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Last time, we saw that these
electron configurations that you

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have been writing down are
nothing other than a shorthand

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way of writing down the wave
functions for each electron in a

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multi-electron atom within the
one-electron wave approximation,

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where we let every electron in
the system, or in the atom have

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its own wave function.
And, as an approximation,

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we gave it a hydrogen atom wave
function.

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But we also saw that at most,
we gave two electrons a

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hydrogen atom wave function,
the same hydrogen atom wave

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function.
And, of course,

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you already know that the
reason we did that is because of

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this quantum mechanical concept
called spin, quantum mechanical

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phenomenon called spin.
Spin is the intrinsic angular

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momentum.
It is the angular momentum

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built into the particle.
And that angular momentum comes

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in sort of two polarities,
a spin up and a spin down,

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--
-- corresponding to the two

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possible values of the spin
quantum number,

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m sub s,
that we looked at last time.

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m sub s can have the value plus
one-half, or it can have the

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value minus one-half.
We are about to talk,

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00:01:55 --> 00:02:01
now, about how spin was
actually discovered.

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And it was discovered by these
two gentlemen here,

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George Uhlenbeck and Sam
Goudsmit.

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They were really very young
scientists.

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They might have been post-docs
at the time.

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What they were looking at was
the emission spectra from sodium

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atoms.
And this is 1925,

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and so they already knew enough
about the electronic structure

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to anticipate at what frequency
they ought to see emission from

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the sodium atoms.
And so they got a discharge

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going, looked at the emission,
disbursed it,

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and they thought that they
would see some emission at this

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particular frequency.
But instead,

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what they saw was some emission
a little bit lower in frequency

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than they expected and a little
bit higher in energy than what

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they expected.
In spectroscopy,

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this kind of emission,
here, is called the doublet.

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A little lower and a little
higher.

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They looked at these results,
thought about it and said I

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think we can understand those
two lines, those emissions at

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the two frequencies there,
if the electron,

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in particular that extra s
electron in the sodium,

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existed in one of two spin
states.

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This was a revolutionary idea.
They were quite excited about

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it.
They took the results of their

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experiment and their
interpretation to the resident

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established scientist at the
time that was closest to them,

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this guy, Wolfgang Pauli.
Wolfgang Pauli was not a nice

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man.
They showed him the data and

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said, this makes sense if the
electron is in two different

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spin states.
And Wolfgang Pauli said

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rubbish.
You publish that and you will

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wreck your young scientific
careers.

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Uhlenbeck and Goudsmit let
dejectedly.

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No sooner than the door slammed
shut, Pauli sits down and writes

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a paper on the presence of the
fourth quantum number,

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m sub s.
This is one of the best

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well-known travesties of
science, now well-known.

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And it actually took,
however, another three years

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and another gentleman,
Dirac, who actually wrote down

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the relativistic Schršdinger
equation and solved it.

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When you do that,
out drops this fourth quantum

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number, m sub s.
However, Pauli did contribute

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to this problem in the sense
that he worked on the principles

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behind these problems with
electrons being fermions,

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etc., which we won't go into.
Out of that work came something

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called this, the Pauli Exclusion
Principle.

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And the essence of that
principle is that no two

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electrons in the same atom can
have the same electron wave

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function and the same spin.
Or, another way to say that,

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no two electrons can have the
same set of four quantum

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numbers.
For example,

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in our electron configuration
here of neon,

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this electron has the quantum
numbers 1, 0,

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0, plus one-half
for m sub s.

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This electron has the quantum
numbers 1, 0,

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0, minus one-half
for m sub s.

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These two electrons don't have
the same set of four quantum

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numbers, and that is why we
could only put two electrons,

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here, in this 1s state.
Likewise, this electron,

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here, has the quantum numbers
2, 1, minus 1,

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plus one-half.

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This electron has the quantum
numbers 2, 1,

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minus 1, minus one-half.

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That is the Pauli Exclusion
Principle, which prevents us

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from putting more than two
electrons in each one of these

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states, 2s, 2s,
2px, 2pz, or 2py.

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Now, what I want to do is try
to look at the wave functions

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and what they look like for the
electrons in the multi-electron

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atom.
And, to look at their shapes,

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what we are going to do is
we're going to look at the

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radial probability distribution
function.

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Remember what the radial
probability distribution

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function tells us?
It tells us the probability of

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finding the electron between r
and r plus dr.

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I plotted those radial
probability distribution

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functions versus r for each one
of the electrons in the

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different states for this
multi-electron atom,

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argon.
And what we want to do is we

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want to compare and contrast
these wave functions for the

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individual electrons in this
multi-electron atom to those of

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hydrogen.
Well, first the similarities.

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If you look at the radial
probability distribution for the

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1s wave function here,
what you see is that the radial

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probability distribution is zero
at r equals zero,

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as all radial probability
distributions.

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That is not a radial node.
That probability distribution

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00:08:34 --> 00:08:38
increases, goes to a maximum,
and then decays exponentially

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00:08:38 --> 00:08:41
with r.
That is exactly what a 1s wave

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function looks like for a
hydrogen atom.

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If you look at the 2s wave
function, it starts at r equals

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zero, it goes up a bit and then
goes to zero.

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Here is a radial node in the 2s
wave function.

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And then goes back up.
Here is the most probable value

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of r and then decays
exponentially.

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Again, it has the same
structure as the 2s wave

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function in the hydrogen atom.
The similarity is that all of

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these wave functions have the
same kind of basic structure as

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that in a hydrogen atom.
They have the same number of

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nodes, the same number of radial
nodes and the same number of

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angular nodes.
The difference between these

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wave functions and those of the
hydrogen atom is that all of

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these wave functions are closer
in to the nucleus.

