Quantum Theory
Important concepts that led to the development of the quantum theory include:
1924 – Louis De Broglie
Light and electrons exist as both a wave and a particle. When we do an experiment it is
the experimental procedure and equipment that determine which characteristics we
perceive. This idea is referred to as Duality.
Waves that are confined to a space can only exist at certain frequencies. (Slide 1)
Electron orbits can be considered as confined waves at specific frequencies representing
specific energies (E = hf) (Slides 2 and 3)
1925 - Pauli’s Exclusion Principle
No two electrons can have the same set of Quantum numbers describing them. A simpler
way of saying this is 2 electrons can occupy the same space (orbital) only if they are
spinning in opposite directions. (Slide 4)
1926 – Erwin Schrodinger’s Equations
Using the wave-like nature of the electron, Schrodinger was able to devise a set of
equations that predicted the most-likely position of the electron about the nucleus and the
shapes of their orbits. His predicted values matched the values calculated by Bohr in
1913.
1927 – Werner Heisenberg’s Uncertainty Principle
Based on the particle nature of electrons Heisenberg found that it was impossible to
measure both the position and momentum of an electron. The better you measured one of
these values, the less accurate your knowledge of the other value became. This leaves
you with an uncertainty as to the position and momentum of the electron.
These four concepts along with Bohr’s limits on the electrons are the basis of the
Quantum Theory and allow us to predict the most likely orbits for the electrons around
the nucleus.
Quantum Numbers:
4 numbers are used to describe the position of an electron about the nucleus, these
numbers are:
A) Principle (Main) Quantum Number (Energy Level)
B) Angular Momentum Quantum Number (Shape)
C) Magnetic Quantum Number (3D Orientation)
D) Spin Quantum Number (Spin)
Principle Quantum Number:
Indicates the Energy level (orbit) occupied by the electron. The Principle Quantum
Number is represented by n with whole number, positive integers as values (starting at 1).
The lower the number the closer the energy level is to the nucleus (Slide 5).
The distance each energy level is from the nucleus was determined by Bohr in 1913.
Excitation experiment:
An atom whose electrons are as close to the nucleus as possible are said to be in their
ground state.
Ground State – lowest energy state for an atom. (Slide 6)
When an atom is given energy, the energy is absorbed by an electron which jumps to a
higher energy level. The atom at this point is referred to as an excited atom. (Slide 7)
Excited Atom – An atom that has absorbed energy and whose electrons are in higher
energy levels than usual.
The atom loses energy to return to its ground state, and in so doing the energy will be
given off as light. (Slide 8) The amount of energy in the light depends on the distance
the electron had to jump. Each jump produces a specific amount of energy which appears
as a distinct line of color in the emission spectrum. (Slide 9 & 10)
Bohr used these lines to calculate the amount of energy in each jump. Allowing him to
calculate how far each energy level was from the nucleus and the number of electrons in
each energy level.
Since each electron has a wave form associated with it, electrons can only orbit at those
locations where the wave is not subject to negative interference. These are the energy
levels described by the principle quantum number. (Slide 11)
Orbitals:
Each energy level is broken into small sublevels called orbitals that can each hold up to 2
electrons. There are n2 orbitals per energy level, therefore there can be up to 2n2
electrons per energy level.
Angular Momentum Quantum Number:
This number indicates the shape of the orbital, with the number of shapes available at
each energy level being equal to n (the principle quantum number). Therefore:
1st energy level = 1 shape
2nd energy level = 2 shapes
3rd energy level = 3 shapes
Etc.
There are only 4 shapes required at this time to be able to place all of the electrons in the
largest atom. Each is assigned a lower case letter. In order of increasing energy, these
shapes are:
s - orbital = spherical (Slide 12)
p - orbital= pear shaped (like 2 pears placed end to end) (Slide 13)
d – orbital (Slide 14)
f – orbital (Slide 15)
Magnetic Quantum Number:
This number indicates the 3D orientation of the orbital.
Since the s-orbital is spherical there is only one possible orientation (Slide 12)
p – orbitals have 3 possible orientations (Slide 13)
d – orbitals have 5 possible orientations (Slide 14)
f – orbitals have 7 possible orientations (Slide 15)
Spin Quantum Number:
Electrons spin as they orbit the nucleus (similar to the Earth’s rotation as it orbits the
Sun). There are 2 possible directions of spin. The electron’s spin gives it a magnetic
field, if two electrons are spinning in opposite directions, they will have opposite
magnetic fields that will result in a slight magnetic attraction, reducing the electrical
repulsion. This will allow them to fit together in an orbital.
Each electron in the atom can be described by these four numbers, but two electrons
cannot have the exact same 4 numbers.
i.e.
If two atoms are in the same Energy Level, are in the same shape orbital with the same
orientation then they must have different spins.
