Lesson 4: Applying Quantum Mechanics
In the last lesson we discussed a code known as quantum mechanics that would enable us to describe how electrons
are arranged about the nucleus of a particular atom. We stated that knowing this arrangement would enable us to
understand the nature of those atoms, i.e. how stable they were or how reactive they were with other atoms.
In order to learn the "code" of quantum mechanics to our model of the atom, you will use a drawing on a 3 X 5 card
showing the different electron orbitals (figure below and to the right - from now on this will be called the board) and
about 40 small paper disks which will represent electrons (from a hole punch is perfect - from now on these will be
called the discs). The discs will be of two different colors to represent the two opposite spins of the electrons
(remember - the spin quantum number). Draw the card shown below on a 3 X 5 card or half a sheet of paper. You will be
reusing this drawing to learn quantum mechanics so the heavier the paper you use the better. You will reuse this card
many times.
Nucleus
Orbit 1
Nucleus
1s
2s
Orbit 2
2p
3s
Orbit 3
3p
x
y
z
x
y
z
x
y
z
4s
3d
Orbit 4
4p
Orbit 5
5s
4d
Look at the first circle just under the nucleus on your board. To the left of the circle you will see the notation 1s. The
1s indicates that the circle will hold the electrons in the first orbit or shell from the nucleus. This orbit is spherical in
shape (s). Each shape (circle, pear, and rectangular) below the 1s circle is another orbit of with a spherical (s), pear (p),
or (d) shape - sorry, the d orbital has several unique shapes. We will use a rectangle to represent a d-orbit. The
sequence that the s, p, and d orbitals on the board is significant, but we will discuss that later. At this point, realizing that
each object (circle, pear-shaped or rectangular) represents an orbit of a particular shape is important.
The paper cutouts that you have separated into two piles of different colors represent electrons, each color
represents the two different spins of the electrons. These electrons will be placed into the circles, pear-shapes, or
rectangular shapes to represent the electrons in the atom. Let's look at an example!
Example 1: Let's start with an atom of helium. The first thing that you will need to find out is how many electrons
will be found in an atom of helium. Do you recall from a previous chapter how to locate this information? Yes, the
periodic table of elements (that chart with all the element symbols and names) has within each element block the
atomic number which tells us the number of electrons or protons and generally the number of neutrons found in an
atom of that particular element. Recall that the symbol for helium is He and can be found in the upper right hand corner
of the periodic table. There you will see that the atomic number for helium is 2. This means that an atom of helium will
have 2 electrons, 2 protons, and generally, 2 neutrons. The number of electrons (2) is important to us right now. See the
diagram below help you locate helium.
2
He
Helium is found in the upper right corner
of the Periodic Table. The location of the numbers
2 and 4.0026 may vary depending on the periodic
table you use.
Helium
4.0026
Now, take two of your discs (electrons), one of each color. The first rule of how electrons fill the orbits so that
electrons begin filling the orbits nearest the nucleus first and proceed outward (downward on your paper). So start filling
up orbitals from the top down, and completely fill an orbital before moving down to the next. The first orbit filled is the
1s orbit, then the 2s orbit, and then the three 2p orbits and so on. This is exactly how you will place your discs
(electrons) on the drawing: from the top down (or to be more precise, from the inside of the atom outward).
Now, let's get back to our example of helium. You found from the periodic table of elements that each atom of
helium has two electrons traveling in its electron cloud. Following the "fill rule" above, you should place the two paper
cutouts (one of each color) you are holding into the first (top) 1s circle. Recall that within each orbit you will find two (or
less) electrons, hence we placed both of the electrons that an atom of helium has in the first orbit. Look at the diagram
below to see what your drawing should look like. Note that we placed one chip of each color in the 1s orbit.
Again - You may be wondering why you are using two colors of electrons (discs). The answer is that the two colors of
cutouts represent the two possible spins of those electrons: clockwise and counter-clockwise (the spin quantum
number).
This circle (orbit) is filled
Nucleus
- it has 2 electrons and
Your drawing for the element
cannot hold any more.
1s
helium (He). Note that we
Notice the rest of the
used 2 cutouts representing
shapes are empty
2s
because helium only has
the 2 electrons found
two electrons so these
in one atom of helium
2p
orbits don't exist for He
2
He
Helium
3s
x
y
z
3p
4.0026
4s
x
y
z
x
y
z
3d
4p
5s
4d
Each circle, pearshape, or rectangle
represents an "orbit"
that can hold up to
two electrons
By looking at your Board, you can describe helium as having two electrons in its electron cloud and
they are both found traveling in a spherical-shaped orbit on the first energy level from the nucleus. This is called
"reading" your board.
At this point in our discussion, we haven't yet explored what this information means regarding the reactivity or
stability of helium. You will find a considerable amount of material in the next lessons which will help you make
inferences regarding reactivity or stability of each element. For now, let's look at another example.
