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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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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 Schrödinger
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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increases, goes to a maximum,
and then decays exponentially

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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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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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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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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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the 3s is lower,
the 2p is lower,

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00:14:08,000 --> 00:14:10,000
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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00:14:32,000 --> 00:14:35,000
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

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or is broken,
as we say.

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Likewise, the 3s state is lower
in energy than the 3p state,

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00:15:08,000 --> 00:15:12,000
than the 3d state.
The degeneracy in those states

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is lifted, or it is broken.
That is now what we have to

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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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And the reason for this has to
do with the phenomenon called

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shielding.
We have to talk about this

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phenomenon, shielding,
and we have to talk about how

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00:15:39,000 --> 00:15:43,000
that leads to a concept called
effective charge.

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00:15:43,000 --> 00:15:47,000
But to do that,
I am going to do the following.

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00:15:47,000 --> 00:15:53,000
I am going to realize that each
one of these energies here,

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E sub nl,
so now these energies are

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labeled by both the principle
quantum number and the angular

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00:16:02,000 --> 00:16:08,000
momentum quantum number.
I am going to realize that

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00:16:08,000 --> 00:16:13,000
these energies here physically
are minus the ionization energy

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00:16:13,000 --> 00:16:17,000
because these energies are minus
the energy it is going to

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require to rip the electron off
from that particular state.

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00:16:22,000 --> 00:16:27,000
And I am going to set those
binding energies equal to a

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00:16:27,000 --> 00:16:31,000
hydrogen atom like energy level
here.

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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

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00:16:57,000 --> 00:17:03,000
and l, they are minus the
ionization energy from that nl

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00:17:03,000 --> 00:17:07,000
state.
And I am going to approximate

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00:17:07,000 --> 00:17:12,000
that as a hydrogen atom kind of
an energy scheme.

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00:17:12,000 --> 00:17:16,000
That is, I am going to set this
equal to R sub H,

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00:17:16,000 --> 00:17:20,000
the Rydberg constant over n
squared.

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00:17:20,000 --> 00:17:24,000
Out here,
there is going to be a Z,

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00:17:24,000 --> 00:17:30,000
but this Z is going to be Z
effective, Zeff.

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00:17:30,000 --> 00:17:34,000
And it is going to be squared,
of course.

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00:17:34,000 --> 00:17:41,000
And that Z is going to depend
on that particular nl state that

236
00:17:41,000 --> 00:17:44,000
you are in.
This Z effective,

237
00:17:44,000 --> 00:17:48,000
here, is the effective nuclear
charge.

238
00:17:48,000 --> 00:17:55,000
It is not the nuclear charge.
It is the effective nuclear

239
00:17:55,000 --> 00:17:56,000
charge.
Why?

240
00:17:56,000 --> 00:18:04,000
Well, because of shielding.
Let's try to explain that.

241
00:18:04,000 --> 00:18:09,000
Let's take helium.
Z is equal to plus 2e.

242
00:18:09,000 --> 00:18:13,000
We have plus 2e here,

243
00:18:13,000 --> 00:18:19,000
and let's do a thought
experiment in that we are going

244
00:18:19,000 --> 00:18:26,000
to take electron number two,
here, and place it kind of

245
00:18:26,000 --> 00:18:32,000
close to this plus 2 charged
nucleus.

246
00:18:32,000 --> 00:18:38,000
And then we are going to have
electron number one way out

247
00:18:38,000 --> 00:18:40,000
here.
In this case,

248
00:18:40,000 --> 00:18:47,000
with electron number one way
out here, the nuclear charge

249
00:18:47,000 --> 00:18:53,000
that electron number one
experiences, because it is so

250
00:18:53,000 --> 00:19:00,000
far out, kind of looks like a
plus one charge.

251
00:19:00,000 --> 00:19:03,000
Because this electron,
on the average,

252
00:19:03,000 --> 00:19:09,000
is canceling one of the
positive charges on the nucleus.

253
00:19:09,000 --> 00:19:15,000
So, the effective charge here
for this electron way out there,

254
00:19:15,000 --> 00:19:18,000
we are going to say,
is plus one.

255
00:19:18,000 --> 00:19:24,000
If that is the
case, well, then the binding

256
00:19:24,000 --> 00:19:30,000
energy of that electron is one
squared times R sub H over n

257
00:19:30,000 --> 00:19:35,000
squared.

258
00:19:35,000 --> 00:19:39,000
We are going to consider n
equal one because we are going

259
00:19:39,000 --> 00:19:43,000
to talk about the ground state
of the helium atom,

260
00:19:43,000 --> 00:19:45,000
here.
But you know what this value is

261
00:19:45,000 --> 00:19:49,000
going to turn out to be.
It is going to turn out to be

262
00:19:49,000 --> 00:19:54,000
minus 2.180x10^-18 joules.
That is the binding energy of

263
00:19:54,000 --> 00:19:58,000
an electron in a hydrogen atom.
This is a thought experiment,

264
00:19:58,000 --> 00:20:01,000
now.
This is helium,

265
00:20:01,000 --> 00:20:06,000
this is just one electron way
out here, and it cannot

266
00:20:06,000 --> 00:20:13,000
discriminate too well between
the nucleus and this electron,

267
00:20:13,000 --> 00:20:18,000
so the overall effective charge
it sees is plus one.

