Answer Key to Accompany
A Study of Electronic Orbitals Using the Program, Orbital Viewer
Tom Kuntzleman, Spring Arbor University
Introduction
Changes occur in the electronic structure of molecules during chemical reactions: atoms transfer
electrons one to another, or share electrons with each other. Thus, electrons are intricately involved
in chemical reactions. As a result, it is useful to have a good understanding of the structure of
electrons within atoms in order to better understand in order to have a good understanding of the
chemical behavior of the elements.
In grade school, you likely learned that the electron in an atom is very much like a small particle that
orbits a very small nucleus – much like the Earth orbits the sun. While this model of the atom is
often useful, it fails to fully describe the behavior of electrons within atoms. In order to accurately
describe electronic behavior, electrons are considered to be waves.
Electron waves, sometimes called orbitals, describe a region of space wherein the electron is likely to
be found. Just like it requires three numbers (x, y and z) to describe the location of a point in three
dimensional space, electron orbitals require 3 numbers (called n, l and ml) to describe their position
around the nucleus. These three numbers, called quantum numbers, are all integers. (There is also a
fourth quantum number that describes the electron spin, but this quantum number is not necessary
to describe the location of the electron around the nucleus and therefore it will not be further
considered in this laboratory).
The first quantum number is called the principal quantum number and is given the symbol n. This
quantum number describes the angular momentum of the electron, and is closely related to the
electron’s energy. The principal quantum number can take on integer values from 1 to infinity:
n = 1, 2, 3 …∞
Equation 1
The second quantum number is called the azimuthal quantum number, and is given the symbol l.
This quantum number describes the shape of the electron wave. For a particular electron wave, the
values that l can take on depend upon the value of n for the electron wave; l can have any integer
value from zero to one less than the value of n:
l = 0, 1, …n – 1
Equation 2
For example, suppose an electron wave has a value of n = 2. For this value of n, there are a total of
two possible values for l, which are 0 or 1.
The final spatial quantum number is called the magnetic quantum number, and is given the symbol
ml. This quantum number describes the orientation of the electron wave in space. For a particular
electron wave, the values that m can take on depend upon the value of l for the electron wave; ml
can have any integer value from – l to l:
ml = – l, -l+1, …l-1, l
Equation 3
Thus, an electron wave with a value of l = 3 can have ml values of -3, -2, -1, 0, 1, 2, or 3.
Based on Equation 1, we see that there are an infinite number of possible electrons waves.
However, Equations 2 and 3 assure us that only certain values of l and ml are possible for a given
value of n. In this assignment, you will explore some of the relationships that exist between the
spatial quantum numbers and how these quantum numbers affect the properties of electron waves.
Materials: Orbital Viewer program, available at http://www.orbitals.com/orb/ov.htm
Procedure:
1. Open Orbital Viewer. Click File  New.
2. On the toolbar, click the = button:
3. An orbital dialog box appears. You will notice that within this orbital dialog box, you can change
a number of parameters, including n, l, ml and number of protons. Currently displayed on the
screen is the orbital for n = 4, l = f, and ml
= 0 attracted to a nucleus with 1 proton.
Click on the left arrow button (indicated
with an arrow in the diagram to the left)
beneath l in the Orbital dialog box. You
will notice that letters, instead of numbers,
are used to designate the value of l. This
convention is often used for l, and you
should be very familiar with this
convention. The letters corresponding to
different numerical values of l are:
Numeric value of l
0
1
2
3
4
5
Letter symbol
s
p
d
f
g
h
The letter designations go alphabetically after 3 = f, with 4 = g, 5 = h, 6 = i, 7 = j and so forth. To
describe the elements in the periodic table, we only need to go as far as l = f.
4. Open the Orbital dialog box by pressing the = button if you have not already done so. Using
the left arrow beneath n in the Orbital dialog box, change the values of n to 1, l to p and ml to 1.
Click done. In the space below, draw a description of what you see.
5. Once again, open the Orbital dialog box by pressing the = button. In the spaces below, write
the values of n, l, and ml displayed:
l: _____s______
n: ____1______
ml: _____0_______
6. Recall that you previously set the values for all three quantum numbers above to be equal to 1
(recall that for l:, p = 1). Why did Orbital viewer automatically set the value of l to be “s” and the
value of ml to be zero when n was set to the value of 1? If you need a hint, see Equations 2 and 3.
