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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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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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d-orbitals for the value of
theta and phi that is found at

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that ligand position.
And so, I am going to put up

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part of the table that we
generated last time.

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And that part will correspond
to ligand position,

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

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five, and six.
Those six ligand positions.

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And two of the d orbitals here,
d z squared and d x

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squared minus y squared,
were found to give

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non-zero values at these six
ligand positions that correspond

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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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nodal planes of xz,
yz, and xy.

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And what we are doing is just
writing down,

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in tabular form,
the relative magnitude of what

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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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six, which are on the big lobes
of d z squared along

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the z-axis.
And we set that to one because

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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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these ligands two through five
that are in the x,y-plane and

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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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what it is along the z-axis.
And then, d x squared minus y

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squared
evaluated at positions one and

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six gave us a zero.
And that is because positions

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one and six fall on the z-axis,
which is actually an

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intersection of two nodal planes
for the d x squared minus y

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squared orbital.
And then, here we have at

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positions two through five,
three-quarters.

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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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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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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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distance, if we are in the
x,y-plane along x or along y,

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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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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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tetrahedral coordination
geometry.

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The tetrahedron is not limited
to being important in organic

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chemistry.
It is also important for

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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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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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chloride ligands and be the
central metal ions,

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something like cobalt two
or nickel two,

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a tetrahedral array
like this is produced.

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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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and ten, all at the same
distance from the center.

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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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orbitals.
And I will start here with

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d(xz), d(yz) and d(xy),
and over here,

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d z squared and d x
squared minus y squared.

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And we need our ligands

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positioned at seven,
eight, nine,

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and ten, which are these
alternating corners of the cube,

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which is the tetrahedral
geometry.

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And what we find,
when we go through and evaluate

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for those theta and phi values
that correspond to these ligand

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positions using our polar
coordinate system and then

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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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00:08:00,000 --> 00:08:02,000
Notice that,
in the case of the octahedron,

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we are interacting with some of
the lobes, like the lobes of d z

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00:08:07,000 --> 00:08:09,000
squared,
the large ones,

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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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The overlap is not going to be
as good.

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And it evaluates to one-third
for all of those positions.

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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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squared has its lobes along x
and y.

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What that means is that ligands
seven, eight,

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nine, and ten actually lie on a
nodal surface of the d x squared

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00:09:17,000 --> 00:09:21,000
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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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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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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00:10:00,000 --> 00:10:05,000
If you set the equation for d z
squared equal to zero

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00:10:05,000 --> 00:10:08,000
and solved for it,
we found this angle here.

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We found that angle.
And you will remember that that

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was the arc cosine of root three
over three.

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It turns out that that value is
exactly half of the tetrahedral

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angle, which is the angel
between seven,

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the center and eight.
And, because of this,

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all ligands,
seven, eight,

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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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octahedral case,
is that z squared has the same

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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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00:11:00,000 --> 00:11:04,000
here.
And xz, yz, and xy all have

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zero.
And over here xz,

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yz, and xy all have
four-thirds.

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And z squared and x
squared minus y squared

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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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of orbitals which we recognize
as xz, yz, and xy.

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And the name that we give to
that triply degenerate set is

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t(2g).
This t(2g) triply degenerate

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set, for this whole set,
is the t(2g) set.

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00:12:58,000 --> 00:13:01,000
If I use that label t(2g),
you should remember the

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00:13:01,000 --> 00:13:04,000
specific d orbitals that
contribute to the t(2g) set and

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00:13:04,000 --> 00:13:08,000
make up the t2g set.
Up here we have the d z squared

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00:13:08,000 --> 00:13:12,000
and the d x squared
minus y squared

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at the same energy,
which are three relative energy

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units here.
And those two together

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constitute the e(g) set.
And I will often write that

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with a star because we are going
to find out later,

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when we do the molecular
orbital theory of coordination

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complexes, that these are
antibonding.

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00:13:31,000 --> 00:13:34,000
And the reason they are
antibonding is because,

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if you go over here and look at
it, if you have a ligand that

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wants to act as a Lewis base
with respect to d z squared,

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00:13:42,000 --> 00:13:45,000
then putting an
electron in d z squared

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00:13:45,000 --> 00:13:49,000
leads to repulsion of
that ligand, generating

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antibonding characters.
You want your Lewis acid to

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have empty orbitals to receive
lone pairs of electrons.

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00:13:58,000 --> 00:14:01,000
And, if instead they are
populated, that corresponds to

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antibonding character,
much like we have talked about

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00:14:05,000 --> 00:14:08,000
for other types of molecular
orbital diagrams.

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00:14:08,000 --> 00:14:12,000
This d-orbital splitting
diagram for Oh is really a

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00:14:12,000 --> 00:14:16,000
simplified molecular orbital
diagram for the molecule in

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00:14:16,000 --> 00:14:20,000
which we just look at those
orbitals that derive from the

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00:14:20,000 --> 00:14:23,000
set of five d orbitals on that
metal ion.

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00:14:23,000 --> 00:14:27,000
Because these are the orbitals,
--

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00:14:27,000 --> 00:14:31,000
-- and, together with the
electrons that occupy them,

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00:14:31,000 --> 00:14:36,000
they control the properties of
color and magnetism that we will

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00:14:36,000 --> 00:14:40,000
be talking about for the rest of
today.

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00:14:40,000 --> 00:14:45,000
And let me remind you here that
we have a name for the magnitude

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00:14:45,000 --> 00:14:49,000
of this splitting,
which is called delta O for

211
00:14:49,000 --> 00:14:52,000
octahedral.
And then, we also have the

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00:14:52,000 --> 00:14:56,000
tetrahedral case.
And how do we get the diagram

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00:14:56,000 --> 00:15:00,000
from this?
We just sum over the

214
00:15:00,000 --> 00:15:04,000
contributions from each of the
ligand positions.