268
00:20:18,000 --> 00:20:21,000
But now, we take the other
case.

269
00:20:21,000 --> 00:20:27,000
The other extreme case is we
are going to bring in electron

270
00:20:27,000 --> 00:20:34,000
one really close to the nucleus.
And electron two is way out

271
00:20:34,000 --> 00:20:40,000
here such that it does not do
anything as far as this electron

272
00:20:40,000 --> 00:20:43,000
is concerned.
And so this electron,

273
00:20:43,000 --> 00:20:47,000
in this case,
is experiencing the total

274
00:20:47,000 --> 00:20:52,000
nuclear charge on the nucleus.
And so we say in this thought

275
00:20:52,000 --> 00:20:56,000
case here, with this electron
really close,

276
00:20:56,000 --> 00:21:01,000
that the effective charge is
equal to plus 2e for this

277
00:21:01,000 --> 00:21:05,000
electron.

278
00:21:05,000 --> 00:21:11,000
Well, if that is the case,
we can calculate the binding

279
00:21:11,000 --> 00:21:17,000
energy of this electron.
And that is going to be 2

280
00:21:17,000 --> 00:21:20,000
squared R sub H over one
squared.

281
00:21:20,000 --> 00:21:26,000
We can plug in R sub H.

282
00:21:26,000 --> 00:21:33,000
And we are going to get minus
8.72x10^-18 joules.

283
00:21:33,000 --> 00:21:38,000
What this is is the binding
energy of an electron in helium

284
00:21:38,000 --> 00:21:43,000
plus.
And so, in this extreme case,

285
00:21:43,000 --> 00:21:48,000
with the electron really close,
that is the binding energy.

286
00:21:48,000 --> 00:21:54,000
With this extreme case,
with the electron way far out,

287
00:21:54,000 --> 00:22:00,000
this is the binding energy.
This is total shielding.

288
00:22:05,000 --> 00:22:11,000
This case is no shielding.
The electron is much more

289
00:22:11,000 --> 00:22:18,000
strongly bound.
The reality is that the binding

290
00:22:18,000 --> 00:22:25,000
energy of an electron in helium
is somewhere in between.

291
00:22:25,000 --> 00:22:33,000
That is, the ionization energy
for helium, for an electron in

292
00:22:33,000 --> 00:22:41,000
now neutral helium here,
is 3.94x10^-18 joules.

293
00:22:41,000 --> 00:22:47,000
Somewhere in between this
extreme case of total shielding

294
00:22:47,000 --> 00:22:52,000
and this extreme case of no
shielding at all.

295
00:22:52,000 --> 00:22:56,000
And that is because,
on the average,

296
00:22:56,000 --> 00:23:03,000
there is another electron in
between that electron and the

297
00:23:03,000 --> 00:23:07,000
nucleus.
And we can calculate the

298
00:23:07,000 --> 00:23:11,000
effective charge,
then, from the experimental

299
00:23:11,000 --> 00:23:16,000
binding energies by just taking
this expression and rearranging

300
00:23:16,000 --> 00:23:19,000
it.
I am going to solve that

301
00:23:19,000 --> 00:23:22,000
expression for the effective
charge, Zeff.

302
00:23:22,000 --> 00:23:26,000
And, when I do that that,
add, of course,

303
00:23:26,000 --> 00:23:31,000
this n squared times the
ionization energy over the

304
00:23:31,000 --> 00:23:36,000
Rydberg constant.

305
00:23:36,000 --> 00:23:42,000
n squared is going to be one
because we are talking about the

306
00:23:42,000 --> 00:23:46,000
ground state.
The ionization energy,

307
00:23:46,000 --> 00:23:52,000
I said, was 3.94x10^-18 joules.
The Rydberg constant is

308
00:23:52,000 --> 00:23:58,000
2.180x10^-18 joules.
In the end, I find an effective

309
00:23:58,000 --> 00:24:04,000
charge here of plus 1.34.

310
00:24:04,000 --> 00:24:09,000
Again, why is that the case?
Well, that is the case because

311
00:24:09,000 --> 00:24:14,000
here is my electron,
here is my helium nucleus and,

312
00:24:14,000 --> 00:24:19,000
just on the average,
between this electron and the

313
00:24:19,000 --> 00:24:23,000
nucleus there is always another
electron around.

314
00:24:23,000 --> 00:24:30,000
This electron kind of partially
shields this nuclear charge.

315
00:24:30,000 --> 00:24:35,000
It partially shields it so that
we calculated,

316
00:24:35,000 --> 00:24:41,000
from using this scheme and
using the experimental

317
00:24:41,000 --> 00:24:47,000
ionization energies,
an effective charge of 1.34.

