Quantum Mechanical Theory
Learning Goal: I will understand the
quantum mechanical model of the atom, be
able to determine an atom’s set of quantum
numbers, electron configuration and orbital
diagram
Dalton’s Atomic Theory (1805)
Experimental
Work
Theoretical
Explanation
Law of definite
Each atom has a
proportions: elements particular combining
combine in a
capacity.
characteristic mass
ratio.
Law of multiple
proportions: there
may be more than
one mass ratio.
Some atoms have
more than one
combining capacity.
Law of conservation
of mass: total mass
remains constant
Atoms are neither
created nor destroyed
in a chemical
reaction.
Atomic Theory
Matter is composed of
indestructible,
indivisible atoms,
which are identical for
one element, but
different from other
elements.
Thomson Atomic Theory (1897)
Experimental Work Theoretical
Explanation
Arrhenius: the electrical
nature of chemical
solutions.
Faraday: quantitative
work with electricity &
solutions.
Crookes: qualitative
studies with the cathode
ray.
Thomson: quantitative
studies with the cathode
ray.
Millikan: charged oil drop
experiment
Atoms may gain or lose
electrons to form ions in
solution.
Atomic Theory
Matter is composed of
atoms that contain
electrons (negative
Particular atom and ions particles) embedded in a
positive material. The
gain or lose a specific
kind of element is
number of electrons.
characterised by the
Electricity is composed of number of electrons in
negatively charged
the atom.
particles.
“The cookie model for
Electrons are a
the atom.”
component of all matter.
Electrons have a specific
fixed electric charge.
Rutherford Atomic Theory (1911)
Experimental
Work
Theoretical
Explanation
Rutherford: A few
positive alpha particles
are deflected at large
angles when fired at
gold foil.
The positive charge in
the atom must be
concentrated in a very
small volume of the
atom.
Most materials are very
stable and do not self
destruct or disintegrate.
Rutherford: Most alpha
particles pass straight
through gold foil.
Atomic Theory
An atom is comprised of
a very tiny nucleus
which contains positive
charges and most of the
mass of the atom. Very
small negative electrons
A very strong force holds occupy most of the
the positive charges
volume of the atom.
together within the
nucleus. (Strong nuclear
force)
Most of the atom is
empty space.
The Limitations of Rutherford’s
Atomic Model
Based on the understanding of physics at the time, for an
electron in motion around a central core:
• If an atom absorbed radiation (light, UV, xrays, etc), radiation must be emitted, so it
was expected that a continuous spectrum of
light energy was being given off
• because of radiation, the electron would lose
energy and its orbit would decrease until it
spiraled into the nucleus, destroying the atom
Rethinking Atomic Structure Based on
the Nature of Energy
Light is one form of electromagnetic radiation, which
travels through space as waves
Electromagnetic waves:
• have frequency, wavelength, and amplitude
• interact with matter in discrete particles called photons
Atomic Spectra
When atoms are excited due to absorption of energy, they emit light as they
lose energy and return to a non-excited state.
Atoms of each element emit light of particular wavelengths called a line
spectrum or emission spectrum.
Each element has a
characteristic line
spectrum.
The Bohr Model of the Hydrogen Atom
Niels Bohr set out to explain the stability of the nuclear model of the atom.
In this model, electrons
• are in circular orbits
• can only exist in certain “allowed” orbits or energy levels (energy of
electrons is quantized)
• do not radiate energy while in one orbit
• can jump between orbits by gaining or losing a specific amount of energy
Bohr’s Atomic Model
A quantum of
energy, equal to the
absorbed quantity,
is released as a
“photon” (EMR)
when the electron
returns to its
ground state.
n=3
Excited State
A quantum of
energy is absorbed
by the electron and
it undergoes
“transition”
e-
n=2
Ground State
n=1
eHydrogen’s
Energy Level Diagram
Energy is added to
the gas sample.
Bohr’s explanation of Spectra
n=3
e-
Emission Line
Spectrum
n=2
IR
n=1
Hydrogen’s
Energy Level Diagram
R O Y G B V UV
Bohr’s Atomic Model Explains the Line
Spectrum of Hydrogen
• Calculated wavelengths of the possible energies of photons that could be
emitted from an excited hydrogen atom (transitions from n = 6, 5, 4, and 3 to n
= 2) corresponded with hydrogen’s visible line spectrum
The Quantum Mechanical Model of the Atom
Today’s quantum mechanical model of the atom incorporates the wave
properties of electrons.
Wave functions, initially described by Erwin Schrodinger, represent a region
in space around a nucleus where an electron will be found. This region of
space is called an atomic orbital
An electron density
diagram represents an
atomic orbital.
The Quantum Mechanical Model of the Atom
Atomic orbitals can be visualized as “fuzzy clouds”
• The higher the density of the “cloud,” the higher the probability of finding an
electron at that point.
