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Let's get started.
We have a lot more chemistry to

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learn today.
And, as you are probably

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becoming aware,
the octahedron plays a very

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important role in the
coordination chemistry of the

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d-block elements.
So we are going to begin today

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also with the octahedron.
And I just want to remind you

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of the approach that we were
taking on Monday to understand

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how d-orbitals split under the
influence of the presence of a

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set of ligands.
And this splitting I am

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referring to is an energy
splitting.

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We are talking once again about
how we can use energy level

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diagrams to understand and
interpret the properties of

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molecules.
And today, in particular,

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we are going to be thinking
about how we can go from the

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angular properties of the d
orbitals all the way to some of

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the magnetic and spectroscopic
or color properties of ions that

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contain these d-block transition
elements.

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And so, you will remember last
time on Monday,

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we looked at placing ligands at
positions one through six on the

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octahedron in reference to our
coordinate system that we had

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chosen, x, y,
and z here, as specified.

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00:01:37 --> 00:01:41
And what we were doing was,
at each of these positions that

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corresponds to a ligand atom,
evaluating each of the

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00:01:46 --> 00:01:51
d-orbitals for the value of
theta and phi that is found at

29
00:01:51 --> 00:01:55
that ligand position.
And so, I am going to put up

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00:01:55 --> 00:02:00
part of the table that we
generated last time.

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00:02:00 --> 00:02:04
And that part will correspond
to ligand position,

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where we have ligands one,
two, three, four,

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00:02:08 --> 00:02:12
five, and six.
Those six ligand positions.

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00:02:12 --> 00:02:18
And two of the d orbitals here,
d z squared and d x

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00:02:18 --> 00:02:24
squared minus y squared,
were found to give

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00:02:24 --> 00:02:30
non-zero values at these six
ligand positions that correspond

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00:02:30 --> 00:02:36
to an octahedral complex.
The other three d-orbitals gave

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zero at those positions because
all six of these ligands lie on

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00:02:41 --> 00:02:44
nodal planes of xz,
yz, and xy.

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00:02:44 --> 00:02:47
And what we are doing is just
writing down,

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00:02:47 --> 00:02:51
in tabular form,
the relative magnitude of what

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00:02:51 --> 00:02:55
you get if you evaluate the
orbital at those ligand

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positions.
We had one in positions one and

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00:02:58 --> 00:03:03
six, which are on the big lobes
of d z squared along

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00:03:03 --> 00:03:08
the z-axis.
And we set that to one because

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00:03:08 --> 00:03:13
that is the largest value that
we get when we evaluate any of

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the orbitals.
And then, relative to that,

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00:03:16 --> 00:03:20
these ligands two through five
that are in the x,y-plane and

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00:03:20 --> 00:03:25
which interact with the torus of
the d z squared

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orbital in the x,y-plane,
have a value of one-quarter of

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00:03:29 --> 00:03:35
what it is along the z-axis.
And then, d x squared minus y

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00:03:35 --> 00:03:40
squared
evaluated at positions one and

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00:03:40 --> 00:03:44
six gave us a zero.
And that is because positions

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00:03:44 --> 00:03:49
one and six fall on the z-axis,
which is actually an

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00:03:49 --> 00:03:54
intersection of two nodal planes
for the d x squared minus y

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00:03:54 --> 00:03:58
squared orbital.
And then, here we have at

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positions two through five,
three-quarters.

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00:04:03 --> 00:04:11


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And if you are having trouble
understanding what these numbers

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mean, remember that we started
out with a sphere,

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00:04:18 --> 00:04:21
so we are interested in seeing
essentially, what the

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probability is in finding that
electron in that particular d

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orbital at that point on the
surface of the sphere.

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00:04:30 --> 00:04:33
And so what this is saying is
that, if you will,

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the radial probability of
finding an electron in d z

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squared along the
torus at some distance from the

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center is one-quarter what it is
along the z-axis at the same

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distance from the center.
And then, at that same

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00:04:50 --> 00:04:54
distance, if we are in the
x,y-plane along x or along y,

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00:04:54 --> 00:04:59
because that is where the lobes
of d x squared minus y squared

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extend,
that probability drops to

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three-quarters relative to the
two big lobes of d z squared

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along z.
You can think of it as telling

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you about the size of the lobes.
And, accordingly,

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00:05:14 --> 00:05:18
the way that these lobes of the
d-orbitals can interact with the

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ligands at these positions on
the surface of the sphere coming

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right out of evaluating the
angular parts of the wave

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function.
And then, there are other

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important coordination
geometries that we have to

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consider.
Because coordination number

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six, which corresponds to the
octahedron, is relatively

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common.
But coordination number four is

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also quite common.
And one of the important

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coordination geometries for
coordination number four is a

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00:05:47 --> 00:05:50
tetrahedral coordination
geometry.

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00:05:50 --> 00:05:54
The tetrahedron is not limited
to being important in organic

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chemistry.
It is also important for

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00:05:57 --> 00:06:02
coordination chemistry.
And you will recall that when

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we look at a tetrahedron we talk
about ligands at alternating

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corners of a cube.
And I will call these ligand

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positions seven,
eight, nine,

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00:06:12 --> 00:06:15
and ten.
And, if we have a tetrahedral

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metal complex with four ligands,
be they water ligands or

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00:06:20 --> 00:06:23
chloride ligands and be the
central metal ions,

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something like cobalt two
or nickel two,

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00:06:27 --> 00:06:33
a tetrahedral array
like this is produced.

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00:06:33 --> 00:06:37
And we can go ahead and
evaluate the d orbital wave

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functions at theta and phi
values that correspond to these

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positions, seven,
eight, nine,

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00:06:45 --> 00:06:49
and ten, all at the same
distance from the center.

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00:06:49 --> 00:06:53
And then we can generate a
similar table,

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where we have our ligand
position and our set of d

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00:06:58 --> 00:07:02
orbitals.
And I will start here with

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00:07:02 --> 00:07:07
d(xz), d(yz) and d(xy),
and over here,

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00:07:07 --> 00:07:14
d z squared and d x
squared minus y squared.