215
00:15:04,000 --> 00:15:08,000
In tetrahedral,
there are only four ligands.

216
00:15:08,000 --> 00:15:13,000
For xz, yz, and zy we have to
sum up four times one-third,

217
00:15:13,000 --> 00:15:17,000
and that gives us,
for that set of three orbitals

218
00:15:17,000 --> 00:15:22,000
using the same energy scale as
over there, three orbitals at a

219
00:15:22,000 --> 00:15:27,000
value of four-thirds and two
orbitals down here at our zero

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00:15:27,000 --> 00:15:32,000
of energy.
And these will be d z squared,

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00:15:32,000 --> 00:15:38,000
located on that
node, and x squared minus y

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00:15:38,000 --> 00:15:44,000
squared.
Whereas, up here we have xz,

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00:15:44,000 --> 00:15:48,000
yz, and zy.
And these get the names e and

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00:15:48,000 --> 00:15:52,000
t(2) in the tetrahedral
geometry.

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00:15:52,000 --> 00:15:58,000
And the splitting between the
levels, the doubly degenerate

226
00:15:58,000 --> 00:16:04,000
and the triply degenerate pair
of energy levels,

227
00:16:04,000 --> 00:16:11,000
is here referred to as delta t.
And from the math of these two

228
00:16:11,000 --> 00:16:18,000
tables and the magnitude of this
splitting being three relative

229
00:16:18,000 --> 00:16:23,000
to this one being four-thirds,
you can easily derive the

230
00:16:23,000 --> 00:16:29,000
relation that delta t is equal
to four-ninths of delta O.

231
00:16:29,000 --> 00:16:35,000
One of the neat things about

232
00:16:35,000 --> 00:16:40,000
this result qualitatively is
that if you have more ligands,

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00:16:40,000 --> 00:16:45,000
six versus four,
you have a larger splitting of

234
00:16:45,000 --> 00:16:49,000
the energy levels.
And that corresponds to putting

235
00:16:49,000 --> 00:16:53,000
more Lewis bases,
more negative charges,

236
00:16:53,000 --> 00:16:56,000
around that sphere that we
talked about,

237
00:16:56,000 --> 00:17:02,000
in the coordination sphere.
And also, if you have

238
00:17:02,000 --> 00:17:08,000
interactions between ligands and
metal d orbitals that are end-on

239
00:17:08,000 --> 00:17:12,000
sigma symmetry,
directed at each other,

240
00:17:12,000 --> 00:17:17,000
you get bigger splitting
because that leads to better

241
00:17:17,000 --> 00:17:23,000
overlap than the type of oblique
overlap that you get if you have

242
00:17:23,000 --> 00:17:29,000
a ligand at a position such as
seven relative to an orbital

243
00:17:29,000 --> 00:17:33,000
like d(xz).
You can now see that to fully

244
00:17:33,000 --> 00:17:37,000
appreciate this,
you really have to have a very

245
00:17:37,000 --> 00:17:40,000
good grasp of the nodal
properties of the d orbitals.

246
00:17:40,000 --> 00:17:44,000
This is why I am going to
re-emphasize that you should

247
00:17:44,000 --> 00:17:48,000
spend some time on the computer
visualizing these things and

248
00:17:48,000 --> 00:17:52,000
rotating them around and seeing
where these ligand positions for

249
00:17:52,000 --> 00:17:56,000
these different coordination
geometries are with respect to

250
00:17:56,000 --> 00:18:00,000
each of the five d orbitals.
That is quite important.

251
00:18:00,000 --> 00:18:04,000
Also, you are going to see that
you could consider other

252
00:18:04,000 --> 00:18:08,000
possible coordination geometries
that involve ligands at these

253
00:18:08,000 --> 00:18:12,000
positions or at other positions.
For coordination number five,

254
00:18:12,000 --> 00:18:16,000
you could have a trigonal
bipyramid coordination geometry.

255
00:18:16,000 --> 00:18:20,000
And you could go ahead and
evaluate the d orbitals at the

256
00:18:20,000 --> 00:18:23,000
relevant positions.
You would already have three of

257
00:18:23,000 --> 00:18:27,000
them here because in a trigonal
bipyramid, you would have

258
00:18:27,000 --> 00:18:31,000
ligands at positions one and
six, and two.

259
00:18:31,000 --> 00:18:34,000
And then two more back here at
120 degrees to two,

260
00:18:34,000 --> 00:18:37,000
but in the x,y-plane.
So you can do a trigonal

261
00:18:37,000 --> 00:18:40,000
bipyramid the same way and
generate the d-orbital splitting

262
00:18:40,000 --> 00:18:43,000
diagram for it.
You can also go ahead and do a

263
00:18:43,000 --> 00:18:47,000
d-orbital splitting diagram for
a different geometry

264
00:18:47,000 --> 00:18:50,000
corresponding to coordination
number four, which would be the

265
00:18:50,000 --> 00:18:53,000
square planar coordination
geometry.

266
00:18:53,000 --> 00:18:55,000
If you just took ligand
positions two,

267
00:18:55,000 --> 00:18:58,000
three, four,
and five you could generate a

268
00:18:58,000 --> 00:19:01,000
diagram.
And that diagram would be

269
00:19:01,000 --> 00:19:05,000
relevant for systems that do
have square planar coordination

270
00:19:05,000 --> 00:19:09,000
around a metal center,
and these are actually fairly

271
00:19:09,000 --> 00:19:12,000
common.
Those are the four coordination

272
00:19:12,000 --> 00:19:14,000
geometries, octahedral,
tetrahedral,

273
00:19:14,000 --> 00:19:17,000
trigonal bipyramid,
and square planar that are all

274
00:19:17,000 --> 00:19:22,000
pretty commonly encountered and
that you can actually interpret

275
00:19:22,000 --> 00:19:25,000
pretty nicely using the
d-orbital splitting diagrams.

