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Properties of Light: Wavelength (l, lambda) and Frequency (n, nu)
long wavelength
Low frequency
Units:
Wavelength
Units:
Frequency
m or nm
short wavelength
1/s or s-1 or Hz
high frequency
Electromagnetic Spectrum: all types of “light”-colored light is only
a small portion of the spectrum.
Blue
Green
Yellow
Orange
High Frequency
Short Wavelength
High Energy
What does ROY G BIV mean?
Red
Low Frequency
Long Wavelength
Low Energy
Remember that Wavelength (l, lambda) is the distance between consecutive peaks or
troughs in a wave.
Also remember that Frequency (n, nu) is the number of wavelengths that pass a given
point in one second.
Since all electromagnetic radiation travels at the same speed (speed of light), the
frequency and wavelength of a given type of light are inversely proportional.
Converting from Wavelength to Frequency
ln = constant
(confirms that l and n are inversely related)
In this case the constant is called the speed of light (c).
c is the speed of light
ln = c
c = 3.00X108 m/s
Wavelength and Frequency
What is the frequency of light that has a wavelength of 7.32X10-6 m?
Given:
l = 7.32X10-6 m
ln = c
n =
n=?
c
n =
l
ln = c
c = 3.00X108 m/s
Click
(3.00X108
m/sto
)
−6
(7.32X10 m)
see Solution
n = 4.10X1013 1/s
What is the wavelength of light that has a frequency of 4.83X105 1/s?
Given:
n = 4.83X105 1/s
ln = c
l =
l=?
c
n
c = 3.00X108 m/s
l =
Click
(3.00X108
m/sto
)
(4.83X105 1/s)
ln = c
see Solution
l = 621 m
4
When an atom is given extra energy, electrons will gain energy and become
“excited”.
When an electron that is “excited” loses energy it gives off light. According to the
Rutherford model of the atom, the electrons could be anywhere outside of the
nucleus. If this were true, excited atoms would be expected to give off the entire
electromagnetic spectrum of light when they lose energy.
When tested, atoms were found to give off very specific colors of light (not the entire
spectrum). This is called the emission spectrum of an atom.
Niels Bohr proposed the Bohr model (also called the planetary model) of the atom.
In this model, electrons can only be positioned at very specific distances from the
nucleus (called orbits). When an electron gains or loses energy, it will move from
one orbit to another which means that an electron can only gain or lose very specific
amounts of energy.
Bohr’s model explains the emission spectrum of hydrogen, but does not work for
larger atoms. This is still a very big step forward.
E1, E2, E3, E4, E5, E6 Represent energies of electrons (or energy levels) in an atom.
The arrows represent electron transitions which would give off light. The length of
the arrow is proportional to the energy of light given off.
Energy
E6
E5
E4
E3
E2
When electrons move from a high
energy level to a lower energy
level, they give off light. The
energy of the light depends upon
the difference in energy between
where the electron begins and ends.
The greater the difference, the
greater the energy of the light.
E1
Lyman
series
Balmer
series
Paschen
series
Lyman series is so energetic that the light is in the ultraviolet range.
Balmer series is less energetic than Lyman and is in the visible range (colored light).
Paschen series is very low in energy and is in the infrared range.
Lyman Series:
n7 → n1
n6 → n1
n5 → n1
n4 → n1
n3 → n1
n2 → n1
Paschen Series:
n7 → n3
n6 → n3
n5 → n3
n4 → n3
Highest Energy (ultraviolet)
Higher energy than visible light
(longer drop)
Lowest Energy (ultraviolet)
The larger the “drop”, the higher the
energy of the light emitted.
Highest Energy (infrared)
Lower energy than visible light
(shorter drop)
Lowest Energy (infrared)
Visible light (Balmer Series) is in between Lyman and Paschen
The Energy of a photon of light is directly proportional to the frequency of the light.
(E/n = constant)
The Energy of a photon of light is inversely proportional to the wavelength of the light.
(El = constant)
E
n
= h
Since
or
n =
E = hn
c
l
where “h” is Plank’s constant
E = hn will become
h = Plank’s Constant = 6.626X10-34 J·s
c = speed of light = 3.00X108 m/s
E=
hc
l
Energy of light (or electron transitions)
What is the energy of light that has a wavelength of 3.92X10-6 m?
