Quantum Numbers
After the quantum revolution started, many different ideas came about. Through countless experiments and theories, it was eventually determined that in order to fully describe an electron
around an atom, four quantum numbers were needed. These quantum numbers are basically the
electron’s ―zip code‖. The four quantum numbers are:
n, l, m , m
l
s
Each of these quantum numbers can have different values according to the rules listed below:
Shells: n = 1, 2, 3…. (integer value starting at 1)
n tells us the basic energy level and distance from the nucleus. The higher the value of n, the
higher the energy and the further out the electron is from the nucleus. It is important to realize
that the energy levels get successively closer together the further you go out from the nucleus.
This will play an important role later in the concepts.
n=3
n=2
n=4
n=1
n=5
Subshell: l = 0….. (n – 1)
(from 0 up to the value of n-1)
The subshell tells us the shape of the electron cloud or how many nodal planes are in the electron cloud. Each subshell has a number, shape, and a letter associated with it. The shape of the
cloud represents the probability of finding an electron in that space around the atom:
l =0
s subshells
0 nodal planes
l
For an ―s‖ orbital, = 0, and has the general shape of a sphere. At successive energy levels,
the spheres get bigger and bigger but the shape is still the same.
l =1
p subshells
l
1 nodal plane
For a ―p‖ orbital, = 1, and has the general shape of a dumbbell. It has 1 nodal plane, which is
a space of zero probability of finding the electron:
l =2
d subshells
2 nodal planes
l
For a ―d‖ orbital, = 2, and has the general shape of a cloverleaf. It has 2 nodal planes, which
bisect the probability twice, leading to the general four-leaf clover shape:
l =3
f subshells
3 nodal planes
l
For an ―f‖ orbital, = 3, and the shapes get quite complex. Imagine bisecting each of the cloverleafs above through the middle again:
http://pubs.usgs.gov/
of/2005/1219/
Orbital: m = -l …. 0 …. +l
l
l
l
(integer values from – to + )
The orbitals tells us the orientation of the electron cloud or how it is arranged around a central
point (the nucleus) in an x, y, z plane. The number of orientations is based upon the subshell or
l value.
l=0
m
then according to the rules above, l= 0 only
Since m = can only have a single value (0 in this case) there is only one orientation the
electron cloud can have around the nucleus. This makes total sense, though, because the
shape of an s orbital is a sphere. No matter what way you turn it, it is still a sphere;
there is no difference.
l
l=1
m
then according to the rules above, l= -1 0 +1
Since m = has three possible values (-1, 0, +1) that means there are three possible orientations of the orbital around the nucleus. These are in the x, y, and z planes respectively:
l
l=2
m
then according to the rules above, l= -2 -1 0 +1
+2
Since m = has five possible values (-2, -1, 0, +1, +2) that means there are five possible
orientations of the orbital around the nucleus. These get pretty complicated:
l
l=3
m
then according to the rules above, l= -3 -2 -1 0 +1 +2 +3
Since m = has seven possible values that means there are seven possible orientations of
the orbital around the nucleus. These are simply insane:
l
Spin: m = + 1/2 or — 1/2
s
The last thing we need to fully describe an electron is to know which way it is spinning.
Against an arbitrarily chosen plane, the electron is either spinning ―clockwise‖ or ―counterclockwise‖. These are labeled as either +1/2 or –1/2. The reason it is ―1/2‖ is well beyond the
scope of this class. Just go with it.
http://hcper71011.blogspot.com/2011/02/quantum-numbers.html
Summary:
Name
Shell
Letter
n
Value
1, 2, 3….
Significance
Tells energy and distance from nucleus
Subshell
l
0, 1…. (n-1)
l=0 s
l=1 p
l=2 d
l=3 f
Tells shape of cloud (# of nodal planes)
Spherical
Dumb bell
Cloverleaf
Split cloverleaf
Orbital
ml
-l...0…+l
Tells orientation of cloud around nucleus
Spin
ms
+1/2 or –1/2
Tells which way electron is spinning
Significance
So what do all these numbers mean? What do they have anything to do with Chemistry? They
seem pretty darn arbitrary and confusing. Where is this going?
Before I answer that, let me bring up one more concept that will help. There is another rule
called the Pauli Exclusion Principle that says that no 2 electrons in the same atom can have the
exact same 4 quantum numbers. As long as at least one of the numbers is different, it is a legal
arrangement. Let’s start at the first energy level, n = 1 and see where it takes us.
n=1
l=0
(0 is the only possibility due to the previous rules)
n=1
l=0
m=0
(because l = 0, ml can only be zero
l
n=1
l=0
m=0
m = +1/2 or –1/2
l
s
(ms can only ever be +1/2 or –1/2)
Now let’s examine what we have. The Pauli Exclusion Principle says that as long as we can
write a unique set of 4 numbers, we have an electron. How many electrons do we have here at
the first energy level?