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For example,
if you looked here at what the

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most probable value of r is for
that 1s electron in argon,

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it is 0.1 a nought.

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What is the most probable value
for r in the 1s state of

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hydrogen?
a nought.

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This is ten times closer.
This is much closer to the

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nucleus than it is in the
hydrogen atom.

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00:10:05 --> 00:10:11
And if we went and compared the
most probable values for 2s,

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2p for those of hydrogen and
3s, 3p for those of hydrogen,

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we would find that all of these
are much closer into the

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nucleus.
Why?

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Because the nucleus has a
larger positive charge on it.

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The Coulomb interaction here is
the charge on the electron times

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the charge on the nucleus.
For argon, that charge is plus

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18 times e.
That greater attractive

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interaction holds those
electrons in closer to the

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nucleus.
That is the bottom line.

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That is how these differ.
The structure is the same,

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node structure is the same,
they are just all closer into

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the nucleus because of that
greater attractive interaction.

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00:11:05 --> 00:11:08
Now that I have this radial
probability distribution

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function up here,
I also want to use it to just

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illustrate a concept that I
think you already know.

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That is this concept of a
shell.

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You know about the n equals 1
shell, n equals 2 shell,

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n equals 3 shell.
And the word shell also denotes

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some kind of spatial
information.

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00:11:30 --> 00:11:35
I want to show you how the
spatial information is depicted,

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here, on this graph.
I want you to see that for the

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n equals 3 states,
3s and 3p, well,

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the most probable value is not
exactly in the same place,

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but it is in the same place as
when you compare it to the 2s

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and the 2p.
You can see how well the n

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equals 3 shell is separated in
space from the n equals 2 shell.

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Again, the most probable value
for 2s and 2p are not exactly in

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the same place.
We saw that 2p is actually a

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little closer than to 2s,
but in terms of comparing it to

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where the most probable values
are for 3s and 3p,

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that is much closer in.
And so this graph,

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here, gives you an idea of the
spatial information that is

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denoted when we talk about
shells.

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The n equals 3 shell is further
out, the n equals 2 shell closer

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in, and the n equals 1 shell
even closer in.

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I think that is a concept that
you mostly know.

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Here, you see it on the radial
probability distribution.

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Now, we have taken a quick look
at those wave functions.

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Now, it is time to actually
look at the energies of the

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states.
We have not done that yet.

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On the left here,
I show an energy level diagram

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for the hydrogen atom.
We saw this before.

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Here is the n equals 1 state.
Here at n equals 2,

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we have four degenerate states.
Here at n equals 3,

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we have 9 degenerate states.
Here at n equals 4,

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we have 16 degenerate states.
But the difference between the

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hydrogen atom and any other
multi-electron atom,

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starting with helium,
are two differences.

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One is that the energies of
these states in the

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multi-electron atom are all
lower than they are in the

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hydrogen atom.
That is, the 1s state here is

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lower in energy than the 1s
state in the hydrogen.

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The 2s is lower than the 2s
state in hydrogen,

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00:14:05 --> 00:14:08
the 3s is lower,
the 2p is lower,

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the 3p is lower,
etc.

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The energies of those states
are all lower.

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Why?
Because of the charge on the

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nucleus.
That potential energy of

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interaction is greater because
the charge on the nucleus is

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larger.
That greater potential energy

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of interaction lowers the energy
of the states.

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It makes those electrons more
strongly bound.

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Starting with helium,
all of these energies are lower

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than those in the hydrogen atom.
That is the first difference.

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The second difference is,
you can see,

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now, that the 2s state is lower
in energy than the 2p state.

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The degeneracy between 2s and
2p in a hydrogen atom is lifted

205
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or is broken,
as we say.

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00:15:03 --> 00:15:08
Likewise, the 3s state is lower
in energy than the 3p state,

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00:15:08 --> 00:15:12
than the 3d state.
The degeneracy in those states

208
00:15:12 --> 00:15:17
is lifted, or it is broken.
That is now what we have to

209
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talk about, why that is the
case.

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Why is 2s, for example,
lower in energy than 2p?

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00:15:24 --> 00:15:30
And the reason for this has to
do with the phenomenon called

212
00:15:30 --> 00:15:34
shielding.
We have to talk about this

213
00:15:34 --> 00:15:39
phenomenon, shielding,
and we have to talk about how

214
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that leads to a concept called
effective charge.

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00:15:43 --> 00:15:47
But to do that,
I am going to do the following.

216
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I am going to realize that each
one of these energies here,

217
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E sub nl,
so now these energies are

218
00:15:57 --> 00:16:02
labeled by both the principle
quantum number and the angular

219
00:16:02 --> 00:16:08
momentum quantum number.
I am going to realize that

220
00:16:08 --> 00:16:13
these energies here physically
are minus the ionization energy

221
00:16:13 --> 00:16:17
because these energies are minus
the energy it is going to

222
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require to rip the electron off
from that particular state.

223
00:16:22 --> 00:16:27
And I am going to set those
binding energies equal to a

224
00:16:27 --> 00:16:31
hydrogen atom like energy level
here.

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00:16:31 --> 00:16:34
And now I am going to use the
board and explain that just a

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little bit more.

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I said these energies,
which are now a function of n

229
00:16:57 --> 00:17:03
and l, they are minus the
ionization energy from that nl

230
00:17:03 --> 00:17:07
state.
And I am going to approximate

231
00:17:07 --> 00:17:12
that as a hydrogen atom kind of
an energy scheme.

232
00:17:12 --> 00:17:16
That is, I am going to set this
equal to R sub H,

233
00:17:16 --> 00:17:20
the Rydberg constant over n
squared.