Given:
l = 3.92X10-6 m
E=
hc
E=
l
E =?
−
(6.626X10 34
Js)(3.00X108 m/s)
Click to see Solution
−6
(3.92X10 m)
E = 5.07X10-20 J
What is the energy of light that has a frequency of 7.34X107 1/s?
Given:
n = 7.34X107 1/s
E =?
E = (6.626X10−34 Js)(7.34X107 1/s)
Click to see Solution
E = hn
E = 4.89X10-26 J
What is the wavelength of light having an energy of 8.93X10-20 J?
Given:
E=
E = 8.93X10-20 J
hc
l
l=
hc
E
l =?
l=
−
(6.626X10 34
Js)(3.00X108
m/s)
Click
to see Solution
−20
(8.93X10
J)
-6
l = 2.23X10 m
9
Scientists studied light and found out that it can act like a wave in some and
that it can act like a particle in others.
Scientists also studied electrons and found out that they act like particles in
some experiments and that they act like waves in other.
When the mathematics of waves are applied to electrons, the quantum
mechanical model of the atom is produced. In this model, electrons have
specific wavelengths which is related to their energy.
Instead of saying that an electron is in a specific orbit (a specific path around
the nucleus), it is said that an electron is in a specific orbital. An orbital is a
region in space where an electron will be 90% of the time. The energy of the
electron determines the shape and the size of the orbital.
Bohr Model: Electrons would follow specific
orbits (paths around the nucleus) and could
not be in any other locations.
Quantum Mechanical Model: Electrons will exist in orbitals (a volume of space
around the nucleus where the electron will be 90% of the time) and electrons will
have specific energies (wavelengths) but they can move anywhere within the
orbital. Orbitals will have different shapes based on their energy (see following
slides).
Summary of atomic theory in historical order:
Dalton: atom is like a solid ball
JJ Thompson: Plum Pudding-atom has soft positive stuff with small electrons in it
Rutherford: Nuclear-atom has small dense nucleus where most of the mass is
Bohr: Planetary-electrons in orbit around nucleus
Schrödinger: Quantum-electrons have wavelike properties
Orbitals:
The shape of an orbital depends upon the energy of the electrons in it.
The energy of an electron primarily depends upon two quantum numbers:
n
is the principle quantum number (like an energy level Bohr’s
model). On the periodic table, the period number also identifies the
principle energy level.
n determines the size of an orbital (larger values of n mean higher energy
electrons and larger orbitals-i.e. electrons farther from the nucleus)
l
is the angular momentum quantum number
l determines the shape of an orbital (larger values of l mean higher energy
and more complex shapes for the orbitals-i.e. s, p, d, and f orbitals are each
more complex and higher energy than the previous one)
When “l” = 0, the shape of an orbital is a sphere
These orbitals are called “s” orbitals.
The value of “n” determines the size of the sphere
There is only ONE “s” orbital
in a given energy level.
A single orbital (of any type)
can hold a maximum of two
electrons.
When “l” = 1, the shape of an orbital is a “dumbbell”
These orbitals are called “p” orbitals and there are three of them for
each energy level.
px
py
pz
There are THREE “p” orbitals in a given energy level.
When “l” = 2, the shape of an orbital is a “ double dumbbell”
These orbitals are called “d” orbitals and there are five of them for each
energy level.
dxz
dyz
dxy
dx2 – y2
There are FIVE “d” orbitals in a given energy level.
dz2
When “l” = 3, the shape of an orbital is a “ quadruple dumbbell”
These orbitals are called “f” orbitals and there are seven of them for
each energy level.
There are SEVEN “f” orbitals in a given energy level.
The Good News: Learning a few basic ideas about the periodic table will
make it easy to use the important parts of Quantum Mechanics for this class!
