The Electron is a What!?
By solving the Schrödinger equation (Hѱ = Eѱ), we obtain a set of mathematical equations,
called wave functions (ѱ), which describe the probability of finding electrons at certain energy
levels within an atom. The equation is a great deal more complicated than this, but if we keep it
simple we can get some sense of what Schrodinger is telling us. The ‘H’ variable is described as
an operator that that provides information to the electron which, in the end, produces a
wavefunction (ѱ) or a specific wave pattern for that wave. I think of it as the kinds of conditions
that would ultimately lead to the different kinds of waves (i.e. surfable versus non-surfable) we
see when we are at the beach.
That’s about the best I can do when I try to envision how this equation works. Electrons have
different wave properties depending upon where they are located. Change the wave properties and
you alter the wave pattern.
A wave function for an electron in an atom is called an atomic orbital; this atomic orbital
describes a region of space in which there is a high probability of finding the electron. Energy
changes within an atom are the result of an electron changing from a wave pattern with one energy
to a wave pattern with a different energy (usually accompanied by the absorption or emission of a
photon of light; a quantum jump).
Each electron in an atom is described by four different quantum numbers. Think of them as the
number in an address where electrons have spent most of their time. The first three (n, l, ml) specify
the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that
orbital.
1. Principal Quantum Number (n) where n = {1, 2, 3,…, ∞}. The principle quantum number
specifies the energy of an electron and the size of the orbital (the distance from the nucleus of
the peak in a radial probability distribution plot).
All orbitals that have the same value of n are said to be in the same shell (energy level). For a
hydrogen atom with n = 1, the electron is in its ground state; if the electron quantum leaps to
the n = 2 orbital, it is in an excited state. The total number of orbitals for a given n value is n2.
The Aufbau principle states that electrons fill orbitals starting at the lowest available
(possible) energy level before filling higher levels. They are much like you or I. They try to
park as close to doors of a shopping mall as they can. If n = 1 has a vacancy that’s where the
electron is going.
2. Angular Momentum (Secondary, Azimuthal) Quantum Number(l): l ={0,1,2..., n-1}. The
Azimuthal number specifies the shape of an orbital with a particular principal quantum
number. This secondary quantum number (l) divides the shells into smaller groups of orbitals
called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion
with n:
l
0
1
2
3
4
5
...
Letter
s
p
d
f
g
h
...
The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so
on. The value of l also has a slight effect on the energy of the subshell; the energy of the
subshell increases with l (s < p < d < f). [ed note: this may seem a bit confusing at this point,
but it is the key to solving your investigational question]
3. Magnetic QuantumNumber(ml): ml ={-l,...,0,...,+l}. The Magnetic Quantum Number
specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number
divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in
each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so
on.
The following images are computer representations of the different orientations the orbitals can
have. These images will make more sense after you’ve completed the investigation. Well…not
really; you will just be as confused as everyone else. Welcome to the World of the Quantum.
Now you know why Albert Einstein wasn’t very fond of the idea.
.
4. Spin Quantum Number (ms): ms = +½ or -½. This part of the address specifies
the orientation of the spin axis of an electron. An electron can spin in only one of two
directions (sometimes called up and down or clockwise or counterclockwise).
Because an electron spins, it creates a magnetic field [ed note: which is why the electron
was deflected in the magnetic field generated in the Thomson experiment], which can be
oriented in one of two directions. For two electrons in the same orbital, the spins must be
opposite to each other; the spins are said to be paired. These substances are not attracted
to magnets and are said to be diamagnetic. Atoms with more electrons that spin in one
direction than another contain unpaired electrons. These substances are weakly attracted
to magnets and are said to be paramagnetic.
The Pauli exclusion principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in
the same atom can have identical values for all four of their quantum numbers. What this means
is that no more than two electrons can occupy the same orbital, and that two electrons in the same
orbital must have opposite spins; no exceptions. The rule does allow for an orbital to have only
one electron; but the ‘no vacancy’ sign goes up after the second electron moves in.
Let’s review some key aspects of atomic structure:
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atoms have a nucleus at their core and are surrounded by electrons, ethe nucleus is comprised of protons, p+ and neutrons, n0
protons are positively charged, +1 unit
neutrons have no charge
electrons are much smaller and negatively charged, -1 unit
the number of protons in the nucleus of an atom is given by a unique atomic
number, Z which defines the type of atom
the total number of protons and neutrons in the nucleus is given by the mass number, A
a neutral atom must have the same number of electrons surrounding the nucleus as there
are protons in the nucleus, so Z also defines the number of electrons in a neutral atom.
in a collection of neutral atoms each may have a different numbers of neutrons. The
different atoms are known as isotopes of that element.
We also know that electrons can be found in the energy levels surrounding the nucleus. We know
from earlier classes that depending upon the element and the numbers of protons and electrons it
possesses, there are 2 electrons in the first energy level, 8 electrons in the second, 18 in the third
and 32 in the fourth. This investigation addresses the question as why that is true. What is it,
given what we are now learning about the quantum addresses of electrons leads us to conclude that
n = {2,8,18,32,…}? Keep in mind that there are certain aspect of this investigation we do not need
to concern ourselves with. We do not need to worry about terms such as Eigenfunction nor should
you need to deal with the mathematical expressions described by Schrodinger. Concern yourselves
only with working with n = {1,2,3,4} and the controls of the simulation that adjust for l = n – 1
where l = {0,1,2,3} or l = {s, p, d, f }.
What can you report about the orbitals in each of the sublevel? How many are there in each
sublevel? What do they look like? What about the numbers of sublevels, orbitals, and electrons
in each of those categories? How do these quantum numbers located in each of the energy levels
and associated electrons generate the set of n = {2,8,18,32,…}?
Based upon your investigation, how would you complete the following table?
energy Level (n )
sublevel,
Orbitals
Numbers of
Total numbers of
l =n–1
ml = {-l, l }
electrons / l
electrons/ n
1
2
2
8
3
18
4
32