318
00:24:47,000 --> 00:24:54,000
So now, we sort of understand
shielding and effective charge.

319
00:24:54,000 --> 00:25:00,000
That is a square root,
thank you.

320
00:25:00,000 --> 00:25:04,000
All right.
Now, we have to use this idea

321
00:25:04,000 --> 00:25:11,000
of effective charge and
shielding to understand why the

322
00:25:11,000 --> 00:25:16,000
2s state is lower in energy than
the 2p state.

323
00:25:16,000 --> 00:25:20,000
And here comes that.

324
00:25:30,000 --> 00:25:33,000
Let's think about the lithium
atom.

325
00:25:33,000 --> 00:25:39,000
The electron confirmation of
lithium is 1s 2 2s 1.

326
00:25:39,000 --> 00:25:43,000
The electron configuration of

327
00:25:43,000 --> 00:25:47,000
lithium is not 1s 2 2p 1.

328
00:25:47,000 --> 00:25:50,000
Why?
Because 2s is lower in energy

329
00:25:50,000 --> 00:25:54,000
than 2p.
But why is that the case?

330
00:25:54,000 --> 00:25:59,000
Well, to look at that,
we have to look at the radial

331
00:25:59,000 --> 00:26:06,000
probability distribution
functions for 2s and 2p.

332
00:26:06,000 --> 00:26:11,000
Here is a radial probability
distribution function for 2s.

333
00:26:11,000 --> 00:26:17,000
Here is the radial probability
distribution function for 2p.

334
00:26:22,000 --> 00:26:26,000
Now, you have to do a thought
experiment, here.

335
00:26:26,000 --> 00:26:30,000
In this 2s radial probability
distribution function,

336
00:26:30,000 --> 00:26:36,000
what you see is that there is
some finite probability of the

337
00:26:36,000 --> 00:26:42,000
electron in the 2s state being
really close to the nucleus.

338
00:26:42,000 --> 00:26:47,000
For the sake of the argument,
here, I am going to say that

339
00:26:47,000 --> 00:26:53,000
for this part of the probability
distribution function,

340
00:26:53,000 --> 00:26:58,000
the effective charge Zeff is
going to be plus 3e.

341
00:26:58,000 --> 00:27:02,000
I mean, that is an exaggeration

342
00:27:02,000 --> 00:27:05,000
because obviously we have some s
electrons.

343
00:27:05,000 --> 00:27:09,000
But for the sake of this
argument, I am going to say the

344
00:27:09,000 --> 00:27:14,000
effective charge for this part
of the probability distribution

345
00:27:14,000 --> 00:27:17,000
function is plus 3e.
For this part of the

346
00:27:17,000 --> 00:27:21,000
probability distribution
function for 2s that is much

347
00:27:21,000 --> 00:27:24,000
further out.
And I have those 1s electrons

348
00:27:24,000 --> 00:27:27,000
closer in.
So, for the sake of this

349
00:27:27,000 --> 00:27:31,000
argument, I am going to say that
these s electrons completely

350
00:27:31,000 --> 00:27:37,000
shield the nuclear charge.
And so, the effective charge

351
00:27:37,000 --> 00:27:42,000
for this part of the
distribution is plus 1e.

352
00:27:42,000 --> 00:27:46,000
Now, what about 2p?
Well, in the case of 2p,

353
00:27:46,000 --> 00:27:51,000
you can see that this 2p wave
function, although it is a

354
00:27:51,000 --> 00:27:56,000
little bit closer in,
for the most part it is about

355
00:27:56,000 --> 00:28:02,000
in the same place as the second
lobe here of the 2s wave

356
00:28:02,000 --> 00:28:06,000
function.
We are going to say that this

357
00:28:06,000 --> 00:28:09,000
effective charge is plus 1e,

358
00:28:09,000 --> 00:28:12,000
just like this lobe of the 2s
wave function.

359
00:28:12,000 --> 00:28:15,000
But now, to get the kind of
total effective charge,

360
00:28:15,000 --> 00:28:20,000
I am going to have to take the
effective charge and average it

361
00:28:20,000 --> 00:28:23,000
over this probability
distribution function.

362
00:28:23,000 --> 00:28:27,000
And so, since in the 2s
function I am going to average

363
00:28:27,000 --> 00:28:31,000
over a part that has a plus 3
effective charge and a part that

364
00:28:31,000 --> 00:28:38,000
has a plus 1 effective charge.
Well, if I average over that,

365
00:28:38,000 --> 00:28:44,000
that is going to be larger than
the effective charge of this 2p

366
00:28:44,000 --> 00:28:49,000
wave function because the 2p
everywhere is plus one.

367
00:28:49,000 --> 00:28:55,000
The effective charge here for
the 2s is going to be greater,

368
00:28:55,000 --> 00:29:00,000
on the average,
than for the 2p.