276
00:19:25,000 --> 00:19:29,000
Now, one of the properties that
we are going to want to

277
00:19:29,000 --> 00:19:33,000
interpret will be the magnetism.

278
00:19:50,000 --> 00:19:54,000
And, in talking about the
magnetism of coordination

279
00:19:54,000 --> 00:20:00,000
complexes, we are going to start
by mentioning two terms that you

280
00:20:00,000 --> 00:20:04,000
have probably heard,
but I would like to define

281
00:20:04,000 --> 00:20:08,000
explicitly today.
One is paramagnetic.

282
00:20:08,000 --> 00:20:13,000
That term is given to
substances that are attracted

283
00:20:13,000 --> 00:20:17,000
into a magnetic field.
It is somehow a bulk phenomenon

284
00:20:17,000 --> 00:20:22,000
that we are going to relate to
properties of d orbital

285
00:20:22,000 --> 00:20:26,000
splitting diagrams.

286
00:20:35,000 --> 00:20:38,000
And systems can also be
described as diamagnetic.

287
00:20:38,000 --> 00:20:42,000
And there are many other types
of magnetic behavior that we

288
00:20:42,000 --> 00:20:50,000
won't be discussing.
I'm a little too close to the

289
00:20:50,000 --> 00:20:56,000
board, here.
Diamagnetic.

290
00:20:56,000 --> 00:21:02,000
And that means that the
substance is repelled out of or

291
00:21:02,000 --> 00:21:10,000
away from a magnetic field.
And one thing that you should

292
00:21:10,000 --> 00:21:16,000
keep in mind is that the
magnitude of paramagnetism is

293
00:21:16,000 --> 00:21:24,000
usually orders of magnitude
greater than the magnitude of

294
00:21:24,000 --> 00:21:30,000
diamagnetism.
So, this is much larger.

295
00:21:34,000 --> 00:21:37,000
And what this means is if a
substance behaves as a

296
00:21:37,000 --> 00:21:41,000
paramagnet, you can often
neglect its diamagnetism that

297
00:21:41,000 --> 00:21:43,000
also may be present,
also must be present.

298
00:21:43,000 --> 00:21:46,000
The substances that are
paramagnetic are also

299
00:21:46,000 --> 00:21:50,000
diamagnetic, but the
paramagnetism is much larger in

300
00:21:50,000 --> 00:21:53,000
terms of its order of magnitude
and wins out.

301
00:21:53,000 --> 00:21:57,000
If you want to measure one of
these quantities for a system,

302
00:21:57,000 --> 00:22:00,000
that which is far more
difficult to measure is the

303
00:22:00,000 --> 00:22:05,000
diamagnetism because it is a
much smaller number.

304
00:22:05,000 --> 00:22:11,000
And then, I want to bring up
this quantity S.

305
00:22:11,000 --> 00:22:17,000
This is the spin quantum
number.

306
00:22:28,000 --> 00:22:35,000
And you can readily calculate S
from looking at a populated d

307
00:22:35,000 --> 00:22:43,000
orbital spitting diagram because
it will be the total number of

308
00:22:43,000 --> 00:22:50,000
electrons multiplied by
one-half, which is the spin per

309
00:22:50,000 --> 00:22:52,000
electron.

310
00:23:00,000 --> 00:23:03,000
If you are confronted with a
particular coordination complex,

311
00:23:03,000 --> 00:23:07,000
the kind of thought process you
will need to go through is,

312
00:23:07,000 --> 00:23:11,000
what is the coordination
number, what is the coordination

313
00:23:11,000 --> 00:23:14,000
geometry, what does the d
orbital splitting diagram look

314
00:23:14,000 --> 00:23:18,000
like, how many electrons do I
have with which to populate the

315
00:23:18,000 --> 00:23:22,000
d orbital splitting diagram?
After I have figured out how to

316
00:23:22,000 --> 00:23:26,000
populate the d orbital splitting
diagram, how many of those

317
00:23:26,000 --> 00:23:30,000
electrons are unpaired?
That gives me this.

318
00:23:30,000 --> 00:23:33,000
I can then calculate S.
And, if I have S,

319
00:23:33,000 --> 00:23:38,000
I can make a prediction about
the value of something called

320
00:23:38,000 --> 00:23:41,000
the spin-only magnetic moment.

321
00:23:49,000 --> 00:23:54,000
Let me mention one more thing
back here, and that would be the

322
00:23:54,000 --> 00:23:55,000
units.

323
00:24:01,000 --> 00:24:08,000
When we talk about magnetism,
we are going to be using units.

324
00:24:08,000 --> 00:24:12,000
This is called the Bohr
magneton.

325
00:24:12,000 --> 00:24:18,000
It is sometimes written as mu
sub b.

326
00:24:18,000 --> 00:24:24,000
It is also sometimes written as
capital B., capital M.

327
00:24:24,000 --> 00:24:30,000
That is the Bohr
magneton.

328
00:24:30,000 --> 00:24:39,000
And it is equal to
9.2741x10^-24 joules per tesla.

329
00:24:39,000 --> 00:24:47,000
This is the value of the Bohr
magneton.

