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Big Picture
The quantum mechanical model is now the modern and accepted model of the atom. Unlike previous models, the
quantum mechanical model of the atom does not predict the path that an electron takes around the nucleus. Due to
the Heisenberg uncertainty principle, the position of an electron cannot be precisely known. Instead, electrons occupy
orbitals, regions of space where the electrons have the highest probability of existing. To understand the properties of
an atom, there are four quantum numbers that describe an electron’s energy, orbital shape, orientation, and spin state.
No two electrons in a given atom can have the same four quantum numbers.
Key Terms
Chemistry
Quantum Mechanical Model
Orbital: A region of space within the atom where an electron is likely to be found.
Orbital Energy: The amount of energy associated with an electron in a particular orbital.
Quantum Number: A number describing a property of an electron.
Principal (n): Describes the principal energy level of the electron.
Aizmuthal (l): Describes the shape of the electron orbital (s: l=0, p: l=1, d: l=2, etc). This number must be
between 0 and n-1.
Magnetic (ml): Describes the orientation of the electron orbital. This number must be between +n and -n.
Spin (ms): Describes the spin state of the electron (±½).
Pauli Exclusion Principle: Electrons cannot have the same four quantum numbers within the same atom.
Shell: A set of electrons with the same principal quantum number (n).
Subshell: A set of electrons with the same azimuthal quantum number (l).
Quantum Mechanics
In addition to the quantum theory that energy is quantized, the quantum mechanical model of the atom accounts for
two features of quantum mechanics: the wave-particle duality and the Heisenberg uncertainty principle.
Wave-Particle Duality
All moving objects have wavelike behavior. Louis de Broglie (1892-1987) derived an equation to find the wavelength
of the associated matter wave for a moving object.
• For the matter wave to have a measurable wavelength, the particle needs to atomic or subatomic in size.
Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle (proposed by Werner Heisenberg) states:
It is impossible to know both the velocity and the position of a particle at the same time.
The process of making a measurement actually changes what is being measured.
• For larger objects, this effect is very small and can be ignored.
Schrödinger Equation
Edwin Schrödinger treated the electron as a wave when developing his mathematical equation.
• Solutions to the Schrödinger equation are called wave functions.
• The wave functions can be used to calculate the probability of finding an electron at a given point.
• The solutions naturally supported the idea that the energy of the electrons are quantized (electrons
can only be
found in certain energy levels).
An atomic orbital describes the region of space where the electron is likely to be found. Typically it is the region of
space where there is a 90% probability of finding the electron.
• The electron is not contained solely within this region but rather has a high probability of being found within it.
• The uncertainty principle tells us there is no way to know exactly where the electron is, and we cannot know the
path the electron takes.
• Orbitals have different sizes and shapes.
• The orbital energy is the energy of an electron in a particular orbital.
Why do we draw orbitals where there is less than 100% probability of finding the electron? The region of space
where there is a 100% probability of finding the electron would be as large as the universe!
This guide was created by Steven Lai, Rory Runser, and Jin Yu. To learn more
about the student authors, visit http://www.ck12.org/about/about-us/team/
interns.
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v1.1.12.2012
Disclaimer: this study guide was not created to replace
your textbook and is for classroom or individual use only.
Atomic Orbitals
Chemistry
Quantum Mechanical Model
cont .
Quantum Numbers
You are not expected to solve the Schrödinger equation. The equation is so complex, it is impossible to solve the
equation for atoms and ions with more than one electron. Powerful computers are needed to produce very close
approximations to the solutions.
Instead, you need to understand the four quantum numbers that describe the atomic orbitals and the electrons in the
orbitals. The numbers come from the Schrödiner equation. The four numbers have special names. They are:
• Principal quantum number (n)
• Aizmuthal quantum number (l) (also known as the angular momentum quantum number)
• Magnetic quantum number (ml)
• Spin quantum number (ms)
No two electrons within the same atom can have the same four numbers! This is known as the Pauli exclusion
principle.
Principal Quantum Number
The principal quantum number is the main energy level occupied by the electron.
• Can have the integer values n = 1, 2, 3, 4, and so on
• Determines the size of the orbital (larger n, larger orbital)
• As n increases, the energy of the electron increases, and the electron is more likely to be found at a distance farther
away from the nucleus
• n = 1 is lowest in energy, and the electron is most likely close to the nucleus
• Electrons with the same value of n in an atom belong to the same shell
• The maximum number of electrons in each shell is 2n2
• The total number of allowable orbitals in each shell is equal to n2
• Contains one or more sublevels, or subshells
Aizmuthal Quantum Number
The aizmuthal quantum number indicates the overall shape of the orbital.
• Also referred to as energy sublevels
• Can have integer values between 0 and n–1
• Each nth principal energy level has n sublevels available
• Can also be designated by the letters s, p, d, f
• s orbital is spherical
• p orbital is dumbbell-shaped
• d and f orbitals are more complex
• Affects orbital energies (larger l, higher energy)
• Electrons with the same value of l
in an atom belong to the same subshell
Electron Arrangement Within Energy Levels
Principal Quantum
Number (n)
Allowable
Sublevels
Number of
Orbitals per
Sublevel
Number of Orbitals
per Principal Energy
Level
Number of
Electrons per
Sublevel
Number of Electrons
per Principal Energy
Level
1
s
1
1
2
2
2
s
p
1
3
4
2
6
8
3
s
p
d
1
3
5
9
2
6
10
18
4
s
p
d
f
1
3
5
7
16
2
6
10
14
32
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cont .
Chemistry
Quantum Mechanical Model
Quantum Numbers (cont.)
Magnetic Quantum Number
The magnetic quantum number indicates the orientation of the orbital.
• Can have integer values between – l
and +
l
• The number of possible ml values for a subshell is equal to the number of orbitals within the subshell
• s orbital: l = 0, so ml = 0
• It is spherical, so it has only one possible orientation
• p orbital: l = 1, so ml can equal -1, 0, and 1
• The p orbitals have three possible orientation: px aligned along the x-axis, py aligned along the y-axis, and pz
aligned along the z-axis
Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0
• The d orbitals have 5 possible orientations
Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0
• The f orbitals have 7 possible orbitals
Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0
Spin Quantum Number
The spin quantum number indicates the orientation of the magnetic field.
• Electrons spin on their own internal axis, and the spinning creates a magnetic field
• The orientation depends on the direction the electron is spinning
• There are two possible spin quantum numbers: +½ and -½
Each orbital holds a maximum of two electrons and have opposite spins.
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