For any value of n, l can only take on integer values from 0 to n – 1. With n = 1, the only possible
value for l is 0. Because ml can only take on integer values from - l to l , ml has been
automatically set to zero by Orbital Viewer.
7. Can you make an orbital with a value of n = 1 that has any value of l greater than 0 (“s”) or any
value of ml that is greater than 0? Why or why not?
Can you make an orbital with a value of n = 1 that has any value of l greater than 0. Likewise, you
cannot make an orbital with a value of n = 1 that has a value of ml that is greater than 0. See
Equations 2 and 3.
8. How many sets of orbitals can you produce for electron waves that have n = 1? Write out the
possible values of n, l and ml in the chart below:
Electron wave
1
2
3
4
5
n
l
ml
1
All
should be
-------
0
remaining
left
-------
0
spaces
blank
-------
For question 8 above, you should recognize that there is only one possible electron wave with n = 1.
So only one row should be filled in the table above, and the remaining rows should be left blank!
For an orbital with n = 1, l can only have the value of 0 (see Equation 2). If l = 0 (which is the
only possible case when l = 0, ml can only have the value of 0 (Equation 3). Thus, for n = 1, there
is only one possible orbital, and that orbital has the values:
n = 1, l = 0 and ml = 0
or,
n = 1, l = s and ml = 0
A very useful equation that can be used to determine the total number of different orbitals that may
be found for a particular value of n (a particular shell) is:
Total number of orbitals in a shell = n2
(Equation 4)
Thus, using Equation 4 we find that n = 1 shell contains only one orbital. The name of this orbital
is a 1s orbital, because this orbital is comprised of a wave with n = 1 and l = s.
9. How many different orbitals with n = 2 can be produced? Using Equation 4, we see that a total
of 4 waves can be produced when n = 2. Below, the four possible electron waves in the n = 2 shell
and their associated spatial quantum numbers are displayed.
Electron wave
n
l
ml
1
2
3
4
2
2
2
2
0, or s
1, or p
1, or p
1, or p
0
-1
0
1
Number of
nodes
1
1
1
1
Orbital name
2s0
2p-1
2p0
2p+1
10. Use the Orbital Viewer program to construct each of the four electron waves designated in the
chart above. Draw the picture of each electron wave in the space below. As much as you are able,
try to indicate the position of the nodes in your drawings.
2s
2p-1
2p0
2p+1
11. For the three 2p orbitals above, how does changing the value of ml in the drawings above
change the orbital?
It changes the orientation of the orbital.
12. Fill in the table below for the 9 different orbitals that can exist in the n = 3 shell. Construct
each orbital using the Orbital Viewer program. Describe each shape under the column heading
“shape” in the table below. The first orbital has been filled in for you.
Electron
wave
1
2
3
4
5
6
7
8
9
n
l
ml
3
3
3
3
3
3
3
3
3
0, or s
1, or p
1, or p
1, or p
2, or d
2, or d
2, or d
2, or d
2, or d
0
1
0
-1
2
1
0
-1
-2
Number
of nodes
2
2
2
2
2
2
2
2
2
Orbital
name
3s0
3p1
3p0
3p-1
3d2
3d1
3d0
3d-1
3d-2
Shape
spherical
dumbbell
dumbbell
dumbbell
clover leaf
clover leaf
dumbbell with donut middle
clover leaf
clover leaf
13. Fill in the table below for the 16 different orbitals that can exist in the n = 4 shell. Construct
each orbital using the Orbital Viewer program. Describe each shape under the column heading
“shape” in the table below.