330
00:24:47,000 --> 00:24:58,000
And the Bohr magneton is the
unit, also, of our magnetic

331
00:24:58,000 --> 00:25:01,000
moment.

332
00:25:10,000 --> 00:25:14,000
And in particular,
today, I am going to be talking

333
00:25:14,000 --> 00:25:18,000
about the spin-only value of the
magnetic moment.

334
00:25:18,000 --> 00:25:22,000
So we are going to be making
predictions using spin-only

335
00:25:22,000 --> 00:25:26,000
considerations.
And for those considerations,

336
00:25:26,000 --> 00:25:30,000
we need to be able to calculate
S.

337
00:25:30,000 --> 00:25:34,000
And we can do that pretty
quickly after we have it

338
00:25:34,000 --> 00:25:38,000
correctly populated,
correctly chosen d orbital

339
00:25:38,000 --> 00:25:41,000
splitting diagram for the
system.

340
00:25:41,000 --> 00:25:46,000
And so the spin-only value for
the magnetic moment is given by

341
00:25:46,000 --> 00:25:50,000
this formula,
which is mu is equal to 2.00

342
00:25:50,000 --> 00:25:54,000
times the square root of S times
S plus one.

343
00:25:54,000 --> 00:26:00,000
And, in this formula,

344
00:26:00,000 --> 00:26:06,000
this part, this square root of
S times S plus one,

345
00:26:06,000 --> 00:26:13,000
from the quantum mechanics,
is the value of the angular

346
00:26:13,000 --> 00:26:20,000
momentum, the electron spin
angular momentum.

347
00:26:30,000 --> 00:26:34,000
And this value 2.00,
I am calling it 2.00.

348
00:26:34,000 --> 00:26:40,000
It is actually a number that
differs only very slightly from

349
00:26:40,000 --> 00:26:44,000
two.
So we can, for most purposes,

350
00:26:44,000 --> 00:26:48,000
use 2.00.
This is called the gyromagnetic

351
00:26:48,000 --> 00:26:52,000
ratio of the electron.

352
00:27:00,000 --> 00:27:04,000
And, as its name implies,
it is the ratio of the angular

353
00:27:04,000 --> 00:27:09,000
momentum to the magnetic moment
for the electron.

354
00:27:09,000 --> 00:27:14,000
This is our conversion factor
to go back and forth between

355
00:27:14,000 --> 00:27:17,000
angular momentum and magnetic
moment.

356
00:27:17,000 --> 00:27:21,000
And that factor is the
gyromagnetic ratio for the

357
00:27:21,000 --> 00:27:24,000
electron.
You will see this as gamma,

358
00:27:24,000 --> 00:27:28,000
sometimes.
And, for the free electron

359
00:27:28,000 --> 00:27:33,000
value, it is a number very close
to two.

360
00:27:33,000 --> 00:27:37,000
And so, if you think about
different ions that could be at

361
00:27:37,000 --> 00:27:41,000
the center of a coordination
complex, you wonder,

362
00:27:41,000 --> 00:27:46,000
well, what are my possible
values for the spin-only

363
00:27:46,000 --> 00:27:49,000
magnetic moment,
as given by this formula?

364
00:27:49,000 --> 00:27:52,000
Well, we can make a table for
that.

365
00:27:52,000 --> 00:27:56,000
We have the number of electrons
that are unpaired,

366
00:27:56,000 --> 00:27:59,000
remember.
And then, we will be able to

367
00:27:59,000 --> 00:28:04,000
calculate S.
And then we will want to get

368
00:28:04,000 --> 00:28:10,000
mu, which is this spin-only
value calculated according to

369
00:28:10,000 --> 00:28:15,000
that formula.
And ions from the part of the

370
00:28:15,000 --> 00:28:21,000
Periodic Table that we may
consider working with can have

371
00:28:21,000 --> 00:28:24,000
one, two, three,
four, five, six,

372
00:28:24,000 --> 00:28:30,000
up to seven unpaired electrons.
These numbers,

373
00:28:30,000 --> 00:28:33,000
like six and seven,
are more important for the

374
00:28:33,000 --> 00:28:38,000
lanthanide ions in systems with
f-electrons than for systems

375
00:28:38,000 --> 00:28:40,000
with d electrons.
But, nonetheless,

376
00:28:40,000 --> 00:28:44,000
we can easily make these
predictions the same way.

377
00:28:44,000 --> 00:28:48,000
And so S would be one-half for
one electron,

378
00:28:48,000 --> 00:28:51,000
one, then three-halves,
then two for four unpaired

379
00:28:51,000 --> 00:28:56,000
electrons, then five-halves for
five unpaired electrons,

380
00:28:56,000 --> 00:29:00,000
then three, and then
seven-halves finally for seven

381
00:29:00,000 --> 00:29:06,000
unpaired electrons.
And if you put this into this

382
00:29:06,000 --> 00:29:13,000
formula, you would see that for
one unpaired electron,

383
00:29:13,000 --> 00:29:20,000
it would be one-half times
one-half plus one square root

384
00:29:20,000 --> 00:29:25,000
times two.
That would be square root of

385
00:29:25,000 --> 00:29:31,000
three, so this would be a
spin-only magnetic moment of

386
00:00:01,730 --> 00:29:35,000
And then for S equals one,

387
00:29:35,000 --> 00:29:42,000
we actually have a value of
about 2.83.

388
00:29:42,000 --> 00:29:48,000
And then for three unpaired
electrons, we have a value of,

389
00:29:48,000 --> 00:29:55,000
let's say approximately 3.87.
And then for S equals two,

390
00:29:55,000 --> 00:30:02,000
or four unpaired electrons,
we have a value of 4.90.