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00:07:14 --> 00:07:17
And we need our ligands

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00:07:17 --> 00:07:21
positioned at seven,
eight, nine,

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00:07:21 --> 00:07:27
and ten, which are these
alternating corners of the cube,

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which is the tetrahedral
geometry.

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00:07:33 --> 00:07:37
And what we find,
when we go through and evaluate

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00:07:37 --> 00:07:43
for those theta and phi values
that correspond to these ligand

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00:07:43 --> 00:07:48
positions using our polar
coordinate system and then

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00:07:48 --> 00:07:54
scaling it the same way that we
do over here for the octahedron,

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is that all of these values
come out to be one-third.

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00:08:00 --> 00:08:02
Notice that,
in the case of the octahedron,

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00:08:02 --> 00:08:07
we are interacting with some of
the lobes, like the lobes of d z

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squared,
the large ones,

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00:08:09 --> 00:08:13
directly along the axis that
makes it a pure sigma

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interaction, a cylindrically
symmetric interaction.

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A big lobe of d z squared
and a ligand coming

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in along the z-axis.
But if you think of the

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positions of the lobes of the
xz, yz, and xy orbitals relative

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to these ligand positions,
seven, eight,

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nine, and ten,
the interaction is kind of

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oblique.
It is not zero.

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We are not on a node,
but it is pretty small.

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00:08:39 --> 00:08:42
The overlap is not going to be
as good.

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00:08:42 --> 00:08:46
And it evaluates to one-third
for all of those positions.

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00:08:46 --> 00:08:50
What we find is that while xz,
yz, and xy were all zero over

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here for the octahedral case,
over here, for the tetrahedral

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case, they are all the same
again, but they are not at zero.

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They are at one-third.
And then, for x squared minus y

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squared,
remember x squared minus y

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00:09:07 --> 00:09:09
squared has its lobes along x
and y.

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What that means is that ligands
seven, eight,

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00:09:13 --> 00:09:17
nine, and ten actually lie on a
nodal surface of the d x squared

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minus y squared orbital.
And that means that this is

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zero, this is zero,
this is zero,

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00:09:23 --> 00:09:29
and this is zero.
And the amazing thing about the

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math of the tetrahedral geometry
is that these ligands,

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seven, eight,
nine, and ten also lie on the

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00:09:39 --> 00:09:43
nodal surface of d z squared.

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Remember that d z squared looks
like this and has our opposite

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phase torus in the middle and
then has a conical nodal

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surface.
We actually went ahead and --

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00:10:00 --> 00:10:05
If you set the equation for d z
squared equal to zero

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00:10:05 --> 00:10:08
and solved for it,
we found this angle here.

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00:10:08 --> 00:10:13
We found that angle.
And you will remember that that

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00:10:13 --> 00:10:16
was the arc cosine of root three
over three.

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00:10:16 --> 00:10:21
It turns out that that value is
exactly half of the tetrahedral

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00:10:21 --> 00:10:25
angle, which is the angel
between seven,

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00:10:25 --> 00:10:29
the center and eight.
And, because of this,

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00:10:29 --> 00:10:31
all ligands,
seven, eight,

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00:10:31 --> 00:10:35
nine, and ten lie on the
conical nodal surface of d z

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squared,
such that we get zeros here,

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too.
One thing that relates very

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nicely, the orbital picture for
the tetrahedral case for the

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00:10:46 --> 00:10:49
octahedral case,
is that z squared has the same

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00:10:49 --> 00:10:52
energy as x squared minus y
squared.

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When you sum over all the
ligand positions,

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we are going to get three here,
and we are going to get three

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here.
And xz, yz, and xy all have

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zero.
And over here xz,

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00:11:06 --> 00:11:09
yz, and xy all have
four-thirds.

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00:11:09 --> 00:11:14
And z squared and x
squared minus y squared

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00:11:14 --> 00:11:18
both have zero.
And so I will redraw that over

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here.

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What we are doing is drawing up
d orbital splitting diagrams for

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two different possible
coordination geometries for a

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coordination complex.
Let me just make that very

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clear.

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And I have my energy units
here, zero, one,

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two, three.
And this energy level diagram

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looks like this for the
octahedral.

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And we symbolize the octahedral
geometry with this Oh for

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octahedral.
And we have here an energy

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splitting.
We have a triply degenerate set

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00:12:27 --> 00:12:35
of orbitals which we recognize
as xz, yz, and xy.

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00:12:35 --> 00:12:44
And the name that we give to
that triply degenerate set is

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00:12:44 --> 00:12:50
t(2g).
This t(2g) triply degenerate

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00:12:50 --> 00:12:58
set, for this whole set,
is the t(2g) set.

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00:12:58 --> 00:13:01
If I use that label t(2g),
you should remember the

185
00:13:01 --> 00:13:04
specific d orbitals that
contribute to the t(2g) set and

186
00:13:04 --> 00:13:08
make up the t2g set.
Up here we have the d z squared

187
00:13:08 --> 00:13:12
and the d x squared
minus y squared

188
00:13:12 --> 00:13:15
at the same energy,
which are three relative energy

189
00:13:15 --> 00:13:17
units here.
And those two together

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00:13:17 --> 00:13:20
constitute the e(g) set.
And I will often write that

191
00:13:20 --> 00:13:23
with a star because we are going
to find out later,

192
00:13:23 --> 00:13:27
when we do the molecular
orbital theory of coordination

193
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complexes, that these are
antibonding.

194
00:13:31 --> 00:13:34
And the reason they are
antibonding is because,

195
00:13:34 --> 00:13:38
if you go over here and look at
it, if you have a ligand that

196
00:13:38 --> 00:13:42
wants to act as a Lewis base
with respect to d z squared,

197
00:13:42 --> 00:13:45
then putting an
electron in d z squared

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00:13:45 --> 00:13:49
leads to repulsion of
that ligand, generating

199
00:13:49 --> 00:13:53
antibonding characters.
You want your Lewis acid to

200
00:13:53 --> 00:13:58
have empty orbitals to receive
lone pairs of electrons.

201
00:13:58 --> 00:14:01
And, if instead they are
populated, that corresponds to

202
00:14:01 --> 00:14:05
antibonding character,
much like we have talked about

203
00:14:05 --> 00:14:08
for other types of molecular
orbital diagrams.