Here is a summary of the effect of these four Quantum Numbers on the number of
electrons in each Energy Level. (Slides 16 & 17)
1st Energy Level has an s-orbital with 1 orientation = 2 e2nd Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6e(total of 8e-)
3rd Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6ed-orbital with 5 orientations = 10e(total of 18e-)
4th Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6ed-orbital with 5 orientations = 10ef-orbital with 7 orientations = 14e(total of 32e-)
5th Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6ed-orbital with 5 orientations = 10ef-orbital with 7 orientations = 14e(total of 32e-)
6th Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6ed-orbital with 5 orientations = 10e(total of 18e-)
7th Energy Level has an s-orbital with 1 orientation = 2ep-orbital with 3 orientations = 6e(total of 8e-)
Total electrons accounted for = 118
Rules for electron placement:
Aufbau Principle:
An electron will occupy the lowest energy orbital available to it.
Both energy level and shape affect the amount of energy, resulting in an overlapping of
orbitals.
As you can see from this diagram the orbitals would be arranged in the following order
based on increasing energy. (Slide 18)
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, etc. notice the overlap of orbitals after the 3d orbital.
(Slide 19)
There are 2 ways to keep the order of energy straight:
1) On the diagram below start at the top of the first line, follow it down through the
orbitals listing them in the order you pass through them. When you get to the bottom of a
line, start at the top of the next line. Continue until you run out of electrons. (Slide 20)
2) The periodic table is arranged by orbital, as illustrated below. (Slide 21)
You can use this as a guide to the order of fill, by counting following the order of atomic
numbers.
Pauli’s Exclusion Principle:
No 2 electrons in the same atom can have the same set of 4 quantum numbers.
Example: Helium has 2 electrons based on the Aufbau principle they must both be in the
first energy level which only has one orbital with one orientation,
Electron #1 n = 1, Shape = s, one possible orientation, spin = ↑
Electron #2 n = 1, Shape = s, one possible orientation, spin = ↓
Since the Aufbau principle requires the first three numbers to be the same, the spins must
be different.
Since there is no other set of numbers for the first energy level, this energy level is filled
and the next electron must go into the next energy level.
Another way to state this principle is that 2 electrons can only occupy the same orbital if
they are spinning in opposite directions. (Slides 22 and 23)
Hund’s Rule:
Orbitals of equal energy are occupied by 1 electron each before any orbital is occupied by
a second electron. All unpaired electrons must spin in the same direction.
Example:
The p-orbital has 3 orientations. All three orientations are at the same energy. If there
were 3 electrons to be placed in the p-orbital, then each would have a different
orientation, as illustrated below: (Slide 24)
If there were a fourth electron to be placed in this set of orbitals it would pair with one of
these, spinning in the opposite direction. (Slide 25)
These are all equivalent in energy.
Placing electrons in their proper positions in the atom.
Electron Configuration Notation: (Slide 26)
The electron configuration gives the number of electrons in each energy level and orbital
shape (sublevel)
Example: Hydrogen has 1 electron in the 1st energy level s-sublevel. This would be
written as: 1𝑠1
Keep in mind the following limits:
s-sublevel = 2 electrons
d-sublevel = 10 electrons
p-sublevel = 6 electrons
f-sublevel = 14 electrons
Example 2: Lithium has 3 electrons. 2 in the first energy level, s-sublevel, one in the
second energy level, s-sublevel
1𝑠 2 2𝑠1
The electron configuration must be written in the order of increasing energy.
Example 3: Zirconium has 40 electrons
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d2
Orbital Notation:
Sometimes it is helpful to see not just the placement of the electrons, but the spins of each
electron. This is done by drawing an orbital notation.
Using the electron configuration as a guide, draw a horizontal line for each orbital’s
orientation, the electrons are indicated by an up or down arrow drawn above the line.
The orbital notation drawn must obey all of the rules.
Examples:
1) Carbon has 6 electrons, its electron configuration is 1s2 2s2 2p2.
The s-sublevels have only one orientation; the p-sublevels have 3 orientations.
The unfilled orbitals must obey Hund’s rule:
Do not pair electrons until there is at least one electron in each orbital of equal energy
(orientation). All unpaired electrons spin in the same direction.
2) Nickel has 28 electrons, its electron configuration is 1s2 2s2 2p6 3s2 3p6 4s2 3d8
Noble Gas Notation:
Bohr discovered that an atom with 8 electrons in its outermost energy (filled s and p
sublevels) is uniquely stable. He further found that atoms that do not have this
arrangement will gain or lose electrons to emulate atoms with this arrangement. It is this
tendency that determines the chemical and physical properties of an element.
The elements in the last column on the periodic table, the noble gases, have 8 electrons in
their outermost orbits. This makes them unreactive. Since the other elements are trying
to be like the noble gases and only the outer most electrons matter in determining
chemical properties it is convenient to use a simpler form of the electron configuration
that shows the closest noble gas with a lower atomic number than the element being used,
followed by any additional electron sublevels with their electrons.
Examples:
1) Sodium has 11 electrons and is the element after neon. The electron configuration for
sodium is:
This can be simplified as [Ne] 3s1
2) Scandium has 21 electrons. The noble gas below scandium is Argon which has a filled
3p sublevel. The noble gas notation for scandium would be: [Ar] 4s2 3d1