• The cloud has no definite boundary.
• The region where an electron will spend 90 percent of its time is depicted by
drawing a circle.
The circle does not
represent a real
boundary.
Heisenberg’s Uncertainty principle
Heisenberg
• Electrons
behave as both waves
“Uncertainty
Principle”
and particles
behave as determined
both waves and particles.
• Electrons
• Heisenberg
that it is
impossible to know BOTH the
• Heisenberg determined that it is impossible to
W.
W.
exact position and velocity of an
Heisenberg know BOTH the exact position and momentum of
Heisenberg
electron.
1901 - 1976 an electron.
1901-1976
• He observed that on cannot simultaneously define
the position and momentum (= m•v) of an
electron.
Quantum Mechanics
Quantu
• Wave function – mathematical
description of an orbital in an atom
where an electron of a certain
energy is likely to be found.
Schro
aw
• Electron probability density –
indicates regions around the
nucleus with the greatest probability
of finding an electron.
▫ 3D shapes of the atomic orbitals
▫ “cloud of electron density”
He de
E.
E.Schrodinger
Schrodinger
1887 - 1961
1887-1961
Solut
WA
The s
des
wh
Quantum Numbers Describe Orbitals
Electrons in the quantum mechanical model of the atom are described using
quantum numbers.
Three quantum numbers describe the distribution of electrons in the atom and a
fourth describes the behaviour of each electron.
Symbols for the four quantum numbers:
n
l
ml
ms
The Principle Quantum Number, n
•
•
•
•
•
n=1
n=2
n=3
n=4
Is the first quantum number
Describes the energy level, or shell, of an orbital
All orbitals with the same n value are in the same shell
The larger the n value, the larger the size of the shell
Values can range from n = 1 to n = ∞
first shell
second shell
third shell
fourth shell
The Orbital-Shape Quantum Number, l
•
•
•
•
•
Is the second quantum number
Describes the shape of an orbital
Refers to energy sublevels, or subshells
Values depend on the value of n. They are positive integers from 0 to (n – 1)
Each value is identified by a letter
l = 0 orbital
l = 1 orbital
l = 2 orbital
l = 3 orbital
s
p
d
f
An energy sublevel is
identified by combining n
with the orbital letter. For
example, n = 2, l = 1: 2p
sublevel
Shapes of Orbitals
• The act of measuring or identifying an electron requires
us to interfere with its motion and energy.
▫ This interference may result in an inaccuracy in identifying
the very characteristic we intended to measure.
• The act of measuring the location of an electron requires
the absorption or release of its energy, which in turn
affects its speed and maybe location.
▫ Heisenburg’s uncertainty principle indicates that it is
impossible to know the exact position and speed of a
particle at the same time.
• Such an understanding reinforces a sense of probability in
determining the location of an electron.
• Schrodinger’s wave equations can be used to predict
the likelihood of “finding” and electron at a specific
location. These probabilities can be used to plot an
electron probability density.
▫ The second quantum number, l, indicates the variety in
“shape” of the orbital. (s – sharp, p – principal, d –
diffuse, f – fundamental)
 Recent theory suggests that there may be g-orbitals!?!
• An orbital is associated with a size, three dimensional
shape and orientation around the nucleus.
▫ Together the size, shape and position represent the
probability of finding a specific electron at that
location.
Shape of orbitals
• The probability
density plot can
be assessed in
multiple
dimensions to
generate a 3-d
density cloud or
“shape” for the
orbital.
Shape of orbitals
s-orbital
p-orbital
d-orbital
Shape of orbitals
• The culmination of all of the electrons produces a combined effect
from all involved orbitals to generate a unique “orbital shape” for
each atom. For example, the structure of Ne . . .
Shape of orbitals
• The diagram we used to represent oxygen is;
-
-
8
Protons
-
-
16
8
O
Shape of orbitals

The diagram we might currently use to
represent oxygen is;
TYPES OF ORBITALS: S
Types of orbitals: s
en l• When
= 0,l the
orbital
is
called
s
= 0, the orbital is called s
012
Types of Orbitals (l)
• When l = 1, the orbital is called p
p orbitals
• The p sublevel has 3 orbitals
• The three p orbitals lie 90 degrees apart in
space
P ORBITALS
• They are designated px, py, pz for the axis
• The p sublevel has 3 orbitals
• The three p orbitals lie 90 o apart in space
d orbitals
• When l = 2, the orbital is called d
• d sublevel has 5 orbitals
f orbitals
• What are the values for ml when l = 3?
• When l = 3, the orbital is called f
The Magnetic Quantum Number, ml
•
•
Is the third quantum number
Indicates the orientation of the
orbital in space
• Named because in a magnetic
field, these different orientations
have different energies
• For a given l there are (2l +1)
values for ml
• How do we know p orbitals have
3 sublevels?