We will use the periodic like a legal “cheat sheet” throughout this class-you
have access to a periodic table on all quizzes and tests in this class
We will use an element’s position on the periodic table to identify its:
electron configuration
valence electrons
Lewis dot structure
orbital diagram
quantum numbers
relative energy of all orbitals
Key
n
1
2
3
4
5
6
7
elements with last electron in an “s” orbital
elements with last electron in an “p” orbital
elements with last electron in an “d” orbital
elements with last electron in an “f” orbital
n is the principle energy level
n-1
n-2
Key
n
1
2
3
4
5
6
7
Last electron has quantum number “l” = 0
Last electron has quantum number “l” = 1
Last electron has quantum number “l” = 2
Last electron has quantum number “l” = 3
n is the principle energy level
n-1
n-2
Electron Configuration Practice
Example:
Sulfur
S
1s2, 2s2, 2p6, 3s2, 3p4
Valence Electrons:
Electrons in the outermost orbitals-highest energy electrons in the
atom.
The valence electrons are the electrons that are taking part in chemical
reactions of an element.
Electron Dot Structures:
Diagrams showing the number of electrons in the valence shell using an
element symbol and “dots” to represent electrons. Note: electrons in
“d” or “f” orbitals are not shown in dot structures.
Examples:
Orbital Diagrams: box diagrams used to record where electrons are
located in an atom.
Use the number of boxes to identify which type of orbital is being filled
A single box is “s”
Three boxes is “p”
Five boxes is “d”
Seven boxes is “f”
Remember: each box represents a single orbital, and each orbital can contain a
maximum of 2 electrons (Pauli Exclusion Principle).
Orbital Diagrams:
Put arrows into boxes for each electron of a given type that is present in an
orbital. For example: C has an electron configuration of 1s2, 2s2, 2p2
1s
2s
2p
A total of 6 electrons need to be placed in the orbital diagram.
Electrons must fill the lowest energy orbital before a higher energy orbital can
be filled. Each box (orbital) can only contain up to 2 electrons and must have
opposite spin (Pauli Exclusion Principle).
If two or more orbitals are of equal energy, then electrons can not pair up until
each orbital has one electron in it (Hund’s Rule).
Fill in the orbital diagram for the valence shell of sulfur.
3s
3p
S is 1s2, 2s2, 2p6, 3s2, 3p4
outer
shell
Fill in the orbital diagram for the valence shell of bromine.
Br is 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d10, 4p5
outer
shell
4s
3d
4p
Note: all of the lower energy orbitals in these diagrams would have 2 electrons (arrows)
Noble-Gas Configurations:
Noble-Gases: He, Ne, Ar, Kr, Xe, Rn
Each of these elements has an entirely full outer shell of
electrons. This makes them very stable-unreactive, inert.
To shorten electron configurations of other elements, we can
replace part of the electron configuration with a noble-gas
symbol in brackets and then show just the outer shell electrons.
For example:
Br: 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d10, 4p5
electrons in Ar
Therefore:
Br: [Ar] 4s2, 3d10, 4p5
We do not really have to memorize the order of energy for all of the orbitals.
Instead we use our understanding of the positions on the periodic table to
copy down the correct order of energies from the periodic table.
Every time the number of protons in a nucleus increases (atomic number),
the number of electrons increases as well. The atomic numbers on the
periodic table tell us where the element is located and we can use that to
identify the order of orbital energies because the order of orbital energies is
the same as the order of orbitals for an electron configuration.
For example: If we write the electron configuration for Ni (element 28),
we get: 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d8
The order of energy for these orbitals is: 1s, 2s, 2p, 3s, 3p, 4s, 3d
We will use this same technique to learn quantum numbers!
Orbital energies in
single electron
system.
Bohr Model
Orbital energies in
multi electron
systems.
Quantum Model
Bohr’s theory did not predict different energies within the same principle
energy level for multi electron systems. Quantum mechanics does! Note that
4s should be between 3p and 3d but was not included because of the
comparison being used.
Quantum Numbers are solutions to the Schrödinger equation and tell us
about the relative energy of the electrons. T he solutions to the Schrödinger
equation determine how the atoms on the periodic table will be arranged, and the
quantum numbers are a type of short hand that we use to describe this.
n is the principle quantum number and is related to the major
energy levels of the electron. n must be a positive integer!