Electron
wave
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
n
l
ml
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
0, or s
1, or p
1, or p
1, or p
2, or d
2, or d
2, or d
2, or d
2, or d
3, or f
3, or f
3, or f
3, or f
3, or f
3, or f
3, or f
0
1
0
-1
2
1
0
-1
-2
3
2
1
0
-1
-2
-3
Number
of nodes
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Orbital
name
4s0
4p1
4p0
4p-1
4d2
4d1
4d0
4d-1
4d-2
4f3
4f2
4f1
4f0
4f-1
4f-2
4f-3
Shape
spherical
dumbbell
dumbbell
dumbbell
exploded clover leaf
exploded clover leaf
double dumbbell/donut middle
exploded clover leaf
exploded clover leaf
six lobes
eight lobes
six lobes
dumbbell / double donut
six lobes
eight lobes
six lobes
14. In the space below, describe the relationship between the value of l and the shape of an orbital.
Revisiting the tables and drawings you completed above will be useful.
s – spherical, p – dumbbell, d- clover leaf, f – many lobed
15. The shape of an orbital (the value of l ) for a particular shell is sometimes called a subshell of an
orbital. For example, we often talk about the 3d subshell. Fill in the table below, which indicates
how many different orbitals can exist in a particular subshell. Revisiting the tables you filled in
above might be helpful.
subshell
s
p
d
f
Number of different orbitals in subshell
1
3
5
7
Now that we have introduced the idea of a subshell, we can relate the terms shell, subshell and
orbital to sets of quantum numbers. These relationships are outlined as follows: Orbitals with the
same value of n belong to the same shell. Orbitals with the same value of n AND l belong to the
same subshell. If the value of n AND l AND ml are all assigned, then you have identified one
particular orbital.
16. It turns out that two electrons can reside in a single electron orbital. This fact arises from a
property of electrons called spin which will be discussed in class later. Based on the information
that two electrons can exist in a particular orbital and the number of different orbitals in a subshell,
fill in the table below, which indicates the total number of electrons that can exist in a particular
subshell:
subshell
Total number of electrons that can reside in subshell
s
2
p
6
d
10
f
14
17. There is an equation that describes the total number of electrons that can exist in a particular
subshell. It’s tricky. This equation only depends upon integers and the quantum number l. What is
the equation?
Number of electrons = 4l + 2
18. There is an interesting and useful connection between the number of electrons that can reside in
a subshell and the periodic table. Answering the questions below, while keeping in mind your
answers to number 16 above, may help you see this connection. Use the periodic table on the back
of this lab manual to answer the following questions:
a. How many columns of main block elements exist on the periodic table to the left of the transition
elements?
2
b. How many columns of main block elements exist on the periodic table to the right of the
transition elements?
6
c. How many transition elements exist in any single row of the periodic table?
10
d. How many elements exist in the Lanthanide series? How many in the Actinide series?
14
19. Based on your answer to question 17 above identify each element below as an s-block, p-block,
d-block or f-block element.
___s__ calcium
__p___ aluminum
___d___ iron
___p___ carbon
___f___ uranium
__p__ fluorine
___s__ potassium
___d__ zinc
20. If n2 (Equation 4) is the total number of orbitals that each shell contains, and if two electrons
can fit into each orbital, then:
Total number of electrons that can exist in a particular shell = 2n2
Fill in the chart below:
Shell
1
2
3
4
5
6
7
Total possible number of
different orbitals in a shell
1
4
9
16
25
36
49
(Equation 5)
Total number of electrons
possible in shell
2
8
18
32
50
72
98
Additional Questions:
1. Is it possible to construct a 2d orbital using Orbital Viewer? Why or why not? If so, be sure to
include a drawing of a 2d orbital in the space below.
No. For a 2d orbital, n = 2 and l = 2 which violates Equation 2.
2. Use Orbital Viewer to construct a 2s orbital with 1 proton in the nucleus. Then use Orbital
Viewer to construct a 2s orbital with 10 protons in the nucleus. How do the two compare? Explain
any differences.
The orbital with 1 proton in the nucleus is larger than the orbital with 10 protons in the nucleus.
The greater positive charge in the 10-proton nucleus greatly attracts the electron cloud towards
itself, making this orbital very small relative to the 2s orbital that surrounds the 1-proton nucleus.
3. It is important to know the shapes of orbitals with different values of l. Use Orbital Viewer to
construct a few different s, p and d orbitals. Draw the shape of each orbital type below:
s
p
d
Reference: Manthey, David, Orbital Viewer, http://www.orbitals.com/orb/ov.htm