234
00:17:20 --> 00:17:24
Out here,
there is going to be a Z,

235
00:17:24 --> 00:17:30
but this Z is going to be Z
effective, Zeff.

236
00:17:30 --> 00:17:34
And it is going to be squared,
of course.

237
00:17:34 --> 00:17:41
And that Z is going to depend
on that particular nl state that

238
00:17:41 --> 00:17:44
you are in.
This Z effective,

239
00:17:44 --> 00:17:48
here, is the effective nuclear
charge.

240
00:17:48 --> 00:17:55
It is not the nuclear charge.
It is the effective nuclear

241
00:17:55 --> 00:17:56
charge.
Why?

242
00:17:56 --> 00:18:04
Well, because of shielding.
Let's try to explain that.

243
00:18:04 --> 00:18:09
Let's take helium.
Z is equal to plus 2e.

244
00:18:09 --> 00:18:13
We have plus 2e here,

245
00:18:13 --> 00:18:19
and let's do a thought
experiment in that we are going

246
00:18:19 --> 00:18:26
to take electron number two,
here, and place it kind of

247
00:18:26 --> 00:18:32
close to this plus 2 charged
nucleus.

248
00:18:32 --> 00:18:38
And then we are going to have
electron number one way out

249
00:18:38 --> 00:18:40
here.
In this case,

250
00:18:40 --> 00:18:47
with electron number one way
out here, the nuclear charge

251
00:18:47 --> 00:18:53
that electron number one
experiences, because it is so

252
00:18:53 --> 00:19:00
far out, kind of looks like a
plus one charge.

253
00:19:00 --> 00:19:03
Because this electron,
on the average,

254
00:19:03 --> 00:19:09
is canceling one of the
positive charges on the nucleus.

255
00:19:09 --> 00:19:15
So, the effective charge here
for this electron way out there,

256
00:19:15 --> 00:19:18
we are going to say,
is plus one.

257
00:19:18 --> 00:19:24
If that is the
case, well, then the binding

258
00:19:24 --> 00:19:30
energy of that electron is one
squared times R sub H over n

259
00:19:30 --> 00:19:35
squared.

260
00:19:35 --> 00:19:39
We are going to consider n
equal one because we are going

261
00:19:39 --> 00:19:43
to talk about the ground state
of the helium atom,

262
00:19:43 --> 00:19:45
here.
But you know what this value is

263
00:19:45 --> 00:19:49
going to turn out to be.
It is going to turn out to be

264
00:19:49 --> 00:19:54
minus 2.180x10^-18 joules.
That is the binding energy of

265
00:19:54 --> 00:19:58
an electron in a hydrogen atom.
This is a thought experiment,

266
00:19:58 --> 00:20:01
now.
This is helium,

267
00:20:01 --> 00:20:06
this is just one electron way
out here, and it cannot

268
00:20:06 --> 00:20:13
discriminate too well between
the nucleus and this electron,

269
00:20:13 --> 00:20:18
so the overall effective charge
it sees is plus one.

270
00:20:18 --> 00:20:21
But now, we take the other
case.

271
00:20:21 --> 00:20:27
The other extreme case is we
are going to bring in electron

272
00:20:27 --> 00:20:34
one really close to the nucleus.
And electron two is way out

273
00:20:34 --> 00:20:40
here such that it does not do
anything as far as this electron

274
00:20:40 --> 00:20:43
is concerned.
And so this electron,

275
00:20:43 --> 00:20:47
in this case,
is experiencing the total

276
00:20:47 --> 00:20:52
nuclear charge on the nucleus.
And so we say in this thought

277
00:20:52 --> 00:20:56
case here, with this electron
really close,

278
00:20:56 --> 00:21:01
that the effective charge is
equal to plus 2e for this

279
00:21:01 --> 00:21:05
electron.

280
00:21:05 --> 00:21:11
Well, if that is the case,
we can calculate the binding

281
00:21:11 --> 00:21:17
energy of this electron.
And that is going to be 2

282
00:21:17 --> 00:21:20
squared R sub H over one
squared.

283
00:21:20 --> 00:21:26
We can plug in R sub H.

284
00:21:26 --> 00:21:33
And we are going to get minus
8.72x10^-18 joules.

285
00:21:33 --> 00:21:38
What this is is the binding
energy of an electron in helium

286
00:21:38 --> 00:21:43
plus.
And so, in this extreme case,

287
00:21:43 --> 00:21:48
with the electron really close,
that is the binding energy.

288
00:21:48 --> 00:21:54
With this extreme case,
with the electron way far out,

289
00:21:54 --> 00:22:00
this is the binding energy.
This is total shielding.

290
00:22:00 --> 00:22:05


291
00:22:05 --> 00:22:11
This case is no shielding.
The electron is much more

292
00:22:11 --> 00:22:18
strongly bound.
The reality is that the binding

293
00:22:18 --> 00:22:25
energy of an electron in helium
is somewhere in between.

294
00:22:25 --> 00:22:33
That is, the ionization energy
for helium, for an electron in

295
00:22:33 --> 00:22:41
now neutral helium here,
is 3.94x10^-18 joules.

296
00:22:41 --> 00:22:47
Somewhere in between this
extreme case of total shielding

297
00:22:47 --> 00:22:52
and this extreme case of no
shielding at all.

298
00:22:52 --> 00:22:56
And that is because,
on the average,

299
00:22:56 --> 00:23:03
there is another electron in
between that electron and the

300
00:23:03 --> 00:23:07
nucleus.
And we can calculate the

301
00:23:07 --> 00:23:11
effective charge,
then, from the experimental

302
00:23:11 --> 00:23:16
binding energies by just taking
this expression and rearranging

303
00:23:16 --> 00:23:19
it.
I am going to solve that

304
00:23:19 --> 00:23:22
expression for the effective
charge, Zeff.