▫ Magnetic number represents the
sublevels
▫ ml = -l to +l
▫ So for the p orbital l = 1,
possible sublevels are -1, 0, 1
s, p, and d orbitals have
characteristic shapes.
The Spin Quantum Number, ms
•
Is the fourth quantum number
•
Specifies the orientation of the axis of electron spin
•
Two
possible
values:
–½ spin they
• e- spin
in
their
orbits
and+½
asorthey
•
e- spin
in their orbits
and as they spin they generate a magnetic field
generate
a magnetic
field
•
e- can spin either clockwise or counterclockwise
• e- can spin either clockwise or counterclockwise
Belle River District High
School
To summarize:
Identifying Electrons Using Sets of
Quantum Numbers
According to the Pauli exclusion principle:
• an orbital can have a maximum of two electrons
• two electrons in an orbital must have opposite spins
No two electrons of an atom have the same set of four quantum numbers.
Summary of Quantum Numbers
Principal
quantum
number, n: the
main electron
energy level or
shell (n )
Secondary
quantum number,
l: the electron
sublevels or
subshells
(0 to n-1)
Magnetic
quantum number,
ml: the orientation
of the sublevel
(-l to +l)
Spin quantum
number, ms: the
electron spin
(-1/2 to +1/2)
Orbital
orientation
Electron Spin
0
0
1
0
0
-1,0,+1
+1/2,-1/2
+1/2,-1/2
+1/2,-1/2
0
1
2
0
-1,0,+1
-2,-1,0,+1,+2
+1/2,-1/2
+1/2,-1/2
+1/2,-1/2
Energy shell Orbital shape
1
2
3
Try it!
• Write a set of quantum numbers for
the 2 electrons in a 1s orbtial
1s
Tells us the principal
quantum number (n) is 1
n=1
l=0
ml = 0
ms = +1/2
Tells us the shape of the
orbital (l) is s
n=1
l=0
ml = 0
ms = -1/2
Try some more!
• Write a set of quantum numbers for
an electron in a 3p orbital
3p
Tells us the shape
of the orbital (l) is p
Tells us the principal
quantum number (n) is 3
n=3
l=1
ml = -1
ms = +1/2
n=3
l=1
ml = 0
ms = +1/2
n=3
l=1
ml = +1
ms = +1/2
n=3
l=1
ml = -1
ms = -1/2
n=3
l=1
ml = 0
ms = -1/2
n=3
l=1
ml = +1
ms = -1/2
Representing Electrons:
Electron Configurations and Orbital Diagrams
The electron configuration for an atom shows the number and arrangement of
its electrons, in the ground state.
The electron
configuration for
hydrogen
An orbital diagram uses boxes or lines to represent orbitals at each n and
shows electron spin.
Orbital diagrams often accompany
electron configurations.
Identify the atom.
Three rules are used to build Energy
Level Diagrams:
▫ Aufbau principle
▫ Pauli Exclusion Principle
▫ Hund’s Rule
Aufbau Principle
(German for “building up’)
• Electrons occupy orbitals of lower
energy first.
-Pauli Exclusion Principle
(Wolfgang Pauli, Austria, 1900-1958)
• An orbital can hold only two electrons and they must
have opposite spin.
• Electron Spin Quantum Number (ms):
+1/2, -1/2
Hund’s Rule
In a set of orbitals, the electrons will fill the orbitals in a
way that would give the maximum number of parallel
spins (maximum number of unpaired electrons).
one electron is placed in each orbital at the same energy level
before the second electron is placed
Analogy: Students could fill each seat of a school bus,
one person at a time, before doubling up.
Conventions for Creating Energy-Level Diagrams
• Circles or squares are used to represent the orbitals
• Arrows are used to represent the electrons
▫ up represents one electron rotation (clockwise) while
down the other (counter clockwise)
▫ there is no convention as to which one you must start
with
Conventions for Creating Energy-Level Diagrams
O
(z = 8)
1s
2s
2p
3s
3p
P
(z = 15)
1s
2s
2p
3s
3p
1s
2s
2p
3s
3p
Ar
(z = 18)
Conventions for Creating Energy-Level
Diagrams
•Energy-level diagrams may be written in a vertical manner
to exemplify the energy level subtleties.
6e10e14e2e-
32e-
4d
6e10e2e-
18e-
3d
6e10e2e-
18e-
5d
6s
5p
5s
4p
4s
3p
3s
2p
2s
1s
4f
Order of filling orbitals
6p
6e2e-
8e-
6e2e-
2e-
•As the number
of energy levels
and orbitals
increase, so too
does the
complexity of
the energy-level
diagram.