391
00:30:02,000 --> 00:30:06,000
For the spin-only magnetic
moment, if you go through with

392
00:30:06,000 --> 00:30:09,000
these values of S,
these numbers of unpaired

393
00:30:09,000 --> 00:30:12,000
electrons.
Let's just make sure we are

394
00:30:12,000 --> 00:30:14,000
very clear on that.

395
00:30:20,000 --> 00:30:26,000
And then 5.92,
6.93, 7.94 approximately.

396
00:30:26,000 --> 00:30:34,000
You can check those numbers,
but they are approximately

397
00:30:34,000 --> 00:30:38,000
right.
And in each case what you are

398
00:30:38,000 --> 00:30:42,000
seeing is the spin-only
prediction for the magnetic

399
00:30:42,000 --> 00:30:46,000
moment is always,
in units of Bohr magnetons,

400
00:30:46,000 --> 00:30:51,000
a little less than one plus the
number of unpaired electrons.

401
00:30:51,000 --> 00:30:55,000
It is not like you have four
unpaired electrons,

402
00:30:55,000 --> 00:31:00,000
so your spin-only magnetic
moment is around four.

403
00:31:00,000 --> 00:31:03,000
It is actually a little less
than five.

404
00:31:03,000 --> 00:31:08,000
And that is because of the
quantum mechanics of the

405
00:31:08,000 --> 00:31:15,000
electron spin angular momentum.
And this kind of consideration,

406
00:31:15,000 --> 00:31:20,000
here, leads you into thinking
about what happens with

407
00:31:20,000 --> 00:31:25,000
different d^n counts for a
particular type of metal

408
00:31:25,000 --> 00:31:27,000
complex.

409
00:31:32,000 --> 00:31:40,000
This means that we are going to
attack and address the high

410
00:31:40,000 --> 00:31:44,000
spin/low spin problem.

411
00:31:56,000 --> 00:31:59,000
These systems are not so
terribly difficult,

412
00:31:59,000 --> 00:32:01,000
but let's face it;
--

413
00:32:01,000 --> 00:32:04,000
-- once you have been presented
with a particular coordination

414
00:32:04,000 --> 00:32:07,000
complex, and you think you know
the geometry,

415
00:32:07,000 --> 00:32:11,000
and you think you know the d
orbital splitting diagram that

416
00:32:11,000 --> 00:32:15,000
corresponds to that geometry,
and you think you know how many

417
00:32:15,000 --> 00:32:18,000
d electrons go into that
diagram, there is one more

418
00:32:18,000 --> 00:32:20,000
thing.
And that is sometimes you have

419
00:32:20,000 --> 00:32:25,000
to figure out if it is high spin
or if it is low spin.

420
00:32:25,000 --> 00:32:29,000
And what that means is for a
particular d^n count,

421
00:32:29,000 --> 00:32:33,000
there might be two possible
choices over here,

422
00:32:33,000 --> 00:32:38,000
depending on how the electrons
occupy the orbitals.

423
00:32:38,000 --> 00:32:41,000
And so let's look at an example
of this.

424
00:32:41,000 --> 00:32:46,000
Here is energy.
Again, we are really focusing a

425
00:32:46,000 --> 00:32:51,000
lot on energy level diagrams.
Sometimes we focus on molecular

426
00:32:51,000 --> 00:32:56,000
orbital diagrams.
Right now, we are using these d

427
00:32:56,000 --> 00:33:03,000
orbital splitting diagrams.
And I will give you an example

428
00:33:03,000 --> 00:33:07,000
that is octahedral.
And this is a species that we

429
00:33:07,000 --> 00:33:12,000
looked at in a recent
demonstration in class,

430
00:33:12,000 --> 00:33:18,000
which is hexaaquo meaning six
waters around an iron two plus

431
00:33:18,000 --> 00:33:21,000
in an octahedral array.

432
00:33:21,000 --> 00:33:27,000
This will be our metal complex
for which we will draw the d

433
00:33:27,000 --> 00:33:33,000
orbital splitting diagram.
It is an octahedral complex,

434
00:33:33,000 --> 00:33:37,000
so we draw our diagram like
this.

435
00:33:37,000 --> 00:33:42,000
And of course,
we have t(2g) and e(g)* levels

436
00:33:42,000 --> 00:33:46,000
that we can populate with our d
electrons.

437
00:33:46,000 --> 00:33:52,000
And we are going to need to
figure out how many d electrons

438
00:33:52,000 --> 00:34:00,000
we have to put into this diagram
for the octahedral system.

439
00:34:00,000 --> 00:34:04,000
The oxidation state of the iron
here is +2.

440
00:34:04,000 --> 00:34:10,000
And you can tell that because
the way you tell oxidation state

441
00:34:10,000 --> 00:34:16,000
is to take into account both the
charge on the ligands.

442
00:34:16,000 --> 00:34:20,000
Chlorides, for example,
are minus one.

443
00:34:20,000 --> 00:34:26,000
Because water is a neutral
molecule, it is simply neutral.

444
00:34:26,000 --> 00:34:32,000
And then, you have to also take
into account the charge on the

445
00:34:32,000 --> 00:34:36,000
whole ion.
So this is iron two plus.

446
00:34:36,000 --> 00:34:38,000
The iron in the plus two

447
00:34:38,000 --> 00:34:40,000
oxidation state with six neutral
ligands.

448
00:34:40,000 --> 00:34:43,000
If I had six NH three
ligands in the same charge on

449
00:34:43,000 --> 00:34:46,000
the ion, then it would still be
iron plus two.