305
00:23:22 --> 00:23:26
And, when I do that that,
add, of course,

306
00:23:26 --> 00:23:31
this n squared times the
ionization energy over the

307
00:23:31 --> 00:23:36
Rydberg constant.

308
00:23:36 --> 00:23:42
n squared is going to be one
because we are talking about the

309
00:23:42 --> 00:23:46
ground state.
The ionization energy,

310
00:23:46 --> 00:23:52
I said, was 3.94x10^-18 joules.
The Rydberg constant is

311
00:23:52 --> 00:23:58
2.180x10^-18 joules.
In the end, I find an effective

312
00:23:58 --> 00:24:04
charge here of plus 1.34.

313
00:24:04 --> 00:24:09
Again, why is that the case?
Well, that is the case because

314
00:24:09 --> 00:24:14
here is my electron,
here is my helium nucleus and,

315
00:24:14 --> 00:24:19
just on the average,
between this electron and the

316
00:24:19 --> 00:24:23
nucleus there is always another
electron around.

317
00:24:23 --> 00:24:30
This electron kind of partially
shields this nuclear charge.

318
00:24:30 --> 00:24:35
It partially shields it so that
we calculated,

319
00:24:35 --> 00:24:41
from using this scheme and
using the experimental

320
00:24:41 --> 00:24:47
ionization energies,
an effective charge of 1.34.

321
00:24:47 --> 00:24:54
So now, we sort of understand
shielding and effective charge.

322
00:24:54 --> 00:25:00
That is a square root,
thank you.

323
00:25:00 --> 00:25:04
All right.
Now, we have to use this idea

324
00:25:04 --> 00:25:11
of effective charge and
shielding to understand why the

325
00:25:11 --> 00:25:16
2s state is lower in energy than
the 2p state.

326
00:25:16 --> 00:25:20
And here comes that.

327
00:25:20 --> 00:25:30


328
00:25:30 --> 00:25:33
Let's think about the lithium
atom.

329
00:25:33 --> 00:25:39
The electron confirmation of
lithium is 1s 2 2s 1.

330
00:25:39 --> 00:25:43
The electron configuration of

331
00:25:43 --> 00:25:47
lithium is not 1s 2 2p 1.

332
00:25:47 --> 00:25:50
Why?
Because 2s is lower in energy

333
00:25:50 --> 00:25:54
than 2p.
But why is that the case?

334
00:25:54 --> 00:25:59
Well, to look at that,
we have to look at the radial

335
00:25:59 --> 00:26:06
probability distribution
functions for 2s and 2p.

336
00:26:06 --> 00:26:11
Here is a radial probability
distribution function for 2s.

337
00:26:11 --> 00:26:17
Here is the radial probability
distribution function for 2p.

338
00:26:17 --> 00:26:22


339
00:26:22 --> 00:26:26
Now, you have to do a thought
experiment, here.

340
00:26:26 --> 00:26:30
In this 2s radial probability
distribution function,

341
00:26:30 --> 00:26:36
what you see is that there is
some finite probability of the

342
00:26:36 --> 00:26:42
electron in the 2s state being
really close to the nucleus.

343
00:26:42 --> 00:26:47
For the sake of the argument,
here, I am going to say that

344
00:26:47 --> 00:26:53
for this part of the probability
distribution function,

345
00:26:53 --> 00:26:58
the effective charge Zeff is
going to be plus 3e.

346
00:26:58 --> 00:27:02
I mean, that is an exaggeration

347
00:27:02 --> 00:27:05
because obviously we have some s
electrons.

348
00:27:05 --> 00:27:09
But for the sake of this
argument, I am going to say the

349
00:27:09 --> 00:27:14
effective charge for this part
of the probability distribution

350
00:27:14 --> 00:27:17
function is plus 3e.
For this part of the

351
00:27:17 --> 00:27:21
probability distribution
function for 2s that is much

352
00:27:21 --> 00:27:24
further out.
And I have those 1s electrons

353
00:27:24 --> 00:27:27
closer in.
So, for the sake of this

354
00:27:27 --> 00:27:31
argument, I am going to say that
these s electrons completely

355
00:27:31 --> 00:27:37
shield the nuclear charge.
And so, the effective charge

356
00:27:37 --> 00:27:42
for this part of the
distribution is plus 1e.

357
00:27:42 --> 00:27:46
Now, what about 2p?
Well, in the case of 2p,

358
00:27:46 --> 00:27:51
you can see that this 2p wave
function, although it is a

359
00:27:51 --> 00:27:56
little bit closer in,
for the most part it is about

360
00:27:56 --> 00:28:02
in the same place as the second
lobe here of the 2s wave

361
00:28:02 --> 00:28:06
function.
We are going to say that this

362
00:28:06 --> 00:28:09
effective charge is plus 1e,

363
00:28:09 --> 00:28:12
just like this lobe of the 2s
wave function.

364
00:28:12 --> 00:28:15
But now, to get the kind of
total effective charge,

365
00:28:15 --> 00:28:20
I am going to have to take the
effective charge and average it

366
00:28:20 --> 00:28:23
over this probability
distribution function.

367
00:28:23 --> 00:28:27
And so, since in the 2s
function I am going to average

368
00:28:27 --> 00:28:31
over a part that has a plus 3
effective charge and a part that

369
00:28:31 --> 00:28:38
has a plus 1 effective charge.
Well, if I average over that,

370
00:28:38 --> 00:28:44
that is going to be larger than
the effective charge of this 2p

371
00:28:44 --> 00:28:49
wave function because the 2p
everywhere is plus one.