204
00:14:08 --> 00:14:12
This d-orbital splitting
diagram for Oh is really a

205
00:14:12 --> 00:14:16
simplified molecular orbital
diagram for the molecule in

206
00:14:16 --> 00:14:20
which we just look at those
orbitals that derive from the

207
00:14:20 --> 00:14:23
set of five d orbitals on that
metal ion.

208
00:14:23 --> 00:14:27
Because these are the orbitals,
--

209
00:14:27 --> 00:14:31
-- and, together with the
electrons that occupy them,

210
00:14:31 --> 00:14:36
they control the properties of
color and magnetism that we will

211
00:14:36 --> 00:14:40
be talking about for the rest of
today.

212
00:14:40 --> 00:14:45
And let me remind you here that
we have a name for the magnitude

213
00:14:45 --> 00:14:49
of this splitting,
which is called delta O for

214
00:14:49 --> 00:14:52
octahedral.
And then, we also have the

215
00:14:52 --> 00:14:56
tetrahedral case.
And how do we get the diagram

216
00:14:56 --> 00:15:00
from this?
We just sum over the

217
00:15:00 --> 00:15:04
contributions from each of the
ligand positions.

218
00:15:04 --> 00:15:08
In tetrahedral,
there are only four ligands.

219
00:15:08 --> 00:15:13
For xz, yz, and zy we have to
sum up four times one-third,

220
00:15:13 --> 00:15:17
and that gives us,
for that set of three orbitals

221
00:15:17 --> 00:15:22
using the same energy scale as
over there, three orbitals at a

222
00:15:22 --> 00:15:27
value of four-thirds and two
orbitals down here at our zero

223
00:15:27 --> 00:15:32
of energy.
And these will be d z squared,

224
00:15:32 --> 00:15:38
located on that
node, and x squared minus y

225
00:15:38 --> 00:15:44
squared.
Whereas, up here we have xz,

226
00:15:44 --> 00:15:48
yz, and zy.
And these get the names e and

227
00:15:48 --> 00:15:52
t(2) in the tetrahedral
geometry.

228
00:15:52 --> 00:15:58
And the splitting between the
levels, the doubly degenerate

229
00:15:58 --> 00:16:04
and the triply degenerate pair
of energy levels,

230
00:16:04 --> 00:16:11
is here referred to as delta t.
And from the math of these two

231
00:16:11 --> 00:16:18
tables and the magnitude of this
splitting being three relative

232
00:16:18 --> 00:16:23
to this one being four-thirds,
you can easily derive the

233
00:16:23 --> 00:16:29
relation that delta t is equal
to four-ninths of delta O.

234
00:16:29 --> 00:16:35
One of the neat things about

235
00:16:35 --> 00:16:40
this result qualitatively is
that if you have more ligands,

236
00:16:40 --> 00:16:45
six versus four,
you have a larger splitting of

237
00:16:45 --> 00:16:49
the energy levels.
And that corresponds to putting

238
00:16:49 --> 00:16:53
more Lewis bases,
more negative charges,

239
00:16:53 --> 00:16:56
around that sphere that we
talked about,

240
00:16:56 --> 00:17:02
in the coordination sphere.
And also, if you have

241
00:17:02 --> 00:17:08
interactions between ligands and
metal d orbitals that are end-on

242
00:17:08 --> 00:17:12
sigma symmetry,
directed at each other,

243
00:17:12 --> 00:17:17
you get bigger splitting
because that leads to better

244
00:17:17 --> 00:17:23
overlap than the type of oblique
overlap that you get if you have

245
00:17:23 --> 00:17:29
a ligand at a position such as
seven relative to an orbital

246
00:17:29 --> 00:17:33
like d(xz).
You can now see that to fully

247
00:17:33 --> 00:17:37
appreciate this,
you really have to have a very

248
00:17:37 --> 00:17:40
good grasp of the nodal
properties of the d orbitals.

249
00:17:40 --> 00:17:44
This is why I am going to
re-emphasize that you should

250
00:17:44 --> 00:17:48
spend some time on the computer
visualizing these things and

251
00:17:48 --> 00:17:52
rotating them around and seeing
where these ligand positions for

252
00:17:52 --> 00:17:56
these different coordination
geometries are with respect to

253
00:17:56 --> 00:18:00
each of the five d orbitals.
That is quite important.

254
00:18:00 --> 00:18:04
Also, you are going to see that
you could consider other

255
00:18:04 --> 00:18:08
possible coordination geometries
that involve ligands at these

256
00:18:08 --> 00:18:12
positions or at other positions.
For coordination number five,

257
00:18:12 --> 00:18:16
you could have a trigonal
bipyramid coordination geometry.

258
00:18:16 --> 00:18:20
And you could go ahead and
evaluate the d orbitals at the

259
00:18:20 --> 00:18:23
relevant positions.
You would already have three of

260
00:18:23 --> 00:18:27
them here because in a trigonal
bipyramid, you would have

261
00:18:27 --> 00:18:31
ligands at positions one and
six, and two.

262
00:18:31 --> 00:18:34
And then two more back here at
120 degrees to two,

263
00:18:34 --> 00:18:37
but in the x,y-plane.
So you can do a trigonal

264
00:18:37 --> 00:18:40
bipyramid the same way and
generate the d-orbital splitting

265
00:18:40 --> 00:18:43
diagram for it.
You can also go ahead and do a

266
00:18:43 --> 00:18:47
d-orbital splitting diagram for
a different geometry

267
00:18:47 --> 00:18:50
corresponding to coordination
number four, which would be the

268
00:18:50 --> 00:18:53
square planar coordination
geometry.

269
00:18:53 --> 00:18:55
If you just took ligand
positions two,

270
00:18:55 --> 00:18:58
three, four,
and five you could generate a

271
00:18:58 --> 00:19:01
diagram.
And that diagram would be

272
00:19:01 --> 00:19:05
relevant for systems that do
have square planar coordination

273
00:19:05 --> 00:19:09
around a metal center,
and these are actually fairly

274
00:19:09 --> 00:19:12
common.
Those are the four coordination

275
00:19:12 --> 00:19:14
geometries, octahedral,
tetrahedral,

276
00:19:14 --> 00:19:17
trigonal bipyramid,
and square planar that are all

277
00:19:17 --> 00:19:22
pretty commonly encountered and
that you can actually interpret

278
00:19:22 --> 00:19:25
pretty nicely using the
d-orbital splitting diagrams.