450
00:34:46,000 --> 00:34:50,000
And you are going to get some
practice figuring out oxidation

451
00:34:50,000 --> 00:34:53,000
state because if you cannot do
that correctly you cannot

452
00:34:53,000 --> 00:34:56,000
identify the number of d
electrons to put into the

453
00:34:56,000 --> 00:34:58,000
diagram correctly.
And this is iron plus two.

454
00:34:58,000 --> 00:35:02,000
You need to know that iron is

455
00:35:02,000 --> 00:35:05,000
in Group 8 of the periodic
table.

456
00:35:05,000 --> 00:35:11,000
The equation for number of d
electrons here is eight minus

457
00:35:11,000 --> 00:35:15,000
two equals six.
If it is in Group 8,

458
00:35:15,000 --> 00:35:20,000
that means that an iron atom
has eight valance electrons,

459
00:35:20,000 --> 00:35:26,000
the same way that carbon has
four valance electrons or oxygen

460
00:35:26,000 --> 00:35:32,000
has six valance electrons.
And in Group 8 of the periodic

461
00:35:32,000 --> 00:35:36,000
table, we are subtracting two
from the eight valance electrons

462
00:35:36,000 --> 00:35:40,000
because there is a two plus
charge on the system.

463
00:35:40,000 --> 00:35:43,000
That leaves six valance
electrons on the iron.

464
00:35:43,000 --> 00:35:48,000
And that means this is what we
call a d six system.

465
00:35:48,000 --> 00:35:51,000
By convention,
we are considering all the

466
00:35:51,000 --> 00:35:55,000
valance electrons that remain
after the metal has been

467
00:35:55,000 --> 00:35:59,000
oxidized to plus two as d
electrons, meaning they are

468
00:35:59,000 --> 00:36:04,000
going to go into our d orbital
splitting diagram.

469
00:36:04,000 --> 00:36:08,000
Now, the reason we do this,
in fact, is because in ions

470
00:36:08,000 --> 00:36:13,000
like this, the d orbitals are
the orbitals on the metal,

471
00:36:13,000 --> 00:36:17,000
on the iron,
that are the lowest in energy.

472
00:36:17,000 --> 00:36:22,000
After, in the case of iron,
iron has principle quantum

473
00:36:22,000 --> 00:36:26,000
number three.
There is also a 4s and a 4p set

474
00:36:26,000 --> 00:36:31,000
of orbitals available to iron,
but they are much higher in

475
00:36:31,000 --> 00:36:35,000
energy.
We are simplifying the system

476
00:36:35,000 --> 00:36:40,000
and just looking at what happens
with the five d orbitals,

477
00:36:40,000 --> 00:36:45,000
which are the lowest valance
energy orbitals available to the

478
00:36:45,000 --> 00:36:48,000
iron.
Just like a carbon has a 2s and

479
00:36:48,000 --> 00:36:53,000
a set of three 2p orbitals,
an iron has a set of five 3d

480
00:36:53,000 --> 00:36:56,000
orbitals, a 4s,
and three 4p orbitals.

481
00:36:56,000 --> 00:37:00,000
But those s and p ones are
higher.

482
00:37:00,000 --> 00:37:04,000
We are just focusing on our d
orbitals, here,

483
00:37:04,000 --> 00:37:10,000
using our d orbital only
splitting diagrams with which we

484
00:37:10,000 --> 00:37:14,000
have six electrons for
population of a diagram.

485
00:37:14,000 --> 00:37:20,000
And so what you see immediately
is you will identify with the

486
00:37:20,000 --> 00:37:26,000
notion that there are two ways
to populate this diagram with

487
00:37:26,000 --> 00:37:29,000
six electrons.

488
00:37:32,000 --> 00:37:34,000
Because we can do this.

489
00:37:37,000 --> 00:37:42,000
Here, we are following Hund's
rule of maximum multiplicity,

490
00:37:42,000 --> 00:37:47,000
trying not to pair up any
electrons until we run out of

491
00:37:47,000 --> 00:37:50,000
spots for them,
and so we have to.

492
00:37:50,000 --> 00:37:54,000
There is one way to populate
the diagram.

493
00:37:54,000 --> 00:38:00,000
Two electrons in e(g)*.
Four electrons in t(2g).

494
00:38:00,000 --> 00:38:04,000
One of them,
spin inverted relative to the

495
00:38:04,000 --> 00:38:10,000
other five because you cannot
put two electrons in the same

496
00:38:10,000 --> 00:38:14,000
orbital unless they have
different spins.

497
00:38:14,000 --> 00:38:18,000
And, finally,
we could populate the diagram

498
00:38:18,000 --> 00:38:22,000
this way.
The case on the left is what we

499
00:38:22,000 --> 00:38:25,000
call high spin.

500
00:38:30,000 --> 00:38:35,000
The case on the right is what
we call low spin.

501
00:38:40,000 --> 00:38:44,000
And, as you think about it,
you will realize that the high

502
00:38:44,000 --> 00:38:49,000
spin/low spin problem only
arises for certain d^n counts.

503
00:38:49,000 --> 00:38:51,000
It doesn't arise,
for example,

504
00:38:51,000 --> 00:38:56,000
for d one because we
would just put one electron down

505
00:38:56,000 --> 00:39:00,000
here in t(2g) and that is not a
problem.

506
00:39:00,000 --> 00:39:04,000
And you will see that,
as you run through the d one

507
00:39:04,000 --> 00:39:07,000
all the way up to the d
ten.

508
00:39:07,000 --> 00:39:12,000
d^n counts can go from d^0 to
d^10, and that is the limit of

509
00:39:12,000 --> 00:39:15,000
the possibilities here.
Zero, one, two,

510
00:39:15,000 --> 00:39:17,000
three, four,
five, six, seven,

511
00:39:17,000 --> 00:39:19,000
eight, nine,
and ten.