372
00:28:49 --> 00:28:55
The effective charge here for
the 2s is going to be greater,

373
00:28:55 --> 00:29:00
on the average,
than for the 2p.

374
00:29:00 --> 00:29:04
Because of this part of the
probability distribution

375
00:29:04 --> 00:29:09
function, this part that is
really close to the nucleus,

376
00:29:09 --> 00:29:14
where the electrons can feel
more of the nuclear charge or

377
00:29:14 --> 00:29:19
experience more of the nuclear
charge than they could if they

378
00:29:19 --> 00:29:23
were a 2p electron.
Therefore, since that 2s

379
00:29:23 --> 00:29:27
electron has a greater effective
charge, here,

380
00:29:27 --> 00:29:33
what that is going to mean is
that the binding energy of the

381
00:29:33 --> 00:29:38
2s state is going to be lower in
energy.

382
00:29:38 --> 00:29:43
It is going to be more negative
than the binding energy of the

383
00:29:43 --> 00:29:47
2p state.
The same thing for 3s and 3p.

384
00:29:47 --> 00:29:52
The same reasoning there.
And it is all because of this

385
00:29:52 --> 00:29:57
little part of the probability
distribution function.

386
00:29:57 --> 00:30:01
Now we can understand why the
2s is, in fact,

387
00:30:01 --> 00:30:06
lower in energy than the 2p.
Yeah?

388
00:30:06 --> 00:30:12


389
00:30:12 --> 00:30:14
Because the 3s,
although it does,

390
00:30:14 --> 00:30:18
in fact, have this,
it is a little bit further out

391
00:30:18 --> 00:30:21
here, enough to make for this to
compensate.

392
00:30:21 --> 00:30:24
But, again, 3s is lower than
3p.

393
00:30:24 --> 00:30:30
We are just going to compare 3s
and 3p within the same shell.

394
00:30:30 --> 00:30:38


395
00:30:38 --> 00:30:42
Yes, it does have to do with
the net area underneath.

396
00:30:42 --> 00:30:47
If the probability was much
higher, and it is not that much

397
00:30:47 --> 00:30:52
higher, but if it were then you
are right, it could cancel it

398
00:30:52 --> 00:30:53
out.

399
00:30:53 --> 00:30:58


400
00:30:58 --> 00:31:02
Now, therefore,
we are ready to write the

401
00:31:02 --> 00:31:09
electron configurations of all
the atoms on the Periodic Table.

402
00:31:09 --> 00:31:15
And you already know that we
use the Aufbau Principle to do

403
00:31:15 --> 00:31:18
that.
Aufbau means building up.

404
00:31:18 --> 00:31:23
What do you do?
You take all the allowed states

405
00:31:23 --> 00:31:29
and order them according to
their energy.

406
00:31:29 --> 00:31:33
Most strongly bound or most
negative energy goes on the

407
00:31:33 --> 00:31:37
bottom, and next most negative
energy, on and on.

408
00:31:37 --> 00:31:41
And we will talk about a
pneumonic for remembering those

409
00:31:41 --> 00:31:45
energies in a moment.
We use the Aufbau principle.

410
00:31:45 --> 00:31:50
We start with the lowest energy
state and we put an electron in

411
00:31:50 --> 00:31:52
for hydrogen.
There it is,

412
00:31:52 --> 00:31:55
the 1s state.
For helium, well,

413
00:31:55 --> 00:31:59
we also put that into the 1s
state.

414
00:31:59 --> 00:32:04
Except, as we fill these
states, we have to heed the

415
00:32:04 --> 00:32:08
Pauli Exclusion Principle.
This electron,

416
00:32:08 --> 00:32:12
here, goes in with the opposite
spin.

417
00:32:12 --> 00:32:15
Next lithium,
2s electron.

418
00:32:15 --> 00:32:19
For beryllium,
another 2s electron,

419
00:32:19 --> 00:32:22
opposite spin.
And then for nitrogen,

420
00:32:22 --> 00:32:29
electron, here,
has to go into the 2p state.

421
00:32:29 --> 00:32:33
Now, it does not matter whether
you put it in 2px,

422
00:32:33 --> 00:32:37
2py or 2pz.
They are all the same energy.

423
00:32:37 --> 00:32:42
The next electron,
carbon, what are we going to do

424
00:32:42 --> 00:32:44
here?
Well, here we have to obey

425
00:32:44 --> 00:32:50
something called Hund's Rule.
And Hund's Rule says that when

426
00:32:50 --> 00:32:55
electrons are added to states of
the same energy,

427
00:32:55 --> 00:32:59
and that is what we are doing
here in the 2p,

428
00:32:59 --> 00:33:04
a single electron enters each
state before a second electron

429
00:33:04 --> 00:33:11
enters any state.
And, those single electrons

430
00:33:11 --> 00:33:17
have to go in so that the
resulting spins are parallel.

431
00:33:17 --> 00:33:20
That is, they have the same
spin.

432
00:33:20 --> 00:33:24
Do we put in an electron like
this?

433
00:33:24 --> 00:33:27
No.
Do we put it in like that?

434
00:33:27 --> 00:33:32
No.
Do we put it in like this?

435
00:33:32 --> 00:33:37
Yeah, according to Hund's Rule.
And then the next electron has

436
00:33:37 --> 00:33:43
to go into that other empty p
state before any of these states

437
00:33:43 --> 00:33:46
double up.
Then we keep doubling them up.

438
00:33:46 --> 00:33:51
The next electron has to go
into the next state 3s.

439
00:33:51 --> 00:33:55
Again, we double them up.
Now we are to the 3p state

440
00:33:55 --> 00:34:00
again.
One electron goes into 2px.