369
00:29:00,000 --> 00:29:04,000
Because of this part of the
probability distribution

370
00:29:04,000 --> 00:29:09,000
function, this part that is
really close to the nucleus,

371
00:29:09,000 --> 00:29:14,000
where the electrons can feel
more of the nuclear charge or

372
00:29:14,000 --> 00:29:19,000
experience more of the nuclear
charge than they could if they

373
00:29:19,000 --> 00:29:23,000
were a 2p electron.
Therefore, since that 2s

374
00:29:23,000 --> 00:29:27,000
electron has a greater effective
charge, here,

375
00:29:27,000 --> 00:29:33,000
what that is going to mean is
that the binding energy of the

376
00:29:33,000 --> 00:29:38,000
2s state is going to be lower in
energy.

377
00:29:38,000 --> 00:29:43,000
It is going to be more negative
than the binding energy of the

378
00:29:43,000 --> 00:29:47,000
2p state.
The same thing for 3s and 3p.

379
00:29:47,000 --> 00:29:52,000
The same reasoning there.
And it is all because of this

380
00:29:52,000 --> 00:29:57,000
little part of the probability
distribution function.

381
00:29:57,000 --> 00:30:01,000
Now we can understand why the
2s is, in fact,

382
00:30:01,000 --> 00:30:06,000
lower in energy than the 2p.
Yeah?

383
00:30:12,000 --> 00:30:14,000
Because the 3s,
although it does,

384
00:30:14,000 --> 00:30:18,000
in fact, have this,
it is a little bit further out

385
00:30:18,000 --> 00:30:21,000
here, enough to make for this to
compensate.

386
00:30:21,000 --> 00:30:24,000
But, again, 3s is lower than
3p.

387
00:30:24,000 --> 00:30:30,000
We are just going to compare 3s
and 3p within the same shell.

388
00:30:38,000 --> 00:30:42,000
Yes, it does have to do with
the net area underneath.

389
00:30:42,000 --> 00:30:47,000
If the probability was much
higher, and it is not that much

390
00:30:47,000 --> 00:30:52,000
higher, but if it were then you
are right, it could cancel it

391
00:30:52,000 --> 00:30:53,000
out.

392
00:30:58,000 --> 00:31:02,000
Now, therefore,
we are ready to write the

393
00:31:02,000 --> 00:31:09,000
electron configurations of all
the atoms on the Periodic Table.

394
00:31:09,000 --> 00:31:15,000
And you already know that we
use the Aufbau Principle to do

395
00:31:15,000 --> 00:31:18,000
that.
Aufbau means building up.

396
00:31:18,000 --> 00:31:23,000
What do you do?
You take all the allowed states

397
00:31:23,000 --> 00:31:29,000
and order them according to
their energy.

398
00:31:29,000 --> 00:31:33,000
Most strongly bound or most
negative energy goes on the

399
00:31:33,000 --> 00:31:37,000
bottom, and next most negative
energy, on and on.

400
00:31:37,000 --> 00:31:41,000
And we will talk about a
pneumonic for remembering those

401
00:31:41,000 --> 00:31:45,000
energies in a moment.
We use the Aufbau principle.

402
00:31:45,000 --> 00:31:50,000
We start with the lowest energy
state and we put an electron in

403
00:31:50,000 --> 00:31:52,000
for hydrogen.
There it is,

404
00:31:52,000 --> 00:31:55,000
the 1s state.
For helium, well,

405
00:31:55,000 --> 00:31:59,000
we also put that into the 1s
state.

406
00:31:59,000 --> 00:32:04,000
Except, as we fill these
states, we have to heed the

407
00:32:04,000 --> 00:32:08,000
Pauli Exclusion Principle.
This electron,

408
00:32:08,000 --> 00:32:12,000
here, goes in with the opposite
spin.

409
00:32:12,000 --> 00:32:15,000
Next lithium,
2s electron.

410
00:32:15,000 --> 00:32:19,000
For beryllium,
another 2s electron,

411
00:32:19,000 --> 00:32:22,000
opposite spin.
And then for nitrogen,

412
00:32:22,000 --> 00:32:29,000
electron, here,
has to go into the 2p state.

413
00:32:29,000 --> 00:32:33,000
Now, it does not matter whether
you put it in 2px,

414
00:32:33,000 --> 00:32:37,000
2py or 2pz.
They are all the same energy.

415
00:32:37,000 --> 00:32:42,000
The next electron,
carbon, what are we going to do

416
00:32:42,000 --> 00:32:44,000
here?
Well, here we have to obey

417
00:32:44,000 --> 00:32:50,000
something called Hund's Rule.
And Hund's Rule says that when

418
00:32:50,000 --> 00:32:55,000
electrons are added to states of
the same energy,

419
00:32:55,000 --> 00:32:59,000
and that is what we are doing
here in the 2p,

420
00:32:59,000 --> 00:33:04,000
a single electron enters each
state before a second electron

421
00:33:04,000 --> 00:33:11,000
enters any state.
And, those single electrons

422
00:33:11,000 --> 00:33:17,000
have to go in so that the
resulting spins are parallel.

423
00:33:17,000 --> 00:33:20,000
That is, they have the same
spin.