We have 2 sets of 4 quantum numbers here (1, 0, 0, +1/2) and (1, 0, 0, -1/2) so the first
n=1
energy level has 2 electrons in it.
l=0
m=0
m = +1/2
l
s
First energy level = 2 electrons
n=1
l=0
m=0
m = -1/2
l
s
I’m sure you’re still lost but stick with me here. Let’s examine the 2nd energy level, n = 2.
n=2
l=0
n=2
l=1
If n = 2, then l could = 0 or l could = 1
n=2
l=0
m=0
n=2
l=1
m = -1, 0, +1
If n = 2 and l = 0, then m = 0
l
If n = 2 and l = 1, then m = -1, 0, +1
l
l
n=2
l=0
m=0
m = +1/2 or –1/2
l
n=2
l=1
m = -1, 0, +1
m = +1/2 or –1/2
ms = +1/2 or –1/2 always
l
l
s
s
Now how many sets of 4 quantum numbers can we get here?
n
2
n
2
n
2
n
2
l
0
l
0
l
1
l
1
ml
0
ml
0
ml
-1
ml
-1
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
n
2
n
2
l
1
l
1
ml
0
ml
0
ms
+1/2
ms
-1/2
n
2
n
2
l
1
l
1
ml
+1
ml
+1
ms
+1/2
ms
-1/2
There are 2
electrons here
Second energy level
A group of 2 electrons
A group of 6 electrons
Total of 8 electrons
Do you get it yet? Probably not so let’s go one more level...
There are 6
electrons here
You still probably don’t see where this is going but bear with me for one more level.
n=3
l=0
n=3
l=1
n=3
l=2
n=3
l=0
m=0
n=3
l=1
m = -1, 0, +1
n=3
l=2
m = -2, -1, 0, +1, +2
n=3
l=0
m=0
m = +1/2 or –1/2
n=3
l=1
m = -1, 0, +1
m = +1/2 or –1/2
n=3
l=2
m = -2, -1, 0, +1, +2
m = +1/2 or –1/2
l
l
l
l
l
s
l
s
s
Now how many sets of 4 quantum numbers can we get here?
n
3
n
3
n
3
n
3
n
3
n
3
n
3
n
3
l
0
l
0
l
1
l
1
l
2
l
2
l
2
l
2
ml
0
ml
0
ml
-1
ml
-1
ml
-2
ml
-2
ml
+2
ml
+2
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
n
3
n
3
n
3
n
3
n
3
n
3
l
1
l
1
l
2
l
2
l
2
l
2
ml
0
ml
0
ml
-1
ml
-1
ml
+1
ml
+1
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
ms
+1/2
ms
-1/2
n
3
n
3
n
3
l
1
l
1
l
2
ml
+1
ml
+1
ml
0
ms
+1/2
ms
-1/2
ms
+1/2
There are 2
electrons here
There are 6
electrons here
There are 10
electrons here
Third energy level
A group of 2 electrons
A group of 6 electrons
A group of 10 electrons
Total of 18 electrons
I know, I know. What does this have to do with ANYTHING???!!!! GET TO THE POINT!
Let’s recap what we learned in the past three pages:
1st Energy Level:
2nd Energy Level:
3rd Energy Level:
2 electrons
2 and 6 electrons to make 8 total
2 and 6 and 10 to make 18 total
Now let’s look at the Periodic Table of Elements.
1
1
2
2
3
3
4
5
4
5
6
6
7
7
But instead of looking at the whole thing, let’s look at it row by row; or more to the point, energy level by energy level:
1
1
1st Energy Level:
2 elements
Does this sound familiar? It should. If we look at the listing above, we see the same thing.
Does this continue?
2
2
2nd Energy Level:
2 elements and 6 elements
Holy cow! Our quantum model predicted exactly what we see here.
What is the pattern here? If we continue the trend and apply it to the Periodic table we see that
the table is broken into the groups we see here. Groups of 2, 6, 10, and 14:
1st Energy Level:
2nd Energy Level:
3rd Energy Level:
4th Energy Level:
2
2 and 6
2 and 6 and 10
2 and 6 and 10 and 14
Group of 10
Group of 2
Group of 6
Group of 14
http://farm2.static.flickr.com/1006/1234245996_4c658ef54c.jpg
These are sections of the Periodic Table that contain elements that behave similarly to elements
in the rest of the section. Thus, we are going to designate each section with a letter to signify
what part of the table it is from. This is just like our baseball ticket that told us where to sit!