441
00:34:00 --> 00:34:04
The next electron will go into
either 2py or 2pz,

442
00:34:04 --> 00:34:09
according to Hund's Rule,
and the spins remain parallel

443
00:34:09 --> 00:34:12
to get the lowest energy
configuration.

444
00:34:12 --> 00:34:17
And then the next electron 2pz.
And then we start doubling up.

445
00:34:17 --> 00:34:21
And we keep going.
You keep going in that way so

446
00:34:21 --> 00:34:26
that you write the electron
configuration of all the atoms

447
00:34:26 --> 00:34:32
in the Periodic Table.
Let's look at these electron

448
00:34:32 --> 00:34:36
configurations kind of quickly
here.

449
00:34:36 --> 00:34:41
Let's start with the third
period, the third row here,

450
00:34:41 --> 00:34:45
sodium going across here to
argon.

451
00:34:45 --> 00:34:49
Here is the electron
configuration for sodium.

452
00:34:49 --> 00:34:55
Notice here that I don't mind
if you write that electron

453
00:34:55 --> 00:34:59
configuration as 2p 6
instead of 2px,

454
00:34:59 --> 00:35:04
2py, 2pz.
Because you cannot tell the

455
00:35:04 --> 00:35:06
difference between x,
y, and z, anyway,

456
00:35:06 --> 00:35:09
if you are not in a magnetic
field.

457
00:35:09 --> 00:35:13
These electrons here in sodium
that make up the inert gas

458
00:35:13 --> 00:35:17
configuration of argon,
of course, are the core

459
00:35:17 --> 00:35:19
electrons.
When we talk about core

460
00:35:19 --> 00:35:24
electrons, we are talking about
the electrons that make up the

461
00:35:24 --> 00:35:30
nearest rare gas configuration.
And then the valance electron,

462
00:35:30 --> 00:35:35
well, is the electrons that are
beyond the nearest but lowest

463
00:35:35 --> 00:35:39
inert gas configurations.
And also, when you are writing

464
00:35:39 --> 00:35:43
these electron configurations,
say, for sodium,

465
00:35:43 --> 00:35:48
you can write it as the neon
configuration and then just show

466
00:35:48 --> 00:35:51
the valence electron, 3s 1.

467
00:35:51 --> 00:35:56
As we go across that third
period here everything is very

468
00:35:56 --> 00:36:01
normal.
3s fills up first and then the

469
00:36:01 --> 00:36:05
3p's fill up.
Now we get to the fourth

470
00:36:05 --> 00:36:09
period, from potassium to
krypton.

471
00:36:09 --> 00:36:14
Fourth period,
what happens here is that those

472
00:36:14 --> 00:36:19
first two electrons go into the
s states.

473
00:36:19 --> 00:36:23
They do not go into the 3d
states.

474
00:36:23 --> 00:36:29
They don't because those s
states are lower in energy than

475
00:36:29 --> 00:36:34
the 3d states.
And so these electron

476
00:36:34 --> 00:36:38
configurations are argon 4s 1,

477
00:36:38 --> 00:36:43
argon 4s 2 **[Ar]4s^2**.
And then, once we fill those 4s

478
00:36:43 --> 00:36:46
states, we start filling the 3d
states.

479
00:36:46 --> 00:36:49
Here is scandium,
here is titanium,

480
00:36:49 --> 00:36:52
here is vanadium,
everything is normal,

481
00:36:52 --> 00:36:55
3d 1, 3d 2,

482
00:36:55 --> 00:37:00
3d 3,
and then we get to chromium.

483
00:37:00 --> 00:37:03
We have an exception here,
chromium.

484
00:37:03 --> 00:37:07
Chromium is not what you would
expect.

485
00:37:07 --> 00:37:11
It is not 4s 2 3d 4.

486
00:37:11 --> 00:37:14
Chromium is 4s 1 3d 5.

487
00:37:14 --> 00:37:19
There is no way for you to know
that a priori,

488
00:37:19 --> 00:37:24
unless you do a very
sophisticated calculation,

489
00:37:24 --> 00:37:30
but this is experimentally
observed that this is the

490
00:37:30 --> 00:37:33
configuration.
Why?

491
00:37:33 --> 00:37:39
Because it is lower in total
energy than this configuration.

492
00:37:39 --> 00:37:45
It turns out that there is some
extra stability in having a half

493
00:37:45 --> 00:37:51
filled 4s shell and a half
filled 3d sub-shell compared to

494
00:37:51 --> 00:37:58
having a filled 4s shell and a
less than half filled 3d shell.

495
00:37:58 --> 00:38:03
This is an exception that you
do have to know.

496
00:38:03 --> 00:38:08
But after chromium,
here, manganese behaves well.

497
00:38:08 --> 00:38:12
Iron behaves well.
Cobalt okay.

498
00:38:12 --> 00:38:16
Nickel okay.
And now we get to copper,

499
00:38:16 --> 00:38:22
and we have a problem.
It is not what you would

500
00:38:22 --> 00:38:25
expect.
It is not 4s 2 3d 9.

501
00:38:25 --> 00:38:29
Instead, it is 4s 1 3d 10.

502
00:38:29 --> 00:38:35
*Again, this configuration is

503
00:38:35 --> 00:38:38
something you could not have
predicted a priori.

504
00:38:38 --> 00:38:43
If you do a sophisticated
calculation you can see it,

505
00:38:43 --> 00:38:46
but this is also experimentally
observed.

506
00:38:46 --> 00:38:49
We know this to be the
configuration.

507
00:38:49 --> 00:38:53
This is not the configuration.
This is the lower energy

508
00:38:53 --> 00:38:56
configuration.
Here is another exception that

509
00:38:56 --> 00:39:00
you have to know,
copper.