424
00:33:20,000 --> 00:33:24,000
Do we put in an electron like
this?

425
00:33:24,000 --> 00:33:27,000
No.
Do we put it in like that?

426
00:33:27,000 --> 00:33:32,000
No.
Do we put it in like this?

427
00:33:32,000 --> 00:33:37,000
Yeah, according to Hund's Rule.
And then the next electron has

428
00:33:37,000 --> 00:33:43,000
to go into that other empty p
state before any of these states

429
00:33:43,000 --> 00:33:46,000
double up.
Then we keep doubling them up.

430
00:33:46,000 --> 00:33:51,000
The next electron has to go
into the next state 3s.

431
00:33:51,000 --> 00:33:55,000
Again, we double them up.
Now we are to the 3p state

432
00:33:55,000 --> 00:34:00,000
again.
One electron goes into 2px.

433
00:34:00,000 --> 00:34:04,000
The next electron will go into
either 2py or 2pz,

434
00:34:04,000 --> 00:34:09,000
according to Hund's Rule,
and the spins remain parallel

435
00:34:09,000 --> 00:34:12,000
to get the lowest energy
configuration.

436
00:34:12,000 --> 00:34:17,000
And then the next electron 2pz.
And then we start doubling up.

437
00:34:17,000 --> 00:34:21,000
And we keep going.
You keep going in that way so

438
00:34:21,000 --> 00:34:26,000
that you write the electron
configuration of all the atoms

439
00:34:26,000 --> 00:34:32,000
in the Periodic Table.
Let's look at these electron

440
00:34:32,000 --> 00:34:36,000
configurations kind of quickly
here.

441
00:34:36,000 --> 00:34:41,000
Let's start with the third
period, the third row here,

442
00:34:41,000 --> 00:34:45,000
sodium going across here to
argon.

443
00:34:45,000 --> 00:34:49,000
Here is the electron
configuration for sodium.

444
00:34:49,000 --> 00:34:55,000
Notice here that I don't mind
if you write that electron

445
00:34:55,000 --> 00:34:59,000
configuration as 2p 6
instead of 2px,

446
00:34:59,000 --> 00:35:04,000
2py, 2pz.
Because you cannot tell the

447
00:35:04,000 --> 00:35:06,000
difference between x,
y, and z, anyway,

448
00:35:06,000 --> 00:35:09,000
if you are not in a magnetic
field.

449
00:35:09,000 --> 00:35:13,000
These electrons here in sodium
that make up the inert gas

450
00:35:13,000 --> 00:35:17,000
configuration of argon,
of course, are the core

451
00:35:17,000 --> 00:35:19,000
electrons.
When we talk about core

452
00:35:19,000 --> 00:35:24,000
electrons, we are talking about
the electrons that make up the

453
00:35:24,000 --> 00:35:30,000
nearest rare gas configuration.
And then the valance electron,

454
00:35:30,000 --> 00:35:35,000
well, is the electrons that are
beyond the nearest but lowest

455
00:35:35,000 --> 00:35:39,000
inert gas configurations.
And also, when you are writing

456
00:35:39,000 --> 00:35:43,000
these electron configurations,
say, for sodium,

457
00:35:43,000 --> 00:35:48,000
you can write it as the neon
configuration and then just show

458
00:35:48,000 --> 00:35:51,000
the valence electron, 3s 1.

459
00:35:51,000 --> 00:35:56,000
As we go across that third
period here everything is very

460
00:35:56,000 --> 00:36:01,000
normal.
3s fills up first and then the

461
00:36:01,000 --> 00:36:05,000
3p's fill up.
Now we get to the fourth

462
00:36:05,000 --> 00:36:09,000
period, from potassium to
krypton.

463
00:36:09,000 --> 00:36:14,000
Fourth period,
what happens here is that those

464
00:36:14,000 --> 00:36:19,000
first two electrons go into the
s states.

465
00:36:19,000 --> 00:36:23,000
They do not go into the 3d
states.

466
00:36:23,000 --> 00:36:29,000
They don't because those s
states are lower in energy than

467
00:36:29,000 --> 00:36:34,000
the 3d states.
And so these electron

468
00:36:34,000 --> 00:36:38,000
configurations are argon 4s 1,

469
00:36:38,000 --> 00:36:43,000
argon 4s 2 **[Ar]4s^2**.
And then, once we fill those 4s

470
00:36:43,000 --> 00:36:46,000
states, we start filling the 3d
states.

471
00:36:46,000 --> 00:36:49,000
Here is scandium,
here is titanium,

472
00:36:49,000 --> 00:36:52,000
here is vanadium,
everything is normal,

473
00:36:52,000 --> 00:36:55,000
3d 1, 3d 2,

474
00:36:55,000 --> 00:37:00,000
3d 3,
and then we get to chromium.

475
00:37:00,000 --> 00:37:03,000
We have an exception here,
chromium.

476
00:37:03,000 --> 00:37:07,000
Chromium is not what you would
expect.

477
00:37:07,000 --> 00:37:11,000
It is not 4s 2 3d 4.