Example 2: Describe the arrangement of electrons found in an atom of carbon. Find carbon on your periodic table of
elements. The symbol for carbon is C. If you look on the right-hand side of the table you will see the element carbon on
the first row of elements going across the top of the table.
The atomic number for carbon is 6 meaning that an atom of carbon has 6 electrons, 6 protons and, generally, 6
neutrons. We are concerned with the number of electrons which is 6. Take 6 discs, representing the 6 electrons,
Nucleus
6
1s
C
2s
2p
x
y
x
y
z
3s
Carbon
12.0115
3p
4s
z
three of each of the two colors. Using the first fill rule, place two electrons in the 1s circle, then two more electrons in
the next circle (the 2s circle). Now you come to a series of three pear-shapes at the 2p level. Remember in Lesson 3
when we discussed the magnetic quantum number (the quantum number which tells us the orientation in space in
which the orbits lie)? We stated that the pear-shaped orbits (p) were the only shaped orbits for which knowing an
orientation in space would be helpful when describing the arrangement of electrons in the electron cloud.
Now, look back at your Board at the 2p level and beneath each of the three circles you will see an x, y or z. These
notations represent each of the three possible orientations in space in which you might find electrons traveling in the
pear-shaped orbits. Refer to Lesson 3 if you would like to review the concepts we discussed regarding the magnetic
quantum number.
Let's return to our example of carbon. We have used four electrons so far (in the 1s and 2s circles) and have two
remaining. The second fill rule states that one electron is placed in each orientation position (x, y and z) and if necessary,
a second electron is placed in each position. In other words, you place one of your doo-wops in the 2px circle, the next
doo-wop in the 2py circle and, if necessary, one more disc in the 2pz circle. Then, if necessary, you go back and place a
second disc in each circle. The diagram on this page shows you the sequence of filling the p orbits.
2p
x
y
z
In the case of carbon, we have two remaining discs to place on the board. Place one disc in the 2px circle and the last
disc in the 2py circle. Note that it does not matter which color you place in the circle first, as long as you have one disc of
each color to represent the clockwise and counter-clockwise spins of the electrons. Look at the diagram below to see
how you should have filled your Board to represent the arrangement of electrons for carbon.
Nucleus
1s
2p
2s
x
y
Note that when filling electrons on
the pear-shaped (p) orbits as in this
example of carbon, one electron
goes into each direction orbit
before a second electron fills the circle.
z
2p
x
y
x
y
z
3s
3p
4s
z
Notice there are 6 discs. How many electrons
are there in Carbon (C)? Notice each circle is filled
with discs (2) before the one above it is started.
To complete this example, let's make a statement regarding the arrangement of the electrons found in an atom of
carbon. We can say by looking at our Board that each atom of carbon contains six electrons: two electrons are found in
the spherical-shaped (1s) orbit on the first energy level, two are found in the spherical shaped (2s) orbit on the second
energy level and two are found in the pear-shaped (2p) orbits on the second energy level - one in the x orientation and
one in the y orientation.
Example 3: Describe the arrangement of electrons found in an atom of magnesium (Mg). Find magnesium on your
periodic table of elements. Its symbol is Mg (do not get confused with manganese whose symbol is Mn!) and can be
found on the left side of the table in the second column from the left. See the diagram below to assist you in finding
magnesium on the periodic table of elements.
12
Mg
Magnesium
24.31
You should have two electrons in the 1s orbit, two electrons in the 2s orbit, and then by following the second fill rule,
you should place one electron in the 2px orbit, one in the 2py orbit, one in the 2pz orbit and then return to place a
second electron (of different spin/color) in the 2px orbit, a second electron in the 2py orbit and a third in the 2pz orbit.
Were you able to follow that sequence? Look at the diagram on the next page to help you follow the sequence of
placing the electrons in the p orbits.
You should still have two electrons left in your hand. They will fill the next orbit which is the 3s orbit. Look on the next
page to see how your completed arrangement of electrons in an atom of magnesium should appear on your Board.
Nucleus
1s
Filled Board for the element
magnesium (Mg). Note that a total of
12 discs were used representing
the 12 electrons in an atom of magnesium.
2s
2p
x
y
x
y
z
3s
3p
4s
z
Based upon your Board you can "read" your board and say that magnesium has 12 electrons in its electron cloud with
two electrons in the spherical shaped 1s orbit on the first energy level, two electrons in the 2s orbit on the second
energy level, a total of six electrons divided among the three p orbits (x, y and z) in the 2p orbitals and two electrons in
the 3s orbit at the third energy level.
Problems
1. What do the circles, pear-shaped, and rectangular objects represent?
2. What shape are the s and p orbitals?
3. Is the order that you place the discs important? Why?
4. What do the two different colored discs represent?
5. Which orbital is furthest from the nucleus, the orbital at the top of the card or the bottom
of the card?
6. Draw the card and location of the discs for the following elements:
a. potassium
b. oxygen
c. hydrogen