279
00:19:25 --> 00:19:29
Now, one of the properties that
we are going to want to

280
00:19:29 --> 00:19:33
interpret will be the magnetism.

281
00:19:33 --> 00:19:50


282
00:19:50 --> 00:19:54
And, in talking about the
magnetism of coordination

283
00:19:54 --> 00:20:00
complexes, we are going to start
by mentioning two terms that you

284
00:20:00 --> 00:20:04
have probably heard,
but I would like to define

285
00:20:04 --> 00:20:08
explicitly today.
One is paramagnetic.

286
00:20:08 --> 00:20:13
That term is given to
substances that are attracted

287
00:20:13 --> 00:20:17
into a magnetic field.
It is somehow a bulk phenomenon

288
00:20:17 --> 00:20:22
that we are going to relate to
properties of d orbital

289
00:20:22 --> 00:20:26
splitting diagrams.

290
00:20:26 --> 00:20:35


291
00:20:35 --> 00:20:38
And systems can also be
described as diamagnetic.

292
00:20:38 --> 00:20:42
And there are many other types
of magnetic behavior that we

293
00:20:42 --> 00:20:50
won't be discussing.
I'm a little too close to the

294
00:20:50 --> 00:20:56
board, here.
Diamagnetic.

295
00:20:56 --> 00:21:02
And that means that the
substance is repelled out of or

296
00:21:02 --> 00:21:10
away from a magnetic field.
And one thing that you should

297
00:21:10 --> 00:21:16
keep in mind is that the
magnitude of paramagnetism is

298
00:21:16 --> 00:21:24
usually orders of magnitude
greater than the magnitude of

299
00:21:24 --> 00:21:30
diamagnetism.
So, this is much larger.

300
00:21:30 --> 00:21:34


301
00:21:34 --> 00:21:37
And what this means is if a
substance behaves as a

302
00:21:37 --> 00:21:41
paramagnet, you can often
neglect its diamagnetism that

303
00:21:41 --> 00:21:43
also may be present,
also must be present.

304
00:21:43 --> 00:21:46
The substances that are
paramagnetic are also

305
00:21:46 --> 00:21:50
diamagnetic, but the
paramagnetism is much larger in

306
00:21:50 --> 00:21:53
terms of its order of magnitude
and wins out.

307
00:21:53 --> 00:21:57
If you want to measure one of
these quantities for a system,

308
00:21:57 --> 00:22:00
that which is far more
difficult to measure is the

309
00:22:00 --> 00:22:05
diamagnetism because it is a
much smaller number.

310
00:22:05 --> 00:22:11
And then, I want to bring up
this quantity S.

311
00:22:11 --> 00:22:17
This is the spin quantum
number.

312
00:22:17 --> 00:22:28


313
00:22:28 --> 00:22:35
And you can readily calculate S
from looking at a populated d

314
00:22:35 --> 00:22:43
orbital spitting diagram because
it will be the total number of

315
00:22:43 --> 00:22:50
electrons multiplied by
one-half, which is the spin per

316
00:22:50 --> 00:22:52
electron.

317
00:22:52 --> 00:23:00


318
00:23:00 --> 00:23:03
If you are confronted with a
particular coordination complex,

319
00:23:03 --> 00:23:07
the kind of thought process you
will need to go through is,

320
00:23:07 --> 00:23:11
what is the coordination
number, what is the coordination

321
00:23:11 --> 00:23:14
geometry, what does the d
orbital splitting diagram look

322
00:23:14 --> 00:23:18
like, how many electrons do I
have with which to populate the

323
00:23:18 --> 00:23:22
d orbital splitting diagram?
After I have figured out how to

324
00:23:22 --> 00:23:26
populate the d orbital splitting
diagram, how many of those

325
00:23:26 --> 00:23:30
electrons are unpaired?
That gives me this.

326
00:23:30 --> 00:23:33
I can then calculate S.
And, if I have S,

327
00:23:33 --> 00:23:38
I can make a prediction about
the value of something called

328
00:23:38 --> 00:23:41
the spin-only magnetic moment.

329
00:23:41 --> 00:23:49


330
00:23:49 --> 00:23:54
Let me mention one more thing
back here, and that would be the

331
00:23:54 --> 00:23:55
units.

332
00:23:55 --> 00:24:01


333
00:24:01 --> 00:24:08
When we talk about magnetism,
we are going to be using units.

334
00:24:08 --> 00:24:12
This is called the Bohr
magneton.

335
00:24:12 --> 00:24:18
It is sometimes written as mu
sub b.

336
00:24:18 --> 00:24:24
It is also sometimes written as
capital B., capital M.

337
00:24:24 --> 00:24:30
That is the Bohr
magneton.

338
00:24:30 --> 00:24:39
And it is equal to
9.2741x10^-24 joules per tesla.

339
00:24:39 --> 00:24:47
This is the value of the Bohr
magneton.

340
00:24:47 --> 00:24:58
And the Bohr magneton is the
unit, also, of our magnetic

341
00:24:58 --> 00:25:01
moment.

342
00:25:01 --> 00:25:10


343
00:25:10 --> 00:25:14
And in particular,
today, I am going to be talking

344
00:25:14 --> 00:25:18
about the spin-only value of the
magnetic moment.

345
00:25:18 --> 00:25:22
So we are going to be making
predictions using spin-only

346
00:25:22 --> 00:25:26
considerations.
And for those considerations,

347
00:25:26 --> 00:25:30
we need to be able to calculate
S.

348
00:25:30 --> 00:25:34
And we can do that pretty
quickly after we have it

349
00:25:34 --> 00:25:38
correctly populated,
correctly chosen d orbital

350
00:25:38 --> 00:25:41
splitting diagram for the
system.

351
00:25:41 --> 00:25:46
And so the spin-only value for
the magnetic moment is given by

352
00:25:46 --> 00:25:50
this formula,
which is mu is equal to 2.00

353
00:25:50 --> 00:25:54
times the square root of S times
S plus one.