478
00:37:11,000 --> 00:37:14,000
Chromium is 4s 1 3d 5.

479
00:37:14,000 --> 00:37:19,000
There is no way for you to know
that a priori,

480
00:37:19,000 --> 00:37:24,000
unless you do a very
sophisticated calculation,

481
00:37:24,000 --> 00:37:30,000
but this is experimentally
observed that this is the

482
00:37:30,000 --> 00:37:33,000
configuration.
Why?

483
00:37:33,000 --> 00:37:39,000
Because it is lower in total
energy than this configuration.

484
00:37:39,000 --> 00:37:45,000
It turns out that there is some
extra stability in having a half

485
00:37:45,000 --> 00:37:51,000
filled 4s shell and a half
filled 3d sub-shell compared to

486
00:37:51,000 --> 00:37:58,000
having a filled 4s shell and a
less than half filled 3d shell.

487
00:37:58,000 --> 00:38:03,000
This is an exception that you
do have to know.

488
00:38:03,000 --> 00:38:08,000
But after chromium,
here, manganese behaves well.

489
00:38:08,000 --> 00:38:12,000
Iron behaves well.
Cobalt okay.

490
00:38:12,000 --> 00:38:16,000
Nickel okay.
And now we get to copper,

491
00:38:16,000 --> 00:38:22,000
and we have a problem.
It is not what you would

492
00:38:22,000 --> 00:38:25,000
expect.
It is not 4s 2 3d 9.

493
00:38:25,000 --> 00:38:29,000
Instead, it is 4s 1 3d 10.

494
00:38:29,000 --> 00:38:35,000
*Again, this configuration is

495
00:38:35,000 --> 00:38:38,000
something you could not have
predicted a priori.

496
00:38:38,000 --> 00:38:43,000
If you do a sophisticated
calculation you can see it,

497
00:38:43,000 --> 00:38:46,000
but this is also experimentally
observed.

498
00:38:46,000 --> 00:38:49,000
We know this to be the
configuration.

499
00:38:49,000 --> 00:38:53,000
This is not the configuration.
This is the lower energy

500
00:38:53,000 --> 00:38:56,000
configuration.
Here is another exception that

501
00:38:56,000 --> 00:39:00,000
you have to know,
copper.

502
00:39:00,000 --> 00:39:05,000
And then after copper here,
zinc, things follow a pattern

503
00:39:05,000 --> 00:39:08,000
again.
The next electron,

504
00:39:08,000 --> 00:00:10,000
then, just fills up the 4s 2 3d

505
00:39:13,000 --> 00:39:17,000
Now, at gallium,
all of those are filled.

506
00:39:17,000 --> 00:39:20,000
We start filling up the 4p
states.

507
00:39:20,000 --> 00:39:24,000
Everything is fine until we get
to krypton.

508
00:39:24,000 --> 00:39:30,000
So, you have to know chromium
and copper.

509
00:39:30,000 --> 00:39:34,000
Now, the fifth row here,
starting with rubidium,

510
00:39:34,000 --> 00:39:37,000
strontium, ytterbium,
zirconium, etc.

511
00:39:37,000 --> 00:39:42,000
all the way across here.
Starting with rubidium,

512
00:39:42,000 --> 00:39:45,000
again, the electrons go into
the 5s shell.

513
00:39:45,000 --> 00:39:50,000
They do not go into the 4d.
That is because with rubidium

514
00:39:50,000 --> 00:39:54,000
and strontium,
5s is actually lower than 4d.

515
00:39:54,000 --> 00:40:00,000
And it is only once you fill up
the 5s states that you start

516
00:40:00,000 --> 00:40:05,000
filling up the 4d states right
here.

517
00:40:05,000 --> 00:40:10,000
Now, there are exceptions along
this fifth row.

518
00:40:10,000 --> 00:40:14,000
The two exceptions,
molybdenum and silver,

519
00:40:14,000 --> 00:40:20,000
are the same kind of exceptions
as chromium and copper.

520
00:40:20,000 --> 00:40:26,000
You have to know the exceptions
for silver and copper and

521
00:40:26,000 --> 00:40:32,000
molybdenum and chromium.
The same identical kind of

522
00:40:32,000 --> 00:40:35,000
exception as in the fourth row
here.

523
00:40:35,000 --> 00:40:40,000
There are other exceptions
along this fifth row.

524
00:40:40,000 --> 00:40:45,000
You do not have to know those.
There is really no way,

525
00:40:45,000 --> 00:40:50,000
a priori, for you to know that.
Again, it is an experimental

526
00:40:50,000 --> 00:40:53,000
observation.
A sophisticated hard

527
00:40:53,000 --> 00:40:59,000
calculation will also show that.
And then, once you are done

528
00:40:59,000 --> 00:41:02,000
with cadmium,
well, then the 5p's start

529
00:41:02,000 --> 00:41:07,000
filling and everything is normal
and you get to the xenon inner

530
00:41:07,000 --> 00:41:09,000
gas configuration.

