Name: ______________________________________________________ Date: ____________________________ Period: ________
Why Study Quantum?
Quantum mechanics is a mathematical theory that can be used to predict chemical properties. Since
nowadays we have better computational methods and faster computers, we can calculate properties for systems
that previously were too difficult to isolate experimentally. With this advanced technology, the renewed
interest in and continued study of quantum chemistry may result in a quantitative understanding of transition
states, prediction of alternate reaction pathways, and hopefully even prediction of new chemical concepts!
The Quantum Mechanical Model of the Atom
Most definitions of quantum theory and quantum mechanics offer the same description for both. These
definitions essentially describe quantum theory as a theory in which both energy and matter have
characteristics of waves under some conditions and characteristics of particles under other conditions.
Quantum theory suggests that energy comes in discrete packages called quanta (or, in the case of
electromagnetic radiation, photons). Quantum theory has some mathematical development, often referred to as
quantum mechanics, that offers explanations for the behavior of electrons inside the electron clouds of atoms.
The wave-particle duality of electrons within the electron cloud limits our ability to measure both the energy
and the position of an electron simultaneously. The more accurately we measure either the energy or the
position of an electron, the less we know about the other. The fact that we cannot accurately know both the
position and the momentum of an electron at the same time causes an inability to predict a trajectory for an
electron. Consequently, electron behavior is described differently than the behavior of normal sized particles.
We use statistical descriptions that show, not the electron path, but the region where it is most likely to be
found. Since it is the electron in the electron cloud of an atom that determines its chemical behavior, the
quantum mechanics description of electron configuration is necessary to understanding chemistry.
Reading Comprehension Review A
1. What does “wave-particle duality” mean?
2. Why can’t we describe an electron’s behavior by discussing its path as it orbits around the nucleus?
Schröndinger Equation to Describe Electron Movement
The most common way to describe electrons in atoms according to quantum mechanics is to solve the
Schrodinger equation for the energy states of the electrons within the electron cloud. When the electron is in
these states, its energy is well-defined but its position is not. The position is described by a probability
distribution map called an orbital.
Schrodinger found that the values that solve the equation to show the stable energy levels for an electron in an
electron cloud produce a wave function that described a standing wave. These solutions to Schrödinger’s
equation involve four special numbers called quantum numbers. (Three of the numbers, , , and , come
from Schrödinger’s equation, and the fourth one comes from an extension of the theory). These four numbers
completely describe the energy of an electron. Each electron has exactly four quantum numbers, and no two
electrons have the same four numbers. The statement that no two electrons can have the same four quantum
numbers is known as the Pauli exclusion principal.
The principal quantum number, , is a positive
integer
that indicates the main energy level
of an electron within an atom. According to quantum
mechanics, every principal energy level has one or more
sub-levels within it. The number of sub-levels in a given
energy level is equal to the number assigned to that energy
level. That is, principal energy level 1 will have 1 sublevel, principal energy level 2 will have two sub-levels,
principal energy level 3 will have three sub-levels, and so
on.
Table 1 Number of Sub-Levels and Electrons by
Principal Quantum Number
Principal
Number of
Total Number
Quantum Number Sub-Levels
of Electrons
1
1
2
2
2
8
3
3
18
4
4
32
In any energy level, the maximum number of electrons possible is 2n2. Therefore, the maximum number of
electrons that can occupy the first energy level is 2 (2 x 12). For energy level 2, the maximum number of
electrons is 8 (2 x 22), and for the 3rd energy level, the maximum number of electrons is 18 (2 x 32). Table 1
lists the number of sub-levels and electrons for the first four principal quantum numbers.
Reading Comprehension Review B
1. What is the Pauli Exclusion Principle? Why is it important?
2. What does the principal quantum number (n) tell us? What is the principal quantum number of Sulfur?
3. How many sublevels could be in energy level 5? How many electrons could occupy the 5th energy
level?
