Daniel L. Reger
Scott R. Goode
David W. Ball
www.cengage.com/chemistry/reger
Chapter 7
Electronic Structure
Waves
• Waves are periodic disturbances –
they repeat at regular intervals of time
and distance.
Properties of Waves
• Wavelength (l) is the distance between
one peak and the next.
• Frequency (n) is the number of waves
that pass a fixed point each second.
Electromagnetic Radiation
• Light or electromagnetic radiation
consists of oscillating electric and
magnetic fields.
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  ln  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.
Kinds of Electromagnetic Radiation
• Visible light is only a very small portion
of the electromagnetic spectrum.
• Other names for regions are gamma rays,
x rays, ultraviolet, infrared, microwaves,
radar, and radio waves.
Quantization of Energy
• In 1900, Max Planck proposed that there
is a smallest unit of energy, called a
quantum. The energy of a quantum is
E  hn
where h is Planck’s constant, 6.626×1034 J·s.
The Photoelectric Effect
• The photoelectric effect: the process in
which electrons are ejected from a metal
when it is exposed to light.
• No electrons are ejected by light with a
frequency lower than a threshold frequency,
n0.
• At frequencies higher than n0, kinetic energy
of ejected electron is hn – hn0.
Photoelectric Effect (cont.)
• Einstein suggested an explanation by
assuming light is a stream of particles
called photons.
• The energy of each photon is given by
Planck’s equation, E = hn.
• The minimum energy needed to free an
electron is hn0.
• Law of conservation of energy means
that the kinetic energy of ejected
electron is hn – hn0.
Dual Nature of Light?
• Is light a particle, or is it a wave?
• 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.
• Gaseous atoms produce a line spectrum
– one that contains light only at specific
wavelengths and not at others.
Line Spectra of Some Elements
The Rydberg Equation
• Study of the spectrum of hydrogen,
the simplest element, show that the
wavelengths of lines of light can be
calculated using the Rydberg
equation:
 1
1 
 RH  2  2 
l
 n1 n2 
1
• n1 and n2 are whole numbers and RH
= 1.097×107 m-1.
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
• 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:
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 hn.
• This assumption leads directly to the
Rydberg equation.
Hydrogen Atom Energy Diagram
Matter as Waves
• Louis de Broglie proposed that matter
might be viewed as waves as well as
particles.
• de Broglie suggested that the
wavelength of matter is given by
h
h
l 
p mv
where h is Planck’s constant, p is
momentum, m is mass, and v is velocity.
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.
• What is the wavelength of a marathon runner
moving at a speed of 5 m/s?
(mass of the runner is 52 kg)
Uncertainty
• Heisenberg showed that the more precisely the
momentum of a particle is known, the less precisely is
its position known:
(x) (mv) 
h
4
Cannot know precisely where and with what momentum
an electron is.
New ideas for determining this information based on
probability
Quantum Mechanics was born
Standing Waves
• 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
• The de Broglie wave of an electron in a
hydrogen atom must be a standing wave,
restricting its wavelength to values of l =
2r/n, with n being an integer.
• This leads directly to quantized angular
momentum, one of Bohr’s assumptions.
Schrödinger Wave Equation
• The 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.
• 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.
Quantum Numbers in the H Atom
• The solution of the Schrödinger equation
produces quantum numbers that
describe the characteristics of the electron
wave.
• Three quantum numbers, represented by
n, l, and ml, describe the distribution of
the electron in three dimensional space.
• An atomic orbital is a wave function of
the electron for specific values of n, l, and
ml.
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.
Angular Momentum Quantum Number, l
• The angular momentum quantum
number, l, is associated with the shape of
the orbital.
• Allowed values: 0 and all positive integers up
to n-1.
• The l quantum number can never equal or
exceed the value of n.
• A subshell is all possible orbitals that have
the same values of both n and l.
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
• The magnetic quantum number,
ml, indicates the orientation of the
atomic orbital in space.
• Allowed values: all whole numbers from
–l to l, including 0.
• A wave function described by all
three quantum numbers (n, l, ml) is
called an orbital.
Allowed Combinations of n, l, ml
n
l
ml
# orbitals
1
0
0
1
2
0
1
0
-1, 0, +1
1
3
3
0
1
2
0
-1, 0, +1
-2, -1, 0, +1, +2
1
3
5
4
0
1
2
3
0
-1, 0, +1
-2, -1, 0, +1, +2
-3, -2, -1, 0, +1, +2, +3
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
Electron Spin
• An electron behaves as a small
magnet that is visualized as coming
from the electron spinning.
• The electron spin quantum number,
ms, has two allowed values: +1/2 and
-1/2.
Electron Density Diagrams
• Different densities of dots or colors are
used to represent the probability of
finding the electron in space.
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 Atom 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 are
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.).
Effective Nuclear Charge
• In multielectron atoms, the energy
dependence on nuclear charge must
be modified to account for
interelectronic repulsions.
• The effective nuclear charge 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 of
3+.
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.
Multielectron Energy Level Diagram
• Within any principal
shell, the energy
increases in the
order of the l
quantum number:
4s < 4p < 4d < 4f.
Increasing Energy Order
• Based on experimental observations,
subshells are usually occupied in the
order
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p
< 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s
< 5f < 6d
Electrons in Multielectron Atoms
• Each electron in a multielectron atom
can be described by hydrogen-like wave
functions by assigning values to the four
quantum numbers n, l, ml, and ms.
• These wavefunctions differ from those in
the hydrogen atom because of
interelectronic repulsions.
• The energy of these wave functions
depends on both n and l.
Pauli Exclusion Principle
• The Pauli Exclusion Principle: no
two electrons in the same atom can
have the same set of four quantum
numbers.
• A difference in only one of the four
quantum numbers means that the sets
are different.
The Aufbau Principle
• The aufbau principle: as electrons
are added to an atom one at a time,
they are assigned the quantum
numbers of the lowest energy orbital
that is available.
• The resulting atom is in its lowest
energy state, called the ground state.
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.
↑↓
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”).
Electron Configurations of Elements
• Hydrogen contains one electron in
the 1s subshell.
1s1
↑
• Helium has two electrons in the 1s
subshell.
1s2
↑↓
Electron Configurations of Elements
• Lithium has three electrons.
1s2 2s1
↑↓
↑
• Beryllium has four electrons.
1s2 2s2
↑↓
↑↓
• Boron has five electrons.
1s2 2s2 2p1
↑↓
↑↓
↑
Orbital Diagram of Carbon
• Carbon, with six electrons, has the
electron configuration of 1s2 2s2 2p2.
• The lowest energy arrangement of
electrons in degenerate (same-energy)
orbitals is given by Hund’s rule: one
electron occupies each degenerate orbital
with the same spin before a second
electron is placed in an orbital.
↑↓
↑↓
↑
↑
Other Elements in the Second Period
• N 1s2 2s2 2p3
↑↓
↑↓
↑
↑
↑
• O 1s2 2s2 2p4
↑↓
↑↓
↑↓ ↑
↑
• F
1s2 2s2 2p5
↑↓
↑↓
↑↓ ↑↓ ↑
• Ne 1s2 2s2 2p6
↑↓
↑↓
↑↓ ↑↓ ↑↓
Electron Configurations of Heavier Atoms
• Heavier atoms follow aufbau principle
in organization of electrons.
• Because their electron configurations
can get long, larger atoms can use an
abbreviated electron configuration,
using a noble gas to represent core
electrons.
Fe: 1s2 2s2 2p6 3s2 3p6 4s2 3d6 → [Ar] 4s2 3d6
Ar
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.