510
00:39:00 --> 00:39:05
And then after copper here,
zinc, things follow a pattern

511
00:39:05 --> 00:39:08
again.
The next electron,

512
00:39:08 --> 10.
then, just fills up the 4s 2 3d

513
10. --> 00:39:13


514
00:39:13 --> 00:39:17
Now, at gallium,
all of those are filled.

515
00:39:17 --> 00:39:20
We start filling up the 4p
states.

516
00:39:20 --> 00:39:24
Everything is fine until we get
to krypton.

517
00:39:24 --> 00:39:30
So, you have to know chromium
and copper.

518
00:39:30 --> 00:39:34
Now, the fifth row here,
starting with rubidium,

519
00:39:34 --> 00:39:37
strontium, ytterbium,
zirconium, etc.

520
00:39:37 --> 00:39:42
all the way across here.
Starting with rubidium,

521
00:39:42 --> 00:39:45
again, the electrons go into
the 5s shell.

522
00:39:45 --> 00:39:50
They do not go into the 4d.
That is because with rubidium

523
00:39:50 --> 00:39:54
and strontium,
5s is actually lower than 4d.

524
00:39:54 --> 00:40:00
And it is only once you fill up
the 5s states that you start

525
00:40:00 --> 00:40:05
filling up the 4d states right
here.

526
00:40:05 --> 00:40:10
Now, there are exceptions along
this fifth row.

527
00:40:10 --> 00:40:14
The two exceptions,
molybdenum and silver,

528
00:40:14 --> 00:40:20
are the same kind of exceptions
as chromium and copper.

529
00:40:20 --> 00:40:26
You have to know the exceptions
for silver and copper and

530
00:40:26 --> 00:40:32
molybdenum and chromium.
The same identical kind of

531
00:40:32 --> 00:40:35
exception as in the fourth row
here.

532
00:40:35 --> 00:40:40
There are other exceptions
along this fifth row.

533
00:40:40 --> 00:40:45
You do not have to know those.
There is really no way,

534
00:40:45 --> 00:40:50
a priori, for you to know that.
Again, it is an experimental

535
00:40:50 --> 00:40:53
observation.
A sophisticated hard

536
00:40:53 --> 00:40:59
calculation will also show that.
And then, once you are done

537
00:40:59 --> 00:41:02
with cadmium,
well, then the 5p's start

538
00:41:02 --> 00:41:07
filling and everything is normal
and you get to the xenon inner

539
00:41:07 --> 00:41:09
gas configuration.

540
00:41:09 --> 00:41:17


541
00:41:17 --> 00:41:22
How do you remember what the
ordering is of these states,

542
00:41:22 --> 00:41:26
the energy ordering?
Well, here is a pneumonic that

543
00:41:26 --> 00:41:30
maybe some of you have seen
before.

544
00:41:30 --> 00:41:35
Start out and write 1s,
then write 2s right below it

545
00:41:35 --> 00:41:40
and then 2p to the side of it.
And then write 3s,

546
00:41:40 --> 00:41:43
3p, 3d.
And then write 4s,

547
00:41:43 --> 00:41:46
4p, 4d, 4f.
And then write 5s,

548
00:41:46 --> 00:41:50
5p, 5d, 5f, and 6s,
6p, 6d, and 7s,

549
00:41:50 --> 00:41:53
7p.
And now, to get the energy

550
00:41:53 --> 00:41:59
ordering we are going to draw
diagonals.

551
00:41:59 --> 00:42:02
Well, first of all,
the 1s is the lowest energy

552
00:42:02 --> 00:42:08
state, and then the next highest
energy state is the 2s state,

553
00:42:08 --> 00:42:12
and then the next highest
energy state is the 2p.

554
00:42:12 --> 00:42:14
We are going to draw a
diagonal.

555
00:42:14 --> 00:42:18
The next state to fill is 2p.
The next one is 3s.

556
00:42:18 --> 00:42:22
Now we are going to draw a
diagonal again.

557
00:42:22 --> 00:42:26
The next one to fill is 3p.
The next one to fill is 4s.

558
00:42:26 --> 00:42:32
Draw another diagonal.
The next one to fill is 3d,

559
00:42:32 --> 00:42:34
4p, 5s.
4d, 5p, 6s.

560
00:42:34 --> 00:42:38
4f, 5d, 6p, 7s.
5s, 6d, 7p, and that is all

561
00:42:38 --> 00:42:45
that is going to be important.
That is one way to remember the

562
00:42:45 --> 00:42:51
relative energy orderings here.
And you do have to be able to

563
00:42:51 --> 00:42:57
write this down on an exam.
You will have a Periodic Table,

564
00:42:57 --> 00:43:04
but it won't have the electron
configurations on it.

565
00:43:04 --> 00:43:10
Now, I am going to tell you
something that sometimes people

566
00:43:10 --> 00:43:15
find a little bit confusing.
That is, the electron

567
00:43:15 --> 00:43:21
configuration of ions.
What I am about to say has no

568
00:43:21 --> 00:43:28
effect on what I just said about
how to write the electron

569
00:43:28 --> 00:43:35
configuration for neutrals.
This does not affect anything

570
00:43:35 --> 00:43:40
in your writing down the
electron configuration for

571
00:43:40 --> 00:43:43
neutrals.
This is for ions.

572
00:43:43 --> 00:43:49
The point I want to make is
that if you actually look at the

573
00:43:49 --> 00:43:56
energies of the individual 3d
states and 4s states across that

574
00:43:56 --> 00:44:02
fourth row, this is what they
look like.

575
00:44:02 --> 00:44:04
For example,
at potassium,

576
00:44:04 --> 00:44:08
that 4s state is lower in
energy than the 3d state.