354
00:25:54 --> 00:26:00
And, in this formula,

355
00:26:00 --> 00:26:06
this part, this square root of
S times S plus one,

356
00:26:06 --> 00:26:13
from the quantum mechanics,
is the value of the angular

357
00:26:13 --> 00:26:20
momentum, the electron spin
angular momentum.

358
00:26:20 --> 00:26:30


359
00:26:30 --> 00:26:34
And this value 2.00,
I am calling it 2.00.

360
00:26:34 --> 00:26:40
It is actually a number that
differs only very slightly from

361
00:26:40 --> 00:26:44
two.
So we can, for most purposes,

362
00:26:44 --> 00:26:48
use 2.00.
This is called the gyromagnetic

363
00:26:48 --> 00:26:52
ratio of the electron.

364
00:26:52 --> 00:27:00


365
00:27:00 --> 00:27:04
And, as its name implies,
it is the ratio of the angular

366
00:27:04 --> 00:27:09
momentum to the magnetic moment
for the electron.

367
00:27:09 --> 00:27:14
This is our conversion factor
to go back and forth between

368
00:27:14 --> 00:27:17
angular momentum and magnetic
moment.

369
00:27:17 --> 00:27:21
And that factor is the
gyromagnetic ratio for the

370
00:27:21 --> 00:27:24
electron.
You will see this as gamma,

371
00:27:24 --> 00:27:28
sometimes.
And, for the free electron

372
00:27:28 --> 00:27:33
value, it is a number very close
to two.

373
00:27:33 --> 00:27:37
And so, if you think about
different ions that could be at

374
00:27:37 --> 00:27:41
the center of a coordination
complex, you wonder,

375
00:27:41 --> 00:27:46
well, what are my possible
values for the spin-only

376
00:27:46 --> 00:27:49
magnetic moment,
as given by this formula?

377
00:27:49 --> 00:27:52
Well, we can make a table for
that.

378
00:27:52 --> 00:27:56
We have the number of electrons
that are unpaired,

379
00:27:56 --> 00:27:59
remember.
And then, we will be able to

380
00:27:59 --> 00:28:04
calculate S.
And then we will want to get

381
00:28:04 --> 00:28:10
mu, which is this spin-only
value calculated according to

382
00:28:10 --> 00:28:15
that formula.
And ions from the part of the

383
00:28:15 --> 00:28:21
Periodic Table that we may
consider working with can have

384
00:28:21 --> 00:28:24
one, two, three,
four, five, six,

385
00:28:24 --> 00:28:30
up to seven unpaired electrons.
These numbers,

386
00:28:30 --> 00:28:33
like six and seven,
are more important for the

387
00:28:33 --> 00:28:38
lanthanide ions in systems with
f-electrons than for systems

388
00:28:38 --> 00:28:40
with d electrons.
But, nonetheless,

389
00:28:40 --> 00:28:44
we can easily make these
predictions the same way.

390
00:28:44 --> 00:28:48
And so S would be one-half for
one electron,

391
00:28:48 --> 00:28:51
one, then three-halves,
then two for four unpaired

392
00:28:51 --> 00:28:56
electrons, then five-halves for
five unpaired electrons,

393
00:28:56 --> 00:29:00
then three, and then
seven-halves finally for seven

394
00:29:00 --> 00:29:06
unpaired electrons.
And if you put this into this

395
00:29:06 --> 00:29:13
formula, you would see that for
one unpaired electron,

396
00:29:13 --> 00:29:20
it would be one-half times
one-half plus one square root

397
00:29:20 --> 00:29:25
times two.
That would be square root of

398
00:29:25 --> 00:29:31
three, so this would be a
spin-only magnetic moment of

399
00:29:31.776 --> 1.73.


400
1.73. --> 00:29:35
And then for S equals one,

401
00:29:35 --> 00:29:42
we actually have a value of
about 2.83.

402
00:29:42 --> 00:29:48
And then for three unpaired
electrons, we have a value of,

403
00:29:48 --> 00:29:55
let's say approximately 3.87.
And then for S equals two,

404
00:29:55 --> 00:30:02
or four unpaired electrons,
we have a value of 4.90.

405
00:30:02 --> 00:30:06
For the spin-only magnetic
moment, if you go through with

406
00:30:06 --> 00:30:09
these values of S,
these numbers of unpaired

407
00:30:09 --> 00:30:12
electrons.
Let's just make sure we are

408
00:30:12 --> 00:30:14
very clear on that.

409
00:30:14 --> 00:30:20


410
00:30:20 --> 00:30:26
And then 5.92,
6.93, 7.94 approximately.

411
00:30:26 --> 00:30:34
You can check those numbers,
but they are approximately

412
00:30:34 --> 00:30:38
right.
And in each case what you are

413
00:30:38 --> 00:30:42
seeing is the spin-only
prediction for the magnetic

414
00:30:42 --> 00:30:46
moment is always,
in units of Bohr magnetons,

415
00:30:46 --> 00:30:51
a little less than one plus the
number of unpaired electrons.

416
00:30:51 --> 00:30:55
It is not like you have four
unpaired electrons,

417
00:30:55 --> 00:31:00
so your spin-only magnetic
moment is around four.

418
00:31:00 --> 00:31:03
It is actually a little less
than five.

419
00:31:03 --> 00:31:08
And that is because of the
quantum mechanics of the

420
00:31:08 --> 00:31:15
electron spin angular momentum.
And this kind of consideration,

421
00:31:15 --> 00:31:20
here, leads you into thinking
about what happens with

422
00:31:20 --> 00:31:25
different d^n counts for a
particular type of metal

423
00:31:25 --> 00:31:27
complex.

424
00:31:27 --> 00:31:32


425
00:31:32 --> 00:31:40
This means that we are going to
attack and address the high

426
00:31:40 --> 00:31:44
spin/low spin problem.