512
00:39:19,000 --> 00:39:23,000
Here I have chosen a d^6 case
to illustrate the high spin

513
00:39:23,000 --> 00:39:28,000
versus low spin dichotomy.
And you might wonder when do

514
00:39:28,000 --> 00:39:32,000
you --
Well, let me get back to that

515
00:39:32,000 --> 00:39:36,000
in a moment.
And let me just say here that

516
00:39:36,000 --> 00:39:40,000
we have four unpaired electrons
in d^6 high spin.

517
00:39:40,000 --> 00:39:43,000
We have one,
two, three, four.

518
00:39:43,000 --> 00:39:46,000
That makes this an s equals two
case.

519
00:39:46,000 --> 00:39:50,000
And, over here,
we have s equals zero.

520
00:39:50,000 --> 00:39:55,000
And so this one on the right,
this low spin case is s equals

521
00:39:55,000 --> 00:39:58,000
zero.
We would say that this is not

522
00:39:58,000 --> 00:40:03,000
paramagnetic.
It is going to be only

523
00:40:03,000 --> 00:40:07,000
diamagnetic.
It has no unpaired electrons to

524
00:40:07,000 --> 00:40:12,000
give it a magnetic moment
according to the spin-only

525
00:40:12,000 --> 00:40:15,000
formula that we looked at over
here.

526
00:40:15,000 --> 00:40:20,000
On the other hand,
over here, we have an s equals

527
00:40:20,000 --> 00:40:25,000
two state that predicts a
magnetic moment of 4.90 Bohr

528
00:40:25,000 --> 00:40:28,000
magnetons.
And experimentally,

529
00:40:28,000 --> 00:40:35,000
mu was found for this system to
be 5.1 Bohr magnetons.

530
00:40:35,000 --> 00:40:40,000
So we say this one turned out
to be high spin.

531
00:40:40,000 --> 00:40:45,000
And in a little while,
we will be talking more about

532
00:40:45,000 --> 00:40:52,000
what are the physical factors
that determine whether a system

533
00:40:52,000 --> 00:40:58,000
is low spin or high spin.
Remember, we have this delta O

534
00:40:58,000 --> 00:41:02,000
here.
And there is not one value for

535
00:41:02,000 --> 00:41:07,000
delta O, and this is something I
am going to emphasize again in a

536
00:41:07,000 --> 00:41:10,000
moment.
Delta O varies according to a

537
00:41:10,000 --> 00:41:14,000
number of factors,
and these factors include the

538
00:41:14,000 --> 00:41:19,000
specific nature of the ligands.
It includes the charge on the

539
00:41:19,000 --> 00:41:22,000
metal center.
It includes the specific metal

540
00:41:22,000 --> 00:41:25,000
that you are using.
And at some point,

541
00:41:25,000 --> 00:41:30,000
if delta O gets bigger and
bigger and bigger and bigger and

542
00:41:30,000 --> 00:41:35,000
exceeds what we call the pairing
energy --

543
00:41:47,000 --> 00:41:50,000
This pairing energy is
something I will talk more about

544
00:41:50,000 --> 00:41:53,000
later, but it is the energy
required to put two electrons in

545
00:41:53,000 --> 00:41:55,000
the same d orbital.

546
00:42:10,000 --> 00:42:15,000
If you get to a condition where
delta O is greater than this

547
00:42:15,000 --> 00:42:20,000
pairing energy,
in other words,

548
00:42:20,000 --> 00:42:25,000
there is a big splitting
between t(2g) and e(g)*,

549
00:42:25,000 --> 00:42:31,000
then the electrons will prefer
to pair up down here in the

550
00:42:31,000 --> 00:42:35,000
t(2g).
Whereas, if you think of it

551
00:42:35,000 --> 00:42:39,000
going to the limit of an
infinitely small delta O,

552
00:42:39,000 --> 00:42:44,000
they would have S equals two
because the gap between t(2g)

553
00:42:44,000 --> 00:42:47,000
and e(g)* in the limit of
vanishing delta O,

554
00:42:47,000 --> 00:42:51,000
for example,
when ligands go to an infinite

555
00:42:51,000 --> 00:42:55,000
distance away from the metal
center, they would then,

556
00:42:55,000 --> 00:43:00,000
in that limit,
have all five the same energy.

557
00:43:00,000 --> 00:43:03,000
And so your S would naturally
equal two.

558
00:43:03,000 --> 00:43:09,000
If delta O is much greater than
the pairing energy giving a

559
00:43:09,000 --> 00:43:14,000
large value of delta O,
then that favors low spin.

560
00:43:14,000 --> 00:43:20,000
Factors that give rise to large
values of delta O favor low

561
00:43:20,000 --> 00:43:26,000
spin, and I will talk about a
few more factors like that in a

562
00:43:26,000 --> 00:43:28,000
moment.

563
00:43:36,000 --> 00:43:40,000
Here is a brief introduction to
the magnetic properties of

564
00:43:40,000 --> 00:43:45,000
coordination complexes.
And I want to show that we can

565
00:43:45,000 --> 00:43:49,000
use the same types of diagrams
to talk about the colors of

566
00:43:49,000 --> 00:43:54,000
transition metal complexes.
We looked at the color change

567
00:43:54,000 --> 00:43:58,000
in that demonstration I showed
you.