8e-
2e-
•The diagram
indicates nicely
the order in
which the
orbitals are
filled
Energy-Level Diagrams for ions
▫ For anions – add the proper number of electrons using
regular conventions
S 2(z = 16)
1s
2s
2p
3s
3p
▫ For cations – remove the correct number of ions in the
proper manner
Al 3+
(z = 13)
1s
2s
2p
3s
3p
Electron configurations
• The energy-level diagram is the best way to visualize
the energy relationships between electrons. However it
may prove cumbersome.
• Electron configurations show the same information in
a more concise manner.
Electron configurations
Cl
(z = 17)
1s
2s
2p
3s
3p
becomes . . .
Cl: 1s2 2s2 2p6 3s2 3p5
In the shorthand form of electron
configurations, the previous noble gas structure
is used to reflect the lower energy level
electrons. So . . .
Cl: 1s2 2s2 2p6 3s2 3p5
becomes . . .
Cl: [Ne] 3s2 3p5
Anomalies in Electron Configurations
•Completely filled orbitals are more stable than half filled
orbitals which are more stable than partially filled orbitals.
>
>
•As the principle energy level increases the difference between
each orbital’s energy level comes to the point where the dorbitals energies are very close to the s-orbital’s.
•In situations like this the “s-” and “d-orbitals” may be treated
as a similar energy level and the application of Hund’s rule may
be applied over the range of orbitals rather than the single
second quantum orbital.
•The stability of having each orbital containing one electron is
greater than having a partially filled group of orbitals and a
filled orbital of only minimally less energy.
Anomalies in Electron Configurations
•Vanadium has the electron configuration of –
V (z=23) [Ar] 4s2 3d3
•Chromium has one extra electron and conventions would predict it to be
added to the next open d-orbital
Cr (z=24) [Ar] 4s2 3d4
•However, the stability of having one electron in each orbital of similar takes
president and the true electron configuration becomes
Cr (z=24) [Ar] 4s1 3d5
n=4
n=3
s
p
d
Explaining the multivalent ions
•Many transitional metals have the ability to have multiple charges as ions
and the explanation for this behaviour has yet to be explained.
•As orbitals are being filled there are varying levels of stability due to the
interaction of forces or electrostatic repulsion between electrons and force
associates with the magnetic field due to the electron’s spin.
•Filled orbitals are most stable because the electrostatic repulsion is balanced
against the magnetic attraction.
eNorth
Magnetic
field
e-
Electrostatic
repulsion
Magnetic attraction
ee-
South
Magnetic
field
Explaining the multivalent ions
•Orbitals that are completely filled, like Noble gases, have the most stable
structure due to the balanced forces between electrostatic repulsion and
magnetic attraction.
•Transitional metals often have partially filled d-orbitals and complete sorbitals with similar energy levels. Electrons are lost to achieve the best
combination of stability.
•Best stability – completely filled orbitals (2 electrons/orbital)
Electrostatic repulsion and Magnetic attraction
•Next best stability – half filled orbitals (1 electron/orbital)
Electrostatic repulsion with minimal crowding
Co: [Ar] 4s2 3d7
Co: [Ar]
Co2+:
[Ar]
Co3+:
[Ar]
- Filled 4s and half filled 3d
- Half filled 4s & 3d less stable
than Co2+ so less common ion
Explaining Magnetism
•Some materials exhibit strong magnetic properties naturally these are
referred to as being ferromagnetic.
•As moving charges have magnetic fields so too do spinning electrons. Those
spinning in one direction would have one magnetic polarity as those spinning
in the opposite direction the opposing magnetic pole.
•For example – clockwise – south pole and counter clockwise –north pole
•Those atoms that have a number of similarly spinning electrons would have
similar magnetic fields.
•If these “magnetic atoms” were free to align themselves with neighbouring
atoms of similar characteristics they would create regions of magnetism in
the material called domains.
•The arrangement of these domains in a material results in a magnet.
Atoms themselves have magnetic properties due
to the spin of the atom’s electrons.
Groups of atoms join so that their magnetic fields
are all going in the same direction
These areas of atoms are called “domains”
When an unmagnetized substance is placed in a magnetic
field, the substance can become magnetized.
This happens when the spinning electrons line up in the
same direction.
Explaining Magnetism
Iron, nickel and cobalt are such naturally occurring atoms.
Iron, nickel and cobalt are small enough atoms that they can realign
themselves due to the magnetic properties of their surrounding and thereby
create domains.
There are other such atoms that have similar electron configurations but
limited ability to migrate. Hence, they are reduced in their magnetic
properties.
Explaining Magnetism
Ferromagnetic – materials with strong magnetic
properties. Their presence increases a magnetic field
substantially.
Paramagnetic – materials with weak magnetic properties.
Their presence only slightly strengthens a magnetic field.
Diamagnetic – materials that have reduced magnetic
properties. Their presence weakens a magnetic field.