427
00:31:44 --> 00:31:56


428
00:31:56 --> 00:31:59
These systems are not so
terribly difficult,

429
00:31:59 --> 00:32:01
but let's face it;
--

430
00:32:01 --> 00:32:04
-- once you have been presented
with a particular coordination

431
00:32:04 --> 00:32:07
complex, and you think you know
the geometry,

432
00:32:07 --> 00:32:11
and you think you know the d
orbital splitting diagram that

433
00:32:11 --> 00:32:15
corresponds to that geometry,
and you think you know how many

434
00:32:15 --> 00:32:18
d electrons go into that
diagram, there is one more

435
00:32:18 --> 00:32:20
thing.
And that is sometimes you have

436
00:32:20 --> 00:32:25
to figure out if it is high spin
or if it is low spin.

437
00:32:25 --> 00:32:29
And what that means is for a
particular d^n count,

438
00:32:29 --> 00:32:33
there might be two possible
choices over here,

439
00:32:33 --> 00:32:38
depending on how the electrons
occupy the orbitals.

440
00:32:38 --> 00:32:41
And so let's look at an example
of this.

441
00:32:41 --> 00:32:46
Here is energy.
Again, we are really focusing a

442
00:32:46 --> 00:32:51
lot on energy level diagrams.
Sometimes we focus on molecular

443
00:32:51 --> 00:32:56
orbital diagrams.
Right now, we are using these d

444
00:32:56 --> 00:33:03
orbital splitting diagrams.
And I will give you an example

445
00:33:03 --> 00:33:07
that is octahedral.
And this is a species that we

446
00:33:07 --> 00:33:12
looked at in a recent
demonstration in class,

447
00:33:12 --> 00:33:18
which is hexaaquo meaning six
waters around an iron two plus

448
00:33:18 --> 00:33:21
in an octahedral array.

449
00:33:21 --> 00:33:27
This will be our metal complex
for which we will draw the d

450
00:33:27 --> 00:33:33
orbital splitting diagram.
It is an octahedral complex,

451
00:33:33 --> 00:33:37
so we draw our diagram like
this.

452
00:33:37 --> 00:33:42
And of course,
we have t(2g) and e(g)* levels

453
00:33:42 --> 00:33:46
that we can populate with our d
electrons.

454
00:33:46 --> 00:33:52
And we are going to need to
figure out how many d electrons

455
00:33:52 --> 00:34:00
we have to put into this diagram
for the octahedral system.

456
00:34:00 --> 00:34:04
The oxidation state of the iron
here is +2.

457
00:34:04 --> 00:34:10
And you can tell that because
the way you tell oxidation state

458
00:34:10 --> 00:34:16
is to take into account both the
charge on the ligands.

459
00:34:16 --> 00:34:20
Chlorides, for example,
are minus one.

460
00:34:20 --> 00:34:26
Because water is a neutral
molecule, it is simply neutral.

461
00:34:26 --> 00:34:32
And then, you have to also take
into account the charge on the

462
00:34:32 --> 00:34:36
whole ion.
So this is iron two plus.

463
00:34:36 --> 00:34:38
The iron in the plus two

464
00:34:38 --> 00:34:40
oxidation state with six neutral
ligands.

465
00:34:40 --> 00:34:43
If I had six NH three
ligands in the same charge on

466
00:34:43 --> 00:34:46
the ion, then it would still be
iron plus two.

467
00:34:46 --> 00:34:50
And you are going to get some
practice figuring out oxidation

468
00:34:50 --> 00:34:53
state because if you cannot do
that correctly you cannot

469
00:34:53 --> 00:34:56
identify the number of d
electrons to put into the

470
00:34:56 --> 00:34:58
diagram correctly.
And this is iron plus two.

471
00:34:58 --> 00:35:02
You need to know that iron is

472
00:35:02 --> 00:35:05
in Group 8 of the periodic
table.

473
00:35:05 --> 00:35:11
The equation for number of d
electrons here is eight minus

474
00:35:11 --> 00:35:15
two equals six.
If it is in Group 8,

475
00:35:15 --> 00:35:20
that means that an iron atom
has eight valance electrons,

476
00:35:20 --> 00:35:26
the same way that carbon has
four valance electrons or oxygen

477
00:35:26 --> 00:35:32
has six valance electrons.
And in Group 8 of the periodic

478
00:35:32 --> 00:35:36
table, we are subtracting two
from the eight valance electrons

479
00:35:36 --> 00:35:40
because there is a two plus
charge on the system.

480
00:35:40 --> 00:35:43
That leaves six valance
electrons on the iron.

481
00:35:43 --> 00:35:48
And that means this is what we
call a d six system.

482
00:35:48 --> 00:35:51
By convention,
we are considering all the

483
00:35:51 --> 00:35:55
valance electrons that remain
after the metal has been

484
00:35:55 --> 00:35:59
oxidized to plus two as d
electrons, meaning they are

485
00:35:59 --> 00:36:04
going to go into our d orbital
splitting diagram.

486
00:36:04 --> 00:36:08
Now, the reason we do this,
in fact, is because in ions

487
00:36:08 --> 00:36:13
like this, the d orbitals are
the orbitals on the metal,

488
00:36:13 --> 00:36:17
on the iron,
that are the lowest in energy.

489
00:36:17 --> 00:36:22
After, in the case of iron,
iron has principle quantum

490
00:36:22 --> 00:36:26
number three.
There is also a 4s and a 4p set

491
00:36:26 --> 00:36:31
of orbitals available to iron,
but they are much higher in

492
00:36:31 --> 00:36:35
energy.
We are simplifying the system

493
00:36:35 --> 00:36:40
and just looking at what happens
with the five d orbitals,

494
00:36:40 --> 00:36:45
which are the lowest valance
energy orbitals available to the

495
00:36:45 --> 00:36:48
iron.
Just like a carbon has a 2s and

496
00:36:48 --> 00:36:53
a set of three 2p orbitals,
an iron has a set of five 3d

497
00:36:53 --> 00:36:56
orbitals, a 4s,
and three 4p orbitals.

498
00:36:56 --> 00:37:00
But those s and p ones are
higher.

499
00:37:00 --> 00:37:04
We are just focusing on our d
orbitals, here,

500
00:37:04 --> 00:37:10
using our d orbital only
splitting diagrams with which we

501
00:37:10 --> 00:37:14
have six electrons for
population of a diagram.

502
00:37:14 --> 00:37:20
And so what you see immediately
is you will identify with the

503
00:37:20 --> 00:37:26
notion that there are two ways
to populate this diagram with

504
00:37:26 --> 00:37:29
six electrons.

