons released.
Photoelectric Effect (cont.)
• Einstein applied Planck’s equation to the photoelectric
effect, and suggested that light can behave as a stream
of particles (photons) in addition to behaving as a wave.
• The energy of each photon is given by Planck’s equation,
E = h.
• The minimum energy needed to free an electron is h0.
• The energy of the photon (hv) required to dislodge an
electron had to be equal to the energy required to
separate the electron from the solid (hv0), and the KE of
the electron (conservation of energy).
Dual Nature of Light
Wave-particle duality
• Light has both particle and wave properties,
depending on the property.
• Particle behavior, wave behavior no longer
considered to be exclusive from each other.
Spectra
• A spectrum is a graph of light intensity as a function of
wavelength or frequency.
• The light emitted by heated objects is a continuous
spectrum; light of all wavelengths is present.
• When energy in the form of heat or electric discharge is added
to sample of gaseous atoms (excitation), the atoms emit some
of the added energy in the form of light (releasing photons).
• This produces a line spectrum that contains light only at
specific wavelengths and not at others (see figure on next
slide), and let’s us identify specific elements based on
these spectra.
Line Spectra of Some Elements
The Rydberg Equation
• In 1890, J. R. Rydberg studied the spectrum of
hydrogen, the simplest element, and determined that the
wavelengths of lines of the spectrum could be calculated
using the Rydberg equation:
• n1 and n2 are positive integers (n1 < n2) and RH =
1.097×107 m-1.
• At the time, Rydberg did NOT know the meaning of the
whole numbers for n1 and n2, but determined numbers
empirically.
Example: Rydberg Equation
• Calculate the wavelength (in nm) of the line in the hydrogen
atom spectrum for which n1 = 2 and n2 = 3.
The Bohr Model of Hydrogen
• Once relationship between light energy and frequency was
established, the spectra suggested that electrons were allowed only
in certain discrete energy levels.
• In 1911, Niels Bohr proposed a model for the hydrogen atom, based
on its spectrum.
• Bohr assumed:
• that the electron followed a circular orbit about the nucleus; and
• that the angular momentum of the electron was quantized.
• Using these assumptions, he found that the energy of the electron
was quantized, and the allowed energy levels could be calculated
using the following equation:
2 2me4  1 
B
18
En  


,
B

-2.18

10
J

2
2 
2
h
n
n 
Bohr Model and the Rydberg Equation
• Assume that when one electron transfers from
one orbit to another, energy must be added or
removed by a single photon with energy h, and
this energy is exactly equal to the difference
between energies of 2 allowed levels.
• This assumption leads directly to the Rydberg
equation.
Hydrogen Atom Energy Diagram
Hydrogen spectra have
been collected using
different types of EMR,
resulting in different
series describing
excitation of hydrogen
electron, and transitions
between allowed levels.
We now know that
electrons can absorb a
photon whose E is
equivalent to difference
between any 2 levels.
Do particles exhibit wave properties?
• 1923, French physicist Louis de Broglie showed that electrons form
standing waves.
• Only certain discrete rotational frequencies about the nucleus of an
atom were allowed.
• These quantized orbits correspond to discrete energy levels.
De Broglie Relation
(wavelength of a particle)
Wavelength is inversely
proportional to mass
h