568
00:43:58,000 --> 00:44:02,000
And one of the really
fascinating aspects of

569
00:44:02,000 --> 00:44:08,000
transition metal chemistry is
the colors that arise because of

570
00:44:08,000 --> 00:44:13,000
absorption of a photon
coincident with promotion of an

571
00:44:13,000 --> 00:44:19,000
electron from t(2g) to e(g)*.
And so we will talk briefly now

572
00:44:19,000 --> 00:44:25,000
about electronic / absorption
spectroscopy.

573
00:44:40,000 --> 00:44:48,000
And the most important thing I
want to impart to you here is a

574
00:44:48,000 --> 00:44:55,000
selection rule wherein delta S
is equal to zero.

575
00:44:58,000 --> 00:45:02,000
This is the spin selection
rule.

576
00:45:10,000 --> 00:45:14,000
And this means that when a
photon comes in and interacts

577
00:45:14,000 --> 00:45:19,000
with d electrons in a system of
the type we are discussing,

578
00:45:19,000 --> 00:45:24,000
that photon can be absorbed
coincident with promotion of an

579
00:45:24,000 --> 00:45:29,000
electron from t(2g) to e(g)*,
as long as it doesn't violate

580
00:45:29,000 --> 00:45:34,000
this spin selection rule.
And that means that the number

581
00:45:34,000 --> 00:45:39,000
of unpaired electrons should be
the same in the system before

582
00:45:39,000 --> 00:45:43,000
the photon is taken up,
as it is after.

583
00:45:43,000 --> 00:45:47,000
And so here is an example of
that, a simple example.

584
00:45:47,000 --> 00:45:52,000
Here is an octahedral system,
titanium with six waters and a

585
00:45:52,000 --> 00:45:56,000
three plus charge.

586
00:45:56,000 --> 00:46:01,000
Titanium is in Group 4 of the
periodic table.

587
00:46:01,000 --> 00:46:05,000
Our water ligands are neutral,
with three plus charges to be

588
00:46:05,000 --> 00:46:07,000
subtracted from four.
We have a d^1 system.

589
00:46:07,000 --> 00:46:11,000
This is one of the simplest
cases for talking about

590
00:46:11,000 --> 00:46:14,000
electronic absorption
spectroscopy and is the first

591
00:46:14,000 --> 00:46:17,000
thing that you encounter in your
textbook.

592
00:46:17,000 --> 00:46:21,000
By the way, let me just remind
you that right now you should be

593
00:46:21,000 --> 00:46:25,000
reading Chapter 16 in your
textbook, that deals with

594
00:46:25,000 --> 00:46:28,000
coordination chemistry,
electronic structure of d-block

595
00:46:28,000 --> 00:46:33,000
elements, colors and magnetism,
and so forth.

596
00:46:33,000 --> 00:46:38,000
And the ground state,
here, is one in which t(2g) has

597
00:46:38,000 --> 00:46:45,000
a single electron in it for this
d^1 ion and e(g)* is vacant.

598
00:46:45,000 --> 00:46:50,000
And, if that system,
in the case of titanium three

599
00:46:50,000 --> 00:46:55,000
aquo in aqueous solution,
is a pale blue color,

600
00:46:55,000 --> 00:47:01,000
a pretty pale blue --
If a photon comes in of the

601
00:47:01,000 --> 00:47:07,000
appropriate energy delta O to
promote the electron into e(g)*,

602
00:47:07,000 --> 00:47:12,000
then that transforms this
system into an excited state,

603
00:47:12,000 --> 00:47:16,000
as follows, with a
configuration (t two g) zero and

604
00:47:16,000 --> 00:47:19,000
(e g star) one.

605
00:47:19,000 --> 00:47:24,000
These diagrams,
just like the molecular orbital

606
00:47:24,000 --> 00:47:28,000
diagrams we developed for
diatomic molecules,

607
00:47:28,000 --> 00:47:33,000
can be associated with
configurations.

608
00:47:33,000 --> 00:47:38,000
And this transition can be
written as( t two g) one (e g

609
00:47:38,000 --> 00:47:44,000
star) zero,
photon in, electron promoted

610
00:47:44,000 --> 00:47:50,000
into e g star,
t two g zero and e g star one

611
00:47:50,000 --> 00:47:55,000
now corresponding
to the excited

612
00:47:55,000 --> 00:47:58,000
state.
And this excited state is

613
00:47:58,000 --> 00:48:04,000
produced in a vibrationally
excited manner.

614
00:48:04,000 --> 00:48:08,000
And it and the ground state
interact with the environment in

615
00:48:08,000 --> 00:48:11,000
different ways,
such that the absorption

616
00:48:11,000 --> 00:48:15,000
spectrum for a system of this
type gives rise to not a sharp,

617
00:48:15,000 --> 00:48:18,000
but a broad peak.
And, in fact,

618
00:48:18,000 --> 00:48:21,000
the spectrum would look
something like this.

619
00:48:21,000 --> 00:48:24,000
Going from 200 to
nanometers in wavelength,

620
00:48:24,000 --> 00:48:28,000
you would see a profile like
this with one broad peak

621
00:48:28,000 --> 00:48:32,000
somewhere in the red,
such that we would be behind

622
00:48:32,000 --> 00:48:36,000
the pale blue color of these
solutions of aqueous titanium

623
00:48:36,000 --> 00:48:42,000
three. So this
would be an absorption

624
00:48:42,000 --> 00:48:45,000
spectrum, and this is
wavelength, down here.

625
00:48:45,000 --> 00:48:49,000
And this is the peak for what
we call this d-d transition,

626
00:48:49,000 --> 00:48:53,000
from t(2g) to e(g)*.
There are a lot more subtleties

627
00:48:53,000 --> 00:48:57,000
on this, which is where I will
begin on Monday.

628
00:48:57,325 --> 00:49:00,000
Have a great weekend.