As n increases, the size of the orbital increases and the electron
spends more time farther from the nucleus. Farther from the
nucleus is higher energy.
Consequence of “n”: periods (horizontal rows or energy levels) on
the periodic table. These currently range from 1 to 7.
Quantum Numbers
l is the angular momentum quantum number and is related to
shape of the atomic orbital. l must be a whole number which
can range from 0 to (n-1)!
For example: when n = 1,
when n = 2,
when n = 3,
when n = 4,
l must be 0
l is either 0 or 1
l is either 0, 1, or 2
l is either 0, 1, 2, or 3
When l = 0, s orbital.
When l = 1, p orbital.
When l = 2, d orbital.
When l = 3, f orbital.
Consequence of “l”: number of types of orbitals in a given period
of the periodic table (only 1 in period 1, 2 in period 2, etc….).
Quantum Numbers
ml is the magnetic quantum number and is related to the
orientation in space of the atomic orbital. ml must be an
integer which can range from -l through 0 to +l
For example, when l = 0, ml = 0
1 value (only one s orbital)
when l = 1, ml = -1, 0, 1
3 values (three p orbitals)
when l = 2, ml = -2, -1, 0, 1, 2
5 values (five d orbitals)
when l = 3, ml = -3, -2, -1, 0, 1, 2, 3
7 values (seven f orbitals)
Consequence of “ml”: number of orbitals for each type of orbital
Quantum Numbers
ms is the spin quantum number and is related to the direction
of “spin” of the electron. ms must be either + ½ or - ½
Pauli Exclusion Principle: No two electrons in an atom can
have the same set of four quantum numbers. Another way of
saying this is that two electrons can only occupy the same
orbital if they have opposite spin. Two electrons in the same
orbital must have the same first three quantum numbers and ms
must be + ½ for one and - ½ for the other.
Consequence of “ms”: only 2 electrons allowed in any single orbital
One way of thinking about “spin”
The gold arrows represent magnetic fields created by the
“spinning” electrical charge. Notice that the arrows point in
opposite directions-therefore the electrons are spinning in
opposite directions (one clockwise and one counterclockwise).
General Rule: electrons must have lowest possible energy
First electron in an atom
Quantum Numbers
n = 1
l =
0
ml = 0
ms = +½
Quantum Numbers are like addresses for the electrons in an
atom. Each electron needs to have its own unique address.
Why would it be wrong for the First electron in an atom to have the
quantum numbers below?
n = 3
l =
2
ml = 0
ms = +½
General Rule: electrons must have lowest possible energy
1st electron in an atom
Quantum Numbers n = 1
1
Only
Electron
in
H
(1s
)
l =
0
ml = 0
First Electron in He (1s1)
ms = +½
2nd electron in an atom
n = 1
l =
0
ml = 0
ms = -½
Second Electron in He (1s2)
Helium has 2 electrons and each electron must have a different set of
quantum numbers and must have the lowest energy possible.
This is one expression of the Pauli Exclusion Principle.
We always want the lowest energy. For the first 2 electrons, n =1, l = 0, ml = 0
and the Pauli Exclusion Principle, tells us that only 2 electrons can have these
values.
If we keep adding electrons to an atom, where will the third and fourth electron
have to go?
n will have to increase to 2. When it does, l could be 0 or 1 (0 is lower in energy)
and we would use l = 0 first. This gives:
3rd
n =
l =
ml =
ms =
2
0
0
+½
4th
n =
l =
ml =
ms =
2
0
0
-½
since l = 0, “s” orbital being filled
2s1 then 2s2
No 2 electrons have the same set of quantum numbers!
If we keep adding electrons to an atom, where will the 5th through 10th electrons
have to go?
n can still be 2 because we have not used l = 1 yet. When l = 1, ml can range
from: -1 to 0 to 1
This gives:
5th
6th
n = 2
l = 1
ml = -1
ms = +½
n = 2
l = 1
ml = 0
ms = +½
7th
n = 2
l = 1
ml = 1
ms = +½
8th
n = 2
l = 1
ml = -1
ms = -½
9th
n = 2
l = 1
ml = 0
ms = -½
since l = 1, “p” orbitals being filled
2p1, then 2p2, and so on … 2p3, 2p4, 2p5, 2p6
10th
n = 2
l = 1
ml = 1
ms = -½
Summary of Quantum Number meanings.