m
momentum
• 1927, Davidson (USA), Germer (USA) and Thomson (UK) demonstrated
that beam of electrons could be diffracted by crystal (wave properties).
Example: de Broglie Wavelength
• At room temperature, the average speed of an electron
is 1.3×105 m/s. The mass of the electron is about
9.11×10-31 kg. Calculate the wavelength of the electron
under these conditions.
h = 6.626×10-34 J·s
Standing Waves
• De Broiglie’s equation suggested that electron
wave must be a standing wave (stays in a constant
position)
• The vibration of a string is restricted to certain
wavelengths because the ends of the string cannot
move.
de Broglie Waves in the H Atom
• Since the de Broglie wave of an electron in a hydrogen atom
must be a standing wave, its wavelengths are restricted to
values of  = 2r/n, with n being an integer, and 2r
representing the circumference of a Bohr orbit, based on
quantized angular momentum.
• If the electron were not in a standing wave, it would partially
cancel itself with each successive orbit (b), until the amplitude
was zero.
Schrödinger Wave Equation
• Erwin Schrödinger (1887-1961) developed the current model to
describe an electron wave (but we are saving the gory details for
P-chem!).
• This wave function (Y) gives the amplitude of the electron wave
at any point in space.
• Y2 gives the probability of finding the electron at any point in
space, with the electron acting like a charged cloud surrounding
the atom.
• There are many acceptable wave functions for the electron in a
hydrogen (or any other) atom.
• The energy of each wave function can be calculated, and these
are identical to the energies from the Bohr model of hydrogen.
• This model lets us understand properties of electrons in atoms
other than hydrogen.
Heisenberg’s Uncertainty Principle
• QM allows us to make statistical projections, but
won’t allow us to describe the electron in the
hydrogen atom as moving in an orbit.
• 1927, Werner Heisenberg demonstrated that it is
not possible to know simultaneously the precise
position and momentum of a particle (e.g., an
electron).
(x)(px )  h 4
p  m
(x)( x )  h 4 m
The less mass an object (e.g.,
particle) has, the more uncertainty
Quantum Numbers in the H Atom
• The solution of the Schrödinger equation produces
quantum numbers that describe the characteristics of
the electron wave function.
• According to QM each electron in atom is described by 4
quantum numbers, (n, l, ml, ms).
• The quantum numbers n, l, and ml, describe the
distribution of the electron in three dimensional space
(i.e., the atomic orbital).
• The atomic orbitals have definite shapes and
orientations.
• Each orbital can contain only 2 electrons, with different
spin values ( the quantum number ms).
The Principal Quantum Number, n
• The principal quantum number, n, provides
information about the energy and the distance of
the electron from the nucleus.
• Allowed value of n are 1, 2, 3, 4, …
• The larger the value of n, the greater the
average distance of the electron from the
nucleus.
• The term principal shell (or just shell) refers to
all atomic orbitals that have the same value of n.
Principle quantum number (n)
• Provides information about the energy and the
distance of the electron from the nucleus.
• n is a positive integer value (1, 2, 3, 4 …).
• Energy of electron is most dependent upon this
number.
• Orbital size increases as n increases.
• Orbitals of the same quantum state n belong to
the same shell (3s and 3p).
Letter
n
K
1
L
2
M
3
N
4
…
…
Angular momentum quantum number (l)
• Distinguishes orbitals of given n having different
shapes.
• l can have any integer value from 0 to n – 1, but
cannot be equal to n.
• Within each n shell, there are n different kinds of
orbitals with distinct shapes.
• Orbitals of the same state n but different state l
are subshells.
Notations for Subshells
• To identify a subshell, values for both n and l must
be assigned, in that order.
• The value of l is represented by a letter:
l
0
1
2
3
4
5
etc.
letter
s
p
d
f
g
h
etc.
• Thus, a 3p subshell has n = 3, l = 1.
• A 2s subshell has n = 2, l = 0.
Magnetic quantum number (ml)
• Distinguishes orbitals of given n and l but having
different orientation in space.
• ml has 2l + 1 degenerate orbitals in each l subshell.
• Allowed values for ml range from -l to +l.
• Example: For l = 1 (p subshell), ml = -1, 0, +1.
• These are 3 different orbitals in p subshell with the same
shape but different spatial orientations.
Atomic Orbitals
s subshell (l=0)
1 orbital
(ml = 0)
p subshell (l=1)
3 orbitals
(ml = -1, 0, +1)
d subshell (l=2)
5 orbitals
(ml = -2, -1, 0, +1, +2)
http://www.chemcomp.com/journal/molorbs.htm
Allowed Combinations of n, l, ml
• l can have any integer value from 0 to n – 1, but cannot be
equal to n.
• ml has 2l + 1 degenerate orbitals in each l subshell.
n
1
2
2
3
3
3
4
4
4
4
l
0
0
1
0
1
2
0
1
2
3
ml*
0
0
-1, 0, +1
0
-1, 0, +1
-2, -1, 0, +1, +2
0
-1, 0, +1
-2, -1, 0, +1, +2
-3, -2, -1, 0, +1, +2, +2
Subshell Orbitals in
Notation Subshell
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
1
1
3
1
3
5
1
3
5
7
Example: Quantum Numbers
• Give the notation for each of the following orbitals if
it is allowed. If it is not allowed, explain why.
(a) n = 4, l = 1, ml = 0
(b) n = 2, l = 2, ml = -1
(c) n = 5, l = 3, ml = +3
Test Your Skill
• For each of the following subshells, give the
value of the n and the l quantum numbers.
(a) 2s
(b) 3d
(c) 4p
Test Your Skill
• For each of the following subshells, give the
value of the n and the l quantum numbers.
(a) 2s
(b) 3d
(c) 4p
Answers:
(a) n = 2, l = 0