531
00:41:17,000 --> 00:41:22,000
How do you remember what the
ordering is of these states,

532
00:41:22,000 --> 00:41:26,000
the energy ordering?
Well, here is a pneumonic that

533
00:41:26,000 --> 00:41:30,000
maybe some of you have seen
before.

534
00:41:30,000 --> 00:41:35,000
Start out and write 1s,
then write 2s right below it

535
00:41:35,000 --> 00:41:40,000
and then 2p to the side of it.
And then write 3s,

536
00:41:40,000 --> 00:41:43,000
3p, 3d.
And then write 4s,

537
00:41:43,000 --> 00:41:46,000
4p, 4d, 4f.
And then write 5s,

538
00:41:46,000 --> 00:41:50,000
5p, 5d, 5f, and 6s,
6p, 6d, and 7s,

539
00:41:50,000 --> 00:41:53,000
7p.
And now, to get the energy

540
00:41:53,000 --> 00:41:59,000
ordering we are going to draw
diagonals.

541
00:41:59,000 --> 00:42:02,000
Well, first of all,
the 1s is the lowest energy

542
00:42:02,000 --> 00:42:08,000
state, and then the next highest
energy state is the 2s state,

543
00:42:08,000 --> 00:42:12,000
and then the next highest
energy state is the 2p.

544
00:42:12,000 --> 00:42:14,000
We are going to draw a
diagonal.

545
00:42:14,000 --> 00:42:18,000
The next state to fill is 2p.
The next one is 3s.

546
00:42:18,000 --> 00:42:22,000
Now we are going to draw a
diagonal again.

547
00:42:22,000 --> 00:42:26,000
The next one to fill is 3p.
The next one to fill is 4s.

548
00:42:26,000 --> 00:42:32,000
Draw another diagonal.
The next one to fill is 3d,

549
00:42:32,000 --> 00:42:34,000
4p, 5s.
4d, 5p, 6s.

550
00:42:34,000 --> 00:42:38,000
4f, 5d, 6p, 7s.
5s, 6d, 7p, and that is all

551
00:42:38,000 --> 00:42:45,000
that is going to be important.
That is one way to remember the

552
00:42:45,000 --> 00:42:51,000
relative energy orderings here.
And you do have to be able to

553
00:42:51,000 --> 00:42:57,000
write this down on an exam.
You will have a Periodic Table,

554
00:42:57,000 --> 00:43:04,000
but it won't have the electron
configurations on it.

555
00:43:04,000 --> 00:43:10,000
Now, I am going to tell you
something that sometimes people

556
00:43:10,000 --> 00:43:15,000
find a little bit confusing.
That is, the electron

557
00:43:15,000 --> 00:43:21,000
configuration of ions.
What I am about to say has no

558
00:43:21,000 --> 00:43:28,000
effect on what I just said about
how to write the electron

559
00:43:28,000 --> 00:43:35,000
configuration for neutrals.
This does not affect anything

560
00:43:35,000 --> 00:43:40,000
in your writing down the
electron configuration for

561
00:43:40,000 --> 00:43:43,000
neutrals.
This is for ions.

562
00:43:43,000 --> 00:43:49,000
The point I want to make is
that if you actually look at the

563
00:43:49,000 --> 00:43:56,000
energies of the individual 3d
states and 4s states across that

564
00:43:56,000 --> 00:44:02,000
fourth row, this is what they
look like.

565
00:44:02,000 --> 00:44:04,000
For example,
at potassium,

566
00:44:04,000 --> 00:44:08,000
that 4s state is lower in
energy than the 3d state.

567
00:44:08,000 --> 00:44:14,000
That is why we put the electron
in the 4s state when we wrote

568
00:44:14,000 --> 00:44:17,000
the neutral.
The same thing for calcium.

569
00:44:17,000 --> 00:44:22,000
That is why we put that
electron into the 4s state and

570
00:44:22,000 --> 00:44:25,000
not the 3s state.
Lo and behold,

571
00:44:25,000 --> 00:44:30,000
right here at scandium,
Z equals 21.

572
00:44:30,000 --> 00:44:34,000
What happens is that the 3d
state actually drops below in

573
00:44:34,000 --> 00:44:39,000
energy than the 4s state.
These are the energies of the

574
00:44:39,000 --> 00:44:44,000
individual state now,
and that continues all the way

575
00:44:44,000 --> 00:44:48,000
across the Periodic Table.
However, that does not affect

576
00:44:48,000 --> 00:44:53,000
how you write the electron
configuration of the neutrals.

577
00:44:53,000 --> 00:44:57,000
For example,
if you are writing the electron

578
00:44:57,000 --> 00:45:02,000
configuration for titanium,
here it is.

579
00:45:02,000 --> 00:45:07,000
It is the argon core,
4s 2 3d 2.

580
00:45:07,000 --> 00:45:12,000
And, by the way,
I don't care whether you write

581
00:45:12,000 --> 00:45:17,000
3d 2 4s 2 or 4s 2 3d 2.