Number of orbitals in a given energy level = n2
where n is the energy level
Maximum number of electrons in a given energy level = 2n2
Imagine that electrons live in a fancy hotel with the following rules.
1) Electrons must go the lowest energy bedroom available. Lower floors are
lower in energy.
2) The maximum number of electrons in a bedroom is 2.
3) Smaller suites are lower in energy than larger suites on the same floor. There
are four types of suites: 1 bedroom, 3 bedrooms, 5 bedrooms, and 7 bedrooms.
4) Electrons will not share a bedroom until each bedroom in that suite has one
electron in it.
5) Electrons can not move into higher energy suites until the lower energy
suites are full.
6) The 5 bedroom suites are so much higher in energy than the 3 bedroom
suites, that they appear one floor above their real floor number. The 7 bedroom
suites appear 2 floors above their real floor number for the same reason.
A diagram of our electron “hotel”
6th floor
5d
4f
6s
5th floor
5p
4d
5s
4th floor
4p
3d
4s
3rd floor
3p
3s
2nd floor
2p
2s
1st floor
1s
6p
Electron Configurations: like a bookkeeping system for where
electrons will be located in an atom in its ground state.
Electrons fill into the lowest possible energy orbitals using
rules much like those we used for the electron “hotel”.
The pattern can easily be remembered by using the periodic
table to full effect.
Start at element 1 and follow the atomic numbers. Fill in all
electrons according to the Period Number (principle quantum
number “n”) and the number of electrons in the orbital type
(s, p, d, and f are the orbital types-from the angular quantum
number (l)).
The technical name for this is called the aufbau principle
d
block
(n-1)
(n)
p block
s
block
(n-2)
f block
Designation for last electron of each of the first 18 elements.
s1
s2
p1
p2
p3
p4
p5
Notice: Elements in the same group have the same type
of electron as the last electron (except He).
He is in the last group because it has a full outer shell!
p6
Designation for last electron of each of elements 19 through 36.
Notice the two exceptions to normal electron configurations.
Orbital Diagrams:
In the first energy level ( n = 1), l can only be 0
and ml must also be 0. Therefore, there is only an “s” orbital
in the first energy level.
In the second energy level ( n = 2), l can be 0 or 1
and ml can be -1, 0, or 1. When l is 0, an “s” orbital is present,
and when l is 1, three “p” orbitals are present.
In the third energy level ( n = 3), l can be 0, 1, or 2
and ml can be -2, -1, 0, 1, or 2. When l is 0, an “s” orbital is
present, when l is 1, three “p” orbitals are present, and when l is 2,
five “d” orbitals are present.
Why only one “s” orbital in any energy level?
When l = 0, ml must also = 0 (only value), so one “s” orbital.
Why three “p” orbitals in any energy level above 1?
When l = 1, ml can be -1, 0, or 1 (three values), so three “p” orbitals.
Why five “d” orbitals in any energy level above 2?
When l = 2, ml can be –2, -1, 0, 1, or 2 (five values), so five “d” orbitals.
Why seven “f” orbitals in any energy level above 3?
When l = 3, ml can be –3, –2, -1, 0, 1, 2, or 3 (seven values), so seven “f”
orbitals.
s
p
d
f
s
p
d
f
Complete the orbital diagram for N is based on the electron configuration.
N 1s2, 2s2, 2p3
1s
2s
2p
Since we are often most interested in the outer shell (valence) electrons, we
sometimes shorten the orbital diagram to include only the outer shell electrons.
So the orbital diagram for N could simply be written as:
2s
2p
With the understanding that all lower energy level orbitals are full (2 e- each).
Examples:
Complete the orbital diagram for the valence electrons of Br.
4s
3d
4p
Complete the orbital diagram for the valence electrons of Fe.
4s
3d
4p
Complete the orbital diagram for the valence electrons of Sn.
__ s
__ d
__ p