(b) n = 3, l = 2
(c) n = 4, l = 1
Spin quantum number (ms)
• There are two possible orientations of the spin axis of an electron:
+ ½ or -½.
• Spin is an intrinsic property that generates a circulating
electromagnetic field (i.e., electron behaves like a magnet with
north and south poles).
Spin is important because it plays a
critical role in electron pairing in
orbitals (opposite spins are paired).
Pauli Exclusion Principle
No two electrons in a single atom
can have the same four quantum
numbers; if n, l, and ml are the
same, ms must be different such that
the electrons have opposite spins.
Visualizing Orbitals: Electron Density Diagrams
• Different densities of dots or colors are used to
represent the probability of finding the electron in
space.
Visualizing Orbitals: Contour Diagrams
• In a contour diagram, a surface is drawn that
encloses some fraction of the electron probability
(usually 90%).
Shapes of p Orbitals
• p orbitals (l = 1) have two lobes of electron density
on opposite sides of the nucleus.
Orientation of the p Orbitals
• There are three p orbitals in each principal shell
with an n of 2 or greater, one for each value of ml.
• They are mutually perpendicular, with one
each directed along the x, y, and z axes.
Shapes of the d Orbitals
• The d orbitals have four lobes where the electron
density is high.
• The dz2 orbital is mathematically equivalent
to the other d orbitals, in spite of its
different appearance.
Energies of Hydrogen Atomic Orbitals
• The energies of the
hydrogen atom
orbitals depend only
on the value of the n
quantum number.
• The s, p, d, and f
orbitals in any
principal shell have
the same energies.
Other One-Electron Systems
• The energy of a one-electron species also depends
on the value of n, and is given by the equation
Z 2B
2.18  1018 Z 2
En   2  
joules
2
n
n
where Z is the charge on the nucleus.
• This equation applies to all one-electron species (H,
He+, Li2+, etc.).
• Please see example problem 7.7, p. 269.
Effective Nuclear Charge
• In multielectron atoms, the energy dependence
on nuclear charge must be modified to account
for interelectronic repulsions (electrons repulse
each other to an extent, as they all have
negative charges).
• The effective nuclear charge (Zeff) is a
weighted average of the nuclear charge that
affects an electron in the atom, after correction
for the shielding by inner electrons and
interelectronic repulsions.
Effective Nuclear Charge
• Electron shielding is the
result of the influence of
inner electrons on the
effective nuclear charge.
• The effective nuclear
charge that affects the
outer electron in a lithium
atom is considerably less
than the full nuclear
charge in the nucleus.
http://crescentok.com/staff/jaskew/isr/tigerchem/element/family10.htm
Energy Dependence on l
• The 2s electron
penetrates the electron
density of the 1s
electrons more than the
2p electrons, giving it a
higher effective nuclear
charge and a lower
energy.
2p
3s
1s
2s
Multielectron Energy Level Diagram
• Within any principal shell,
the energy increases in
the order of the l
quantum number: 4s <
4p < 4d < 4f.
Orbital Diagrams
• An orbital diagram represents each orbital with
a box, with orbitals in the same subshell in
connected boxes; electrons are shown as
arrows in the boxes, pointing up or down to
indicate their spins.
• Two electrons in the same orbital must have
opposite spins.
↑↓
Why do atoms fill orbitals the way they do?
Aufbau Principle
Electrons fill orbitals starting at the lowest
available (possible) energy states before
filling higher states (e.g. 1s before 2s).
Sometimes a low energy subshell has lower
energy than upper subshell of preceding
shell (e.g., 4s fills before 3d).
Pauli exclusion principle
QM principle that no two identical fermions
(particles with half-integer spin) may occupy
the same quantum state simultaneously
(why paired electrons have different spin).
Hund's rule
Every orbital in a subshell is singly occupied
with one electron before any one orbital is
doubly occupied, and all electrons in singly
occupied orbitals have the same spin.
2p
Energy
2s
1s
How to Get Aufbau from Periodic Table
1
1
2
For n=2, can only have 2
subshells (s, p)
3
2
3
4
3
4
5
4
5
6
5
6
7
6
7
4
5
For n=3, can only have 3
subshells (s, p, d)
Electron Configuration
• An electron configuration lists the occupied
subshells using the usual notation (1s, 2p, etc.).
Each subshell is followed by a superscripted
number giving the number of electrons present
in that subshell.
• Two electrons in the 2s subshell would be 2s2
(spoken as “two-ess-two”).
• Four electrons in the 3p subshell would be
3p4 (“three-pea-four”).
Other Elements in the Second Period
• N 1s2 2s2 2p3
↑↓
↑↓
↑
↑
↑
• O 1s2 2s2 2p4
↑↓
↑↓
↑↓ ↑
↑
• F
1s2 2s2 2p5
↑↓
↑↓
↑↓ ↑↓ ↑
• Ne 1s2 2s2 2p6
↑↓
↑↓
↑↓ ↑↓ ↑↓
Electronic Configuration
Z
Symbol Electronic Configuration
Short Version
4
Be
1s22s2
[He]2s2
17
Cl
1s 22s 22p 63s 23p 5
[Ne]3s 23p 5
39
Y
1s 22s 22p 63s 23p 64s 23d 104p 65s 24d 1
[Kr]5s 24d 1
Noble gas configurations can
be used to write electronic
configurations in an
abbreviated form in which the
noble gas symbol enclosed in
brackets is used to represent
all electrons found in the
noble gas configuration.
Seager SL, Slabaugh MR, Chemistry for Today: General, Organic and Biochemistry, 7 th Edition, 2011
Anomalous Electron Configurations
• The electron configurations for some atoms do
not strictly follow the aufbau principle; they are
anomalous.
• Cannot predict which ones will be anomalous.
• Example: Ag predicted to be
[Kr] 5s2 4d9; instead, it is
[Kr] 5s1 4d10.