The largest known atom contains slightly more than 100 electrons. Quantum mechanics sets no limit as to how
many energy levels exist, but no more than 7 principal energy levels are needed to describe all the electrons of
all the known atoms. Each energy level can have as many sub-levels as the principal quantum number, as
discussed above, and each sub-level is identified by a letter. Beginning with the lowest energy sub-level, the
sub-levels are identified by the letters s, p, d, f, g, h, i, and so on. Every energy level will have an s sub-level,
but only energy levels 2 and above will have p sub-levels. Similarly, d sub-levels occur in energy level 3 and
above, and f sub-levels occur in energy level 4 and
above. Energy level 5 could have a fifth sub-energy level
named g, but all the known atoms can have their
electrons described without ever using the g sub-level.
Therefore, we often say there are only four sub-energy
levels, although theoretically there can be more than four
sub-levels.
The principal energy levels and sub-levels are shown in
the following diagram. The principal energy levels and
sub-levels that we use to describe electrons are in red.
The energy levels and sub-levels in black are
theoretically present but are never used for known atoms.
Reading Comprehension Review C
1. Why do we say we only have 4 sub-energy levels? Why don’t we need to use g, h and i?
2. What would make us need the g sub-energy level in the future?
Quantum mechanics also tells us how many orbitals are in each sub-level. In Bohr’s model, an orbit was a
circular path that the electron followed around the nucleus. In quantum mechanics, an orbital is defined as an
area in the electron cloud where the probability of finding the electron is high. The number of orbitals in an
energy level is equal to the square of the principal quantum number. Hence, energy level 1 will have 1 orbital (1
2
), energy level 2 will have 4 orbitals (2 2 ), energy level 3 will have 9 orbitals (3 2 ), and energy level 4 will
have 16 orbitals (4 2 ).
The sub-level has only one orbital. Each of the sub-levels has three orbitals. The sub-levels have five
orbitals, and the sub-levels have seven orbitals. As a result, the single orbital in energy level 1 is the orbital.
The four orbitals in energy level 2 are a single orbital and three
orbitals. The nine orbitals in energy level
3 are a single
orbital, three
orbitals, and five
orbitals. The sixteen orbitals in energy level 4 would be
the ________________________________________________________________ (complete the table below).
Table 2 shows the relationship
between (the principal quantum
number), the number of orbitals,
and the maximum number of
electrons in a principal energy
level. Theoretically, the number of
orbitals and number of electrons
continue to increase for higher
values of . However, no atom
actually has more than 32 electrons
in any of its principal levels.
Principal Energy Number of Orbitals
Present
Level (n)
Total Number Maximum Number
of Orbitals (n 2 ) of Electrons (2n 2 )
1
1
2
2
4
8
3
9
18
4
16
Each orbital will also have a probability pattern that is
determined by interpreting Schrödinger's equation. The
3-dimensional probability pattern for the single orbital
in the sub-level is a sphere. The probability patterns
for the three orbitals in the sub-levels are shown
below – independently and all three together.
The probability patterns for the five orbitals are more complicated, and the seven orbitals are even more
complicated.
You should keep in mind that no matter how complicated the
probability pattern is, each shape represents a single orbital, and the
entire probability pattern is the result of the various positions that
either one or two electrons can take.
Reading Comprehension Review D
1. In reflecting on this article, what do you think learning about
“orbital diagrams” and “electron configurations” will entail?
This article activity is adapted by Miss Scott from several resources:
Iyengar, Srinivasan S. "Atomic and Molecular Quantum Theory." C561. Indiana University, 2008. Web. 2015
"Quantum Mechanical Model of the Atom." CK-12 Science. CK-12 Foundation, 26 June 2013. Web. 27 Feb. 2015.
Vocabulary Review
In the table provided, identify any words you aren’t familiar with. Using what you know about prefixes and
suffixes, make an “educated guess” about each word’s meaning. Look up the word in the dictionary and write
down its definition in your own words.
Vocabulary Word(s)
Educated Guess
Dictionary Definition (in your own words)
Concept Review
To summarize, explain the following concepts (and why/how they are important to the concept of quantum
theory) in your own words. Include any other concepts you think are important.
1. Wave-Particle Duality
2. Schrondinger Equation
3. Pauli Exclusion Principle
4. Bohr’s Model
5. Energy Levels
6. Quantum Numbers
7. Orbitals