505
00:37:29 --> 00:37:32


506
00:37:32 --> 00:37:34
Because we can do this.

507
00:37:34 --> 00:37:37


508
00:37:37 --> 00:37:42
Here, we are following Hund's
rule of maximum multiplicity,

509
00:37:42 --> 00:37:47
trying not to pair up any
electrons until we run out of

510
00:37:47 --> 00:37:50
spots for them,
and so we have to.

511
00:37:50 --> 00:37:54
There is one way to populate
the diagram.

512
00:37:54 --> 00:38:00
Two electrons in e(g)*.
Four electrons in t(2g).

513
00:38:00 --> 00:38:04
One of them,
spin inverted relative to the

514
00:38:04 --> 00:38:10
other five because you cannot
put two electrons in the same

515
00:38:10 --> 00:38:14
orbital unless they have
different spins.

516
00:38:14 --> 00:38:18
And, finally,
we could populate the diagram

517
00:38:18 --> 00:38:22
this way.
The case on the left is what we

518
00:38:22 --> 00:38:25
call high spin.

519
00:38:25 --> 00:38:30


520
00:38:30 --> 00:38:35
The case on the right is what
we call low spin.

521
00:38:35 --> 00:38:40


522
00:38:40 --> 00:38:44
And, as you think about it,
you will realize that the high

523
00:38:44 --> 00:38:49
spin/low spin problem only
arises for certain d^n counts.

524
00:38:49 --> 00:38:51
It doesn't arise,
for example,

525
00:38:51 --> 00:38:56
for d one because we
would just put one electron down

526
00:38:56 --> 00:39:00
here in t(2g) and that is not a
problem.

527
00:39:00 --> 00:39:04
And you will see that,
as you run through the d one

528
00:39:04 --> 00:39:07
all the way up to the d
ten.

529
00:39:07 --> 00:39:12
d^n counts can go from d^0 to
d^10, and that is the limit of

530
00:39:12 --> 00:39:15
the possibilities here.
Zero, one, two,

531
00:39:15 --> 00:39:17
three, four,
five, six, seven,

532
00:39:17 --> 00:39:19
eight, nine,
and ten.

533
00:39:19 --> 00:39:23
Here I have chosen a d^6 case
to illustrate the high spin

534
00:39:23 --> 00:39:28
versus low spin dichotomy.
And you might wonder when do

535
00:39:28 --> 00:39:32
you --
Well, let me get back to that

536
00:39:32 --> 00:39:36
in a moment.
And let me just say here that

537
00:39:36 --> 00:39:40
we have four unpaired electrons
in d^6 high spin.

538
00:39:40 --> 00:39:43
We have one,
two, three, four.

539
00:39:43 --> 00:39:46
That makes this an s equals two
case.

540
00:39:46 --> 00:39:50
And, over here,
we have s equals zero.

541
00:39:50 --> 00:39:55
And so this one on the right,
this low spin case is s equals

542
00:39:55 --> 00:39:58
zero.
We would say that this is not

543
00:39:58 --> 00:40:03
paramagnetic.
It is going to be only

544
00:40:03 --> 00:40:07
diamagnetic.
It has no unpaired electrons to

545
00:40:07 --> 00:40:12
give it a magnetic moment
according to the spin-only

546
00:40:12 --> 00:40:15
formula that we looked at over
here.

547
00:40:15 --> 00:40:20
On the other hand,
over here, we have an s equals

548
00:40:20 --> 00:40:25
two state that predicts a
magnetic moment of 4.90 Bohr

549
00:40:25 --> 00:40:28
magnetons.
And experimentally,

550
00:40:28 --> 00:40:35
mu was found for this system to
be 5.1 Bohr magnetons.

551
00:40:35 --> 00:40:40
So we say this one turned out
to be high spin.

552
00:40:40 --> 00:40:45
And in a little while,
we will be talking more about

553
00:40:45 --> 00:40:52
what are the physical factors
that determine whether a system

554
00:40:52 --> 00:40:58
is low spin or high spin.
Remember, we have this delta O

555
00:40:58 --> 00:41:02
here.
And there is not one value for

556
00:41:02 --> 00:41:07
delta O, and this is something I
am going to emphasize again in a

557
00:41:07 --> 00:41:10
moment.
Delta O varies according to a

558
00:41:10 --> 00:41:14
number of factors,
and these factors include the

559
00:41:14 --> 00:41:19
specific nature of the ligands.
It includes the charge on the

560
00:41:19 --> 00:41:22
metal center.
It includes the specific metal

561
00:41:22 --> 00:41:25
that you are using.
And at some point,

562
00:41:25 --> 00:41:30
if delta O gets bigger and
bigger and bigger and bigger and

563
00:41:30 --> 00:41:35
exceeds what we call the pairing
energy --

564
00:41:35 --> 00:41:47


565
00:41:47 --> 00:41:50
This pairing energy is
something I will talk more about

566
00:41:50 --> 00:41:53
later, but it is the energy
required to put two electrons in

567
00:41:53 --> 00:41:55
the same d orbital.

568
00:41:55 --> 00:42:10


569
00:42:10 --> 00:42:15
If you get to a condition where
delta O is greater than this

570
00:42:15 --> 00:42:20
pairing energy,
in other words,

571
00:42:20 --> 00:42:25
there is a big splitting
between t(2g) and e(g)*,

572
00:42:25 --> 00:42:31
then the electrons will prefer
to pair up down here in the

573
00:42:31 --> 00:42:35
t(2g).
Whereas, if you think of it

574
00:42:35 --> 00:42:39
going to the limit of an
infinitely small delta O,

575
00:42:39 --> 00:42:44
they would have S equals two
because the gap between t(2g)

576
00:42:44 --> 00:42:47
and e(g)* in the limit of
vanishing delta O,

577
00:42:47 --> 00:42:51
for example,
when ligands go to an infinite

578
00:42:51 --> 00:42:55
distance away from the metal
center, they would then,

579
00:42:55 --> 00:43:00
in that limit,
have all five the same energy.

580
00:43:00 --> 00:43:03
And so your S would naturally
equal two.