577
00:44:08 --> 00:44:14
That is why we put the electron
in the 4s state when we wrote

578
00:44:14 --> 00:44:17
the neutral.
The same thing for calcium.

579
00:44:17 --> 00:44:22
That is why we put that
electron into the 4s state and

580
00:44:22 --> 00:44:25
not the 3s state.
Lo and behold,

581
00:44:25 --> 00:44:30
right here at scandium,
Z equals 21.

582
00:44:30 --> 00:44:34
What happens is that the 3d
state actually drops below in

583
00:44:34 --> 00:44:39
energy than the 4s state.
These are the energies of the

584
00:44:39 --> 00:44:44
individual state now,
and that continues all the way

585
00:44:44 --> 00:44:48
across the Periodic Table.
However, that does not affect

586
00:44:48 --> 00:44:53
how you write the electron
configuration of the neutrals.

587
00:44:53 --> 00:44:57
For example,
if you are writing the electron

588
00:44:57 --> 00:45:02
configuration for titanium,
here it is.

589
00:45:02 --> 00:45:07
It is the argon core,
4s 2 3d 2.

590
00:45:07 --> 00:45:12
And, by the way,
I don't care whether you write

591
00:45:12 --> 00:45:17
3d 2 4s 2 or 4s 2 3d 2.

592
00:45:17 --> 00:45:21
You can write them in either
order.

593
00:45:21 --> 00:45:26
Now, you might say,
well, why is this the electron

594
00:45:26 --> 00:45:31
configuration if at this value
for Z, Z equals 22,

595
00:45:31 --> 00:45:38
titanium, the 3d state is lower
than the 4s state?

596
00:45:38 --> 00:45:43
Why don't these 4s electrons
just hop into the 3d states?

597
00:45:43 --> 00:45:47
Well, they don't do that
because this electron

598
00:45:47 --> 00:45:52
configuration actually minimizes
the electron repulsions.

599
00:45:52 --> 00:45:56
If these four electrons were in
the 3d state,

600
00:45:56 --> 00:46:02
well, then the repulsive
interactions would be greater.

601
00:46:02 --> 00:46:05
Because they are in the same
state now.

602
00:46:05 --> 00:46:08
And, therefore,
the entire energy of the atom

603
00:46:08 --> 00:46:12
will be larger.
And it is the entire energy,

604
00:46:12 --> 00:46:17
the total energy of the atom
that is important when we look

605
00:46:17 --> 00:46:22
at the electron configurations.
In this particular case,

606
00:46:22 --> 00:46:26
if we look at the individual d
states and the 4s states,

607
00:46:26 --> 00:46:31
yeah, the d states are lower in
energy.

608
00:46:31 --> 00:46:36
But what is important is when
we sum up all of the energies of

609
00:46:36 --> 00:46:38
the interactions,
what is lower?

610
00:46:38 --> 00:46:42
What I am saying to you is that
when we do that,

611
00:46:42 --> 00:46:46
when we sum up all of the
interaction energies,

612
00:46:46 --> 00:46:50
this still is the lower energy
configuration,

613
00:46:50 --> 00:46:54
even though the 3d electrons at
this value of Z,

614
00:46:54 --> 00:47:00
those 3d states are lower in
energy than the 4s states.

615
00:47:00 --> 00:47:05
That is why we do not have this
hoping over into the 3d states

616
00:47:05 --> 00:47:09
for the neutral.
Now, here comes the ion

617
00:47:09 --> 00:47:13
configuration.
If you have this configuration

618
00:47:13 --> 00:47:18
for the neutral,
and now you ionize titanium to

619
00:47:18 --> 00:47:23
make titanium plus,
the electron configuration is

620
00:47:23 --> 00:47:28
argon 3d 2.
That is, it is the 4s electrons

621
00:47:28 --> 00:47:33
that come off,
that are plucked out.

622
00:47:33 --> 00:47:37
They are plucked out because
they are the higher energy

623
00:47:37 --> 00:47:41
electrons now.
This is the electron

624
00:47:41.127 --> 2.
configuration for titanium plus

625
2. --> 00:47:45


626
00:47:45 --> 00:47:50
We are going to pull out those
higher energy electrons,

627
00:47:50 --> 00:47:55
which are the 4s electrons.
Again, this affects only the

628
00:47:55 --> 00:47:59
ion configuration.
The same thing happens here on

629
00:47:59 --> 00:48:04
the fifth row.
The fifth row starting here

630
00:48:04 --> 00:48:09
with rubidium and strontium,
5s is lower in energy than 4d.

631
00:48:09 --> 00:48:14
That is why that rubidium
electron went in the 5s state.

632
00:48:14 --> 00:48:17
The same thing with the
strontium electron,

633
00:48:17 --> 00:48:22
it went in the 5s state.
But right here at ytterbium,

634
00:48:22 --> 00:48:26
the 4d state goes below in
energy the 5s state.

635
00:48:26 --> 00:48:30
Again, that does not affect how
you write the electron

636
00:48:30 --> 00:48:35
configuration of a neutral.
For example,

637
00:48:35 --> 00:48:42
silver, which is way out here
and is one of those exceptions,

638
00:48:42 --> 00:48:47
if you ionize it,
if you pluck off an electron,

639
00:48:47 --> 00:48:50
which electron is going to go?
5s.

640
00:48:50 --> 00:48:57
And so the silver plus one
configuration is the

641
00:48:57 --> 00:49:02
krypton core, 4d 10.

642
00:49:02 --> 00:49:07
It is that 5s electron that is
going to disappear.

643
00:49:07 --> 00:49:13
This is important.
Do not let it confuse you with

644
00:49:13 --> 00:49:19
the writing the electron
configurations of the neutrals.

645
00:49:19.276 --> 49:22
Okie-dokie.
See you Friday.