582
00:45:17,000 --> 00:45:21,000
You can write them in either
order.

583
00:45:21,000 --> 00:45:26,000
Now, you might say,
well, why is this the electron

584
00:45:26,000 --> 00:45:31,000
configuration if at this value
for Z, Z equals 22,

585
00:45:31,000 --> 00:45:38,000
titanium, the 3d state is lower
than the 4s state?

586
00:45:38,000 --> 00:45:43,000
Why don't these 4s electrons
just hop into the 3d states?

587
00:45:43,000 --> 00:45:47,000
Well, they don't do that
because this electron

588
00:45:47,000 --> 00:45:52,000
configuration actually minimizes
the electron repulsions.

589
00:45:52,000 --> 00:45:56,000
If these four electrons were in
the 3d state,

590
00:45:56,000 --> 00:46:02,000
well, then the repulsive
interactions would be greater.

591
00:46:02,000 --> 00:46:05,000
Because they are in the same
state now.

592
00:46:05,000 --> 00:46:08,000
And, therefore,
the entire energy of the atom

593
00:46:08,000 --> 00:46:12,000
will be larger.
And it is the entire energy,

594
00:46:12,000 --> 00:46:17,000
the total energy of the atom
that is important when we look

595
00:46:17,000 --> 00:46:22,000
at the electron configurations.
In this particular case,

596
00:46:22,000 --> 00:46:26,000
if we look at the individual d
states and the 4s states,

597
00:46:26,000 --> 00:46:31,000
yeah, the d states are lower in
energy.

598
00:46:31,000 --> 00:46:36,000
But what is important is when
we sum up all of the energies of

599
00:46:36,000 --> 00:46:38,000
the interactions,
what is lower?

600
00:46:38,000 --> 00:46:42,000
What I am saying to you is that
when we do that,

601
00:46:42,000 --> 00:46:46,000
when we sum up all of the
interaction energies,

602
00:46:46,000 --> 00:46:50,000
this still is the lower energy
configuration,

603
00:46:50,000 --> 00:46:54,000
even though the 3d electrons at
this value of Z,

604
00:46:54,000 --> 00:47:00,000
those 3d states are lower in
energy than the 4s states.

605
00:47:00,000 --> 00:47:05,000
That is why we do not have this
hoping over into the 3d states

606
00:47:05,000 --> 00:47:09,000
for the neutral.
Now, here comes the ion

607
00:47:09,000 --> 00:47:13,000
configuration.
If you have this configuration

608
00:47:13,000 --> 00:47:18,000
for the neutral,
and now you ionize titanium to

609
00:47:18,000 --> 00:47:23,000
make titanium plus,
the electron configuration is

610
00:47:23,000 --> 00:47:28,000
argon 3d 2.
That is, it is the 4s electrons

611
00:47:28,000 --> 00:47:33,000
that come off,
that are plucked out.

612
00:47:33,000 --> 00:47:37,000
They are plucked out because
they are the higher energy

613
00:47:37,000 --> 00:47:41,000
electrons now.
This is the electron

614
00:47:41,127 --> 00:00:02,000
configuration for titanium plus

615
00:47:45,000 --> 00:47:50,000
We are going to pull out those
higher energy electrons,

616
00:47:50,000 --> 00:47:55,000
which are the 4s electrons.
Again, this affects only the

617
00:47:55,000 --> 00:47:59,000
ion configuration.
The same thing happens here on

618
00:47:59,000 --> 00:48:04,000
the fifth row.
The fifth row starting here

619
00:48:04,000 --> 00:48:09,000
with rubidium and strontium,
5s is lower in energy than 4d.

620
00:48:09,000 --> 00:48:14,000
That is why that rubidium
electron went in the 5s state.

621
00:48:14,000 --> 00:48:17,000
The same thing with the
strontium electron,

622
00:48:17,000 --> 00:48:22,000
it went in the 5s state.
But right here at ytterbium,

623
00:48:22,000 --> 00:48:26,000
the 4d state goes below in
energy the 5s state.

624
00:48:26,000 --> 00:48:30,000
Again, that does not affect how
you write the electron

625
00:48:30,000 --> 00:48:35,000
configuration of a neutral.
For example,

626
00:48:35,000 --> 00:48:42,000
silver, which is way out here
and is one of those exceptions,

627
00:48:42,000 --> 00:48:47,000
if you ionize it,
if you pluck off an electron,

628
00:48:47,000 --> 00:48:50,000
which electron is going to go?
5s.

629
00:48:50,000 --> 00:48:57,000
And so the silver plus one
configuration is the

630
00:48:57,000 --> 00:49:02,000
krypton core, 4d 10.

631
00:49:02,000 --> 00:49:07,000
It is that 5s electron that is
going to disappear.

632
00:49:07,000 --> 00:49:13,000
This is important.
Do not let it confuse you with

633
00:49:13,000 --> 00:49:19,000
the writing the electron
configurations of the neutrals.

634
00:49:19,276 --> 00:49:22,000
Okie-dokie.
See you Friday.