581
00:43:03 --> 00:43:09
If delta O is much greater than
the pairing energy giving a

582
00:43:09 --> 00:43:14
large value of delta O,
then that favors low spin.

583
00:43:14 --> 00:43:20
Factors that give rise to large
values of delta O favor low

584
00:43:20 --> 00:43:26
spin, and I will talk about a
few more factors like that in a

585
00:43:26 --> 00:43:28
moment.

586
00:43:28 --> 00:43:36


587
00:43:36 --> 00:43:40
Here is a brief introduction to
the magnetic properties of

588
00:43:40 --> 00:43:45
coordination complexes.
And I want to show that we can

589
00:43:45 --> 00:43:49
use the same types of diagrams
to talk about the colors of

590
00:43:49 --> 00:43:54
transition metal complexes.
We looked at the color change

591
00:43:54 --> 00:43:58
in that demonstration I showed
you.

592
00:43:58 --> 00:44:02
And one of the really
fascinating aspects of

593
00:44:02 --> 00:44:08
transition metal chemistry is
the colors that arise because of

594
00:44:08 --> 00:44:13
absorption of a photon
coincident with promotion of an

595
00:44:13 --> 00:44:19
electron from t(2g) to e(g)*.
And so we will talk briefly now

596
00:44:19 --> 00:44:25
about electronic / absorption
spectroscopy.

597
00:44:25 --> 00:44:40


598
00:44:40 --> 00:44:48
And the most important thing I
want to impart to you here is a

599
00:44:48 --> 00:44:55
selection rule wherein delta S
is equal to zero.

600
00:44:55 --> 00:44:58


601
00:44:58 --> 00:45:02
This is the spin selection
rule.

602
00:45:02 --> 00:45:10


603
00:45:10 --> 00:45:14
And this means that when a
photon comes in and interacts

604
00:45:14 --> 00:45:19
with d electrons in a system of
the type we are discussing,

605
00:45:19 --> 00:45:24
that photon can be absorbed
coincident with promotion of an

606
00:45:24 --> 00:45:29
electron from t(2g) to e(g)*,
as long as it doesn't violate

607
00:45:29 --> 00:45:34
this spin selection rule.
And that means that the number

608
00:45:34 --> 00:45:39
of unpaired electrons should be
the same in the system before

609
00:45:39 --> 00:45:43
the photon is taken up,
as it is after.

610
00:45:43 --> 00:45:47
And so here is an example of
that, a simple example.

611
00:45:47 --> 00:45:52
Here is an octahedral system,
titanium with six waters and a

612
00:45:52 --> 00:45:56
three plus charge.

613
00:45:56 --> 00:46:01
Titanium is in Group 4 of the
periodic table.

614
00:46:01 --> 00:46:05
Our water ligands are neutral,
with three plus charges to be

615
00:46:05 --> 00:46:07
subtracted from four.
We have a d^1 system.

616
00:46:07 --> 00:46:11
This is one of the simplest
cases for talking about

617
00:46:11 --> 00:46:14
electronic absorption
spectroscopy and is the first

618
00:46:14 --> 00:46:17
thing that you encounter in your
textbook.

619
00:46:17 --> 00:46:21
By the way, let me just remind
you that right now you should be

620
00:46:21 --> 00:46:25
reading Chapter 16 in your
textbook, that deals with

621
00:46:25 --> 00:46:28
coordination chemistry,
electronic structure of d-block

622
00:46:28 --> 00:46:33
elements, colors and magnetism,
and so forth.

623
00:46:33 --> 00:46:38
And the ground state,
here, is one in which t(2g) has

624
00:46:38 --> 00:46:45
a single electron in it for this
d^1 ion and e(g)* is vacant.

625
00:46:45 --> 00:46:50
And, if that system,
in the case of titanium three

626
00:46:50 --> 00:46:55
aquo in aqueous solution,
is a pale blue color,

627
00:46:55 --> 00:47:01
a pretty pale blue --
If a photon comes in of the

628
00:47:01 --> 00:47:07
appropriate energy delta O to
promote the electron into e(g)*,

629
00:47:07 --> 00:47:12
then that transforms this
system into an excited state,

630
00:47:12 --> 00:47:16
as follows, with a
configuration (t two g) zero and

631
00:47:16 --> 00:47:19
(e g star) one.

632
00:47:19 --> 00:47:24
These diagrams,
just like the molecular orbital

633
00:47:24 --> 00:47:28
diagrams we developed for
diatomic molecules,

634
00:47:28 --> 00:47:33
can be associated with
configurations.

635
00:47:33 --> 00:47:38
And this transition can be
written as( t two g) one (e g

636
00:47:38 --> 00:47:44
star) zero,
photon in, electron promoted

637
00:47:44 --> 00:47:50
into e g star,
t two g zero and e g star one

638
00:47:50 --> 00:47:55
now corresponding
to the excited

639
00:47:55 --> 00:47:58
state.
And this excited state is

640
00:47:58 --> 00:48:04
produced in a vibrationally
excited manner.

641
00:48:04 --> 00:48:08
And it and the ground state
interact with the environment in

642
00:48:08 --> 00:48:11
different ways,
such that the absorption

643
00:48:11 --> 00:48:15
spectrum for a system of this
type gives rise to not a sharp,

644
00:48:15 --> 00:48:18
but a broad peak.
And, in fact,

645
00:48:18 --> 00:48:21
the spectrum would look
something like this.

646
00:48:21 --> 00:48:24
Going from 200 to
nanometers in wavelength,

647
00:48:24 --> 00:48:28
you would see a profile like
this with one broad peak

648
00:48:28 --> 00:48:32
somewhere in the red,
such that we would be behind

649
00:48:32 --> 00:48:36
the pale blue color of these
solutions of aqueous titanium

650
00:48:36 --> 00:48:42
three. So this
would be an absorption

651
00:48:42 --> 00:48:45
spectrum, and this is
wavelength, down here.

652
00:48:45 --> 00:48:49
And this is the peak for what
we call this d-d transition,

653
00:48:49 --> 00:48:53
from t(2g) to e(g)*.
There are a lot more subtleties

654
00:48:53 --> 00:48:57
on this, which is where I will
begin on Monday.

655
00:48:57.325 --> 49:00
Have a great weekend.